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author:
- |
(for HotQCD Collaboration)[^1]\
Physics Department, Brookhaven National Laboratory, Upton, NY 11973
title: 'Chiral transition temperature and aspects of deconfinement in [2+1]{} flavor QCD with the HISQ/tree action'
---
Introduction {#sec: intro}
============
In these proceedings we follow up on the line of work on 2+1 flavor QCD thermodynamics with improved staggered fermions by the HotQCD collaboration. The setup of simulations and some preliminary results have been reported earlier, e.g. [@Jamaica; @lat10; @soeldner_lat10; @prasad_panic11] and the continuum extrapolation for chiral $T_c$ at the physical quark masses is presented in [@hotqcd2]. Full data set includes several lines of constant physics down to the light quark mass $m_l=m_s/20$ for asqtad and $m_l=m_s/40$ for HISQ/tree[^2]. The lattice spacings cover the range of temperatures $T=130-440$ MeV with $N_\tau=6$, 8 and 12 for HISQ/tree and $T=148-304$ MeV with $N_\tau=8$ and 12 for asqtad.
Performing the continuum limit requires control over cutoff effects. In improved staggered discretization schemes the leading $O(a^2)$ errors at low temperature (coarse lattices) originate from violations of taste symmetry, that distort the hadron spectrum. For this reason it is important to perform simulations on fine enough lattices (large $N_\tau$ in finite-temperature setup) and/or use actions with the smallest discretization effects. Analysis of the discretization effects for asqtad and HISQ/tree used in this study is presented in Ref. [@lat10].
Chiral transition
=================
For vanishing light quark masses there is a chiral phase transition which is expected to be of second order and in the $O(4)$ universality class [@rob_o4]. However, universal scaling allows to define pseudo-critical temperatures for the chiral transition even for non-zero light quark masses, provided they are small enough. For staggered fermions that preserve only a part of the chiral symmetry there is a complication: in the chiral limit at finite lattice spacing the relevant universality class is $O(2)$ rather than $O(4)$. Fortunately, in the numerical analysis the differences between $O(2)$ and $O(4)$ universality classes are small so when referring to scaling we will use the term $O(N)$ scaling. Previous studies with the p4 action provided evidence for $O(N)$ scaling [@rbcbi09; @rbcbi10]. Similar analysis for the asqtad and HISQ/tree action establishing if the $O(N)$ scaling is applicable is performed in [@hotqcd2] and explained below.
The order parameter for the chiral transition is the chiral condensate $$M_b \equiv \frac{m_s \langle \bar{\psi}\psi \rangle_l}{T^4} \; .
\label{order}$$ Its temperature and quark mass dependence near the critical temperature can be parametrized by a universal scaling function $f_G$ and a regular function $f_{M,reg}$ that describes corrections to scaling: $$M_b(T,m_l,m_s) = h^{1/\delta} f_G(t/h^{1/\beta\delta}) + f_{M,reg}(T,H),
\,\,\,\,
t = \frac{1}{t_0} \left( \frac{T-T_c^0}{T_c^0} \right),\,\,\,\,
h= \frac{1}{h_0} H,\,\,\,\,H=\frac{m_l}{m_s}
\label{order_scaling}$$ and $T_c^0$ is the critical temperature in the chiral limit. The pseudo-critical temperature can be defined as the peak position of the chiral susceptibility $$\chi_{m,l} = \frac{\partial}{\partial m_l}\langle \bar{\psi}\psi \rangle_l
\label{suscept}$$ whose scaling behavior is also described by $f_G$ and $f_{M,reg}$ as $$\frac{\chi_{m,l}}{T^2} = \frac{T^2}{m_s^2}
\left( \frac{1}{h_0}
h^{1/\delta -1} f_\chi(z) + \frac{\partial f_{M,reg}(T,H)}{\partial H}
\right),\,\,\,\,
f_{\chi}(z)=\frac{1}{\delta} [f_G(z)-\frac{z}{\beta} f_G'(z)],
\,\,\,\,z=\frac{t}{h^{1/\beta\delta}}.
\label{chiralsuscept}$$
The singular function $f_G$ is well studied in spin models and has been parametrized for $O(2)$ and $O(4)$ groups. For the regular part we consider leading-order (linear) dependence in $H$ and quadratic in $T$: $$f_{M,reg}(T,H) = \left( a_0 + a_1 \frac{T-T_c^0}{T_c^0} +
a_2 \left(\frac{T-T_c^0}{T_c^0} \right)^2 \right) H.
\label{eq:freg}$$ Then we are left with 6 parameters to be determined from fitting the data, $T_c^0$, $t_0$, $h_0$, $a_0$, $a_1$ and $a_2$.
We perform simultaneous fits to $M_b$ and $\chi_{m,l}$ for the asqtad action on $N_\tau=8$, 12 and the HISQ/tree action on $N_\tau=6$, 8 and 12. An example of such a fit for HISQ/tree, $N_\tau=8$ is shown in Fig. \[pbp\_and\_chi\].
![ An example of a simultaneous fit to the chiral condensate (left) and susceptibility (right) for HISQ/tree on $N_\tau=8$ lattices. Open symbols indicate the range included in the fit. Dotted black line is an extrapolation to the physical light quark mass. []{data-label="pbp_and_chi"}](hisq_pbpO4_Nt8.eps "fig:"){width="48.00000%"} ![ An example of a simultaneous fit to the chiral condensate (left) and susceptibility (right) for HISQ/tree on $N_\tau=8$ lattices. Open symbols indicate the range included in the fit. Dotted black line is an extrapolation to the physical light quark mass. []{data-label="pbp_and_chi"}](hisq_chiO4_Nt8.eps "fig:"){width="48.00000%"}
Then, performing a combined $1/N_{\tau}^2$ extrapolation of $T_c$ values obtained with the asqtad and HISQ/tree action as shown in Fig. \[tc\_comb\] we obtain $$T_c=( 154 \pm 8 \pm 1)\mbox{ MeV},$$ where the first error is from the fit and the second is the overall error on the lattice scale determination. The fits for asqtad and HISQ/tree are constrained to have a common intercept. (See Ref. [@hotqcd2] for more details on the fitting procedure and analysis of systematic errors.) To present a combined error we add the two errors giving our final value $T_c=154\pm9$ MeV.
{width="48.00000%"} {width="46.40000%"}
Deconfinement aspects of the transition
=======================================
The deconfinement phenomenon in pure gauge theory is governed by breaking of the $Z(N_c)$ symmetry. The order parameter is the renormalized Polyakov loop, obtained from the bare Polyakov loop as $$L_{ren}(T)=z(\beta)^{N_{\tau}} L_{bare}(\beta)=
z(\beta)^{N_{\tau}} \left\langle\frac{1}{N_c} {\rm Tr }
\prod_{x_0=0}^{N_{\tau}-1} U_0(x_0,\vec{x})\right\rangle,$$ where $z(\beta)=\exp(-c(\beta)/2)$. $c(\beta)$ is the additive normalization of the static potential chosen such that it coincides with the string potential at distance $r=1.5r_0$ with $r_0$ being the Sommer scale.
In QCD $Z(N_c)$ symmetry is explicitly broken by dynamical quarks, therefore there is no obvious reason for the Polyakov loop to be sensitive to the singular behavior close to the chiral limit. Indeed, the temperature dependence of the Polyakov loop in pure gauge theory and in QCD is quite different, as one can see from Fig. \[Lren\_absT\]. Also note, that in this purely gluonic observable there is very little sensitivity (through the sea quark loops) to the cut-off effects coming from the fermionic sector. While losing the status of the order parameter in QCD, the Polyakov loop is still a good probe of screening of static color charges in quark-gluon plasma [@okacz02; @digal03].
Other probes of deconfinement are fluctuations and correlations of various charges that can signal liberation of degrees of freedom with quantum numbers of quarks and gluons in the high-temperature phase. Here we consider quadratic fluctuations and correlations of conserved charges: $$\begin{aligned}
\frac{\chi_i(T)}{T^2}=
\left.\frac{1}{T^3 V}\frac{\partial^2 \ln Z(T,\mu_i)}{\partial (\mu_i/T)^2}
\right|_{\mu_i=0},\,\,\,\,\,
\frac{\chi_{11}^{ij}(T)}{T^2}=
\left.\frac{1}{T^3 V}\frac{\partial^2 \ln Z(T,\mu_i,\mu_j)}{\partial (\mu_i/T)
\partial (\mu_j/T)} \right|_{\mu_i=\mu_j=0}.\end{aligned}$$ Fluctuations are also sensitive to the singular part of the free energy density. The light and strange quark number susceptibilities are often considered in connection with deconfinement. These quantities show a rapid rise in the transition region and for the strange quark number susceptibility the behavior is consistent with the Hadron Resonance Gas (HRG) model, as shown in Fig. \[fig:qns\_s\]. For the light quark number susceptibility, a quantity dominated by pions and therefore very sensitive to the taste symmetry, cut-off effects are still significant, and comparing to HRG is more subtle and requires further study, Fig. \[fig:qns\_l\]. In the left panels of Fig. \[fig:qns\_s\] and \[fig:qns\_l\] temperature is set by Sommer scale ($r_1$ in this case).
In part cut-off effects in the observables directly related to hadrons can be accounted for if the temperature scale is set with a hadronic observable. Following the insight by the Budapest-Wuppertal collaboration [@stout; @stout3] we use the kaon decay constant $f_K$ for this purpose. For the strange and light quark number susceptibility the results with $f_K$ scale are shown in the right panels of Fig. \[fig:qns\_s\] and \[fig:qns\_l\].
![ The strange quark number susceptibility for the HISQ/tree action at $m_l=m_s/20$ with $r_1$ (left) and $f_K$ (right) temperature scale. The solid curve in the right panel is from the HRG model. []{data-label="fig:qns_s"}](chiS.eps "fig:"){width="48.00000%"} ![ The strange quark number susceptibility for the HISQ/tree action at $m_l=m_s/20$ with $r_1$ (left) and $f_K$ (right) temperature scale. The solid curve in the right panel is from the HRG model. []{data-label="fig:qns_s"}](chiS_fK.eps "fig:"){width="48.00000%"}
![ The light quark number susceptibility for the HISQ/tree action at $m_l=m_s/20$ with $r_1$ (left) and $f_K$ (right) temperature scale. []{data-label="fig:qns_l"}](chil_hisqonly.eps "fig:"){width="48.00000%"} ![ The light quark number susceptibility for the HISQ/tree action at $m_l=m_s/20$ with $r_1$ (left) and $f_K$ (right) temperature scale. []{data-label="fig:qns_l"}](chil_hisqonly_fK.eps "fig:"){width="48.00000%"}
As one can see, for the strange quark number susceptibility the $f_K$ scale eliminates virtually all cut-off dependence and the data from different lattices are hardly distinguishable. For the light quark number susceptibility some residual cut-off dependence remains. This may be expected for, at least, two reasons: a) $\chi_l$ is very sensitive to taste symmetry breaking (even at our finest, $N_\tau=12$ lattice the root-mean-squared pion mass is still about 200 MeV for the HISQ/tree action [@lat10]), b) there is no a priori reason for the lattice spacing dependence in $\chi_l$, $\chi_s$ to be the same as in $f_K$.
Next, we consider correlations of the electric charge and strangeness. With the $r_1$ scale we observe substantial cut-off dependence, which is largely reduced with $f_K$ scale, compare left and right panel in Fig. \[fig:qns\_QS\]. At high temperatures these correlations are close to the value expected in the non-interacting ideal gas (Stefan-Boltzmann limit), while at low temperatures they are well described by HRG.
Correlations of different quark numbers are also a convenient way to study the deconfinement aspects of the transition. At high temperatures such correlations are expected to be very small, while at low temperatures hadronic degrees of freedom give naturally rise to such correlations. In Fig. \[fig:qns\_us\] we present our results for the light-strange and light-light quark number correlations. We divide them, correspondingly, by the strange and light quark number susceptibility in attempt to get rid of the trivial mass effects. For the strange-light quark number correlations we see a good agreement with the HRG model, while the discrepancies with HRG predictions for $\chi_{ud}/\chi_l$ are likely due to cut-off effects.
![ Electric charge–strangeness correlations with $r_1$ (left) and $f_K$ (right) temperature scale for HISQ/tree. The solid curve in the right panel is from the HRG model. []{data-label="fig:qns_QS"}](chiQS_lat11.eps "fig:"){width="48.00000%"} ![ Electric charge–strangeness correlations with $r_1$ (left) and $f_K$ (right) temperature scale for HISQ/tree. The solid curve in the right panel is from the HRG model. []{data-label="fig:qns_QS"}](chiQS_fK_lat11.eps "fig:"){width="48.00000%"}
{width="48.00000%"} {width="48.00000%"}
Comparing Figs. \[fig:qns\_s\]–\[fig:qns\_us\] we can conclude that quantities, where pions do not contribute, can be reproduced quite well on the lattice, while those, for which pions give dominant contribution, still demonstrate noticeable cut-off effects, that are not completely removed by switching to a hadronic ($f_K$) scale (at least, on the lattices typically in use, $N_\tau=6-12$).
Update on the trace anomaly
===========================
The deconfinement transition is also often defined as the rapid rise in the energy density associated with liberation of new degrees of freedom. The energy density and other thermodynamic quantities are usually obtained by integrating the trace anomaly $(\epsilon-3p)/T^4$. This quantity is under extensive investigation on the lattice. We present an update of the results reported in [@Jamaica] in Fig. \[fig:EoS\]. The solid line is a parametrization of $(\epsilon-3p)/T^4$ derived in Ref. [@PasiPeter] that combines HRG result at low $T$ ( $T<160$ MeV) with the lattice data [@hoteos] at high $T$ ($T>220$ MeV ). From the figure we conclude that the lattice results overshoot the HRG prediction for $T>160$MeV since any cut-off effects could only decrease $(\epsilon-3p)/T^4$ at low $T$.
Conclusions
===========
In this contribution we studied the chiral transition in 2+1 flavor QCD close to the physical point. Using the $O(N)$ scaling we determined the chiral transition temperature in the continuum limit at the physical light quark mass to be $154(9)$ MeV. We also studied the deconfinement phenomenon in terms of the renormalized Polyakov loop as well as in terms of fluctuations and correlations of conserved charges. We concluded that it is difficult to study the deconfinement aspects of the transition in terms of the Polyakov loop, while correlations and fluctuations of conserved charges are much better suited for this purpose. If $f_K$ is used to set the scale we see a good agreement with the HRG model predictions for fluctuations and correlations that do not involve pions in their HRG expansion. For the quantities like $\chi_l$ and $\chi_{ud}$, where the pion contribution is significant, cut-off effects are still too large for a meaningful comparison with the HRG model.
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[^1]: HotQCD Collaboration members are: A. Bazavov, T. Bhattacharya, M. Cheng, N.H. Christ, C. DeTar, H.-T. Ding, S. Gottlieb, R. Gupta, P. Hegde, U.M. Heller, C. Jung, F. Karsch, E. Laermann, L. Levkova, R.D. Mawhinney, S. Mukherjee, P. Petreczky, D. Renfrew, C. Schmidt, R.A. Soltz, W. Soeldner, R. Sugar, D. Toussaint, P. Vranas
[^2]: The lightest mass $m_l=m_s/40$ is used for the analysis of the chiral condensate and susceptibility [@future], while all other quantities are calculated at $m_l=m_s/20$ for HISQ/tree.
| ArXiv |
---
abstract: |
The purpose of this paper is to find optimal estimates for the Green function of a half-space of [*the relativistic $\alpha$-stable process*]{} with parameter $m$ on ${\mathbb{R}^d}$ space. This process has an infinitesimal generator of the form $mI-(m^{2/\alpha}I-\Delta)^{\alpha/2},$ where $0<\alpha<2$, $m>0$, and reduces to the isotropic $\alpha$-stable process for $m=0$ . Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets. The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space. As a byproduct we obtain some improvements of the estimates known for bounded sets.
Our approach combines the recent results obtained in [@ByczRyzMal], where an explicit integral formula for the $m$-resolvent of a half-space was found, with estimates of the transition densities for the killed process on exiting a half-space. The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one. On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic $\alpha$-stable process. For example, for $d\ge3$, it is comparable with the Green function for the isotropic $\alpha$-stable process, provided that the points are close enough.
author:
- |
T. Grzywny, M. Ryznar\
Institute of Mathematics and Computer Sciences,\
Wrocław University of Technology, Poland
title: 'Two-sided optimal bounds for Green functions of half-spaces for relativistic $\alpha$-stable process'
---
**Keywords:** [stable relativistic process, Green function, first exit time from a ball, tail function]{}\
**Mathematics Subject Classifications (2000):** [60J45]{}
Introduction
============
In the paper we deal with some aspects of the potential theory of the $\alpha$-[*stable relativistic process*]{}. That is a L[é]{}vy process on ${\mathbb{R}^d}$ with a generator of the form $$H^{m}_{\alpha}=mI-(m^{2/\alpha} I-\Delta)^{\alpha/2},\quad\ 0<\alpha <2, \ m>0.$$ For $m=0$ the operator above reduces to the generator of the $\alpha$-[*stable rotation invariant (isotropic) L[é]{}vy process*]{} which potential theory was intensively studied in the literature.
For $\alpha=1$ the operator $$H^{m}_{1}=mI-(m^2 I-\Delta)^{1/2}$$ plays a very important role in relativistic quantum mechanics since it corresponds to the kinetic energy of a relativistic particle with mass $m$. Generators of this kind were investigated for example by E. Lieb [@L] in connection with the problem of stability of relativistic matter. An interested reader will find references on this subject e.g. in a recent paper [@KS].
Another reason that the operator $H^{m}_{\alpha}$ is an interesting object of study is its role in the theory of the so-called [*interpolation spaces of Bessel potentials*]{} and its application in harmonic analysis and partial differential equations (see, e.g. [@S] and [@Ho]). This theory is based on [*Bessel potentials*]{} defined as $J_{\alpha}=
(I-\Delta)^{-\alpha/2}$. As Stein pointed out in his monograph [@S], the Bessel potentials exhibit the same [*local*]{} behaviour (as $|x| \to 0$) as the Riesz potentials but the [*global*]{} one (as $|x| \to \infty$) of $J_{\alpha}$ is much more regular. In terms of the relativistic process the potential $J_{\alpha}$ is so-called $1$-resolvent kernel of the semigroup generated by $ H^{1}_{\alpha}$.
In the paper we consider the process killed on exiting the half-space ${\mathbb{H}}=\{x\in {\mathbb{R}^d}: x_d>0 \}$ and examine the behaviour of its Green function $G_{\mathbb{H}}(x,y)$. Contrary to the stable case a closed formula for that Green function is not know and seems to be a very challenging target. Recently in [@ByczRyzMal] an integral formula in terms of the Macdonald functions was found for $G^m_{\mathbb{H}}(x,y)$ - the $m$-resolvent kernel for ${\mathbb{H}}$. As proved in [@ByczRyzMal], for $d\ge 3$, the behaviour of the Green function is equivalent to the behaviour the $m$-resolvent if $|x-y| \to 0$. Our main result establishes optimal bounds for the Green function of ${\mathbb{H}}$. To our best knowledge it is the first result of that type when optimal estimates for unbounded set (different than the whole ${\mathbb{R}^d}$) are derived.
At this point let us mention that the potential theory for bounded sets has been well developed during recent years (see [@CS], [@Ry], [@KL], [@GRy]). Under various assumptions of the regularity of a bounded open set $D$ it was shown that the Green function of $D$ was comparable with its stable counterpart. This comparison allowed to prove the relativistic potential theory shares most of the properties of the stable one if bounded sets are considered. Comparing the potential kernel for the stable process with the potential kernel for the relativistic process (see [@RSV]) we can conclude that such a comparison of Green functions is not generally possible for unbounded sets. Since the relativistic potential kernel (for $d\ge 3$) is asymptotically equivalent (if $|x-y|$ is large) to that of the Brownian motion it may suggest that the Green function of ${\mathbb{H}}$, at least for some part of the range of $x,y$, is comparable with the Green function of ${\mathbb{H}}$ for the Brownian motion. Our main result confirms that suggestion and we prove the comparability for points $x,y$ being away from the boundary and with $|x-y|\ge 1$. For other points our bound is also optimal.
We also thoroughly examine the one-dimensional case and provide optimal estimates for the Green functions for bounded intervals taking into account their length. While for intervals of moderate length (say smaller than $1$) we can use the well known results about comparability of stable and relativistic Green functions, for large intervals we relay on the estimates for half-lines obtained in this paper. Again we show that the Green functions for large intervals are comparable to the Brownian Green functions for most of the range.
The organization of the paper is as follows. In Section 2 we collect all definitions and preliminary results needed for the rest of the paper. The next section is basic for the paper. Here we prove the estimates for the Green function of $(0,\infty)$. Then in Section 4 we apply them to prove the optimal bounds for the tail function of the exit time from $(0,\infty)$ and some other properties of the exit times. These estimates will have a crucial role in examining multidimensional case which was accomplished in Section 5. We conclude the paper with exploring in the last section the one-dimensional case with regard to optimal estimates for bounded intervals.
Preliminaries
=============
Throughout the paper by $c, C, C_1\,\dots$ we denote nonnegative constants which may depend on other constant parameters only. The value of $c$ or $C, C_1\,\ldots$ may change from line to line in a chain of estimates.
The notion $p(u)\approx q(u),\ u \in A$ means that the ratio $p(u)/ q(u),\ u \in A$ is bounded from below and above by positive constants which may depend on other constant parameters only but does not depend on the set $A$.
We present in this section some basic material regarding the $\alpha$-stable relativistic process. For more detailed information, see [@Ry] and [@C]. For questions regarding Markov and strong Markov property, semigroup properties, Schrödinger operators and basic potential theory, the reader is referred to [@ChZ] and [@BG].
We first introduce an appropriate class of subordinating processes. Let $\theta_{\alpha}(t,u)$, $u, t>0$, denote the density function of the strictly $\alpha/2$-stable positive standard subordinator, $0<\alpha<2$, with the Laplace transform $e^{-t
\lambda^{\alpha/2}}$.
Now for $m>0$ we define another subordinating process $T_{\alpha}(t,m)$ modifying the corresponding probability density function in the following way: $$\theta_{\alpha}(t,u,m)= e^{mt}\,\theta_{\alpha}(t,u)\,e^{-m^{2/\alpha}u}, \quad u>0\,.$$ We derive the Laplace transform of $T_{\alpha}(t,m)$ as follows: $$\label{subord2}
E^{0} e^{-\lambda T_{\alpha}(t,m)} =e^{mt}\, e^{-t (\lambda+m^{2/\alpha})^{\alpha/2}}, \quad \lambda\ge -m^{2/\alpha}.$$ Let $B_t$ be the symmetric Brownian motion in ${\mathbb{R}^d}$ with the characteristic function of the form $$\label{brownian}
E^{0}e^{i\xi \cdot B_t} = e^{-t|\xi|^2}\,.$$ Assume that the processes $T_{\alpha}(t,m)$ and $B_t$ are stochastically independent. Then the process $X_t^{\alpha,m}= B_{T_{\alpha}(t,m)}$ is called the $\alpha$-stable relativistic process (with parameter $m$). In the sequel we use the generic notation $X^m_t$ instead of $X_t^{\alpha,m}$. If $m=1$ we write $T_{\alpha}(t)$ instead of $T_{\alpha}(t,m)$ and $X_t$ instead of $X_t^1$. From (\[subord2\]) and (\[brownian\]) it is clear that the characteristic function of $X^m_t$ is of the form $$ E^{0}e^{i\xi \cdot X^m_t } = e^{mt}e^{-t(|\xi|^2+m^{2/\alpha})^{\alpha/2}}\,.$$ Obviously in the case $m=0$ the corresponding process is the standard (rotationally invariant or isotropic) $\alpha$-stable process. $X^m_t$ is a Lévy process (i.e. homogeneous, with independent increments). We always assume that sample paths of the process $X^m_t$ are right-continuous and have left-hand limits (“cadlag”). Then $X^m_t$ is Markov and has the strong Markov property under the so-called standard filtration.
From the form of the Fourier transform we have the following scaling property: $$\label{scaling}
{p}^m_{t}(x)=m^{d/\alpha}{p}^1_{mt}(m^{1/\alpha}x).$$ In terms of one-dimensional distributions of the relativistic process (starting from the point $0$) we obtain $$X^m_t \sim m^{-1/\alpha} X_{mt}\,,$$ where $X_t$ denotes the relativistic $\alpha$-stable process with parameter $m=1$ and “$\sim$” denotes equality of distributions. Because of this scaling property, we usually restrict our attention to the case when $m=1$, if not specified otherwise. When $m=1$ we omit the superscript “$1$”, i.e. we write $p_{t}(x)$ instead of $p_t^1(x)$, etc.
Various potential-theoretic objects in the theory of the process $X_t$ are expressed in terms of modified Bessel functions $K_{\nu}$ of the second kind, called also Macdonald functions. For convenience of the reader we collect here basic information about these functions. $K_\nu,\ \nu\in \mathbb{R} $, the modified Bessel function of the second kind with index $\nu$, is given by the following formula: $$K_\nu(r)= 2^{-1-\nu}r^\nu \int_0^\infty e^{-u}e^{- \frac{r^2}{4u}}u^{-1-\nu}du\,,
\quad r>0.$$ For properties of $K_\nu$ we refer the reader to [@E1]. In the sequel we will use the asymptotic behaviour of $K_\nu$:
$$\begin{aligned}
K_\nu(r)& \cong& \frac{\Gamma(\nu)}{2} \left(\frac{r}{2}\right)^{-\nu},\quad r\to 0^+, \ \nu>0,
\label{asympt0}\\
K_0(r)&\cong& -\log r, \quad r\to 0^+, \label{asympt00}\\
K_\nu(r)&\cong& \frac {\sqrt{\pi}} {\sqrt{2 r}}\,e^{-r}, \quad r\to \infty, \label{asympt_infty}
\end{aligned}$$
where $g(r) \cong f(r) $ denotes that the ratio of $g$ and $f$ tends to $1$. For $\nu <0 $ we have $K_\nu(r)=K_{-\nu}(r)$, which determines the asymptotic behaviour for negative indices.
The $\alpha$-stable relativistic density (with parameter $m=1$) can now be computed in the following way: $$\label{reldensity0}
p_t(x)=\int_0^\infty e^{t}\,\theta_{\alpha}(t,u)\,e^{-u}\, g_u(x) du,$$ where $g_u(x)=\frac1{(4\pi u)^{d/2}}e^{- \frac{|x|^2}{4u}}$ is the Brownian semigroup, defined by [(\[brownian\])]{}.
We also recall the form of the density function $\nu(x)$ of the L[é]{}vy measure of the relativistic $\alpha$-stable process (see [@Ry]): $$\begin{aligned}
\nu(x)&=& \frac \alpha{2\Gamma(1- \frac{\alpha }{2})}\int_0^\infty e^{-u}\, g_u(x)\, u^{-1-\alpha/2} du\label{levymeasure0}\\
&=&
\frac{\alpha 2^{\frac{\alpha -d}{ 2}}}{ \pi^{d/2} \Gamma(1- \frac{\alpha }{2})}
|x|^{ -\frac{d+\alpha }{ 2}}
K_{ \frac{d+\alpha }{ 2}}(|x|)\,.\label{levymeasure}
\end{aligned}$$
In the case $0<\alpha <2$ we have the following useful estimates (see [@Ry] for the proof of the first lemma):
\[transden\_0\] There exists a constant $c=c(\alpha,d)$ such that $$\label{transupper}
\max_{x \in {{\mathbb{R}^d}}} p_t(x) \le c (t^{-d/2} + t^{-d/{\alpha}}) \,.$$
\[transden2\] For any $t>0$ and $x\in {\mathbb{R}^d}$ we have $$p_t(x)\le c(d,\alpha) \left(g_{t}(x/\sqrt{2})+t\nu(x/\sqrt{2})\right)$$ and $$p_t(x)\le \frac{c(d)}{|x|^d}.$$
Notice that for $u,t>0$, $$\theta_{\alpha}(t,u)=t^{-2/\alpha}\theta_{\alpha}(1,t^{-2/\alpha}u) \quad \textrm{and}\quad \theta_{\alpha}(1,u)\le c u^{-1-\alpha/2}.$$ Hence $$\label{thetagora}\theta_{\alpha}(t,u)\le c t u^{-1-\alpha/2},\qquad t,u>0.$$ Using [(\[reldensity0\])]{} we obtain for $t\ge 1$, $$\begin{aligned}
p_t(x)&\le&
e^{-\frac{|x|^{2}}{8t}}(4\pi)^{-\frac{d}{2}}e^{t}\int^{2
t}_{0}\theta_{\alpha}(t,u)e^{-u}u^{-d/2}du\\& & +\; c t
\int^{\infty}_{2t}g_u(x)e^{-\frac{u}{2}}u^{-1-\alpha/2}du\\
&\le& e^{-\frac{|x|^{2}}{8t}}(4\pi)^{-\frac{d}{2}}e^{t}
\int^{\infty}_{0}\theta_{\alpha}(t,u)e^{-u}u^{-d/2}du\\ & & +\; c
t
\int^{\infty}_{0}g_u(x)e^{-\frac{u}{2}}u^{-1-\alpha/2}du\\
&=& e^{-\frac{|x|^{2}}{8t}}p_t(0)+c t\nu(x/\sqrt{2}),
\end{aligned}$$ where we used [(\[levymeasure0\])]{} in the last line. Moreover by Lemma \[transden\_0\] we can estimate $$e^{-\frac{|x|^{2}}{8t}}p_t(0)\le c g_{t}(x/\sqrt{2}),\quad t\ge 1.$$ This completes the proof of the first estimate for $t\ge 1$. Next, for $t\le 1$, applying (\[reldensity0\]), (\[thetagora\]) and (\[levymeasure0\]) we arrive at $$p_t(x)\le c t \int^\infty_0 g_u(x)e^{-u}u^{-1-\alpha/2}du=c t
\nu(x)\le c t \nu(x/\sqrt{2}),$$ which complete the proof the first inequality.
The second bound is true for the transition density of any subordinated Brownian motion. Indeed let us observe that for any $t>0$ and $x\in{\mathbb{R}^d},$ $$g_t(x)\le \left(\frac{d}{2\pi}\right)^{d/2}e^{-\frac{d}{2}}|x|^{-d}.$$ Hence by subordination $$p_t(x)=Eg_{T_\alpha(t)}(x)\le
\left(\frac{d}{2\pi}\right)^{d/2}e^{-\frac{d}{2}} |x|^{-d}.$$
The standard reference book on general potential theory is the monograph [@BG]. For convenience of the reader we collect here the basic information with emphasis on what is known (and needed further on) about the $\alpha$-stable relativistic process.
In general potential theory a very important role is played by $\lambda$-resolvent (potential) kernels, $\lambda>0$ , which are defined as
$$U_\lambda(x,y)=\int_0^\infty e^{-\lambda t} p_t(x-y)dt, \ x,y\in {\mathbb{R}^d}.$$
If the defining integral above is finite for $\lambda=0$, the corresponding kernel is called a potential kernel and will be denoted by $U(x,y)$. For the relativistic process the potential kernel is well defined for $d\ge 3$ but contrary to the stable or Brownian case it is not expressible as an elementary function. Recall that for the isotropic $\alpha$-stable process the potential kernel is equal to $C|x-y|^{\alpha-d}$ for $d>\alpha$ and for the Brownian motion it is $C|x-y|^{2-d}$ for $d\ge 3$, where $C$’s are appropriate constants. One can prove that the relativistic potential kernel could be written as a series involving the Macdonald functions of different orders but this formula does not seem very useful. Nevertheless the asymptotic behaviour of the potential kernel was established in [@G], [@RSV].
$$\label{0-potential}
U(x-y)\approx |x-y|^{\alpha-d}, |x-y|\le 1;\quad
U(x-y)\approx |x-y|^{2-d}, |x-y|\ge 1.$$
Note that they suggest that the process locally behaves like a stable one and globally like a Brownian motion. Despite the fact we do not know any simple form for the potential kernel, a formula for the $1$-potential kernel is known (e.g. see [@ByczRyzMal]): $$ U_1(x)=C(\alpha,d)\,
\frac{K_{(d-\alpha)/2}(|x|)}{ |x|^{(d-\alpha)/2}}\,,$$ where $C(\alpha,d)= \frac{2^{1-(d+\alpha)/2} }{ {\Gamma(\alpha/2)\pi^{d/2}}}$.
The [*first exit time*]{} of an (open) set $D\subset {{\mathbb{R}^d}}$ by the process $X_t$ is defined by the formula $$\tau_{D}=\inf\{t> 0;\, X_t\notin D\}\,.$$
The basic object in potential theory of $X_t$ is the $\lambda$-[*harmonic measure*]{} of the set $D$. It is defined by the formula: $$ P_D^{\lambda}(x,A)=
E^x[\tau_D<\infty; e^{-\lambda \tau_D} {\bf{1}}_A(X_{\tau_D})].$$ The density kernel of the measure $P_D^{\lambda}(x,A)$ (if it exists) is called the $\lambda$-[*Poisson kernel*]{} of the set $D$. If $\lambda=0$ the corresponding kernel will be denoted by $P_D(x,z)$ called [*Poisson kernel*]{} of the set $D$.
Another fundamental object of potential theory is the [*killed process*]{} $X_t^D$ when exiting the set $D$. It is defined in terms of sample paths up to time $\tau_D$. More precisely, we have the following “change of variables” formula: $$E^x f(X_t^D) = E^x[t<\tau_D; f(X_t)]\,,\quad t>0\,.$$ The density function of transition probability of the process $X_t^D$ is denoted by $p_t^{D}$. We have $$p_t^{D}(x,y) = p_t(x-y) -
E^x[t> \tau_D; p_{ t-\tau_D}(X_{\tau_D}-y)] \,, \quad x, y \in {{\mathbb{R}^d}}\,.\label{density100}$$ Obviously, we obtain $$p_t^{D}(x,y) \le p_t(x,y) \,, \quad x, y \in {{\mathbb{R}^d}}\,.$$
$p_t^{D}$ is a strongly contractive semigroup (under composition) and shares most of properties of the semigroup $ p_t$. In particular, it is strongly Feller and symmetric: $ p_t^{D}(x,y) = p_t^{D}(y,x)$.
The $\lambda$-potential of the process $X_t^D$ is called the $\lambda$-[*Green function*]{} and is denoted by $G_D^{\lambda}$. Thus, we have $$G_D^{\lambda}(x,y)= \int_0^{\infty} e^{-\lambda t}\,p_t^{D}(x,y)\,dt\,.$$
If $\lambda=0$ the corresponding kernel will be called [*Green function*]{} of the set $D$ and denoted $G_D(x,y)$.
Integrating (\[density100\]) we obtain for $\lambda >0$, $$G^\lambda_{D}(x,y)=U_\lambda (x,y)-E^x\, e^{-\lambda\tau_{D}}U_\lambda (X_{\tau_{D}},y).$$ Suppose that $D_1\subset D_2$ are two open sets. By the Strong Markov Property $$\begin{aligned}
& &\!\!\!\! G^\lambda_{D_2}(x,y)-G^\lambda_{D_1}(x,y)\nonumber\\
& &\!= E^x\left[e^{-\lambda\tau_{D_1}}
U_\lambda(X_{\tau_{D_1}},y)-e^{-\lambda\tau_{{D_2}}}
U_\lambda(X_{\tau_{D_2}},y)\right]\nonumber\\& &\!=
E^x\left[\tau_{D_1}<\tau_{D_2};e^{-\lambda\tau_{D_1}}
\left(U_\lambda(X_{\tau_{D_1}},y)-e^{-\lambda\tau_{D_2}\circ\theta_{\tau_{D_1}}}U_\lambda(X_{\tau_{D_2}},y)\right)\right]\nonumber\\
& &\!=E^x\left[\tau_{D_1}<\tau_{D_2};e^{-\lambda\tau_{D_1}} \left(U_\lambda(X_{\tau_{D_1}},y)-E^{X_{\tau_{D_1}}}e^{-\lambda\tau_{D_2}}U_\lambda(X_{\tau_{D_2}},y)\right)\right]\nonumber\\
& &\! = E^x\left[\tau_{D_1}<\tau_{D_2};e^{-\lambda\tau_{D_1}}
G^\lambda_{D_2}(X_{\tau_{D_1}},y)\right].\label{greenpot100}\end{aligned}$$
The main purpose of the present paper is to obtain sharp estimates for the Green function for $D={\mathbb{H}}=\{x\in {\mathbb{R}^d}:\, x_d>0\}$. The investigation of Green functions of the relativistic process for unbounded sets seems not to be treated in the literature. For bounded sets there many results obtained in recent years showing that the Green functions for open bounded sets under some assumptions about regularity of their boundary are comparable to their stable counterparts in ${\mathbb{R}^d}$, $d> \alpha$ ([@Ry], [@CS], [@KL]). That is, for $x,y\in D$, $$\label{comparison}
C(D)^{-1} G^{stable}_D(x,y)\le G_D(x,y)\le C(D) G^{stable}_D(x,y),$$ where $G_D^{stable}$ is the corresponding Green function for the isotropic stable process and $C(D)$ is a constant usually dependent on $\textrm{diam} (D)= sup_{x,y\in D}|x-y|$. Unfortunately in all known general bounds of the above type the dependence on the set $D$ in the constant $C(D)$ is not very clear and $C(D)$ grows to $\infty$ with $\textrm{diam} (D)$. The constant also depends on some other characteristics of $D$ as e.g. Lipschitz characteristic of $D$ when $D$ is a Lipschitz set. Therefore it is not possible to use well known exact formulas or estimates for the stable Green functions of regular sets as half-spaces, balls or cones to derive the corresponding optimal estimates for the relativistic process. Even for balls the constants grow to $\infty$ and (\[comparison\]) does not yield any estimate for a half-space in the limiting procedure.
Now suppose that $D$ is a bounded set with a $C^{1,1}$ boundary. It is well known that there is a $\rho>0$ such that for each point $ z\in \partial D$ there are balls $B_z\subset D$, $B^*_z
\subset D^c$ of radius $\rho$ such that $z\in \overline{B_z}
\cap\overline{B_z^*}$. Denote by $\rho_0=\rho_0(D)$ the largest $\rho$ having the above property. Finally let $\gamma=
\textrm{diam}\, D/\rho_0$. However not explicitly stated, the following bound can be deduced from the results proved in [@Ry], for $d>\alpha$, $x,y\in D$: $$\begin{aligned}
\label{comparisonR}
C_1(\gamma)C( \textrm{diam}(D))^{-1} G^{stable}_D(x,y)&\le&G_D(x,y)\nonumber\\
&\le& C_2(\gamma)\,C( \textrm{diam}(D))
G^{stable}_D(x,y),\nonumber\\
\end{aligned}$$ where the constant $C$ can be chosen in such a way that $C(\textrm{diam}(D))=1$ for $\textrm{ diam}(D)\le 1$ and $C(\textrm{diam}(D))$ increases with $\textrm{diam}(D)$. With some extra effort one can prove that the growth is polynomial. The constants $C_1(\gamma), C_2(\gamma)$ can be chosen as continuous with respect to $\gamma$. Note that if $D$ is a ball than we can take absolute constants (depending only on $\alpha$ and $d$) instead of $C_1(\gamma), C_2(\gamma)$.
Hence for “smooth” sets with small or moderate diameter the estimate (\[comparisonR\]) is very satisfactory. For example for balls of small or moderate diameter we obtain very precise estimates using well known results for the isotropic stable process. However, in the case of balls of large size, it would be very interesting to find optimal estimates of the relativistic Green function. Our main result provides optimal estimates for the Green function of the half-space ${\mathbb{H}}$. Also we found optimal estimates for intervals in $\mathbb{R}$. Despite the fact we do not examine Green functions for balls in higher dimensional spaces we provide very precise estimates of the expected first exit time from a ball.
Now we define harmonic and regular harmonic functions. Let $u$ be a Borel measurable function on ${\mathbb{R}^d}$. We say that $u$ is [*harmonic*]{} function in an open set $D\subset {\mathbb{R}^d}$ if $$u(x)=E^xu(X_{\tau_B}), \quad x\in B,$$ for every bounded open set $B$ with the closure $\overline{B}\subset D$. We say that $u$ is [*regular harmonic*]{} if $$u(x)=E^x[\tau_D<\infty; u(X_{\tau_D}))], \quad x\in D.$$
As a result of (\[comparisonR\]) we obtain the following version of the Boundary Harnack Principle (for details see [@Ry] or [@GRy] in the one-dimensional case).
\[BHP\]\[BHP\] Let $D$ be a bounded set with a $C^{1,1}$ boundary. Suppose that $diam\, D \le 4$ and $\rho_0(D)\ge 1$. Let $ z\in \partial D$. If $f$ is a non-negative regular harmonic function on D and $f(x)=0$, $x\in B(z,1)\cap D^c$. Then $$f(x)\approx f(x_0)\delta_D(x)^{\alpha/2},\quad x\in B(z,1/2),$$ where $\delta_D(x)=dist(x, \partial D )$ and $x_0 \in D$ such that $\delta_D(x_0)=1$.
For the purpose of this paper we state the following specialized form of BHP which can be easily deduced from Theorem \[BHP\].
\[BHP1\] Let ${\mathbb{H}}\ni\mathbf{1}=(0,\dots,0,1)$ and let $F= B(0,
\sqrt{2})\cap {\mathbb{H}}$. Suppose that $f$ is a regular nonnegative harmonic on $F$ such that $f(x)=0,\ x\in {\mathbb{H}}^c$. Then for every $
x\in B(0, 1)\cap {\mathbb{H}}$ we have $$f(x)\approx f(\mathbf{1})x_d^{\alpha/2}.$$ Assume that $R\ge 2$. Let $D=B(0, R)$, $z_0=(0,\dots,0,R)$ and $x_0=(0,\dots,0,R-1)$. Let $F= B(z_0,
2)\cap D$. Suppose that $f$ is regular nonnegative harmonic on $F$ such that $f(x)=0,\ x\in D^c$. Then for every $ x\in B(z_0, 1)\cap
D$ we have $$f(x)\approx f(x_0)(R-|x|)^{\alpha/2}.$$
As mentioned above, the one-dimensional case for intervals was treated recently in [@GRy] and since we will need it in the next section we present it in a convenient form of the estimate of the Poisson kernel. Actually in [@GRy] it was shown that the Green function of $(0,R)$ is comparable with the Green function of the corresponding stable process (with uniform constant for $R\le 3$). By standard arguments (see [@Ry]) this implies the lemma below.
\[GrzywnyRyznar\] Assume that $d=1$ and $0<R\le 3$. Let $D=(0,R)$. Then $$P_D ( x, z)\approx \frac {(x(R-x))^{\alpha/2}}{(R(z-R))^{\alpha/2}(z-x)}\ e^{-z}
, \quad x\in D,\ z>R.$$ This implies that $$P^x ( X_{\tau_D}>R)\approx (x/R)^{\alpha/2}, \quad x\in D$$ and $$E^x [X_{\tau_D}>R;X_{\tau_D}]\approx (x/R)^{\alpha/2}((R-x)^{\alpha/2}+x),\quad x\in D.$$ We also have that $$E^x\tau_D \approx {(x(R-x))^{\alpha/2}},\quad x\in D.$$
Obtaining any exact formulas for the Green function or the Poisson kernel even for regular sets seems to be a very hard task but in the recent paper [@ByczRyzMal] the formulas for the 1-Poisson and 1-Green function of ${\mathbb{H}}$ were described explicitly in terms of the Macdonald functions:
\[rel1Poisson\] Let $$E^x[ e^{-\tau_{\mathbb{H}}},X_{\tau_{\mathbb{H}}}\in du]=P_{{\mathbb{H}}}^1(x,u)\,$$ be the 1-Poisson kernel for ${\mathbb{H}}$. Then we have $$ P_{{\mathbb{H}}}^1(x,u)=2\frac {\sin (\alpha \pi/2) }{\pi} (2\pi)^{-d/2}
\left(\frac{x_d}{ -u_d} \right)^{\alpha/2} \,
\frac{ K_{d/2}(|x-u|)}{ |x-u|^{d/2}},$$ where $u_d<0<x_d$. Let $G_{{\mathbb{H}}}^1(x,y)$ be the 1-Green function for ${\mathbb{H}}$ then for $x,y\in {\mathbb{H}}$, $$G_{{\mathbb{H}}}^1(x,y) = \frac{2^{1-\alpha} |x-y|^{\alpha-d/2}} { (2\pi)^{d/2} \Gamma(\alpha/2)^2}
\int_0^{\frac{4 x_d y_d}{ |x-y|^2} } \frac{t^{\frac{\alpha}{ 2} -1}}{ (t+1)^{d/4}}
K_{d/2}(|x-y|(t+1)^{1/2}) dt.$$ Moreover,
$$\begin{aligned}
\label{intgral}
\int_{{\mathbb{H}}}G_{{\mathbb{H}}}^1(x,y)dy&= &1-E^x e^{-\tau_{\mathbb{H}}}\nonumber\\&=& \frac{ 1}{ {\Gamma(\alpha/2)}} \int_0^{x_d}t^{\alpha/2-1} e^{-t}\,dt\,, \ x\in {\mathbb{H}}\,.
\end{aligned}$$
This result will be very useful in our analysis since, as shown in [@ByczRyzMal] the behaviour of the Green function $G_{{\mathbb{H}}}(x,y)$ could be described in terms of the 1-Green function $G_{{\mathbb{H}}}^1(x,y)$ when $x$ and $y$ are close enough.
One of our main tools in establishing the upper bounds of the Green function will be estimates for the tail function $P^x(\tau_{\mathbb{H}}> t)$. We start with the following lemma taken from the Master Thesis of the first author [@G].
\[Grzywny\] There is a constant C such that $$\label{Grzywny1}P^x(\tau_{\mathbb{H}}> t)\le
C \frac{x_d + \ln (t+1)}{t^{1/2}}\,,\quad t\ge 1\,,\, x_d>0.$$
Let $Y_t=X^{(d)}_t$, where $X_t=(X^{(1)}_t,\ldots,X^{(d)}_t)$. By the symmetry of the random variable $Y_t$ we obtain $$\begin{aligned}
P^x (\tau_{\mathbb{H}}>t)&=& P^x (\inf_{s\le t}Y_s >0)\\&=& P^0
(\inf_{s\le t}(-Y_s+x_d) >0)= P^0 (\sup_{s\le t}Y_s <
x_d).\end{aligned}$$ Using a version of the Lévy inequality ([@B], Ch.7, 37.9) we have for any $\varepsilon,y>0$ that $$2 P^0 (Y_t\ge
y+2\varepsilon)-2\sum^n_{k=1}P^0(Y_{\frac{tk}{n}}-Y_{\frac{t(k-1)}{n}}\ge
\varepsilon) \le P^0 (\sup_{k\le n}Y_{\frac{tk}{n}} \ge y).$$ Note that $\sum^n_{k=1}P^0(Y_{\frac{tk}{n}}-Y_{\frac{t(k-1)}{n}}\ge
\varepsilon)=nP^0(Y_{\frac{t}{n}}\ge \varepsilon)\to
t\int^{\infty}_\varepsilon\nu(x)dx$, hence, by symmetry again $$\begin{aligned}
P^0
(\sup_{s\le t}Y_s \ge y)&\ge& 2 P^0 (Y_t\ge
y+2\varepsilon)-2t\int^{\infty}_{\varepsilon}\nu(x)dx\\&=&P^0
(|Y_t|\ge y+2\varepsilon)-2t\int^{\infty}_{\varepsilon}\nu(x)dx
.\end{aligned}$$ This implies that $$P^x (\tau_{\mathbb{H}}>t)=P^0 (\sup_{s\le t}Y_s < x_d)\le P^0 (|Y_t| <x_d+2\varepsilon)
+2t\int^{\infty}_{\varepsilon}\nu(x)dx\,.$$ For $\varepsilon\ge
1$ we obtain from [(\[levymeasure\])]{} and [(\[asympt\_infty\])]{} $$\int^{\infty}_{\varepsilon}\nu(x)dx\le C e^{-\varepsilon}\varepsilon^{-\alpha/2-1}.$$ Lemma \[transden\_0\] implies that the density of $Y(t)$ is bounded by $Ct^{-1/2}$, $t\ge 1$, hence taking $\varepsilon=\frac{3}{2}\ln (t+1)$ we obtain $$P^x (\tau_{\mathbb{H}}>t)\le
C \left(x_d+\ln (t+1)\right) t^{-1/2}.$$
In order to improve the above estimate for $x$ close to the boundary we use Lemma \[GrzywnyRyznar\].
\[Grzywnyimprove\] For $0<x_d<2$ we have $$\label{uboundtail}
P^x(\tau_{\mathbb{H}}> t)\le C x_d^{\alpha/2}\;\ln (t+1)/t^{1/2},\quad t\ge 2,$$ where $C$ is a constant.
It is enough to prove the claim for $d=1$. Let $D=(0,2)$ and assume that $0<x<2$.
By the Strong Markov Property and then by Lemma \[Grzywny\] we obtain for $t\ge 1$: $$\begin{aligned}
P^x(\tau_{(0,\infty)}> 2t)&\le& P^x(\tau_D> t, \tau_{(0,\infty)}> 2t)\\& &
+\,
E^x[\tau_D<\tau_{(0,\infty)}\,; P^{X_{\tau_D}}(\tau_{(0,\infty)}> t)]\\
&\le& P^x(\tau_D> t)\\& &+\,
CE^x [\tau_D<\tau_{(0,\infty)}; X_{\tau_D} + \ln (t+1)]/t^{1/2}\\
&\le& \frac{E^x\tau_D}{t}+
CE^x [X_{\tau_D}>2\,;X_{\tau_D}]/t^{1/2}\\
& &+\, C\ln (t+1)\;P^x (X_{\tau_D}>2)/t^{1/2}\\
&\le&C x^{\alpha/2}\ln (t+1)/t^{1/2}.
\end{aligned}$$ The last inequality follows from Lemma \[GrzywnyRyznar\]. The proof is complete.
The estimates from Lemmas \[Grzywny\], \[Grzywnyimprove\] will be very useful for the estimates of the Green function of the half-line, however they are not optimal. In the sequel we will be able to improve them to be sharp enough and optimal (see Proposition \[optimaltail\]). This will have a great importance in estimating the Green function for a half-space in the $d$-dimensional case.
\[density\] There is a constant $C$ such that for any open set $D$: $$E g_{T_{\alpha}(t)} ^D(x,y) \le p_t^{D}(x,y)
\le C (t^{-d/2}+ t^{-d/\alpha}) P^x(\tau_D> t/3)P^y(\tau_D>
t/3),$$ where $g_t ^D(x,y)$ is the transition probability for the Brownian motion killed on exiting $D$.
We start with the upper bound. Since $p_t^{D}(x,y)$ is a density of a semigroup and $p_t^{D}(x,y)\le \max_{z \in {{\mathbb{R}^d}}} p_t(z)$ then we have $$\begin{aligned}
p_{3t}^{D}(x,y)&=&\int_D \int_D
p_t^{D}(x,z)p^D_t(z,w)p_t^{D}(w,y)dz \,dw\\&\le& \max_{z \in
{{\mathbb{R}^d}}} p_t(z)\int_D p_t^{D}(x,z)dz\int_D p_t^{D}(w,y)dw\\&=&
\max_{z \in {{\mathbb{R}^d}}} p_t(z)P^x(\tau_D> t)P^y(\tau_D>
t),\end{aligned}$$ which proves the upper bound since $\max_{z \in {{\mathbb{R}^d}}} p_t(z)\le
C (t^{-d/2}+ t^{-d/\alpha})$ by Lemma \[transden\_0\].
To get the lower bound we use the subordination of the process to the Brownian motion: $X_t=B_{T_{\alpha}(t)}$. Then $$\begin{aligned}
p_t^{D}(x,y)&=&P^x( B_{T_{\alpha}(t)}\in dy,
B_{T_{\alpha}(s)}\in D, 0 \le s<t )\\&\ge&
P^x( B_{T_{\alpha}(t)}\in dy, B_s\in D, 0 \le s<T_{\alpha}(t)
).\end{aligned}$$ Using the independence of $T_{\alpha}$ and the Brownian motion $B$ we obtain $$P^x( B_{T_{\alpha}(t)}\in dy, B_s\in D, 0 \le s<T_{\alpha}(t)|T_{\alpha}(\cdot) )=
g^D_{T_{\alpha}(t)}(x,y),$$ Integrating we obtain the lower bound.
The following lemma provides a very useful lower bound. Its proof closely follows the approach used in [@RSV], where the bounds on the potential kernels (Green functions for the whole ${\mathbb{R}^d}$) were established for some special subordinated Brownian motions (in particular for our process) for $d \ge 3$.
\[potential\_lower\] For any open set $D\in {{\mathbb{R}^d}}$ we have $$G_D(x,y)\ge \frac2\alpha G_{D}^{gauss}(x,y),$$ where $G_{D}^{gauss}(x,y)$ is the Green function of $D$ for the Brownian motion.
Let $Q(x,y)=\int_0^\infty E g^D_{T_{\alpha}(t)}(x,y)dt$. From the previous lemma it is enough to prove that $Q(x,y)\ge \frac2\alpha G_{D}^{gauss}(x,y).$ We have $$\begin{aligned}
Q(x,y)&=&\int_0^\infty E g_{T_{\alpha}(t)} ^D(x,y)dt\\
&=&\int_0^\infty e^{t}\int_0^\infty g_u^D(x,y)e^{-u}\theta_\alpha(t,u) du dt\\
&=&\int_0^\infty g_u^D(x,y)e^{-u}\int_0^\infty e^{t}\theta_\alpha(t,u)dt du\\
&=&\int_0^\infty g_u^D(x,y)G(u) du,\end{aligned}$$ where $G(u)=e^{-u}\int_0^\infty e^{t}\theta_\alpha(t,u)dt$ is the potential kernel of the subordinator $T_\alpha(t)$. It was proved in [@RSV] that $G(u)$ is a completely monotone (hence decreasing) function and $\inf_{u> 0}G(u)=\lim_{u\to \infty}G(u)=C_\alpha$. We find the constant $C_\alpha$ by taking into account the asymptotics of the Laplace transform of $G(u)$ at the origin: $$\begin{aligned}
\int_0^\infty e^{-\lambda u}G(u) du &=& \int_0^\infty
e^{t}\int_0^\infty e^{-u(1+\lambda)}\theta_\alpha(t,u)du dt \\&=&
\int_0^\infty e^{t}e^{-(1+\lambda)^{\alpha/2} t}dt=
\frac 1 {(1+\lambda)^{\alpha/2} -1}\\&\cong& \frac 2{\lambda \alpha},\quad \lambda\rightarrow 0.\end{aligned}$$ Applying the monotone density theorem we obtain that $C_\alpha= 2/
\alpha$. Thus, since $g_u^D(x,y)\ge 0$, we finally obtain $$\begin{aligned}
Q(x,y)
&=&\int_0^\infty g_u^D(x,y)G(u) du\\
&\ge&\frac2\alpha \int_0^\infty g_u^D(x,y) du = \frac2\alpha
G_{D}^{gauss}(x,y).\end{aligned}$$
At this point let us recall that the exact formulas for the Brownian Green functions are well known for several regular sets as balls or half-spaces (see e.g. [@Ba]). Since some of them will be useful in the sequel we will list them for the future reference. Recall that the Brownian motion we refer to in this paper has its clock running twice faster then the usual Brownian motion. For the half-space ${\mathbb{H}}$, for $d\ge3$, we have that $$\begin{aligned}
G^{gauss}_{\mathbb{H}}(x,y)&=& C(d)\left[\frac1{|x-y|^{d-2}}- \frac1{|x-y^*|^{d-2}}\right]\nonumber\\
\label{gaussGreen} &\approx& \min\left\{\frac{ x_d y_d }{|x-y|^d}, \frac1{|x-y|^{d-2}}\right\}, \quad x,y\in{\mathbb{H}},\end{aligned}$$ where $y^*=(y_1,\dots,y_{d-1}, -y_d)\in {\mathbb{H}}^c$.\
For the half-space ${\mathbb{H}}$, for $d=2$, $$\label{gaussGreen2}G^{gauss}_{\mathbb{H}}(x,y)= \frac1{2\pi} \ln\left(1+4\frac{x_2y_2}{|x-y|^2}\right), \quad x,y\in{\mathbb{H}}.$$ In the one dimensional case $$\label{gaussGreen1}G^{gauss}_{(0,\infty)}(x,y)= x\wedge y, \quad x,y>0.$$ For the finite interval $(0, R)$ we have $$\label{gaussGreen3}G^{gauss}_{(0, R)}(x,y)= \frac {x(R-y) \wedge y(R-x)}R, \quad x, y\in (0,R).$$
\[continuity\]Let $D$ be an open subset of ${\mathbb{H}}$. For fixed $y\in {\mathbb{H}}$ the function $G_{{\mathbb{H}}}(\cdot,y)$ is regular harmonic on $D$ provided $ y \notin
\overline{D}$. The same conclusion holds if ${\mathbb{H}}$ is replaced by an open bounded set.
The proof is standard and is included for completeness. First observe that $G_{{\mathbb{H}}}(z,w)<\infty$ for $z\neq w$, which follows from Lemmas \[transden2\], \[Grzywny\] and \[density\]. Next, applying (\[greenpot100\]) with $D_2={\mathbb{H}}$ and $D_1=D$ we have
$$G^\lambda_{{\mathbb{H}}}(x,y)-G^\lambda_{D}(x,y)= E^x\left[\tau_{D}<\tau_{{\mathbb{H}}};e^{-\lambda\tau_{D}}G^\lambda_{{\mathbb{H}}}(X_{\tau_{D}},y)\right].\label{harmonic100}$$
If $y\notin \overline{D}$ then $\tau_D=0$ and $X_{\tau_D}=y,$ $P^y$- a.s. so $G^\lambda_{D}(x,y)=G^\lambda_D(y,x)=0$. Moreover $E^x G^\lambda_{{\mathbb{H}}}(X_{\tau_{{\mathbb{H}}}},y)=0$, which follows from the fact that $P^x\left(X_{\tau_{{\mathbb{H}}}} \in {\mathbb{H}}^c\setminus ({\mathbb{H}}^c)^r \right)=0$, where $({\mathbb{H}}^c)^r$ is a set of regular points of ${\mathbb{H}}^c$ and for every $z\in ({\mathbb{H}}^c)^r$, $y\in {\mathbb{R}^d}$ we have $G^\lambda_{{\mathbb{H}}}(z,y)=0$ (see [@BG]). This implies that (\[harmonic100\]) can be rewritten as $$G^\lambda_{{\mathbb{H}}}(x,y)= E^x\,e^{-\lambda\tau_{D}}G^\lambda_{{\mathbb{H}}}(X_{\tau_{D}},y).$$ Passing with $\lambda\to 0$ and observing that $G^\lambda_{{\mathbb{H}}}\nearrow G_{{\mathbb{H}}}$ we obtain the conclusion by the monotone convergence theorem.
The same arguments can be applied for any bounded set $F$, since there is a half-space containing $F$, which guarantees that the Green function $G_{F}(x,y)< \infty$ for $x\neq y$.
The estimates below following from Theorem \[rel1Poisson\] were proved in [@BRB]. They turn out to be useful in the next sections.
\[Green1est\] Assume that $d=1$ and $\alpha\ge 1$. When $|x-y|\ge 1\wedge
x\wedge y>0$ we obtain $$G^1_{(0,\infty)}(x,y)\approx \frac{e^{-|x-y|}}{|x-y|^{1-\alpha/2}}(1\wedge x\wedge y)^{\alpha/2},$$ while for $|x-y|<1\wedge x\wedge y$ we obtain $$\begin{aligned}
G^1_{(0,\infty)}(x,y)&\approx& \log \left[2\frac{1\wedge x\wedge y}{|x-y|}\right],\quad \textrm{if }\quad \alpha=1,\\
G^1_{(0,\infty)}(x,y)&\approx& (1\wedge x\wedge
y)^{\alpha-1},\quad \textrm{if }\quad \alpha>1.\end{aligned}$$ In the remaining case, $\alpha<d$, we have $$G^1_{\mathbb{H}}(x,y)\approx \frac{K_{(d-\alpha)/2}(|x-y|)}{|x-y|^{(d-\alpha)/2}}\left[\left(\frac{1\wedge x_d\wedge y_d}{|x-y|\wedge 1}\right)^{\alpha/2}\wedge 1\right].$$
Finally we state some basic scaling properties both for the Poisson kernel and the Green function. The proof employs the scaling property [(\[scaling\])]{} and consists of elementary but tedious calculation hence is omitted.
\[scaling1\] Let $D$ be an open subset of ${\mathbb{R}^d}$ and $P_{D,m}$, $G_{D,m}$ be the Poisson kernel, or the Green function, respectively, for $D$ for the process with parameter $m$. Then $$P_{D,m}(x,u)= m^{d/\alpha}P_{m^{1/\alpha}D}(m^{1/\alpha}x,m^{1/\alpha}u),\quad
x\in D, u\in D^c\,,$$ $$G_{D,m}(x,y)= m^{(d-\alpha)/\alpha}G_{m^{1/\alpha}D}(m^{1/\alpha}x,m^{1/\alpha}y),\quad x\in D, y\in D\,.$$ Thus, if $D$ is a cone with vertex at $0$ we obtain: $$P_{D,m}(x,u)= m^{d/\alpha}P_{D}(m^{1/\alpha}x,m^{1/\alpha}u),\quad x\in D, u\in D^c\,,$$ $$G_{D,m}(x,y)= m^{(d-\alpha)/\alpha}G_{D}(m^{1/\alpha}x,m^{1/\alpha}y), \quad x\in D,
y\in D\,.$$
Due to these scaling properties it is enough to investigate the case $m=1$.
Green function of half-line
===========================
In this section $d=1$ and the half-space ${\mathbb{H}}$ is a half-line, that is ${\mathbb{H}}=(0,\infty)$.
\[trivialbound\] Assume that $|x-y|\le 3$. Then there is $ C = C(\alpha)$ such that $$\left(1\wedge x\wedge y \right)^{\alpha/2}\le C G^1_{{(0,\infty)}}(x,y).$$
We use Theorem \[Green1est\]. First let $\alpha \ge 1$ and $|x-y|\ge 1\wedge x\wedge y>0$, then $$\begin{aligned}
G^1_{(0,\infty)}(x,y)&\ge&
ce^{-|x-y|}{|x-y|^{\alpha/2-1}}(1\wedge x\wedge y)^{\alpha/2}\\
&\ge& ce^{-3}3^{\alpha/2-1}(1 \wedge x\wedge y)^{\alpha/2}.\end{aligned}$$ Suppose that $|x-y|< 1\wedge x\wedge y$. For $\alpha=1$, $$G^1_{(0,\infty)}(x,y)\ge c \log \left[2\frac{1\wedge x\wedge y}{|x-y|}\right]\ge c\log 2\ge c (1\wedge x\wedge y )^{\alpha/2}.$$ For $\alpha>1$, $$G^1_{(0,\infty)}(x,y)\ge c (1\wedge x\wedge y )^{\alpha-1}\ge c (1\wedge x\wedge y )^{\alpha/2}.$$ Next, observe that $ K_{(1-\alpha)/2}(r)/r^{(1-\alpha)/2}$ is decreasing. Therefore for $\alpha<1$ we obtain $$\begin{aligned}
G^1_{(0,\infty)}(x,y)&\ge& c \frac{K_{(1-\alpha)/2}(|x-y|)}{|x-y|^{(1-\alpha)/2}}\left[\left(\frac{1\wedge x\wedge y}{|x-y|\wedge 1}\right)^{\alpha/2}\wedge
1\right]\\
&\ge & c\frac{K_{(1-\alpha)/2}(3)}{3^{(1-\alpha)/2}}\left(1\wedge
x\wedge y\right)^{\alpha/2}.\end{aligned}$$
\[green\]For $x,y>0$, $$G_{(0,\infty)}(x,y)\approx
G^1_{(0,\infty)}(x,y)+(x\wedge y)\vee (x\wedge y)^{\alpha/2}.$$
Throughout the whole proof we assume that $0<x\le y$. The proof will rely on the estimates of $P^x(\tau_{(0,\infty)}> t)$ derived in the previous section and the application of Lemma \[density\].
We proceed to estimate the Green function from above. First we split the integration $$\begin{aligned}
\int_0^\infty p_t^{{(0,\infty)}}(x,y)dt & =&
\int_0^{6} p_t^{{(0,\infty)}}(x,y)dt + \int_{6}^{\infty}
p_t^{{(0,\infty)}}(x,y)dt\\&=& V(x,y)+R(x,y).\end{aligned}$$ We start with the estimation of the second integral. Due to Lemma \[density\], $$\begin{aligned}
R(x,y)&=&3\int_{2}^{\infty}
p_{3t}^{{(0,\infty)}}(x,y)dt\\&\le& c\int_{2}^{\infty}
P^x(\tau_{(0,\infty)}> t)P^y(\tau_{(0,\infty)}> t)\, \frac{dt}{ t^{1/2}}. \end{aligned}$$ First consider the case $y< \sqrt{2}$. Then using [(\[uboundtail\])]{} we have $$R(x,y)\le C (xy)^{\alpha/2} \int_{2}^{\infty} \frac {(\ln t)^2}{t} \frac{dt}{t^{1/2}}
\le C (xy)^{\alpha/2}.$$ If $y\ge \sqrt{2}$, using (\[Grzywny1\]) we estimate $$\begin{aligned}
R(x,y)&\le&c\int_{2}^{\infty}\left(P^y(\tau_{(0,\infty)}>
t)\right)^2\, \frac{dt}{t^{1/2}} \\&=& c \int_{2}^{y^2}
\left(P^y(\tau_{(0,\infty)}> t)\right)^2\, \frac{dt}{t^{1/2}}
+ c \int_{y^2}^\infty \left(P^y(\tau_{(0,\infty)}> t)\right)^2 \frac{dt}{t^{1/2}}\\
&\le& C\int_2^{y^2}\, \frac{dt}{t^{1/2}} +
C\int_{y^2}^\infty \left(\frac{y + \ln t}{t^{1/2}}\right)^2\, \frac{dt}{t^{1/2}}\\
&\le& C y.\end{aligned}$$ Hence $$\label{upperbound}
G_{(0,\infty)}(x,y)\le C \left\{
\begin{array}{ll}
V(x,y)+(x\,y)^{\alpha/2}, & \hbox{$y<1$,} \\
V(x,y)+y, & \hbox{$y\ge 1$.} \\
\end{array}
\right.$$
Let $B=(n+2,\infty)$, $n\in\mathbb{N}$. Now assume that $n< x\le n+1$ and $y\in B $. We claim that $$\label{linBound}
G_{{(0,\infty)}}(x,y)\le Cx,$$ where $C$ depends only on $\alpha$. Observe that, $$\begin{aligned}
\label{greenestlarge11}
\int_0^{\infty}V(x,y)dy&=& \int_{0}^{6}
\int_0^{\infty}p_{(0,\infty)}(t,x,y)dydt\nonumber\\&=&
\int_{0}^{6}P^{x}(\tau_{{(0,\infty)}}>t)dt\le 6.\end{aligned}$$
Consider $h(v)=G_{{(0,\infty)}}(x,v),\ v\in B $. By Lemma \[continuity\] it is regular harmonic on $B$. Hence using the estimate (\[upperbound\]) we obtain $$\begin{aligned}
G_{{(0,\infty)}}(x,y)&=& E^y G_{{(0,\infty)}}(x,X_{\tau_{B}})\\
&=& E^y \left[G_{{(0,\infty)}}(x,X_{\tau_{B}}) ;X_{\tau_{B}}\in (0,n+2)\right]\\
&\le& E^y V(x,X_{\tau_{B}})+ C(n+2).\end{aligned}$$ Integrating $G_{{(0,\infty)}}(v,y)$ with respect to $dv$ and applying (\[greenestlarge11\]) we obtain $$\label{greenintegral}\int_n^{n+1} G_{{(0,\infty)}}(v,y)dv \le 6 + C (n+2).$$ The final argument for proving (\[linBound\]) will use Lemma \[GrzywnyRyznar\]. Take $D=(n-1,n+2),$ and recall that $y> n+2$ and $x \in (n,n+1)$ . Due to Lemma \[continuity\] the Green function $G_{{(0,\infty)}}(u,y)$ is positive regular harmonic on $D$ as a function of $u$. By Harnack’s inequality for harmonic functions on $D$, which follows from Lemma \[GrzywnyRyznar\], we arrive at $$G_{{(0,\infty)}}(x,y)\le C G_{{(0,\infty)}}(u,y), \ \ x,u\in (n,n+1),$$ which together with (\[greenintegral\]) completes the proof of the estimate $$G_{{(0,\infty)}}(x,y) \le C x,\ \ 1<x\le y-2.$$ Combining this with (\[upperbound\]) we obtain $$G_{(0,\infty)}(x,y)\le C( V(x,y)+x),\quad x\ge 1.$$ Since $G^{gauss}_{(0,\infty)}(x,y)= x$ (see (\[gaussGreen2\])), then by Lemma \[potential\_lower\] we have that $$G_{(0,\infty)}(x,y)\ge \frac2\alpha x.$$ Therefore we proved that $$\label{greenestlarge1}
G_{(0,\infty)}(x,y)\approx V(x,y)+x,\quad x\ge 1.$$ To estimate $V(x,y)$ we use $$\begin{aligned}
\label{estimate111}V(x,y)&=&\int_0^{6} p_t^{{(0,\infty)}}(x,y)dt\nonumber \le e^6\int_0^{6}e^{-t} p_t^{{(0,\infty)}}(x,y)dt
\\&\le& e^6\, G^1_{{(0,\infty)}}(x,y).
\end{aligned}$$ Next consider $x<1$ and $y\le 2$. By (\[upperbound\]) and Lemma \[trivialbound\] we get $$G_{{(0,\infty)}}(x,y)\approx G^1_{{(0,\infty)}}(x,y)\approx
G^1_{{(0,\infty)}}(x,y)+ x^{\alpha/2}.$$ Now assume that $x<1$ and $y>2$. Again $G_{{(0,\infty)}}(\cdot,y)$, by Lemma \[continuity\], is regular harmonic on $(0,2)$, hence by BHP (see Lemma \[BHP1\]): $$G_{{(0,\infty)}}(x,y)\approx G_{(0,\infty)}(1,y)x^{\alpha/2}.$$ Due to Theorem \[Green1est\], $G^1_{{(0,\infty)}}(1,y)\le C$ so by (\[greenestlarge1\]) we have $$G_{(0,\infty)}(1,y) \approx 1,$$ which implies $$G_{{(0,\infty)}}(x,y)\approx x^{\alpha/2},\quad x<1, y>2.$$ This completes the proof.
\[Greenhalflineasymp\] Let $x\le y$. Then we have $$G_{(0,\infty)}(x,y)\approx \left\{
\begin{array}{ll}
G^1_{(0,\infty)}(x,y), & \hbox{$x\le 1$, $|x-y|<1$;} \\
G^1_{(0,\infty)}(x,y)+x, & \hbox{$x>1$, $|x-y|<1$ ;} \\
x\vee x^{\alpha/2}, & \hbox{$|x-y|\ge 1$.}
\end{array}
\right.$$
Exit time properties
====================
In this section we derive optimal estimates of the expected value of the exit time from a ball of arbitrary radius. Then, which seems the most important result of this section, we provide optimal estimates of the tail distribution for the exit time from a half-space. That is we improve the bounds obtained in Lemmas \[Grzywny\] and \[Grzywnyimprove\]. They will play a crucial role in the next section, where we deal with the Green function of a half-space in ${\mathbb{R}^d}$. We start with the one-dimensional case.
\[exittime\] For $x\in (0,R)$ we have $$E^x\tau_{(0,R)}\approx (x^{\alpha/2}\vee x)
\left((R-x)^{\alpha/2}\vee (R-x)\right).$$
If $R\le 3$ then from Lemma \[GrzywnyRyznar\] we have $$E^x\tau_{(0,R)}\approx (x(R-x))^{\alpha/2}.$$
Throughout the rest of the proof we suppose that $R>3$. Assume $x\le R/2$. First we prove the upper bound. By Theorem \[green\] and [(\[intgral\])]{} we obtain $$\begin{aligned}
E^x\tau_{(0,R)}
&=&\int^R_0 G_{(0,R)}(x,y)dy\le\int^R_0 G_{(0,\infty)}(x,y)dy\nonumber\\
&\approx& \int^R_0 G^1_{(0,\infty)}(x,y)dy +\int^x_0
(y^{\alpha/2}\vee y) dy + (R-x)(x^{\alpha/2}\vee
x)\nonumber\\&\le&2(\alpha\Gamma(\alpha/2))^{-1}x^{\alpha/2}+x(x^{\alpha/2}\vee
x)+(R-x)(x^{\alpha/2}\vee x)\nonumber\\&\le& c R (x^{\alpha/2}\vee
x). \label{exittimeproof1}\end{aligned}$$
Now, we deal with the lower bound. By Lemma \[potential\_lower\], $$G_{(0,R)}(x,y)\ge \frac2\alpha G^{gauss}_{(0,R)}(x,y).$$ Denote the first exit time of $(0,R)$ for the Brownian motion by $\tau^{gauss}_{(0,R)}$. It is well known that $E^x\tau^{gauss}_{(0,R)}=\frac12\,x\,(R-x)$ (eg. see [@Du]). Then we have $$\begin{aligned}
E^x\tau_{(0,R)} &=&\int_0^R
G_{(0,R)}(x,y)dy \ge \frac2\alpha\int_0^R G^{gauss}_{(0,R)}(x,y)
dy\\ &=& \frac2\alpha
E^x\tau^{gauss}_{(0,R)}=\frac1\alpha\,x\,(R-x).\end{aligned}$$ Hence we get, for $1\le x\le R/2$, $$\label{exittimeproof2}
E^x\tau_{(0,R)}\approx x\,R$$ Let $x<1$. Notice that by the Strong Markov Property $$E^x\tau_{(0,R)}=s(x)+E^x\tau_{(0,2)},$$ where $s(x)=E^x(E^{X_{\tau_{(0,2)}}}\tau_{(0,R)})$ is regular harmonic on the interval $(0,2)$ vanishing on its complement. Therefore by BHP (see Lemma \[BHP1\]) we obtain $$s(x)\approx s(1)x^{\alpha/2}.$$ Moreover due to Lemma \[GrzywnyRyznar\] we have $$E^x\tau_{(0,2)} \approx x^{\alpha/2}.$$ This yields $$E^x\tau_{(0,R)} \approx (s(1) +1)
x^{\alpha/2}.$$ Noting that $s(1)=E^1\tau_{(0,R)}-E^1\tau_{(0,2)}$ and observing that [(\[exittimeproof2\])]{} implies $$(E^1\tau_{(0,R)}-E^1\tau_{(0,2)})+1 \approx R,$$we obtain $$\label{exittimeproof3}E^x\tau_{(0,R)}\approx R x^{\alpha/2},\quad 0< x<1.$$ Putting together [(\[exittimeproof1\])]{}, [(\[exittimeproof2\])]{} and [(\[exittimeproof3\])]{} we obtain $$E^x\tau_{(0,R)}\approx R\, (x^{\alpha/2}\vee x),\quad \textrm{
for } x\le R/2.$$ By symmetry we have $E^x\tau_{(0,R)}=E^{R-x}\tau_{(0,R)}$, which ends the proof.
Now we derive bounds for the expected exit times from balls in the multidimensional case.
\[exittimeRd\] For $x\in B(0,R)=\{v\in {\mathbb{R}^d}:|v|<R\}$ we have $$E^x\tau_{B(0,R)}\approx
\left((R-|x|)^{\alpha/2}\vee (R-|x|)\right)\left(R\vee
R^{\alpha/2}\right).$$
Let $\tau^{stable}_{B(0,R)}$ be the first exit time from $B(0,R)$ for the $\alpha$-stable isotropic process. By the result of Getoor [@Ge] we have $E^x\tau^{stable}_{B(0,R)}=c(R^2-|x|^2)^{\alpha/2}$ for $c=c(\alpha,d)$.
First assume that $R\le 3$. Then from [(\[comparisonR\])]{} we obtain $$E^x\tau_{B(0,R)}\approx E^x\tau^{stable}_{B(0,R)}=c(R^2-|x|^2)^{\alpha/2}\approx (R-|x|)^{\alpha/2}\,R^{\alpha/2},$$ which completes the proof in this case.
Next suppose that $R>3$. Let $z = x/|x|$ if $x\neq 0$ and $z=(1,0,\ldots,0)$ if $x=0$. We now take $S_R= \{v:|\left\langle
z,v\right\rangle|<R\}$. The process $\left\langle
z,X_t\right\rangle$ is the one-dimensional relativistic process (with the same parameter) which starts from $|x|$. Note that $$E^x\tau_{B(0,R)}\le E^x\tau_{S_R}.$$ By the one-dimensional result (see Lemma \[exittime\]) we get the upper bound.
For $|x|\le R-1$ we get the lower bound by using Lemma \[potential\_lower\] and the result for the Brownian motion: $E^x\tau^{gauss}_{B(0,R)}=\frac1{2d}(R^2-|x|^2)$ (see [@Du]). Namely $$\begin{aligned}
E^x\tau_{B(0,R)}&=&\int_{B(0,R)}G_{B(0,R)}(x,y)dy\ge \frac2{\alpha}\int_{B(0,R)}G^{gauss}_{B(0,R)}(x,y)dy\nonumber\\
&=& \frac2{\alpha} E^x\tau^{gauss}_{B(0,R)}= \frac1{\alpha
d}(R^2-|x|^2).\label{lowerextG} \end{aligned}$$ To complete the proof we need to consider $R-1\le |x|\le R$. The conclusion will follow in the usual way from BHP (see Lemma \[BHP1\]) and the bound above for $|x|= R-1$. We may and do assume that $x=(0,\dots,0,|x|)$. Denote $x_0= (0,\dots,0,R-1)$ and $z_0=
(0,\dots,0,R)$. Let $$F= B(0,R)\cap B(z_0,2)$$ and $$s(x)=E^xE^{X(\tau_F)}\tau_{B(0,R)}.$$ Observe that $s(x)$ is a positive regular harmonic function on $F$ satisfying the assumptions of the second part of Lemma \[BHP1\] hence $$s(x)\approx s(x_0) (R-|x|)^{\alpha/2}.$$ Next, by the Strong Markov Property $$\begin{aligned}
E^x\tau_{B(0,R)}&=&s(x)+E^x\tau_{F}\ge s(x)+E^x\tau_{B(x_0,1)}\nonumber\\
&\approx&
s(x_0) (R-|x|)^{\alpha/2}+E^x\tau^{stable}_{B(x_0,1)}\nonumber\\
&\approx&(s(x_0)+1) (R-|x|)^{\alpha/2}\nonumber\\
&=&(E^{x_0}\tau_{B(0,R)}-E^{x_0}\tau_F+1)(R-|x|)^{\alpha/2}\nonumber\\
&\ge& c \,R (R-|x|)^{\alpha/2}\label{exitmefinal} .\end{aligned}$$ The equivalence $E^x\tau_{B(x_0,1)} \approx
E^x\tau^{stable}_{B(x_0,1)}$ follows from (\[comparisonR\]) and $$E^{x_0}\tau_{B(0,R)}-E^{x_0}\tau_F+1 \ge c R$$ follows from (\[lowerextG\]). Combining (\[lowerextG\]) and (\[exitmefinal\]) we arrive at the desired lower bound.
Now, we recall the Ikeda-Watanabe formula [@IW] which provides a relationship between the Green function and the Poisson kernel. Assume $D\subset {\mathbb{R}^d}$ is a nonempty open set and $E$ is a Borel set such that $\textrm{dist}(D,E)>0$, then we have $$\label{Ikeda-WatanabeFormula}P^x(X(\tau_D)\in E, \tau_D<\infty)=\int_D
G_D(x,y)\nu(E-y)dy,\ x\in D.$$ The following generalization of the Ikeda-Watanabe formula was proved in [@KS]: $$\label{Ikeda-WatanabeFormula2}
P^x(X(\tau_D)\in E,t_1<\tau_D<t_2)=\int_D
\int^{t_2}_{t_1}p^D_t(x,y)dt\nu(E-y)dy,$$where $ 0\le t_1<t_2$, $x\in D$. For $D$ which satisfies the outer cone property we have $P^x(X_{\tau_D}\in\partial D, \tau_D<\infty)=0$ (see [@KS]). Therefore the above formulas are true for all sets $E\subset
D^c$ for such $D$. In particular, for sets studied in this paper as balls or half-spaces, the process does not hit the boundary, when exiting a set.
As a consequence of formula [(\[Ikeda-WatanabeFormula\])]{} we have the following lemma which proof is omitted.
\[poissonKernel\] Let $D\subset {\mathbb{R}^d}$ be a bounded open set then $$P_D(x,z)\le E^x \tau_D \sup_{v\in D}\nu(z-v),\ z\in (\overline{D})^c, x\in D.$$ Moreover, if $dist(z,D)\ge 1$ then $$P_D(x,z)\le C E^x \tau_D e^{-dist(z,D)}.$$
\[MRestimate\]For $0<x<R$ we have $$P^x(\tau_{(0,R)}<\tau_{(0,\infty)})
\approx \frac{x^{\alpha/2}\vee x}{R^{\alpha/2}\vee R}.$$
Assume that $R\ge 1$ and $0<x<R$. By Lemma \[continuity\] the function $G_{(0,\infty)}(\cdot,2R)$ is regular harmonic on $(0,R)$, therefore by Remark \[Greenhalflineasymp\] we obtain $$\begin{aligned}
\label{MRp1}C x^{\alpha/2}\vee x &\ge&
G_{(0,\infty)}(x,2R)=
E^xG_{(0,\infty)}(X_{\tau_{(0,R)}},2R)\nonumber\\&\ge& c
E^x[X_{\tau_{(0,R)}}\wedge 2R; X_{\tau_{(0,R)}}>0]\nonumber\\&\ge&
c R P^x(X_{\tau_{(0,R)}}>0).\end{aligned}$$
Let $n\ge 3$, which we specify later. Again $G_{(0,\infty)}(\cdot,nR)$ is regular harmonic on $(0,R)$. Applying Remark \[Greenhalflineasymp\] we have $$\begin{aligned}
\label{MRp2}
G_{(0,\infty)}(x,nR)&=&
E^xG_{(0,\infty)}(X_{\tau_{(0,R)}},nR)\nonumber\\&\le& C\left(n R
P^x(X_{\tau_{(0,R)}}>0) +
E^xG^1_{(0,\infty)}(X_{\tau_{(0,R)}},nR)\right).\nonumber\\\end{aligned}$$ Moreover Lemma \[poissonKernel\] and Theorem \[Green1est\] imply $$\begin{aligned}
& &\!\!\!
E^xG^1_{(0,\infty)}(X_{\tau_{(0,R)}},nR)\nonumber\\& &=\, \int_{R}^\infty G^1_{(0,\infty)}(v,nR)P_{(0,R)}(x,v)dv\nonumber\\
& &=\, \int_{R}^{(n-1)R}
G^1_{(0,\infty)}(v,nR)P_{(0,R)}(x,v)dv\nonumber\\& & \;+\,
\int_{(n-1)R}^\infty G^1_{(0,\infty)}(v,nR)P_{(0,R)}(x,v)dv\nonumber\\
& &\le\, \sup_{R\le v\le (n-1)R}
G^1_{(0,\infty)}(v,nR)P^x(X_{\tau_{(0,R)}}\in (R, (n-1)R)
)\nonumber\\ & &\;+
\sup_{ v\ge {(n-1)R}}P_{(0,R)}(x,v) \int_{(n-1)R}^\infty G^1_{(0,\infty)}(v,nR)dv\nonumber\\
& &\le\, c P^x(X_{\tau_{(0,R)}}>0)+
Ce^{-(n-2)R}E^x\tau_{(0,R)},\label{MRp22}\end{aligned}$$ where $P_{(0,R)}(x,v)$ is the Poisson kernel for $(0,R)$ and by Lemma \[poissonKernel\] it admits $$P_{(0,R)}(x,v)\le C E^x\tau_{(0,R)} e^{-(n-2)R},\quad v\ge (n-1)R.$$ Using the [(\[MRp2\])]{} and [(\[MRp22\])]{} we arrive at $$\label{MRp3}c \, n \,R\,P^x(X_{\tau_{(0,R)}}>0)\ge G_{(0,\infty)}(x,nR)- Ce^{-(n-2)R}E^x\tau_{(0,R)}.$$ By Lemma \[exittime\], $E^x\tau_{(0,R)}\approx R\,
(x^{\alpha/2}\vee x) $, so Remark \[Greenhalflineasymp\] implies $$G_{(0,\infty)}(x,n R)- Ce^{-(n-2)R}E^x\tau_{(0,R)}\ge (c-C R e^{-(n-2)R})\, (x^{\alpha/2}\vee x).$$ Now we pick $n$ independently of $R\ge 1$ and large enough so that\
$c-C R e^{-(n-2)R}\ge c/2$. This yields $$\label{MRp4}
P^x(\tau_{(0,R)}<\tau_{(0,\infty)})\ge c\frac{x^{\alpha/2}\vee
x}{R}.$$ Next, for $R<1$ we use Lemma \[GrzywnyRyznar\] to get $$\label{MRp6}P^x(\tau_{(0,R)}<\tau_{(0,\infty)}) \approx
\frac{x^{\alpha/2}}{R^{\alpha/2}}.$$ Combining [(\[MRp1\])]{}, [(\[MRp4\])]{} and [(\[MRp6\])]{} ends the proof.
Now we can prove the main result of this section.
\[optimaltail\] For $x>0$ and $t\ge 1$, $$P^x(\tau_{(0,\infty)}>t)\approx \frac{x^{\alpha/2}\vee x}{t^{1/2}}\wedge 1.$$
Assume that $t\ge 1$. If $2x\ge t^{1/2}$ the upper bound is trivial. So we may assume that $2x< t^{1/2}$. We have $$P^x(\tau_{(0,\infty)}>t)\le P^x(\tau_{(0,R)}>t)+P^x(\tau_{(0,R)}<\tau_{(0,\infty)}).$$ Let $R>2x$. By Chebyschev’s inequality and Proposition \[exittime\] we obtain $$P^x(\tau_{(0,R)}>t)\le
\frac{E^x\tau_{(0,R)}}{t}\approx \frac{(R\vee
R^{\alpha/2})(x^{\alpha/2}\vee x)}{t}.$$ By the Lemma \[MRestimate\] $$P^x(\tau_{(0,R)}<\tau_{(0,\infty)})\le c \frac{x^{\alpha/2}\vee x}{R^{\alpha/2}\vee R }.$$ Setting $R=t^{1/2}$ we arrive at the upper bound.
Next, let us observe that by Lemma \[density\], $$\begin{aligned}
\label{optailp1}P^x(\tau_{(0,\infty)}>t)&=&\int^{\infty}_0 p^{(0,\infty)}_t(x,y)dy\ge
\int^{\infty}_0 Eg^{(0,\infty)}_{T_\alpha(t)}(x,y)dy\nonumber\\&
=& E\, P^x(\tau^{gauss}_{(0,\infty)}>T_{\alpha}(t)).\end{aligned}$$ Let us observe that by Chebyschev’s inequality and [(\[subord2\])]{} for $\lambda=-1$ we have $$P(T_{\alpha}(t)> 2t)=P(e^{T_{\alpha}(t)}>e^{2t})\le Ee^{T_\alpha(t)}e^{-2t}=e^{t-2t}\le e^{-1}.$$ That is $P(T_{\alpha}(t)\le 2t)\ge 1-e^{-1}$. Taking into account the fact that $P^x(\tau^{gauss}_{(0,\infty)}>t)\approx \frac{x}{t^{1/2}}\wedge
1$ we obtain from [(\[optailp1\])]{}, $$P^x(\tau_{(0,\infty)}>t)\ge c\, E
\left(\frac{x}{T_\alpha(t)^{1/2}}\wedge1\right)
\ge c \left(\frac{x}{t^{1/2}}\wedge
1\right).$$
Now, let $x< 1$ then $$\begin{aligned}
P^x(\tau_{(0,\infty)}>t)&\ge& E^x \left[ X_{\tau_{(0,2)}}>0 ;
P^{X_{\tau_{(0,2)}}}(\tau_{(0,\infty)}>t)\right]\\ &\ge& c E^x
\left[ X_{\tau_{(0,2)}}\ge 2;
\frac{X_{\tau_{(0,2)}}}{t^{1/2}}\wedge1 \right]\\&\ge& c
\left(\frac{1}{t^{1/2}}\wedge 1\right) P^x(X_{\tau_{(0,2)}}\ge
2).\end{aligned}$$ Hence by Lemma \[MRestimate\] we obtain $$P^x(\tau_{(0,\infty)}>t)\ge c x^{\alpha/2}\frac{1}{t^{1/2}},$$ which completes the proof.
\[density1A\] There exists a constant C such that, for $t>0$ and $x,y \ge1$, $$\label{density1A1}p_t^{(0,\infty)}(x,y) \le C (t^{-1/2}+t^{-1/\alpha})
\left(\frac{ x}{t^{1/2}}\wedge 1\right)\left(\frac{
y}{t^{1/2}}\wedge 1 \right).$$ For $t\ge1$ and $x,y>0$, $$c t^{-1/2} \left(\frac{ x}{t^{1/2}}\wedge 1\right)\left(\frac{
y}{t^{1/2}}\wedge 1\right)e^{-\frac{|x-y|^2}{c_1 t}} \le
p_t^{(0,\infty)}(x,y),$$ where $c$, $c_1$ are some constants. Hence, for $x,y,t\ge 1$, satisfying $t\ge |x-y|^2$ we have the optimal bound $$p_t^{(0,\infty)}(x,y)\approx t^{-1/2} \left(\frac{
x}{t^{1/2}}\wedge 1\right)\left(\frac{ y}{t^{1/2}}\wedge 1
\right).$$
The upper bound immediately follows from Lemma \[density\] and Theorem \[optimaltail\].
Pick $0<\beta<1/2$ such that $(1+ 1/\beta)^{\alpha/2}-2=1$ and let $$A_t=\{\omega:\beta t <T_{\alpha}(t)(\omega)<2t\}.$$ To obtain the lower bound we use again Lemma \[density\] to get $$p_t^{(0,\infty)}(x,y) \ge E \left[g^{(0,\infty)}_{T_{\alpha}(t)} (x,y); A_t \right].$$
Next by a classical result $$\begin{aligned}
g^{(0,\infty)}_t(x,y)&=& g_t(x-y)-g_t(x+y)
= g_t(x-y)\left(1-e^{-\frac{xy}{t}}\right)\\
&\ge& g_t(x-y)\left(1\wedge \frac{xy}{t}\right)\ge
g_t(x-y)\left(\frac{ x}{t^{1/2}}\wedge 1\right)\left(\frac{
y}{t^{1/2}}\wedge 1 \right).\end{aligned}$$ Hence $$p_t^{(0,\infty)}(x,y)\ge c t^{-1/2}e^{-\frac{|x-y|^2}{4\beta t}}P(A_t)
\left(\frac{ x}{t^{1/2}}\wedge 1\right)\left(\frac{
y}{t^{1/2}}\wedge 1 \right).$$ Next we estimate $P(A^c_t)$. By Chebyschev’s inequality and by [(\[subord2\])]{} for $\lambda=1/\beta$, $$\begin{aligned}
P(T_{\alpha}(t)<\beta t)&=& P(e^{-(1/\beta)T_{\alpha}(t)}>e^{- t})\le e^{ t}Ee^{-(1/\beta)T_{\alpha}(t)}
\\&=& e^{-((1+ 1/\beta)^{\alpha/2}-2) t}= e^{ -t} .\end{aligned}$$ Similarly by [(\[subord2\])]{} for $\lambda=-1$, $$P(T_{\alpha}(t)> 2 t)= P(e^{T_{\alpha}(t)}>e^{2t})\le e^{-2t}Ee^{T_{\alpha}(t)}
= e^{ -t}.$$ Hence $$P(A^c_t)\le 2e^{- t},$$ which implies $\inf_{t>1}P(A_t)\ge 1-2e^{- 1}$ and this ends the proof.
One of the drawbacks of the inequality in the above Corollary is that the right hand side does not depend on the distance $|x-y|$. The following result will be very useful in the next section and it does take into account the distance $|x-y|$.
\[density1\] Let $x,y\ge 1$ and $|x-y|\ge 1$. Then, for $t\le |x-y|^2$, $$p^{(0,\infty)}_t(x,y)\le
C\left(\frac{xy}{|x-y|^{2}}\wedge1\right)\left(g_{t}\left(c(x-y)\right),
+t\,\nu\left(c(x-y)\right)\right)$$ where $c=8\sqrt{2}$ and $C$ is some constant. Moreover $$\int^{\infty}_1t^{-d/2+1/2}p^{(0,\infty)}_t(x,y)dt\le c(d,\alpha)\frac{xy}{|x-y|^d}.$$
Our arguments are based on the idea of proof of Theorem 4.2 in [@KS]. Throughout the whole proof we assume that $x,y\ge 1$ and $x\le
y-1$. We first consider the case $t\le |x-y|^2/16$. The interval $(0,(x+y)/2)$ we denote by $S$ and $(y-s,y+s)$ by $D(s)$. Let $0<s<1/8$, then $D(s)\in (0,\infty)\setminus S$. By the Strong Markov Property we obtain $$\begin{aligned}
& &\!\!\!\!\!\! \int_{D(s)}p^{(0,\infty)}_t(x,y)dz\nonumber\\&
&\!\!\!=\,P^x(X_t\in D(s),\tau_{{(0,\infty)}}>t)\nonumber\\&
&\!\!\!\le\,
P^x(\tau_S<t, X_{\tau_S}>0, X_t\in D(s))\nonumber\\
& &\!\!\!=\,E^x\left[ \tau_S<t,X_{\tau_S}\in(0,\infty)\setminus S,
P^{X(\tau_S)}(X_{t-\tau_S}\in D(s)) \right]. \label{estdensH3}\end{aligned}$$
Let $A=(y-|x-y|/4,y+|x-y|/4)$ and $B=(0,\infty)\setminus(S\cup
A)$. Observe that $\textrm{dist}(A,S)=|x-y|/4$ and $\textrm{dist}(B,D(s))\ge |x-y|/8$. Because $p_t(x)$ is radially decreasing in $|x|$ we have for $X_{\tau_S}\in B$, $$\begin{aligned}
P^{X(\tau_S)}(X_u\in
D(s))&=&\int_{D(s)}p_u(X_{\tau_S}-z)dz\\
&\le&|D(s)|p_u\left(\frac{x-y}{8}\right)\\&\le&
c|D(s)|\left(g_{u}\left(\frac{x-y}{8\sqrt{2}}\right)+u\nu\left(\frac{x-y}{8\sqrt{2}}\right)\right),\end{aligned}$$ where in the last step we applied Lemma \[transden2\]. Next observe that $g_t(x)$ is an increasing function in $t$ on the interval $(0,x^2/2)$. Hence, for $t\le |x-y|^2/264$, we obtain, for $X_{\tau_S}\in B $, $$P^{X(\tau_S)}(X_{t-\tau_S}\in
D(s))\le c|D(s)|\left(g_{t}\left(\frac{x-y}{8\sqrt{2}}\right)+
t\nu\left(\frac{x-y}{8\sqrt{2}}\right)\right) .$$ Define $F(t,z)=g_{t}(z/(8\sqrt{2}))+ t\nu(z/(8\sqrt{2}))$. Then Proposition \[MRestimate\] and the above estimate yield $$\begin{aligned}
\label{estdensH4}
& &\!\!\!E^x\left[ \tau_S<t,X_{\tau_S}\in
B,P^{X({\tau_S})}(X_{t-\tau_S}\in D(s))
\right]\nonumber\\& & \le\, c|D(s)|F(t,x-y)P^x(
\tau_S<t,X_{\tau_S}\in B)\nonumber\\
& &\le\, c|D(s)|F(t,x-y)P^x( \tau_S<\tau_{(0,\infty)})\nonumber\\
& &\le\, c|D(s)|F(t,x-y)\frac{x}{x+y}\nonumber\\
& &\le\, c|D(s)|F(t,x-y)\frac{xy}{|x-y|^2}.\end{aligned}$$
For the set $A$ we have by [(\[Ikeda-WatanabeFormula2\])]{}, $$\begin{aligned}
& &\!\!\!E^x\left[\tau_S<t,X_{\tau_S}\in
A,P^{X(\tau_S)}(X_{t-\tau_S}\in D(s))
\right] \\
& & = \int_{S}\int^t_0 p^S_r(x,z)\int_A\nu(z-w)P^w(X_{t-r}\in
D(s))dwdrdz.\end{aligned}$$ Moreover $$\begin{aligned}
\label{simple}\int_{A}P^w(X_{t}\in D(s))dw &=& \int_{A}\int_{D(s)}p(t,w,z)dzdw\nonumber\\&=&
\int_{D(s)} \left(\int_{A}p(t,w,z)dw\right)dz\nonumber\\
&=&
\int_{D(s)}P^w(X_t\in A)dw\le |D(s)|.
\end{aligned}$$ Using (\[simple\]) and observing that $\nu(z-w)\le
\nu((x-y)/4), \ w\in A, z\in S $ we obtain $$\begin{aligned}
\label{estdensH5}
& & E^x\left[ \tau_S<t,X_{\tau_S}\in
A,P^{X(\tau_S)}(X_{t-\tau_S}\in D(s))
\right]\nonumber\\& &\le\,c|D(s)|\nu((x-y)/4)\int_S \int^{t}_0
p_S(r,x,z)drdz\nonumber\\
& &=\, c|D(s)|\nu((x-y)/4) \int^{t}_0
P^x(\tau_S>r)dr\nonumber\\
& &\le\,c|D(s)|t\nu((x-y)/4)\nonumber\\
& &\le\,c|D(s)|t\frac{xy}{|x-y|^2}\nu((x-y)/(8\sqrt{2})),\end{aligned}$$ where the last step follows from (\[levymeasure\]) and (\[asympt\_infty\]). Combining [(\[estdensH3\])]{}, [(\[estdensH4\])]{} and [(\[estdensH5\])]{} after dividing by $|D(s)|$ and passing $s\searrow0$ we obtain for $x,y\ge1$, $|x-y|\ge1$ and $|x-y|^2\ge
256t$, $$\label{estdensH6}p^{(0,\infty)}_t(x,y)\le
c\frac{xy}{|x-y|^{2}}\left(g_{t}\left((x-y)/(8\sqrt{2})\right)+
t\nu\left((x-y)/(8\sqrt{2})\right)\right).$$
Next we consider $|x-y|^2\le 256t$. By [(\[density1A1\])]{} we get for $t\ge 1/256$ and $x,y\ge 1$, $$\label{estdensH7}
p^{(0,\infty)}_t(x,y)\le c t^{-1/2}\frac{xy}{t}.$$ Since for $t>|x-y|^2/256\ge 1/256$, $$ct^{-1/2}\le g_{t}((x-y)/(8\sqrt{2}))$$ we obtain $$\label{estdensH14}
p^{(0,\infty)}_t(x,y)\le
c\frac{xy}{t}g_{t}((x-y)/(8\sqrt{2})),\quad t>|x-y|^2/256.$$ The above inequality combined with [(\[estdensH6\])]{}, [(\[estdensH7\])]{} and Lemma \[transden2\] implies the first claim of the theorem.
To prove the second conclusion of the theorem we apply [(\[estdensH6\])]{} for $256t<|x-y|^2$ and [(\[estdensH7\])]{} for $256t\ge
|x-y|^2$ to get $$\begin{aligned}
& &\!\!\! \int^{\infty}_1t^{-(d-1)/2}p^{(0,\infty)}_t(x,y)dt\\
& & \le\, c\frac{xy}{|x-y|^2} \int^{
|x-y|^2/256}_{1/256}t^{-(d-1)/2}\left(g_{t}\left(\frac{x-y}{8\sqrt{2}}\right)+t\nu\left(\frac{x-y}{8\sqrt{2}}\right)\right)dt
\\&&\;+\,c\,xy\int^\infty_{|x-y|^2/256}t^{-d/2-1}dt\\
& &\le\, c\, xy\left(\frac{|x-y|^{2-d}}{|x-y|^2}+\frac{e^{-|x-y|/16}}{|x-y|^{\alpha/2+3}}(1\vee |x-y|^{5-d})+\frac{1}{|x-y|^d}\right)\\
& &\le\, c\frac{xy}{|x-y|^d}.\end{aligned}$$ Note that we used [(\[asympt\_infty\])]{} to estimate the density of the Lévy measure.
Green function of ${\mathbb{H}}\subset {\mathbb{R}^d}$, $d\ge 2$.
=================================================================
In this section we extend our one-dimensional estimates for a half-line to higher dimensions. To achieve this we start with some upper estimates of the transition densities of the killed process.
Note that by subordination we have $$p_t(x)=Eg_{T_{\alpha}(t)}(x),\quad x\in {\mathbb{R}^d}.$$
Let $A_t=\{\omega:\beta t <T_{\alpha}(t)(\omega)<2t\}$ be the set defined in the proof of Corollary \[density1A\]. Let us define $$q_t(x)=E\left(g_{T_{\alpha}(t)}(x);{A_t^c}\right),\quad x\in {\mathbb{R}^d}.$$
In the sequel we will need a simple upper bound of $q(t,x)$. Note that $g_t(x)\le \frac c{|x|^d},\ t>0,$ and this used for $q_t(x)$ yields $$\label{tail}q_t(x)\le \frac c{|x|^d}P(A_t^c) \le \frac C{|x|^d}e^{-2 t}.$$
The next lemma will have a very important role in obtaining the upper bound for the Green function. We introduce the following notation. For $x\in {\mathbb{R}^d}$ we denote ${\bf x}=(x_1,\dots,x_{d-1})$ and by ${\bf g}_t( {\bf x})$ we denote the Brownian semigroup in $\mathbb{R}^{d-1}$.
\[density10\]There is a constant $C=C(d,\alpha)$ such that $$\label{density01}p^{\mathbb{H}}_t(x,y)\le C{\bf
g}_{ 2t}( {\bf x}-{\bf y})p^{(0,\infty)}_t( x_d,y_d) +
q_t(x-y),\quad x, y \in {\mathbb{H}}.$$
For $y \in {\mathbb{H}}$ and $\delta>0$ denote $V=V_y(\delta)= [y,y+\delta]=\times_{i=1}^d[y_i,y_i+\delta]=
{\bf V}\times V_d\subset {\mathbb{H}}$. Then by independence of the subordinator $T_{\alpha}(t)$ and Brownian motion $B_t$ one gets $$\begin{aligned}
& &\!\!\! P^x(X_t\in V, \tau_{\mathbb{H}}>t )\\& &=\, P^x(X_t\in V, \tau_{\mathbb{H}}>t, A_t )+ P^x(X_t\in V, \tau_{\mathbb{H}}>t, A^c_t )\\
& &=\, E \Big[A_t;P^{\bf x}\left( {\bf B}_{T_{\alpha}(t)}\in {\bf
V}|T_{\alpha}(\cdot) \right)\times\\
& & \ \ \times\: P^{ x_d}\left(B^{(d)}d_{T_\alpha(t)}\in V_d, B^{(d)}_{T_\alpha(s)}>0; 0<s<t|T\alpha(\cdot) \right)\Big]\\
& &\ \ +\: P^x(X_t\in V, \tau_{\mathbb{H}}>t, A^c_t )\\
& &\le\, \sup_{\beta t\le u\le 2 t} P^{\bf x}( {\bf B}_{u}\in {\bf
V}) P^{ x_d}(B^{(d)}_{T_\alpha(t)}\in V_d,
B^{(d)}_{T_\alpha(s)}>0: 0<s<t )\\& &\ \ +\: \int_{V} q_t(x-z)dz\\
& &\le\, CP^{\bf x}( {\bf B}_{2 t}\in {\bf V}) P^{
x_d}(X^{(d)}_{t}\in V_d, \tau_{\mathbb{H}}>t)+ \int_{V} q(t,x-z)dz.\end{aligned}$$ After dividing both sides by $|V|$ and passing $ \delta \searrow
0$ we obtain the conclusion.
Note that for any $x,y\in {\mathbb{H}}$ we can estimate ${\bf g}_{ 2t}( {\bf
x}-{\bf y})\le ct^{-(d-1)/2}$ so from [(\[density01\])]{} we deduce that $$\label{density2} p^{\mathbb{H}}_t(x,y)\le
ct^{-(d-1)/2}p^{(0,\infty)}_t(x_d,y_d) + q_t(x-y),$$ which will be well estimated with the help of Theorem \[density1\].
Lemma \[density10\], the estimate (\[tail\]) and Theorem \[density1\] show that for the points $x,y\in{\mathbb{H}}$ away from the boundary such that $|x-y|>2$ the Green functions for the relativistic process and the Brownian motion are comparable. In view of the one-dimensional case this result, proved below, is not surprising.
\[GaussEstimate\] For $|x-y|>2$ and $x_d, y_d\ge 1$ we have $$G_{\mathbb{H}}(x,y) \approx G_{\mathbb{H}}^{gauss}(x,y).$$
The lower bound follows from Lemma \[potential\_lower\].
We claim that the following upper bound holds: $$\label{Gauss10}
G_{\mathbb{H}}(x,y)\le c\frac {x_d y_d}{|x - y|^d}.$$ By [(\[density1A1\])]{}, $$p^{(0,\infty)}_t(x_d,y_d)\le C \frac{x_d y_d} {t^{3/2}},\quad t\ge 1,$$ which together with [(\[density01\])]{} and (\[tail\]) yield the following bound for the transition density $$p^{\mathbb{H}}_t(x,y)\le C {\bf g}_{2t}({\bf y}-{\bf x})\frac{x_d y_d} {t^{3/2}} + ce^{-2t}|x-y|^{-d}, \quad t\ge 1.$$ Integrating it over $(1, \infty)$ we arrive at $$\label{Gauss102}\int_1^\infty p^{\mathbb{H}}_t(x,y)dt\le C \frac{x_d y_d} {|{\bf y}-{\bf x}|^d}+
\frac c{|x-y|^d}\le C_1 \frac{x_d y_d} {|{\bf y}-{\bf x}|^d}.$$
If $|x_d - y_d|\ge 1$ we apply [(\[density2\])]{}, (\[tail\]) and Theorem \[density1\] to arrive at $$\begin{aligned}
\int_1^\infty p^{\mathbb{H}}_t(x,y)dt
&\le& C\int_1^\infty
t^{-(d-1)/2}p^{(0,\infty)}_t(x_d,y_d)dt + \frac c{|x-y|^d}\nonumber\\
&\le& C\frac {x_d y_d}{|x_d -
y_d|^d}.\label{Gauss101}\end{aligned}$$ Next note that by Lemma \[transden2\] we can estimate $$\int_0^1 p^{\mathbb{H}}_t(x,y)dt\le \int_0^1 p_t(x-y)dt\le \frac c{|x-y|^d}.$$ This combined with [(\[Gauss102\])]{} and [(\[Gauss101\])]{} implies [(\[Gauss10\])]{}.
Now let $d\ge3$. Since (see [(\[0-potential\])]{}), $$G_{\mathbb{H}}(x,y)\le C \frac {1}{|x - y|^{d-2}}$$ we have the following bound for $|x-y|>2$, $$G_{\mathbb{H}}(x,y)\le C \min\kl \frac {x_d y_d}{|x - y|^d},\frac{1}{|x-y|^{d-2}}\kr\approx G^{gauss}_{\mathbb{H}}(x,y),$$ where the last equivalence follows from (\[gaussGreen\]). This completes the proof in this case.
Now we finish the proof for $d=2$. By [(\[Gauss10\])]{}, for $\frac{x_2y_2}{|x-y|^2}\le 1$, we have $$G_{\mathbb{H}}(x,y)\le C \frac{x_2y_2}{|x-y|^2} \approx \ln \left(1+4\frac{x_2y_2}{|x-y|^2}\right)={2\pi}G^{gauss}_{\mathbb{H}}(x,y),$$ where the last equality is just (\[gaussGreen2\]). If $\frac{x_2y_2}{|x-y|^2}>1$, using Lemmas \[transden2\] and \[density\] together with Theorem \[optimaltail\] we obtain $$\begin{aligned}
G_{\mathbb{H}}(x,y)&\le&\int^{|x-y|^2}_0p_t(x-y)dt\\& &\; +\,
c\int^{\infty}_{|x-y|^2}t^{-1}P^x(\tau_{\mathbb{H}}>t/3)P^y(\tau_{\mathbb{H}}>t/3)dt\\
&\le&\int^{|x-y|^2}_0\frac{c}{|x-y|^2}dt+C\int^{x_2y_2}_{|x-y|^2}t^{-1}dt
+ Cx_2y_2 \int^\infty_{x_2y_2}t^{-2}dt\\
&\le& c +
C\ln\left(\frac{x_2y_2}{|x-y|^2}\right) \\
&\le& C\ln\left(1+4\frac{x_2y_2}{|x-y|^2}\right)=2C \pi
G^{gauss}_{\mathbb{H}}(x,y),\end{aligned}$$ which completes the proof for $d=2$.
Now we are ready to prove the main result of this section.
\[greenhlafspaceest\] For $d\ge 3$ and $x,y\in {\mathbb{H}}$: $$G_{\mathbb{H}}(x,y)\approx \min\left\{\frac{ (x_d\vee x_d^{\alpha/2})
(y_d \vee y_d^{\alpha/2})}{|x-y|^d}, \frac1{|x-y|^{d-2}}\right\},
\ |x-y|> 3,$$ $$G_{\mathbb{H}}(x,y)\approx
\left[\left(\frac{ x_d\wedge y_d}{|x-y|}\right)^{\alpha/2}
\wedge1\right] \frac{1 }{|x-y|^{d-\alpha}}, \ |x-y|\le 3.$$ For $d= 2$ and $x,y\in {\mathbb{H}}$: $$G_{\mathbb{H}}(x,y)\approx \ln \left(1+ 4\frac{(x_2\vee x_2^{\alpha/2})
(y_2 \vee y_2^{\alpha/2})}{|x-y|^2}\right),\ |x-y|> 3,$$ $$G_{\mathbb{H}}(x,y) \approx
\left[\left(\frac{ x_2\wedge y_2}{|x-y|}\right)^{\alpha/2}
\wedge1\right] \frac{1 }{|x-y|^{2-\alpha}} +\ln(1\vee( x_2\wedge
y_2)), \ |x-y|\le 3.$$
First assume $|x-y|\le 3$. In the paper [@ByczRyzMal] it was proved $$G_{\mathbb{H}}(x,y)\approx G^1_{\mathbb{H}}(x,y), \quad d\ge 3$$ and $$G_{\mathbb{H}}(x,y)\approx G^1_{\mathbb{H}}(x,y)+\ln(1\vee( x_2\wedge y_2)) , \quad d= 2.$$ Since $|x-y|\le 3$ by Theorem \[Green1est\] we obtain $$G_{\mathbb{H}}(x,y)\approx
\left[\left(\frac{x_d\wedge y_d}{|x-y|}\right)^{\alpha/2}
\wedge1\right] \frac{1 }{|x-y|^{d-\alpha}},\quad d\ge 3$$ and $$G_{\mathbb{H}}(x,y)\approx
\left[\left(\frac{x_2\wedge y_2}{|x-y|}\right)^{\alpha/2}
\wedge1\right] \frac{1 }{|x-y|^{2-\alpha}} +\ln(1\vee( x_2\wedge
y_2)),\quad d=2.$$ This yields the bound in the case $|x-y|\le 3$.
We introduce the following notation $\tilde{x}=(x_1,\dots,
x_{d-1}, 1\vee x_d )$, $x^*=(x_1,\dots, x_{d-1}, 0)$. Now assume $|x-y|> 3$ and observe that implies that $|x-y|\approx |\tilde{x}-\tilde{y}|> 2$.
Then if both points are away from the boundary ( $x_d\wedge y_d\ge 1$) we use Lemma \[GaussEstimate\] to have $$G_{\mathbb{H}}(x,y)\approx G^{gauss}_{\mathbb{H}}(x,y)= G^{gauss}_{\mathbb{H}}(\tilde{x},\tilde{y}).$$ Next suppose that $x_d< 1\le y_d$. Let $D(x^*)=B(x^*,\sqrt{2})\cap{\mathbb{H}}$. Then $y\notin D(x^*)$ and $G_{\mathbb{H}}(\cdot,y)$ is a regular harmonic function on $D(x^*)$ vanishing on ${\mathbb{H}}^c$. Hence by BHP (see Lemma \[BHP1\]) and next by Theorem \[GaussEstimate\] we have $$G_{\mathbb{H}}(x,y)\approx x_d^{\alpha/2} G_{\mathbb{H}}(\tilde{x},y)= x_d^{\alpha/2} G_{\mathbb{H}}(\tilde{x},\tilde{y})\approx x_d^{\alpha/2}G^{gauss}_{\mathbb{H}}(\tilde{x},\tilde{y}).$$
A similar argument applies for $x_d, y_d< 1$. Notice that $x
\notin D(y^*)$ and $y \notin D(x^*)$. Hence $G_{\mathbb{H}}(\cdot,y)$ and $G_{\mathbb{H}}(x,\cdot)$ are regular harmonic function on $D(x^*)$ and $D(y^*)$, respectively, vanishing on ${\mathbb{H}}^c$. Hence Lemma \[BHP1\] and Theorem \[GaussEstimate\] imply that $$G_{\mathbb{H}}(x,y)\approx x_d^{\alpha/2}y_d^{\alpha/2}G_{\mathbb{H}}(\tilde{x},\tilde{y})
\approx x_d^{\alpha/2}y_d^{\alpha/2}G^{gauss}_{\mathbb{H}}(\tilde{x},\tilde{y}).$$ Taking into account all cases we have $$G_{\mathbb{H}}(x,y)\approx (1\wedge x_d)^{\alpha/2}(1\wedge y_d)^{\alpha/2}G^{gauss}_{\mathbb{H}}(\tilde{x},\tilde{y}), \quad |x-y|> 3.$$ Applying (\[gaussGreen\]) and (\[gaussGreen2\]) we can rewrite the above bound as $$G_{\mathbb{H}}(x,y)\approx (1\wedge x_d)^{\alpha/2}(1\wedge y_d)^{\alpha/2}\min\left\{\frac{ (x_d\vee 1) (y_d \vee 1) }{|\tilde{x}-\tilde{y}|^d}, \frac1{|\tilde{x}-\tilde{y}|^{d-2}}\right\},$$ for $ d\ge 3$, and $$G_{\mathbb{H}}(x,y)\approx (1\wedge x_d)^{\alpha/2}(1\wedge y_d)^{\alpha/2}
\ln\left(1+4\frac{(x_2\vee 1) (y_2\vee 1)}{|\tilde{x}-\tilde{y}|^2}\right),$$ for $d=2$. Taking into account $|x-y|\approx |\tilde{x}-\tilde{y}|\ge 1$, for $|x-y|> 3$, we finally arrive at $$G_{\mathbb{H}}(x,y)\approx \min\left\{\frac{ (x_d\vee x_d^{\alpha/2})
(y_d \vee y_d^{\alpha/2})}{|x-y|^d}, \frac1{|x-y|^{d-2}}\right\},
\quad d\ge 3$$and $$G_{\mathbb{H}}(x,y)\approx \ln\left(1+4\frac{(x_2^{\alpha/2}\vee x_2) (y_2^{\alpha/2}\vee y_2)}{|x-y|^2}\right), \quad d=2.$$
Now we compare the Green functions for half-space for the relativistic process and for the corresponding stable process, so we recall the formula of the Green function in the stable case (see [@BGR]): $$G^{stable}_{{\mathbb{H}}}(x,y)=C(\alpha,d)|x-y|^{\alpha-d}\int_0^{\frac{4x_dy_d}{|x-y|^2}}\frac{t^{\alpha/2-1}}{(t+1)^{d/2}}dt.$$ One can derive sharp estimates from the above formula. Our results from Theorem \[greenhlafspaceest\], Theorem \[green\] and Theorem \[Green1est\] show that for the points $x,y\in{\mathbb{H}}$ such that $|x-y|\le 2$ the Green functions of the half-space ${\mathbb{H}}$ for the relativistic process and for the corresponding stable process are comparable if $d\ge 3$. If $d=1$ or $d=2$ they are also comparable but we have to assume additionally that the points are near the boundary.
Suppose that $|x-y|\le 2$ then $$G_{\mathbb{H}}(x,y)\approx \left\{\begin{array}{ll}
G^{stable}_{{\mathbb{H}}}(x,y), & \hbox{$d\ge 3$;} \\
G^{stable}_{{\mathbb{H}}}(x,y)+\ln(1\vee(x_2\wedge y_2)), & \hbox{$d=2$;} \\
G^{stable}_{{\mathbb{H}}}(x,y)+(x\wedge y)\vee (x\wedge y)^{\alpha/2}, & \hbox{$d=1$.} \\
\end{array}\right.$$ From the estimates obtained in Theorem \[greenhlafspaceest\] and Theorem \[green\] we can infer that $$G_{\mathbb{H}}(x,y)\ge C( G^{stable}_{{\mathbb{H}}}(x,y)+G^{gauss}_{{\mathbb{H}}}(x,y)),\quad x,y\in{\mathbb{H}}.$$
Green functions for intervals
=============================
In this section we provide optimal estimates for Green functions of bounded intervals. We know that for any interval the Green function is comparable with the corresponding Green function of the symmetric process. That is for the interval $(0,R)$, for $R\le
R_0$, we have $$C(R_0)^{-1} G_{(0,R)}^{stable}(x,y) \le G_{(0,R)}(x,y)\le C(R_0)G_{(0,R)}^{stable}(x,y), \label{intervalcomparison}$$ where $0<x,y < R.$ However, if $R_0$ grows, then the constant $C(R_0)$ tends to $\infty$, so the above bound is not optimal in general case. The aim of this section is to provide optimal bounds for large intervals. We recall known estimates for stable cases: $$\label{greenstable}
G_{(0,R)}^{stable}(x,y)\approx \left\{\begin{array}{ll}
\min\left\{\frac{1}{|x-y|^{1-\alpha}},
\frac{(\delta_R(x)\delta_R(y))^{\alpha/2}}{|x-y|}\right\}, & \hbox{$\alpha<1$,} \\
\ln\left(1+\frac{(\delta_R(x)\delta_R(y))^{1/2}}{|x-y|}\right), & \hbox{$\alpha=1$,} \\
\min\left\{(\delta_R(x)\delta_R(y))^{(\alpha-1)/2},
\frac{(\delta_R(x)\delta_R(y))^{\alpha/2}}{|x-y|}\right\}, & \hbox{$\alpha>1$,} \\
\end{array}\right.$$ where $\delta_R(x)=x\wedge(R-x)$.
We start with the proposition showing that for points $x,y$ in the first half of the interval the Green function of the interval and the Green function of $(0,\infty)$ are comparable.
\[Greenintervalprop1\] Let $R\ge 4$. For $x,y\le R/2+1$ we have $$G_{(0,R)}(x,y)\approx G_{(0,\infty)}(x,y).$$
Throughout the whole proof we assume that $x\le y\le R/2+1$. Notice that it is enough to prove that $$G_{(0,R)}(x,y)\ge c G_{(0,\infty)}(x,y),$$ for $x< 1$ or $|x-y|<1$. Indeed, by Lemma \[potential\_lower\] and Remark \[Greenhalflineasymp\] we obtain for $x\ge1$ and $|x-y|\ge1$, $$G_{(0,R)}(x,y)\ge \frac2\alpha G^{gauss}_{(0,R)}(x,y)=\frac2\alpha
x(1-y/R)\ge \frac1{2\alpha} x\ge c G_{(0,\infty)}(x,y).$$
We claim that $$G_{(0,R)}(x,y)\ge CG^{1}_{(0,\infty)}(x,y), \quad
|x-y|<2.\label{lowerbound100}$$
Applying (\[greenpot100\]) with $D_2=(0,\infty)$ and $D_1=(0,R)$ we have $$\begin{aligned}
& &\!\!\! G^1_{(0,\infty)}(x,y)-G^1_{(0,R)}(x,y)\\& & =\,
E^x\left[\tau_{(0,R)}<\tau_{(0,\infty)};e^{-\tau_{(0,R)}}G^1_{(0,\infty)}(X_{\tau_{(0,R)}},y)\right]\\
&&\le \,\sup_{z\ge
R}G^1_{\mathbb{H}}(z,y)P^x(\tau_{(0,R)}<\tau_{(0,\infty)}).\end{aligned}$$ Next, by Theorem \[Green1est\] and Proposition \[MRestimate\] we obtain $$G^1_{(0,\infty)}(x,y)-G^1_{(0,R)}(x,y)\le C e^{-R/2}R^{-2+\alpha/2}(x^{\alpha/2}\vee x).$$ Hence Lemma \[trivialbound\] yields $$G^1_{(0,\infty)}(x,y)-G^1_{(0,R)}(x,y)\le C e^{-R/2}R^{-1+\alpha/2}G^1_{(0,\infty)}(x,y).$$ This proves (\[lowerbound100\]) for $R>R_0$, if $R_0>4$ is large enough.
To handle the case $4\le R\le R_0$ we apply (\[intervalcomparison\]) together with [(\[greenstable\])]{} and Theorem \[Green1est\] to obtain $$G_{(0,R)}(x,y)\ge c(R_0)G^{stable}_{(0,R)}(x,y)\ge c G^1_{(0,\infty)}(x,y),$$ which ends the proof of (\[lowerbound100\]).
That is, by Theorem \[green\], for $|x-y|<2,\ x\ge 1$, we get $$\begin{aligned}
\label{grintl1}G_{(0,R)}(x,y)&\ge&
c(G^1_{(0,R)}(x,y)+G^{gauss}_{(0,R)}(x,y))\nonumber\\&\ge&
c(G^1_{(0,\infty)}(x,y)+x)\ge c G_{(0,\infty)}(x,y).\end{aligned}$$ Next, for $x< 1$ and $y\ge 2$, by BHP (Lemma \[BHP1\]), $$\begin{aligned}
\label{grintl2}G_{(0,R)}(x,y)&\ge& c G_{(0,R)}(1,y)
x^{\alpha/2}\ge c x^{\alpha/2} G_{(0,\infty)}(1,y)\nonumber\\
&\ge& c G_{(0,\infty)}(x,y) .\end{aligned}$$ Combining [(\[grintl1\])]{} and [(\[grintl2\])]{} give us $$G_{(0,R)}(x,y)\approx G_{(0,\infty)}(x,y),$$ for $x,y\le R/2+1$.
\[Greenintervalprop2\]Let $R\ge 4$. Suppose that $1\le x\le y\le R-1$, and $|x-y|\ge 1$ then $$G_{(0,R)}(x,y)\approx G^{gauss}_{(0,R)}(x,y)=x(R-y)/R .$$
Due to Lemma \[potential\_lower\] we only need to prove upper bound. Suppose that $1\le x\le
y\le R-1$ and $|x-y|\ge 1$.
At first, let additionally $y\le 3/4 R$, then by Remark \[Greenhalflineasymp\], $$\label{grintp1}
G_{(0,R)}(x,y)\le G_{(0,\infty)}(x,y)\le c x\le
4cx(R-y)/R.$$ By symmetry and the above inequality we have $$\begin{aligned}
\label{grintp2}G_{(0,R)}(x,y)&=&G_{(-R,0)}(-x,-y)=G_{(0,R)}(R-x,R-y)\nonumber\\
&\le& 4c(R-y)x/R,\end{aligned}$$ for $R/4\le x\le R/2\le y\le R-1$ and $|x-y|>1$.
Hence it remains to consider the case $1\le x\le R/4$ and $3R/4\le
y\le R-1$. Denote $\eta=\tau_{(0,R/2)}$. Since $G_{(0,R)}(\cdot,y)$ is regular harmonic on $(0,R/2)$, so by Proposition \[Greenintervalprop1\] and Remark \[Greenhalflineasymp\], $$\begin{aligned}
& &\!\!\!G_{(0,R)}(x,y)\nonumber\\&
&=\,E^xG_{(0,R)}(X_{\eta},y)\le
E^x\left[X_{\eta}>R/2;G_{(0,\infty)}(R-X_{\eta},R-y)\right]\nonumber\\
& &\le\,cE^x\left[ X_{\eta}>R/2,|X_{\eta}-y|<1;G^1_{(0,\infty)}(R-X_{\eta},R-y)\right]\nonumber\\
& & \; + \,
cE^x\left[X_{\eta}>R/2;\left((R-X_{\eta})\vee(R-X_{\eta})^{\alpha/2}\right)\wedge
(R-y)\right]\nonumber\\
& & \le \, c(R-y)P^x(\eta<\tau_{(0,\infty)})\nonumber\\& &\;+\,cE^x\left[|X_{\eta}-y|<1;G^1_{(0,\infty)}(R-X_{\eta},R-y)\right]\nonumber\\
& &\le\, c(R-y)x/R \nonumber\\& &\;+\,
cE^x\left[|X_{\eta}-y|<1;G^1_{(0,\infty)}(R-X_{\eta},R-y)\right],\label{grintp3}\end{aligned}$$ where the last inequality is a consequence of Proposition \[MRestimate\]. Moreover, by Lemma \[poissonKernel\] we obtain $$\begin{aligned}
& &
\!\!\!E^x\left[|X_{\eta}-y|<1;G^1_{(0,\infty)}(R-X_{\eta},R-y)\right]\nonumber\\&
& = \,
\int^{y+1}_{y-1}P_{(0,R/2)}(x,z)G^1_{(0,\infty)}(R-z,R-y)dz\nonumber\\
& &\le\,
c E^x\eta\int^{y+1}_{y-1}e^{-(z-R/2)}G^1_{(0,\infty)}(R-z,R-y)dz\nonumber\\
& &\le\, c \frac{R}{2} x e^{-(y-1-R/2)}\int^\infty_0
G^1_{(0,\infty)}(R-z,R-y)dz \nonumber\\& &\le\, c
x(R-y)/R,\label{grintp4}\end{aligned}$$ because $y\ge 3/4R$.
Combining [(\[grintp1\])]{}, [(\[grintp2\])]{}, [(\[grintp3\])]{} and [(\[grintp4\])]{} we obtain $$G_{(0,R)}(x,y)\le c x(R-y)/R=cG^{gauss}_{(0,R)}(x,y).$$
Now we can prove the main result of this section.
Let $R\ge 4$ and $x\le y$ then we have for $|x-y|\le 1$, $$G_{(0,R)}(x,y)\approx \min\{G_{(0,\infty)}(x,y),
G_{(0,\infty)}(R-x,R-y) \}$$ and for $|x-y|> 1$ $$G_{(0,R)}(x,y)\approx \frac {(x^{\alpha/2}\vee
x)((R-y)^{\alpha/2}\vee (R-y))}R.$$
Observe that by symmetry $$\label{symmetry}G_{(0,R)}(x,y)=G_{(0,R)}(R-x,R-y).$$ The case $|x-y|\le 1$ follows immediately from Proposition \[Greenintervalprop1\], Theorem \[green\] and (\[symmetry\]).
For $0<x\le y <R$, $|x-y|> 1$ we define $\tilde{x}= x\vee 1$ and $\tilde{y}= y \wedge (R-1)$. Then we can repeat the arguments used in the proof of Theorem \[greenhlafspaceest\] to arrive at $$\begin{aligned}
G_{(0,R)}(x,y)&\approx& (1\wedge x)^{\alpha/2}(1\wedge (R-y))^{\alpha/2}G^{gauss}_{(0,R)}(\tilde{x},\tilde{y})\\
&=& (1\wedge x)^{\alpha/2}(1\wedge (R-y))^{\alpha/2} \frac{\tilde{x}(R-\tilde{y})}R\\
&=& \frac {(x^{\alpha/2}\vee
x)((R-y)^{\alpha/2}\vee (R-y))}R , \quad |x-y|> 1.\end{aligned}$$ This completes the proof.
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| ArXiv |
---
abstract: 'Let $L$ be a reductive subgroup of a reductive Lie group $G$. Let $G/H$ be a homogeneous space of reductive type. We provide a necessary condition for the properness of the action of $L$ on $G/H$. As an application we give examples of spaces that do not admit standard compact Clifford-Klein forms.'
author:
- Maciej Bocheński and Marek Ogryzek
title: A restriction on proper actions on homogeneous spaces of a reductive type
---
Introduction
============
Let $L$ be a locally compact topological group acting continuously on a locally Hausdorff topological space $M$. This action is called [***proper***]{} if for every compact subset $C \subset M$ the set $$L(C):=\{ g\in L \ | \ g\cdot C \cap C \neq \emptyset \}$$ is compact. In this paper, our main concern is the following question posed by T. Kobayashi [@kob3]
How “large” subgroups of $G$ can act properly
on a homogeneous space $G/H$? **(Q1)**
We restrict our attention to the case where $M=G/H$ is a homogeneous space of reductive type and always assume that $G$ is a linear connected reductive real Lie group with the Lie algebra $\mathfrak{g}.$ Let $H\subset G$ be a closed subgroup of $G$ with finitely many connected components and $\mathfrak{h}$ be the Lie algebra of $H.$
The subgroup $H$ is reductive in $G$ if $\mathfrak{h}$ is reductive in $\mathfrak{g},$ that is, there exists a Cartan involution $\theta $ for which $\theta (\mathfrak{h}) = \mathfrak{h}.$ The space $G/H$ is called the homogeneous space of reductive type. \[def1\]
Note that if $\mathfrak{h}$ is reductive in $\mathfrak{g}$ then $\mathfrak{h}$ is a reductive Lie algebra.
It is natural to ask when does a closed subgroup of $G$ act properly on a space of reductive type $G/H.$ This problem was treated, inter alia, in [@ben], [@bt], [@kas], [@kob2], [@kob4], [@kob1], [@kul] and [@ok]. In [@kob2] one can find a very important criterion for a proper action of a subgroup $L$ reductive in $G.$ To state this criterion we need to introduce some additional notation. Let $\mathfrak{l}$ be the Lie algebra of $L.$ Take a Cartan involution $\theta$ of $\mathfrak{g}.$ We obtain the Cartan decomposition $$\mathfrak{g}=\mathfrak{k} + \mathfrak{p}.
\label{eq1}$$ Choose a maximal abelian subspace $\mathfrak{a}$ in $\mathfrak{p}.$ The subspace $\mathfrak{a}$ is called the ***maximally split abelian subspace*** of $\mathfrak{p}$ and $\text{rank}_{\mathbb{R}}(\mathfrak{g}) := \text{dim} (\mathfrak{a})$ is called the ***real rank*** of $\mathfrak{g}.$ It follows from Definition \[def1\] that $\mathfrak{h}$ and $\mathfrak{l}$ admit Cartan decompositions $$\mathfrak{h}=\mathfrak{k}_{1} + \mathfrak{p}_{1} \ \text{and} \ \mathfrak{l}=\mathfrak{k}_{2} + \mathfrak{p}_{2},$$ given by Cartan involutions $\theta_{1}, \ \theta_{2}$ of $\mathfrak{g}$ such that $\theta_{1} (\mathfrak{h})= \mathfrak{h}$ and $\theta_{2} (\mathfrak{l})= \mathfrak{l}.$ Let $\mathfrak{a}_{1} \subset \mathfrak{p}_{1}$ and $\mathfrak{a}_{2} \subset \mathfrak{p}_{2}$ be maximally split abelian subspaces of $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2},$ respectively. One can show that there exist $a,b \in G$ such that $\mathfrak{a}_{\mathfrak{h}} := \text{\rm Ad}_{a}\mathfrak{a}_{1} \subset \mathfrak{a}$ and $\mathfrak{a}_{\mathfrak{l}} := \text{\rm Ad}_{b}\mathfrak{a}_{2} \subset \mathfrak{a}.$ Denote by $W_{\mathfrak{g}}$ the Weyl group of $\mathfrak{g}.$ In this setting the following holds
The following three conditions are equivalent
1. $L$ acts on $G/H$ properly.
2. $H$ acts on $G/L$ properly.
3. For any $w \in W_{\mathfrak{g}},$ $w\cdot \mathfrak{a}_{\mathfrak{l}} \cap \mathfrak{a}_{\mathfrak{h}} =\{ 0 \}.$
\[twkob\]
Note that the criterion 3. in Theorem \[twkob\] depends on how $L$ and $H$ are embedded in $G$ up to inner-automorphisms. Theorem \[twkob\] provides a partial answer to Q1.
If $L$ acts properly on $G/H$ then $$\text{\rm rank}_{\mathbb{R}}(\mathfrak{l}) + \text{\rm rank}_{\mathbb{R}}(\mathfrak{h}) \leq \text{\rm rank}_{\mathbb{R}} (\mathfrak{g}).$$ \[coko\]
Hence the real rank of $L$ is bounded by a constant which depends on $G/H,$ no matter how $H$ and $L$ are embedded in $G.$ In this paper we find a similar, stronger restriction for Lie groups $G,H,L$ by means of a certain tool which we call the a-hyperbolic rank (see Section 2, Definition \[dd2\] and Table \[tab1\]). In more detail we prove the following
If $L$ acts properly on $G/H$ then $$\mathop{\mathrm{rank}}\nolimits_{\text{\rm a-hyp}}(\mathfrak{l}) + \mathop{\mathrm{rank}}\nolimits_{\text{\rm a-hyp}}(\mathfrak{h}) \leq \mathop{\mathrm{rank}}\nolimits_{\text{\rm a-hyp}} (\mathfrak{g}).$$ \[twgl\]
Recall that a homogeneous space $G/H$ of reductive type admits a ***compact Clifford-Klein form*** if there exists a discrete subgroup $\Gamma \subset G$ such that $\Gamma$ acts properly on $G/H$ and $\Gamma \backslash G/H$ is compact. The space $G/H$ admits a ***standard compact Clifford-Klein form*** in the sense of Kassel-Kobayashi [@kako] if there exists a subgroup $L$ reductive in $G$ such that $L$ acts properly on $G/H$ and $L \backslash G/H$ is compact. In the latter case, for any discrete cocompact subgroup $\Gamma ' \subset L,$ the space $\Gamma ' \backslash G/H$ is a compact Clifford-Klein form. Therefore it follows from Borel’s theorem (see [@bor]) that any homogeneous space of reductive type admitting a standard compact Clifford-Klein form also admits a compact Clifford-Klein form. It is not known if the converse statement holds, but all known reductive homogeneous spaces $G/H$ admitting compact Clifford-Klein forms also admit standard compact Clifford-Klein forms.
As a corollary to Theorem \[twgl\], we get examples of the semisimple symmetric spaces without standard compact Clifford-Klein forms. In particular, we cannot find the first example in the existing literature:
The homogeneous spaces $G/H=SL(2k+1, \mathbb{R})/SO(k-1,k+2)$ and $G/H=SL(2k+1, \mathbb{R})/Sp(k-1,\mathbb{R})$ for $k \geq 5$ do not admit standard compact Clifford-Klein forms. \[co1\]
Let us mention the following results, related to the above corollary.
- T. Kobayashi proved in [@kobadm] that $SL(2k,\mathbb{R})/SO(k,k)$ for $k\geq 1$ and $SL(n,\mathbb{R})/Sp(l,\mathbb{R})$ for $0<2l \leq n-2$ do not admit compact Clifford-Klein forms.
- Y. Benoist proved in [@ben] that $SL(2k+1,\mathbb{R})/SO(k,k+1)$ for $k\geq 1$ does not admit compact Clifford-Klein forms.
- Y. Morita proved recently in [@mor] that $SL(p+q,\mathbb{R})/SO(p,q)$ does not admit compact Clifford-Klein forms if $p$ and $q$ are both odd.
Note that these works are devoted to the problem of existence of compact Clifford-Klein forms on a given homogeneous space (not only standard compact Clifford-Klein forms).
The a-hyperbolic rank and antipodal hyperbolic orbits
=====================================================
Let $\Sigma_{\mathfrak{g}}$ be a system of restricted roots for $\mathfrak{g}$ with respect to $\mathfrak{a}.$ Choose a system of positive roots $\Sigma^{+}_{\mathfrak{g}}$ for $\Sigma_{\mathfrak{g}}.$ Then a fundamental domain of the action of $W_{\mathfrak{g}}$ on $\mathfrak{a}$ can be define as $$\mathfrak{a}^{+} := \{ X\in \mathfrak{a} \ | \ \alpha (X) \geq 0 \ \text{\rm for any} \ \alpha \in \Sigma^{+}_{\mathfrak{g}} \}.$$ Note that $$sX+tY \in \mathfrak{a}^{+},$$ for any $s,t \geq 0$ and $X,Y\in \mathfrak{a}^{+}$. Therefore $\mathfrak{a}^{+}$ is a convex cone in the linear space $\mathfrak{a}.$ Let $w_{0} \in W_{\mathfrak{g}}$ be the longest element. One can show that $$-w_{0}: \mathfrak{a} \rightarrow \mathfrak{a}, \ \ X \mapsto -(w_{0} \cdot X)$$ is an involutive automorphism of $\mathfrak{a}$ preserving $\mathfrak{a}^{+}.$ Let $\mathfrak{b} \subset \mathfrak{a}$ be the subspace of all fixed points of $-w_{0}$ and put $$\mathfrak{b}^{+} := \mathfrak{b} \cap \mathfrak{a}^{+}.$$ Thus $\mathfrak{b}^{+}$ is a convex cone in $\mathfrak{a}.$ We also have $\mathfrak{b} = \text{Span} (\mathfrak{b}^{+}).$
The dimension of $\mathfrak{b}$ is called the a-hyperbolic rank of $\mathfrak{g}$ and is denoted by $$\mathop{\mathrm{rank}}\nolimits_{\text{a-hyp}} (\mathfrak{g}).$$ \[dd2\]
The a-hyperbolic ranks of the simple real Lie algebras can be deduce from Table \[tab1\].
A method of calculation of the a-hyperbolic rank of a simple Lie algebra can be found in [@bt]. The a-hyperbolic rank of a semisimple Lie algebra equals the sum of a-hyperbolic ranks of all its simple parts. For a reductive Lie algebra $\mathfrak{g}$ we put $$\mathop{\mathrm{rank}}\nolimits_{\text{a-hyp}} (\mathfrak{g}) := \mathop{\mathrm{rank}}\nolimits_{\text{a-hyp}}([\mathfrak{g},\mathfrak{g}]).$$
There is a close relation between $\mathfrak{b}^{+}$ and the set of antipodal hyperbolic orbits in $\mathfrak{g}.$ We say that an element $X \in \mathfrak{g}$ is [***hyperbolic***]{}, if $X$ is semisimple (that is, $\mathrm{ad}_{X}$ is diagonalizable) and all eigenvalues of $\mathrm{ad}_{X}$ are real.
An adjoint orbit $O_{X}:=\mathop{\mathrm{Ad}}\nolimits (G)(X)$ is said to be hyperbolic if $X$ (and therefore every element of $O_{X}$) is hyperbolic. An adjoint orbit $O_{Y}$ is antipodal if $-Y\in O_{Y}$ (and therefore for every $Z\in O_{Y},$ $-Z\in O_{Y}$).
There is a bijective correspondence between antipodal hyperbolic orbits $O_{X}$ in $\mathfrak{g}$ and elements $Y \in \mathfrak{b}^{+}.$ This correspondence is given by $$\mathfrak{b}^{+}\ni Y \mapsto O_{Y}.$$ Furthermore, for every hyperbolic orbit $O_{X}$ in $\mathfrak{g}$ the set $O_{X} \cap \mathfrak{a}$ is a single $W_{\mathfrak{g}}$ orbit in $\mathfrak{a}$. \[lma\]
The main result
===============
We need two basic facts from linear algebra.
Let $V_{1},V_{2}$ be vector subspaces of a real linear space $V$ of finite dimension. Then $$\text{\rm dim} (V_{1}+V_{2})= \text{\rm dim} (V_{1}) + \text{\rm dim} (V_{2}) - \text{\rm dim} (V_{1}\cap V_{2}).$$ \[lma1\]
Let $V_{1},...,V_{n}$ be a collection of vector subspaces of a real linear space $V$ of a finite dimension and let $A^{+} \subset V$ be a convex cone. Assume that $$A^{+} \subset \bigcup_{k=1}^{n} V_{k}.$$ Then there exists a number $k,$ such that $A^{+} \subset V_{k}.$ \[lma2\]
We also need the following, technical lemma. Choose a subalgebra $\mathfrak{h}$ reductive in $\mathfrak{g}$ which corresponds to a Lie group $H \subset G.$ Let $\mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}^{+} \subset \mathfrak{a}_{\mathfrak{h}}$ be the convex cone constructed according to the procedure described in the previous subsection (for $[\mathfrak{h},\mathfrak{h}]$).
Let $X\in \mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}^{+}.$ The orbit $O_{X}:=\mathop{\mathrm{Ad}}\nolimits (G)(X)$ is an antipodal hyperbolic orbit in $\mathfrak{g}.$ \[lma3\]
By Lemma \[lma\] the vector $X$ defines an antipodal hyperbolic orbit in $\mathfrak{h}.$ Therefore we can find $h \in H \subset G$ such that $\text{\rm Ad}_{h}(X) = - X$. Since a maximally split abelian subspace $\mathfrak{a} \subset \mathfrak{g}$ consists of vectors for which $\mathrm{ad}$ is diagonalizable with real values and $$X \in \mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}^{+} \subset \mathfrak{a}_{\mathfrak{h}} \subset \mathfrak{a},$$ thus the vector $X$ is hyperbolic in $\mathfrak{g}.$ It follows that $\mathop{\mathrm{Ad}}\nolimits (G)(X)$ is a hyperbolic orbit in $\mathfrak{g}$ and $-X \in \mathop{\mathrm{Ad}}\nolimits (G)(X).$
Now we are ready to give a proof of Theorem \[twgl\].
Assume that $\mathop{\mathrm{rank}}\nolimits_{\text{a-hyp}}(\mathfrak{l}) + \mathop{\mathrm{rank}}\nolimits_{\text{a-hyp}}(\mathfrak{h}) > \mathop{\mathrm{rank}}\nolimits_{\text{a-hyp}} (\mathfrak{g})$ and let $\mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}^{+},$ $\mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]}^{+},$ $\mathfrak{b}^{+}$ be appropriate convex cones. If $X \in \mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}^{+}$ then $O_{X}^{H} := \mathop{\mathrm{Ad}}\nolimits (G)(X)$ is an antipodal hyperbolic orbit in $\mathfrak{h}.$ By Lemma \[lma3\] the orbit $O_{X}^{G} := \mathop{\mathrm{Ad}}\nolimits (G)(X)$ is an antipodal hyperbolic orbit in $\mathfrak{g}.$ By Lemma \[lma\] there exists $Y \in \mathfrak{b}^{+}$ such that $$O_{X}^{G} = O_{Y}^{G} = \mathop{\mathrm{Ad}}\nolimits (G)(Y).$$ Since $\mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}^{+} \subset \mathfrak{a}_{\mathfrak{h}} \subset \mathfrak{a}$ and $\mathfrak{b}^{+} \subset \mathfrak{a}$ thus (according to Lemma \[lma\]) we get $X=w_{1} \cdot Y$ for a certain $w_{1} \in W_{\mathfrak{g}}.$ Therefore $$\mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}^{+} \subset W_{\mathfrak{g}} \cdot \mathfrak{b}^{+} = \bigcup_{w\in W_{\mathfrak{g}}} w \cdot \mathfrak{b}^{+} \subset \bigcup_{w\in W_{\mathfrak{g}}} w \cdot \mathfrak{b}.$$ Analogously $$\mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]}^{+} \subset \bigcup_{w\in W_{\mathfrak{g}}} w \cdot \mathfrak{b}^{+} \subset \bigcup_{w\in W_{\mathfrak{g}}} w \cdot \mathfrak{b}.$$ By Lemma \[lma2\] there exist $w_{\mathfrak{h}},w_{\mathfrak{l}} \in W_{\mathfrak{g}}$ such that $$\mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}^{+} \subset w_{\mathfrak{h}}^{-1} \cdot \mathfrak{b} \ \ \text{\rm and} \ \ \mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]}^{+} \subset w_{\mathfrak{l}}^{-1} \cdot \mathfrak{b},$$ because $W_{\mathfrak{g}}$ acts on $\mathfrak{a}$ by linear transformations. Therefore $$\mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]} \subset w_{\mathfrak{h}}^{-1} \cdot \mathfrak{b} \ \ \text{\rm and} \ \ \mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]} \subset w_{\mathfrak{l}}^{-1} \cdot \mathfrak{b}$$ where $\mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]} := \text{Span}(\mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}^{+})$ and $\mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]} := \text{Span} (\mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]}^{+}).$ We obtain $$w_{\mathfrak{h}} \cdot \mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]} \subset \mathfrak{b} , \ w_{\mathfrak{l}} \cdot \mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]} \subset \mathfrak{b}.$$ By the assumption and Lemma \[lma1\] $$\text{\rm dim}(w_{\mathfrak{h}} \cdot \mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]} \cap w_{\mathfrak{l}} \cdot \mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]}) >0.$$ Choose $0 \neq Y \in w_{\mathfrak{h}} \cdot \mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]} \cap w_{\mathfrak{l}} \cdot \mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]}.$ Then $$w_{\mathfrak{l}} \cdot X_{\mathfrak{l}}=Y= w_{\mathfrak{h}} \cdot X_{\mathfrak{h}} \ \text{\rm for some} \ X_{\mathfrak{h}} \in \mathfrak{b}_{[\mathfrak{h},\mathfrak{h}]}\backslash \{ 0 \} \ \text{\rm and} \ X_{\mathfrak{l}} \in \mathfrak{b}_{[\mathfrak{l},\mathfrak{l}]}\backslash \{ 0 \}.$$ Take $w_{2} := w_{\mathfrak{h}}^{-1}w_{\mathfrak{l}} \in W_{\mathfrak{g}},$ we have $X_{\mathfrak{h}}= w_{2} \cdot X_{\mathfrak{l}}$ and $X_{\mathfrak{h}} \in \mathfrak{a}_{\mathfrak{h}}, \ X_{\mathfrak{l}} \in \mathfrak{a}_{\mathfrak{l}}.$ Thus $0 \neq X_{\mathfrak{h}} \in w_{2} \cdot \mathfrak{a}_{\mathfrak{l}} \cap \mathfrak{a}_{\mathfrak{h}}.$ The assertion follows from Theorem \[twkob\].
We can proceed to a proof of Corollary \[co1\]. For a reductive Lie group $D$ with a Lie algebra $\mathfrak{d}$ with a Cartan decomposition $$\mathfrak{d} = \mathfrak{k}_{\mathfrak{d}} + \mathfrak{p}_{\mathfrak{d}}$$ we define $d(G) := \text{dim} (\mathfrak{p}_{\mathfrak{d}}).$ We will need the following properties
Let $L$ be a subgroup reductive in $G$ acting properly on $G/H.$ The space $L \backslash G /H$ is compact if and only if $$d(L)+d(H)=d(G).$$ \[twkk\]
If $J \subset G$ is a semisimple subgroup then it is reductive in $G.$ \[twy\]
Let $L \subset G$ be a semisimple Lie group acting properly on $G/H=SL(2k+1, \mathbb{R})/SO(k-1,k+2)$ or $G/H=SL(2k+1, \mathbb{R})/Sp(k-1,\mathbb{R}).$ Then $$\text{\rm rank}_{\mathbb{R}}(\mathfrak{l}) \leq 2.$$ \[p1\]
Because $\mathop{\mathrm{rank}}\nolimits_{\text{a-hyp}}(\mathfrak{g}) = 1+ \mathop{\mathrm{rank}}\nolimits_{\text{a-hyp}}(\mathfrak{h})$ thus it follows from Table \[tab1\] and Theorem \[twgl\] that if $L$ is simple then $\text{rank}_{\mathbb{R}}(\mathfrak{l}) \leq 2.$ On the other hand if $L$ is semisimple then each (non-compact) simple part of $\mathfrak{l}$ adds at least $1$ to a-hyperbolic rank of $\mathfrak{l}.$ Thus we also have $\text{rank}_{\mathbb{R}}(\mathfrak{l}) \leq 2.$
*Proof of Corollary \[co1\].* Assume now that $L$ is reductive in $G.$ Since the Lie algebra $\mathfrak{l}$ is reductive therefore $$\mathfrak{l} = \mathfrak{c}_{\mathfrak{l}} + [\mathfrak{l},\mathfrak{l}],$$ where $\mathfrak{c}_{\mathfrak{l}}$ denotes the center of $\mathfrak{l}.$ It follows from Corollary \[coko\] that $$\text{\rm rank}_{\mathbb{R}}(\mathfrak{l}) \leq k+1,
\label{eq2}$$ and by Proposition \[p1\] we have $\text{rank}_{\mathbb{R}}([\mathfrak{l},\mathfrak{l}]) \leq 2.$ Note that $$d(G)-d(H)\geq k^{2} +2k +2.$$ We will show that if $L$ acts properly on $G/H$ and $k\geq 5$ then $$d(L) < k^{2} + 2k +2.
\label{eq4}$$ Let $[\mathfrak{l},\mathfrak{l}] = \mathfrak{k}_{0} + \mathfrak{p}_{0}$ be a Cartan decomposition. From (\[eq2\]) $$d(L) \leq \text{\rm dim} (\mathfrak{c}_{\mathfrak{l}}) + \text{\rm dim} (\mathfrak{p}_{0}) \leq k+1 + \text{\rm dim} (\mathfrak{p}_{0}).
\label{eq7}$$ Also, if $\text{rank}_{\mathbb{R}}([\mathfrak{l},\mathfrak{l}]) =2$ then it follows from Table \[tab1\] that (the only) non-compact simple part of $[\mathfrak{l},\mathfrak{l}]$ is isomorphic to $\mathfrak{sl}(3,\mathbb{R}),$ $\mathfrak{su}^{\ast}(6),$ $\mathfrak{e}_{6}^{\text{IV}}$ or $\mathfrak{sl}(3,\mathbb{C})$ (treated as a simple real Lie algebra). In such case $$\text{\rm dim} (\mathfrak{p}_{0}) < 27.
\label{eq5}$$ Therefore assume that $\text{rank}_{\mathbb{R}} ([\mathfrak{l},\mathfrak{l}])=1$ and let $\mathfrak{s} \subset [\mathfrak{l},\mathfrak{l}]$ be (the only) simple part of a non-compact type. We have $$\text{\rm rank}_{\mathbb{R}} (\mathfrak{s}) =1.
\label{eq3}$$ It follows from Theorem \[twy\] that $\mathfrak{s}$ is reductive in $\mathfrak{g}.$ Therefore $\mathfrak{s}$ admits a Cartan decomposition $$\mathfrak{s} = \mathfrak{k}_{\mathfrak{s}} + \mathfrak{p}_{\mathfrak{s}}$$ compatible with $\mathfrak{g}= \mathfrak{k} + \mathfrak{p},$ that is $\mathfrak{k}_{\mathfrak{s}} \subset \mathfrak{k}.$ We also have $\text{dim}(\mathfrak{p}_{s})=\text{dim}(\mathfrak{p}_{0}).$ Since $\mathfrak{k} = \mathfrak{so}(2k+1)$ we obtain $$\text{\rm rank} (\mathfrak{k}_{s}) \leq \text{\rm rank} (\mathfrak{k}) = k.$$ Using the above condition together with (\[eq3\]) we can check (by a case-by-case study of simple Lie algebras) that $$\text{\rm dim} (\mathfrak{p}_{\mathfrak{s}}) < 4k.
\label{eq6}$$ Now (\[eq7\]), (\[eq5\]) and (\[eq6\]) imply that $$d(L) < 5k+1$$ for $k \geq 6,$ and $d(L)<33$ for $k=5.$ Thus we have showed (\[eq4\]). The assertion follows from Theorem \[twkk\].
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T. Kobayashi, [*Discontinuous groups acting on homogeneous spaces of reductive type*]{}, Representation theory of Lie groups and Lie algebras (Fuji-Kawaguchiko, 1990), World Sci. Publ., River Edge (1992), 59-75.
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Maciej Bocheński
Department of Mathematics and Computer Science,
University of Warmia and Mazury,
Słoneczna 54, 10-710, Olsztyn, Poland.
email: [email protected]
Marek Ogryzek
Department of Geodesy and Land Management,
University of Warmia and Mazury,
Prawocheńskiego 15, 10-720, Olsztyn, Poland.
email: [email protected]
| ArXiv |
---
address: |
Theoretical Division, Los Alamos National Laboratory\
Los Alamos, New Mexico 87545, USA\
E-mail: [email protected] [and]{} [email protected]
author:
- and PETER MÖLLER
title: 'MASSES AND DEFORMATIONS OF NEUTRON-RICH NUCLEI'
---
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Introduction {#intro}
============
The accurate calculation of the ground-state mass and deformation of a nucleus far from stability, such as one of the neutron-rich nuclei considered in this conference, remains one of the most fundamental challenges of nuclear theory. Toward this goal, two major approaches—which also allow the simultaneous calculation of a wide variety of other nuclear properties—have been developed (along with numerous semi-empirical formulas for masses alone).
At the most fundamental level, fully selfconsistent microscopic theories, starting with an underlying nucleon-nucleon interaction, have seen progress in both the nonrelativistic Hartree-Fock approximation and more recently the relativistic mean-field approximation. Although microscopic theories offer great promise for the future, their current accuracies are typically a few Mwhich is insufficient for most practical applications. At the next level of fundamentality, the macroscopic-microscopic method—where the smooth trends are obtained from a macroscopic model and the local fluctuations from a microscopic model—has been used in several recent global calculations that are useful for a broad range of applications.
We will concentrate here on the 1992 version of the finite-range droplet model,$\,$[@MNMS; @MNK]with particular emphasis on its reliability for extrapolations to new regions of nuclei, but will also briefly discuss two other models of this type.$\,$[@APDT; @MS]
Finite-Range Droplet Model {#frdms}
==========================
In the finite-range droplet model, which takes its name from the macroscopic model that is used, the microscopic shell and pairing corrections are calculated from a realistic, diffuse-surface, folded-Yukawa single-particle potential by use of Strutinsky’s method.$\,$[@S]In 1992 we made a new adjustment of the constants of an improved version of this model to 28 fission-barrier heights and to 1654 nuclei with $N,Z \ge 8$ ranging from $^{16}$O to $^{263}$106 whose masses were known experimentally in 1989.$\,$[@A]The resulting microscopic enhancement to binding for even-even nuclei throughout the periodic system is shown in Fig. \[enhab\].
enhab fig1 201 [Calculated additional binding energy of even-even nuclei relative to the macroscopic energy of spherical nuclei, illustrating the crucial role of microscopic corrections.]{}
This model has been used to calculate the ground-state mass, deformation, microscopic correction, odd-proton and odd-neutron spins and parities, proton and neutron pairing gaps, binding energy, one- and two-neutron separation energies, quantities related to $\beta$-delayed one- and two-neutron emission probabilities, $\beta$-decay energy release and half-life with respect to Gamow-Teller decay, one- and two-proton separation energies, and $\alpha$-decay energy release and half-life for 8979 nuclei with $N,Z \ge 8$ ranging from $^{16}$O to $^{339}136$ and extending from the proton drip line to the neutron drip line.$\,$[@MNMS; @MNK]These tabulated quantities are available electronically on the World Wide Web at the Uniform Resource Locator [http://t2.lanl.gov/publications/publications.html]{}.
quad fig2 201 [Calculated quadrupole deformations of even-even nuclei, illustrating the transitions from spherical to deformed nuclei as one moves away from magic numbers.]{}
Ground-State Deformations {#def}
=========================
In our calculations, we specify a general nuclear shape in terms of deviations from a spheroidal shape by use of Nilsson’s $\epsilon$ parameterization.$\,$[@Ni]The ground-state shape is determined by initially minimizing the nuclear potential energy of deformation with respect to the two symmetric shape coordinates $\epsilon_2$ and $\epsilon_4$. During this minimization, we include a prescribed smooth dependence of the higher symmetric deformation $\epsilon_6$ on the two independent coordinates $\epsilon_2$ and $\epsilon_4$. This dependence is determined by minimizing the macroscopic potential energy of $^{240}$Pu with respect to $\epsilon_6$ for fixed values of $\epsilon_2$ and $\epsilon_4$. We then vary separately $\epsilon_6$ and the mass-asymmetric, or octupole, deformation $\epsilon_3$, with $\epsilon_2$ and $\epsilon_4$ held fixed at their previously determined values, to calculate any additional lowering in energy from these two degrees of freedom.
For presentation purposes, it is sometimes more convenient to express the nuclear ground-state shape in terms of the $\beta$ parameterization, where the shape coordinates represent the coefficients in an expansion of the radius vector to the nuclear surface in a series of spherical harmonics. Figures \[quad\] and \[hex\] show our calculated quadrupole and hexadecapole deformations, respectively, in terms of $\beta_2$ and $\beta_4$, which are determined by transforming our calculated shapes from the $\epsilon$ parameterization.
hex fig3 201 [Calculated hexadecapole deformations of even-even nuclei, illustrating the transitions from bulging to indented equatorial regions as one moves from smaller to larger magic numbers.]{}
The inclusion of the $\epsilon_6$ and $\epsilon_3$ shape degrees of freedom is crucial for the isolation of such physical effects as the Coulomb redistribution energy, which arises from a central density depression.$\,$[@MNMS2]As illustrated in Fig. \[eps6\], an independent variation of the symmetric deformation $\epsilon_6$ is important for several regions of nuclei. For even-even nuclei, the maximum reduction in energy relative to that for a prescribed smooth $\epsilon_6$ dependence is 1.28 Mand occurs for $^{252}$Fm. As illustrated in Fig. \[eps3\], the mass-asymmetric deformation $\epsilon_3$ is important for nuclei in a few isolated regions. For even-even nuclei, the maximum reduction in energy relative to that for a symmetric shape is 1.29 Mand occurs for the neutron-rich nucleus $^{194}$Gd. For even-even nuclei close to the valley of $\beta$-stability, the maximum reduction in energy relative to that for a symmetric shape is 1.20 Mand occurs for $^{222}$Ra.
eps6 fig4 198 [Calculated reduction in energy of even-even nuclei arising from an independent variation in $\epsilon_6$, relative to that for shapes with a prescribed smooth $\epsilon_6$ dependence. Note that the sign of the $\epsilon_6$ correction is reversed in this plot for clarity of display.]{}
eps3 fig5 198 [Calculated reduction in energy of even-even nuclei arising from the inclusion of $\epsilon_3$ deformations, relative to that for symmetric shapes. Note that the sign of the $\epsilon_3$ correction is reversed in this plot for clarity of display.]{}
Reliability for Extrapolations to New Regions of Nuclei {#extraps}
=======================================================
For the original 1654 nuclei included in the adjustment, the theoretical error, determined by use of the maximum-likelihood method with no contributions from experimental errors,$\,$[@MNMS; @MNK]is 0.669 MAlthough some large systematic errors exist for light nuclei, they decrease significantly for heavier nuclei.
Between 1989 and 1996, the masses of 371 additional nuclei heavier than $^{16}$O have been measured,$\,$[@AW]$^{\sen}\,$[@H]which provides an ideal opportunity to test the ability of mass models to extrapolate to new regions of nuclei whose masses were not included in the original adjustment. Figure \[frdm\] shows as a function of the number of neutrons from $\beta$-stability the individual deviations between these newly measured masses and those predicted by the 1992 finite-range droplet model. The new nuclei fall into three categories, with the first category corresponding to 273 nuclei lying on both sides of the valley of $\beta$-stability.$\,$[@AW]The second category corresponds to 91 proton-rich nuclei produced by fragmentation of $^{209}$Bi projectiles incident on a thick Be target in the experimental storage ring (ESR) at the Gesellschaft für Schwerionenforschung (GSI) in Darmstadt, Germany.$\,$[@K]The third category corresponds to seven proton-rich superheavy nuclei discovered in the separator for heavy-ion reaction products (SHIP) at GSI whose masses are estimated by adding the highest $\alpha$-decay energy release at each step in the decay chain to known masses.$\,$[@H]This procedure could seriously overestimate the experimental masses of some of the heavier nuclei because different energy releases have been observed in some cases.$\,$[@H]To account for this uncertainty, we have assigned a mass error of 0.5 Mfor each of these seven nuclei. Also, to account for errors of unknown origin, we have included an additional 0.076 Mcontribution$\,$[@N] to the mass errors for each of the 91 nuclei in the second category. The theoretical error of the 1992 finite-range droplet model \[FRDM (1992)\] for all of the 371 newly measured masses is 0.570 MThe reduction in error arises partly because most of the new nuclei are located in the heavy region, where the model is more accurate.
Analogous deviations occur for version 1 of the 1992 extended-Thomas-Fermi Strutinsky-integral \[ETFSI-1 (1992)\] model of Aboussir, Pearson, Dutta, and Tondeur.$\,$[@APDT]In this model, the macroscopic energy is calculated for a Skyrme-like nucleon-nucleon interaction by use of an extended Thomas-Fermi approximation. The shell correction is calculated from single-particle levels corresponding to this same interaction by use of a Strutinsky-integral method, and the pairing correction is calculated for a $\delta$-function pairing interaction by use of the conventional BCS approximation. The constants of the model were determined by adjustments to the ground-state masses of 1492 nuclei with mass number $A \ge 36$, which excludes the troublesome region from $^{16}$O to mass number $A = 35$. The theoretical error corresponding to 1540 nuclei whose masses were known experimentally$\,$[@A] at the time of the original adjustment is 0.733 MThe theoretical error for 366 newly measured masses$\,$[@AW]$^{\sen}\,$[@H] for nuclei with $A \ge 36$ is 0.739 M
Similar results hold for the 1994 Thomas-Fermi \[TF (1994)\] model of Myers and Swiatecki.$\,$[@MS]In this model, the macroscopic energy is calculated for a generalized Seyler-Blanchard nucleon-nucleon interaction by use of the original Thomas-Fermi approximation. For $N,Z \ge 30$ the shell and pairing corrections were taken from the 1992 finite-range droplet model, and for $N,Z \le 29$ a semi-empirical expression was used. The constants of the model were determined by adjustments to the ground-state masses of the same 1654 nuclei with $N,Z \ge 8$ ranging from $^{16}$O to $^{263}$106 whose masses were known experimentally in 1989 that were used in the 1992 finite-range droplet model. The theoretical error corresponding to these 1654 nuclei is 0.640 MThe reduced theoretical error relative to that in the 1992 finite-range droplet model arises primarily from the use of semi-empirical microscopic corrections in the extended troublesome region $N,Z \le 29$ rather than microscopic corrections calculated more fundamentally. The theoretical error for 371 newly measured masses$\,$[@AW]$^{\sen}\,$[@H] is 0.620 M
frdm fig6 195 [Deviations between experimental and calculated masses for 371 new nuclei whose masses were not included in the 1992 adjustment of the finite-range droplet model.$\,$[@MNMS; @MNK]]{}
As summarized in Table \[extrapt\], the theoretical error for the newly measured masses relative to that for the original masses to which the model constants were adjusted [*decreases*]{} by 15% for the FRDM (1992), increases by 1% for the ETFSI-1 (1992) model, and [*decreases*]{} by 3% for the TF (1994) model. These macroscopic-microscopic mass models can therefore be extrapolated to new regions of nuclei with differing amounts of confidence.
[lcccccccc]{}\
& & & & & &\
\
Model & & ${N}_{\rm nuc}$ & Error & & ${N}_{\rm nuc}$ & Error & & Error\
& & & (M& & & (M& & ratio\
\
FRDM (1992) & & 1654 & 0.669 & & 371 & 0.570 & & 0.85\
ETFSI-1 (1992) & & 1540 & 0.733 & & 366 & 0.739 & & 1.01\
TF (1994) & & 1654 & 0.640 & & 371 & 0.620 & & 0.97\
Rock of Metastable Superheavy Nuclei {#rock}
====================================
rohic fig7 201 [Ten recently discovered superheavy nuclei,$\,$[@H+]$^{\sen}\,$[@O]superimposed on a theoretical calculation$\,$[@MNMS; @MNK] of the microscopic corrections to the ground-state masses of nuclei extending from the vicinity of lead to heavy and superheavy nuclei. The heaviest nucleus, whose location on the diagram is indicated by the flag, was produced through a gentle reaction between spherical $^{70}$Zn and $^{208}$Pb nuclei in which a single neutron was emitted.$\,$[@H+]]{}
The heaviest nucleus known to man, $^{277}$112, was discovered$\,$[@H+] in February 1996 at the GSI by use of the gentle fusion reaction $^{70}$Zn + $^{208}$Pb $\rightarrow$ $^1$n + $^{277}$112. It is the latest in a series of about 10 recently discovered nuclei$\,$[@H+]$^{\sen}\,$[@O] lying on a rock of deformed metastable superheavy nuclei predicted to exist$\,$[@MNMS; @MNK; @MN]$^{\sen}\,$[@PS] near the deformed proton magic number at 110 and deformed neutron magic number at 162. These 10 superheavy nuclei are shown in Fig. \[rohic\] as tiny deformed three-dimensional objects. Most of the metastable superheavy nuclei that have been discovered live for only about a thousandth of a second, after which they generally decay by emitting a series of alpha particles. However, the decay products of the most recently discovered nucleus $^{277}$112 show for the first time that nuclei at the center of the predicted rock of stability live longer than 10 seconds.
We have used the macroscopic-microscopic method recently to calculate the fusion barrier for several reactions leading to deformed superheavy nuclei.$\,$[@MNAHM]For the reaction $^{70}$Zn + $^{208}$Pb $\rightarrow$ $^1$n + $^{277}$112, the microscopic shell and pairing corrections associated primarily with the doubly magic $^{208}$Pb target nucleus lower the total potential energy at the touching configuration by about 12 Mrelative to the macroscopic energy. These shell and pairing corrections persist from the touching configuration inward to a position only slightly more deformed than the ground-state shape. The resulting maximum in the fusion barrier is about 2 Mlower than the center-of-mass energy that was used in the GSI experiment that produced $^{277}$112.
One possibility to reach the island of spherical superheavy nuclei near $^{290}$110 that is predicted to lie beyond our present horizon involves the use of prolately deformed targets and projectiles that also possess large negative hexadecapole moments, which leads to large indented equatorial regions.$\,$[@IMNS]
Summary and Conclusion {#sum}
======================
The FRDM (1992) and two other macroscopic-microscopic models have been used recently to calculate the ground-state masses and deformations of nuclei throughout our known chart and beyond, and the FRDM (1992) has also been used to simultaneously calculate a wide variety of other nuclear properties. These models are useful for extrapolating to new regions of nuclei whose masses were not included in the original adjustment. Macroscopic-microscopic models have also correctly predicted the existence and location of a rock of deformed metastable superheavy nuclei near $^{272}$110 that has recently been discovered. Nuclear ground-state masses and deformations will continue to provide an invaluable testing ground for nuclear many-body theories. The future challenge is for fully selfconsistent microscopic theories to predict these quantities with comparable or greater accuracy.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the U. S. Department of Energy.
References {#references .unnumbered}
==========
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P. Möller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, .
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| ArXiv |
---
abstract: 'The fidelity decay in a microwave billiard is considered, where the coupling to an attached antenna is varied. The resulting quantity, coupling fidelity, is experimentally studied for three different terminators of the varied antenna: a hard wall reflection, an open wall reflection, and a 50$\,\Omega$ load, corresponding to a totally open channel. The model description in terms of an effective Hamiltonian with a complex coupling constant is given. Quantitative agreement is found with the theory obtained from a modified VWZ approach \[Verbaarschot et al, Phys. Rep. **129**, 367 (1985)\].'
author:
- 'B. Köber'
- 'U. Kuhl'
- 'H.-J. Stöckmann'
- 'T. Gorin'
- 'D. V. Savin'
- 'T. H. Seligman'
title: Microwave fidelity studies by varying antenna coupling
---
Introduction {#sec:Intro}
============
Fidelity is a standard benchmark in quantum information, and plays a relevant role in discussions on quantum chaos [@gor06c]. The corresponding fidelity amplitude can be interpreted as the overlap between two wave functions obtained from the propagation of the same initial state with two different time evolutions or, alternatively, as the overlap of the initial state with itself after being propagated forward in time with one evolution and backward in time with the other. In the latter case one often speaks of Loschmidt echo. Fidelity contains information on both eigenfunctions and spectra of the original and perturbed systems in a non-trivial way. One can show, however, that there exists a profound relation [@koh08] between fidelity decay and purely spectral universal parametric correlations [@sim93b; @tan95; @ale98; @mar03] in chaotic and disordered systems. The connection holds in quite general settings [@smo08]. Efforts to measure fidelity are therefore very important.
There was an early proposal (without the name fidelity) in quantum optics [@gar97]. Along these lines the perturbation of a kicked rotor was discussed in great detail [@hau05]. A realization of that idea is not available today, although an experiment of this type, but with a more complicated process, was conducted [@and03].
Experiments with microwave cavities or elastic bodies seem to provide good options to study the decay of fidelity [@sch05b], but a difficulty arises. Fidelity implies an integration over the entire space. In two-dimensional microwave billiards the antenna always represents a perturbation, and thus moving the antenna defeats the purpose of a fidelity measurement, as the wave-function taken at any point is that of a slightly different system. In contrast to wave function measurements, in fidelity experiments we are precisely interested in such differences, and thus wave functions measured with moveable antennas [@ste92; @ste95; @kuh07b] or a moveable perturbation body [@sri91; @bog06; @lau07] are not appropriate. In elastic experiments on solid blocks [@lob03b; @gor06b; @lob08] or three-dimensional (3D) microwave billiards the wave function inside the volume seems to be inaccessible anyway [@doer98b; @alt97a]. This leads to the development of the concept of scattering fidelity [@sch05b] which tests the sensitivity of $S$-matrix elements to perturbations. This is also of intrinsic interest since the scattering matrix may be considered as the basic building block at least in the case of quantum theory [@str00; @leh55].
In former studies the scattering fidelity has been investigated in chaotic microwave billiards by considering a perturbation of the billiard interior. It can be shown that in such a case the random character of wave functions causes the scattering fidelity to represent the usual fidelity, provided that appropriate averaging is taken [@sch05b; @hoeh08a]. Scars and parabolic manifolds will obviously change that correspondence, but their effect can be avoided in experiment. Specifically, two different types of interior perturbations were experimentally studied. In the first set of experiments a billiard wall was shifted, realizing the so-called global perturbation [@sch05b; @sch05d], meaning that there is a total rearrangement of both spectrum and eigenfunctions already for moderate perturbation strengths. Good agreement with prediction from random matrix theory (RMT), expecting Gaussian or exponential decay depending on perturbation strength, was found. In the second experiment a small scatterer was shifted inside the billiard, the wave function being influenced only locally [@hoeh08a]. Using the random plane wave conjecture, an algebraic decay was predicted and confirmed experimentally.
Actually, any measurement opens the system. Coupling to the continuum changes drastically the system properties by converting discrete energy levels into unstable resonance states. The latter reveal rich dynamics when the coupling strength to the scattering channels is varied [@sok92], see also [@per00] for relevant microwave studies. Since the time evolution operator is subunitary in this case, there appears the leakage of the norm inside the scattering system [@sav97]. This decay is fully controlled by the degree of system openness and may also be considered as a remote analog of fidelity decay for open systems. In the framework of the scattering fidelity coupling to the continuum can be taken into account naturally.
It seems therefore attractive to study the sensitivity of $S$-matrix elements to perturbations in the coupling between the scattering system and decay channels. This will be the central purpose of the present paper. Experimentally, we realize the system by a flat microwave billiard with two attached antennas and measure the reflection in one antenna while modifying the coupling in another, see Sec. \[sec:exper\] for details on the experimental setup. Section \[sec:theory\] presents a theoretical consideration based on RMT and the effective Hamiltonian approach. In Sec. \[sec:results\] we discuss in detail the experimental results and compare them with the theory. Our main findings are then summarized in the concluding Sec. \[sec:conclusions\].
Experiment {#sec:exper}
==========
The basic principles of billiard experiments with microwave cavities as a paradigm of quantum chaos research are described in detail in [@stoe99]. Therefore, we concentrate on the aspects of relevance to the present study. Reflection and transmission measurements have been performed in a flat resonator, with top and bottom plate parallel to each other. The cavity can be considered as two-dimensional for frequencies $\nu\, <\,\nu_{\rm max} = c/(2h)$, where $h=\rm 8\,mm$ is the height of the resonator.
![\[fig:01\] Geometry of the chaotic Sinai billiard, length $l=\rm 472\,mm$, width $w=\rm 200\,mm$ and a quarter-circle of radius $r =\rm 70\, mm$ where an antenna with different terminations may be introduced at position $c$. $a$ denotes the measuring antenna. The additional elements were inserted to reduce the influence of bouncing balls.](fig1){width=".9\columnwidth"}
The setup, as illustrated on Fig. \[fig:01\], is based on a quarter Sinai shaped billiard. Additional elements were inserted into the billiard to reduce the influence of bouncing-ball resonances. The classical dynamics for the chosen geometry of the billiard is dominantly chaotic. At position $a$ one antenna is fixed and connected to an Agilent 8720ES vector network analyzer (VNA), which was used for measurements in a frequency range from $\rm 2$ to $\rm 18\,GHz$ with a resolution of $\rm 0.1\,MHz$. We measured the reflection $S$-matrix element $S_{aa}$ first for the unperturbed system, which corresponds to the situation, where no additional antenna is inserted at position $c$. Then we perturbed the system by inserting another antenna at position $c$ which was terminated consecutively in three different ways:
1. connection to the VNA (total absorption),\
2. standard open (open end reflection),\
3. standard short (hard wall reflection),\
and again measured the corresponding reflection at antenna $a$ for each case. The connection of antenna $c$ to the VNA corresponds to a termination of antenna $c$ with a $50\,\Omega$ load. The terminators for the cases (b) and (c) have been taken from the standard calibration kit (Agilent 85052C Precision Calibration Kit) being part of our microwave equipment. For case (a) the reflection amplitude $S_{cc}$ was also measured. From this measurement the coupling strength of antenna $c$ can be obtained, see Eq. (\[eq:Tc\]) below. For all four cases we measured 18 different realizations by rotating an ellipse (see Fig. \[fig:01\]) to perform ensemble averages.
An alternative to the coupling of an antenna with variable end is an open wave guide whose coupling to the billiard can be varied by a variable slit. It showed up that, contrary to intuition, for this setup the main effect of the variation of the slit does not correspond to a change of the coupling to the outside, but to a distortion of the wave functions in the billiard, thus corresponding more to the case of a local scattering fidelity [@hoeh08a]. This system is discussed in Appendix \[app:Exp\].
Theory {#sec:theory}
======
Generalized VWZ approach to fidelity {#subsec:VWZ}
------------------------------------
The general case of $M$ scattering channels connected to $N$ levels of the closed cavity can be described in terms of the following effective non-Hermitian Hamiltonian $$\label{eq:Heff}
H_{\mathrm{eff}} = H - i\sum_{k=1}^{M}\lambda_k V_kV_k^{\dag}\,.$$ Here, the internal Hamiltonian $H$ of the closed system is represented by a Hermitian $N\times N$ matrix, whereas $V_k$ are $M$ vectors of length $N$ containing the information on the coupling of the levels to the continuum. The $V_k$ are assumed to be normalized to one, $V_k^{\dag}V_k=1$, and $\lambda_k$ is the coupling constant of channel $k$.
Such an approach was initially developed in nuclear physics [@mah69; @ver85a; @sok89] and since then has been successfully applied to study various aspects of open systems, including wave billiards [@stoe99; @fyo97b; @dit00; @fyo05a]. Usually, the phenomenological coupling constants $\lambda_k$ are considered as real numbers which enter the final expressions via the so-called transmission coefficients. However, in the present case of the antenna variation one has to consider the coupling to the variable antenna, $\lambda_{c}$, as a complex number, see discussion in Sec. \[subsec:heff\] below. For the sake of generality, we will treat all $\lambda_k$ as complex numbers with the only constraint on their real parts $\mathrm{Re}(\lambda_k)\geq0$, due to the causality condition on the $S$-matrix. We note that quite a similar problem of nonzero $\mathrm{Im}(\lambda_k)$ arises in shell-model calculations due to the principle value term of the self-energy operator, cf. [@mah69] and [@ver85a]. This requires proper modification of the theory which we briefly outline below.
According to the general scattering formalism [@mah69; @ver85a], the resonance part of the $S$-matrix at the scattering energy $E$ can be expressed in terms of $H_{\mathrm{eff}}$ as follows: $$\label{eq:s_cc1}
S_{ab}(E) = \delta_{ab} - 2i\sqrt{\mathrm{Re}(\lambda_a)\mathrm{Re}(\lambda_{b})}\, V_a^{\dag}\frac{1}{E-H_{\mathrm{eff}}}V_{b}\,.$$ Being interested in a reflection amplitude in channel $a$, it is possible, following [@fyo05a], to obtain another representation for an arbitrary diagonal element $S_{aa}$. To this end, it is convenient first to single out the contribution to $H_{\mathrm{eff}}$ due to channel $a$ by writing $H_{\mathrm{eff}}=H_{\mathrm{eff}}^{a}-i\lambda_a V_a V_a^{\dag}$, and then treat $V_aV_a^{\dag}$ as a rank 1 perturbation to the term $H_{\mathrm{eff}}^{a}=H-i\sum_{k\neq a}\lambda_k V_k V_k^{\dag}$. The upper index “$a$” for $H_{\mathrm{eff}}^{a}$ denotes that the contributions of all channels save the given one, $a$, are included. In the next step we expand $(E-H_{\mathrm{eff}})^{-1}$ into a power series with respect to $(E-H_{\mathrm{eff}}^a)^{-1}V_aV_a^{\dag}$ and, after summing up the resulting geometric series, obtain the following general relationship (Dyson’s equation) for the corresponding resolvents [@sok89]: $$\begin{aligned}
\label{eq:dyson}
\frac{1}{E-H_{\mathrm{eff}}} &=& \frac{1}{E-H_{\mathrm{eff}}^a} -i\lambda_a\frac{1}{E-H_{\mathrm{eff}}^a}V_a \nonumber \\
&&\times
\frac{1}{1+i\lambda_a V_a^{\dag}\displaystyle\frac{1}{E-H_{\mathrm{eff}}^a}V_a}V_a^{\dag}
\frac{1}{E-H_{\mathrm{eff}}^a}\,.\quad\end{aligned}$$ This identity, being substituted in Eq. (\[eq:s\_cc1\]), yields the following expression for the reflection amplitude $S_{aa}$: $$\label{eq:s_cc2}
S_{aa}(E) = \frac{1 - i\lambda_a^* V_a^{\dag}\displaystyle\frac{1}{E-H_{\mathrm{eff}}^a}V_a}{1 + i\lambda_a V_a^{\dag}\displaystyle\frac{1}{E-H_{\mathrm{eff}}^a}V_a}\,.$$
Representation (\[eq:s\_cc2\]) is very convenient to perform statistical averaging. Adopting RMT approach to model intrinsic chaos, we take $H$ from the Gaussian orthogonal ensemble (GOE) of $N\times N$ random real symmetric matrices which is the appropriate choice for the systems with preserved time-reversal symmetry [@stoe99; @meh91]. The quantities $V_k$ are considered as fixed real $N$-dimensional vectors of unit length. They are also supposed to be mutually orthogonal that ensures the absence of the direct “fast” processes (which could be due to a nondiagonal part of the average $S$-matrix) [@ver85a]. In the limit $N\to\infty$, for finite $M$, the leading term for the average value of the resolvent is well known to be $\langle[(E-H_{\mathrm{eff}}^a)^{-1}]_{nm}\rangle=[(E/2)-i\sqrt{1-E^2/4}]\delta_{nm}$, where the imaginary part accounts for the famous Wigner’s semicircle law. This implies the following result (valid up to the terms of the order of $M/N$) for the average $S$-matrix [@ver85a; @sok89], $$\label{eq:s_aver}
\langle S_{ab} \rangle = \frac{1-\lambda_a^*}{1+\lambda_a}\delta_{ab}$$ Here, we set $E=0$ as usual. As a result, the transmission coefficient takes the following form: $$\label{eq:Tc}
T_a \equiv 1- |\langle S_{aa}\rangle|^2 = \frac{4\,\mathrm{Re}(\lambda_a)}{|1+\lambda_a|^2}\,,\quad a=1,\ldots,M,$$ which is in agreement with the result of Ref. [@ver85a] obtained by a supersymmetry calculation. It is worth noting that in the case of real $\lambda$ one gets $T=\frac{4\lambda}{(1+\lambda)^2} \in [0,1]$. In the case of purely imaginary $\lambda$ corresponding to perfect reflection, the channel is closed, $T=0$.
Coupling fidelity
-----------------
We now proceed with the discussion of the scattering fidelity. Its amplitude is defined in terms of the $S$-matrix elements Fourier transformed into the time domain as follows [@sch05b]: $$\label{eq:f_ab}
f_{ab}(t) = \frac{\langle \hat{S}_{ab}(t)\hat{S}_{ab}^{\prime *}(t)\rangle}{ \sqrt{ \langle\hat{S}_{ab}(t)\hat{S}_{ab}^{*}(t)\rangle\langle \hat{S}_{ab}^{\prime}(t)\hat{S}_{ab}^{\prime *}(t)\rangle }}\,.$$ The prime indicates a change of the effective Hamiltonian of the original system after a small perturbation for the backward time evolution. Definition (\[eq:f\_ab\]) guarantees that $f_{ab}(0)=1$. Furthermore, an overall decay of the correlation functions due to absorption drops out, provided the decay is the same for the parametric cross-correlation functions in the nominator and the autocorrelation functions in the denominator [@sch05b]. The scattering fidelity itself is $$\label{eq:F_ab}
F_{ab}(t) = |f_{ab}(t)|^2\,.$$ Note that in contrast to the original definition of the scattering fidelity, we allow for a change in the channel vectors as well.
As it is explained above, the original idea is to change only the complex coupling strength $\lambda_c$ to one channel $c$, while the measuring is done on one or two different channels $a,b\neq c$. We denote the resulting scattering fidelity by *coupling fidelity*. We present below an exact RMT prediction for this quantity.
The starting point is to apply the convolution theorem for Fourier transforms to Eq. (\[eq:f\_ab\]) and relate it to the parametric cross-correlation function $\hat{C}[S_{ab},S_{ab}^{\prime *}](t)$ of the $S$-matrix elements in the time domain [@note1], $$\label{eq:ss_ft}
\langle \hat{S}_{ab}(t)\hat{S}_{ab}^{\prime *}(t)\rangle = \hat{C}[S_{ab},S_{ab}^{\prime *}](t)\,.$$ We denote the coupling constant for the variable antenna $c$ in forward and backward time evolution by $\lambda$ and $\lambda^\prime$, respectively. (We omit the lower index “c” henceforth.) In the case of unchanged coupling, $\lambda=\lambda^\prime$, the autocorrelation function $\hat{C}[S_{ab},S_{ab}^{*}](t)$ is real and its exact expression is obtained from Verbaarschot-Weidenmüller-Zirnbauer (VWZ) integral [@ver85a] and is given by $$\label{eq:VWZ}
\hat C[S_{ab},S_{ab}^*](t) = \delta_{ab} T_a^2(1-T_a) J_a(t) + (1+\delta_{ab}) T_a T_b P_{ab}(t).$$ It is convenient to use the parametrization of Ref. [@gor02a] to write down the explicit expressions for the functions $J_a(t)$ and $P_{ab}(t)$, as
$$\label{eq:J}
J_a(t) = 4\mathcal{I} \left[\left( \frac{r+T_a x}{1+2T_a r +T_a^2\, x} +
\frac{t-r}{1- T_a(t-r)}\right)^2\right]$$
and $$\begin{aligned}
\label{eq:P}
P_{ab}(t) &=& 2\mathcal{I}\left[
\frac{T_a T_bx^2 + d_{ab}(r)x+ (2r+1)r}{(1+2T_a r + T_a^2 x)(1+2T_b r +T_b^2 x)} \right.\nonumber \\
&& \left. + \frac{(t-r)(r+1-t)}{[1-T_a(t-r)][1-T_b(t-r)]} \right]\,,\end{aligned}$$ where $$x \equiv \frac{2r+1}{2u+1}u^2, \quad d_{ab}(r)\equiv T_aT_b+(T_a+T_b)(r+1)-1$$ and the shorthand $\mathcal{I}$ stands for the integral, $$\begin{aligned}
\label{eq:I}
\mathcal{I}[\cdots] &=& \int_{\max(0,t-1)}^t \!\!{\textrm d} r\int_0^r{\textrm d} u
\frac{(t-r)(r+1-t)}{(2u+1)(t^2-r^2+x)^2} \nonumber \\
& & \times \prod_{k=1}^M\frac{1-T_k (t-r)}{\sqrt{1+2T_k r + T_k^2 x}} [\cdots]\,.\end{aligned}$$ Here and below, $t$ denotes the dimensionless time measured in units of the Heisenberg time $t_H=2\pi\hbar/\Delta$, where $\Delta$ is the mean level spacing.
The calculation of the correlator \[Eq. (\[eq:ss\_ft\])\] in the case of $\lambda \neq \lambda^\prime$ proceeds along the same lines as in [@ver85a], see Appendix \[app:theo\]. The result turns out to be formally given by the same VWZ expression (\[eq:VWZ\]), where the transmission coefficient $T_c$ in the varied channel $c$ has to be substituted by $$\label{eq:Teff}
T_c^{\mathrm{eff}} = \frac{2\left(\lambda+\lambda^{\prime *}\right) }{ \left(1+\lambda\right)\left(1+\lambda^{\prime *}\right)}\,,$$ while performing the integration \[Eq. (\[eq:I\])\]. The quantity $T_c^{\mathrm{eff}}$ may be considered as an effective transmission coefficient due to a parametric variation of the coupling strength in the channel $c$. Only if $\lambda=\lambda^\prime$, $T_c^{\mathrm{eff}}$ becomes equal to the conventional transmission coefficient \[Eq. (\[eq:Tc\])\]. In contrast to Eq. (\[eq:Tc\]), $T_c^{\mathrm{eff}}$ is generally complex and also $T_c^{\mathrm{eff}}\neq1-\langle S_{cc}\rangle \langle S_{cc}^{\prime*}\rangle$. We note, however, that Eq. (\[eq:Teff\]) can be cast in the following form $$\label{eq:Teff2}
T_c^{\mathrm{eff}} = 1 - S_{\mathrm{eff}}' S_{\mathrm{eff}}^*\,,$$ where $$\label{eq:Teff3}
S_{\mathrm{eff}}' = \frac{ 1-\lambda^{\prime *} }{ 1+\lambda }\,, \qquad
S_{\mathrm{eff}}^* = \frac{ 1-\lambda }{ 1+\lambda^{\prime *} }\,.$$ These quantities might be interpreted as the (average) parametric $S$-matrix amplitudes in the varied channel for the forward and backward time evolution, respectively.
The subsequent evaluation of coupling fidelity cannot be done analytically and will be performed numerically.
Effective Hamiltonian description {#subsec:heff}
---------------------------------
The experimental situation shall now be mapped onto the theory derived in the preceding subsection. Though the calculation is straightforward, and similar approaches can be found elsewhere [@stoe02c], it is repeated here for the reader’s convenience. Let us start with the expression of the scattering matrix in terms of Wigner’s reaction matrix: $$\label{eq:s02}
S=\frac{1-i W^\dag GW}{1+i W^\dag GW}\,.$$ $G=(E-H)^{-1}$ is the Green’s function of the closed system and matrix $W=(W_a,W_c)$ contains the information on the coupling. As before, index “$c$” refers here to the antenna with variable coupling, and “$a$” to the measuring antenna. Per definition, the $S$-matrix relates the amplitudes of the incoming ($u$) and outgoing ($v$) waves, $$\label{eq:s03}
S\left(\begin{array}{c}
u_c \\
u_a
\end{array}\right)
=\left(\begin{array}{c}
v_c \\
v_a
\end{array}\right).$$ A termination of antenna $c$ is described by $$\label{eq:s04}
u_c=rv_c\,,\qquad r=e^{-(\alpha-i\varphi)}\,,$$ where $r$ contains the information on the reflection properties of the antenna. For reflection at an antenna with open or closed end we have $\alpha=0$ (as long as the absorption in the antenna can be neglected). The termination of the antenna by a 50$\Omega$ load corresponds to $\alpha\to\infty$.
Making use of Eq. (\[eq:s02\]), one can rewrite Eq. (\[eq:s03\]) as $$\label{eq:s05}
i W^\dag GW\left(\begin{array}{c}
u_c+v_c \\
u_a+v_a
\end{array}\right)
=\left(\begin{array}{c}
u_c-v_c \\
u_a-v_a
\end{array}\right).$$ Substituting relation (\[eq:s04\]) in Eq. (\[eq:s05\]), $u_c$ and $v_c$ can be eliminated, resulting in an equation for $u_a$ and $v_a$, $$\label{eq:s06}
iW_a^\dag\hat{G}W_a(u_a+v_a)=u_a-v_a.$$ Here, we have introduced the modified Green’s function, $\hat{G}$, with the following matrix element $$\label{eq:s09}
W_a^\dag\hat{G}W_a \equiv G_{aa}-G_{ac}\frac{1}{1+i\lambda_T G_{cc}}i\lambda_T G_{ca}\,,$$ where $G_{nm}=W^\dag_n G W_m$ and $\lambda_T$ is the coupling constant of the “terminator,” $$\label{eq:s08}
\lambda_T=\frac{1-r}{1+r}=\tanh\frac{\alpha+i\phi}{2}\,.$$
Equation (\[eq:s06\]) has the same form as Eq. (\[eq:s05\]), but for the measuring antenna only and with the modified Green’s function. Substituting explicit expressions for matrix elements $G_{nm}$, we obtain in a number of elementary steps $$\label{eq:s10}
\hat{G} = G \frac{1}{1+i\lambda_T W_c W_c^\dag G} \equiv \frac{1}{E-H^a_\mathrm{eff}}\,,$$ where $H^a_\mathrm{eff}=H-i\lambda_T W_cW_c^\dag$. Introducing the normalized coupling vector $V=\frac{1}{\sqrt{\lambda_W}}W_c$, where $\lambda_W=W_c^\dag W_c$ is a channel coupling strength, $H^a_\mathrm{eff}$ may be finally written as $$\label{eq:s12a}
H^a_{\rm eff}=H-i\lambda VV^\dag, \quad \lambda=\lambda_T\lambda_W\,.$$ The total coupling constant $\lambda$ is generally complex and takes into account the effects of both the channel coupling ($\lambda_W$) and the terminator ($\lambda_T$). The $2\times2$ scattering matrix \[Eq. (\[eq:s02\])\] for the measuring antenna and the antenna with variable terminator has thus been reduced to a $1\times1$ scattering matrix for the measuring antenna only, $$\label{eq:s12b}
S_{aa}=\frac{1-i W_a^\dag\displaystyle\frac{1}{E-H^a_\mathrm{eff}} W_a}{1+i W_a^\dag \displaystyle\frac{1}{E-H^a_\mathrm{eff}}W_a}.$$ In the case of a single measurement antenna and one antenna with variable coupling, Eq. (\[eq:s12b\]) is equivalent to Eq. (\[eq:s\_cc2\]). Equations (\[eq:s10\])–(\[eq:s12b\]) constitute the main result of this section. They show that the influence of the variable antenna can be taken into account by an appropriate modification of the Hamiltonian.
Two special cases are of particular importance. For the termination of the antenna with a 50$\Omega$ load the outgoing wave is completely absorbed, corresponding to the limit $\alpha\to\infty$. It follows $\lambda_T=\tanh \infty=1$ and $$\label{eq:s13}
H^a_{\rm eff}=H-i\lambda_W VV^\dag\,.$$ In this case the coupling is purely imaginary. For the two cases, where the antenna is terminated by a hard wall or an open reflecting end, we may assume $\alpha=0$, resulting in $\lambda_T=\tanh(i\varphi/2)=i\tan\varphi/2$, and $$\label{eq:s14}
H^a_{\rm eff}=H+\tan\left(\frac{\varphi}{2}\right)\lambda_W VV^\dag\,.$$ In this case the coupling is purely real, and the antenna does not correspond any longer to an open channel but to a scattering center only. This is true, as long as the absorption in the antenna can really be neglected. This becomes questionable, as soon as $\varphi$ approaches $\pi$, corresponding to the excitation of a resonance within the antenna. For this singular situation the perturbative treatment of the antenna coupling applied in the derivation looses its justification.
The value of $\varphi$ depends on the length of the antenna in units of the wave length and thus on frequency. But independently of frequency the difference of the phase shift $\varphi$ for the reflection at the open end (oe) and the hard wall (hw), respectively, is always $\pi$. A phase difference of $\pi$ means a replacement of the tangent by the cotangent in Eq. (\[eq:s14\]), i.e. the coupling constants $\lambda_T$ for the two situations are related via $$\lambda_{T,\rm hw}=1/\lambda_{T,\rm oe}$$ With the above introduced total coupling constant $\lambda=\lambda_T\lambda_W$, this may be alternatively be written as $$\label{eq:s15}
\lambda_{\rm hw}\lambda_{\rm oe}=\lambda_W^2=\lambda_{50\Omega}^2$$ since $\lambda_W$ is the coupling constant for the 50$\Omega$ load, see Eq. (\[eq:s13\]). $\lambda_{\rm hw}$ and $\lambda_{\rm oe}$ denote the total coupling constants for the hard-wall and the open-end reflections. These relations allow for explicit tests of the theory.
Results and Discussion {#sec:results}
======================
In this section we want to discuss the experimental and theoretical results for the coupling fidelity decay under the perturbations (a)–(c) described in Sec. \[sec:exper\]. For all results below the system without the varied antenna, corresponding to $\lambda=0$, is chosen as the reference, whereas for the perturbed system the coupling constant is $\lambda'=\lambda_{50\Omega}$, $\lambda_{\rm oe}$, or $\lambda_{\rm hw}$, depending on the terminator.
 Real part $f_R(t)$ and imaginary part $f_I(t)$ of the fidelity amplitude for three types of perturbation: $\lambda_{\rm 50\Omega}$ (black); $\lambda_{\rm hw}$ (orange, light gray); $\lambda_{\rm oe}$ (green, dark gray) in frequency range $8.0-8.5\rm\,GHz$. The time is given in units of the Heisenberg time $t_H=2\pi\hbar/\Delta$, where $\Delta$ is the mean level spacing. Solid lines show the experimental results. The theoretical curves are dotted for experimental parameter (available only for the 50$\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Tab. \[tab:01\]. The black dotted curve is nearly indistinguishable from the dashed one.](fig2a "fig:"){width=".95\columnwidth"}\
 Real part $f_R(t)$ and imaginary part $f_I(t)$ of the fidelity amplitude for three types of perturbation: $\lambda_{\rm 50\Omega}$ (black); $\lambda_{\rm hw}$ (orange, light gray); $\lambda_{\rm oe}$ (green, dark gray) in frequency range $8.0-8.5\rm\,GHz$. The time is given in units of the Heisenberg time $t_H=2\pi\hbar/\Delta$, where $\Delta$ is the mean level spacing. Solid lines show the experimental results. The theoretical curves are dotted for experimental parameter (available only for the 50$\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Tab. \[tab:01\]. The black dotted curve is nearly indistinguishable from the dashed one.](fig2b "fig:"){width=".95\columnwidth"}
We start with a plot of the complex valued fidelity amplitude for one frequency range, see Fig. \[fig:02\]. The solid lines show the experimental results, derived from the Fourier transform of the measured $S_{aa}(\nu)$ and $S^{\prime}_{aa}(\nu)$ via relation (\[eq:f\_ab\]) for the situations (a) 50$\Omega$ load (black), (b) open-end reflection (green, dark gray) and (c) hard-wall reflection (orange, light gray). For the case (a) we are able to calculate the corresponding theoretical curve (black dotted line) without any free parameter, since the coupling constant $\lambda_W$ can be determined directly from the additional reflection measurement at antenna $c$ (see Sec. \[sec:exper\]) using relation (\[eq:Tc\]). According to Eq. (\[eq:s13\]) we expect a purely imaginary coupling with $\lambda_{\rm 50\Omega}$. For our antenna $\lambda_W$ varies from 0.1 to 0.4 in a range from 6 to $10\,\rm GHz$. Using the experimentally determined $\lambda_W$, one gets already very good agreement between experiment and theory for the $\rm 50\,\Omega$ load without any fit. A fit of $\lambda_W$ to the experimental curves only marginally improves the correspondence.
 Fidelity decay $|f(t)|^2$ for three types of perturbation: $\lambda_{\rm 50\Omega}$ (black); $\lambda_{\rm hw}$ (orange, light gray); $\lambda_{\rm oe}$ (green, dark gray) in three different frequency ranges: (i) $7.2-7.7\rm\,GHz$; (ii) $8.0-8.5\rm\,GHz$; (iii) $8.7-9.2\rm\,GHz$. Solid lines show the experimental results and the theoretical curves are dotted for experimental parameter (available only for the 50$\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Tab. \[tab:01\].](fig3a "fig:"){width=".95\columnwidth"}\
 Fidelity decay $|f(t)|^2$ for three types of perturbation: $\lambda_{\rm 50\Omega}$ (black); $\lambda_{\rm hw}$ (orange, light gray); $\lambda_{\rm oe}$ (green, dark gray) in three different frequency ranges: (i) $7.2-7.7\rm\,GHz$; (ii) $8.0-8.5\rm\,GHz$; (iii) $8.7-9.2\rm\,GHz$. Solid lines show the experimental results and the theoretical curves are dotted for experimental parameter (available only for the 50$\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Tab. \[tab:01\].](fig3b "fig:"){width=".95\columnwidth"}\
 Fidelity decay $|f(t)|^2$ for three types of perturbation: $\lambda_{\rm 50\Omega}$ (black); $\lambda_{\rm hw}$ (orange, light gray); $\lambda_{\rm oe}$ (green, dark gray) in three different frequency ranges: (i) $7.2-7.7\rm\,GHz$; (ii) $8.0-8.5\rm\,GHz$; (iii) $8.7-9.2\rm\,GHz$. Solid lines show the experimental results and the theoretical curves are dotted for experimental parameter (available only for the 50$\Omega$ load case) and dashed for fitting parameter. The corresponding parameter and the transmission coefficient for the measuring antenna $a$ are listed in Tab. \[tab:01\].](fig3c "fig:"){width=".95\columnwidth"}
For the perturbations with reflecting ends the systems are closed. The correct description for this situations is Eq. (\[eq:s14\]). With the total coupling constants $\lambda_{\rm oe}$ and $\lambda_{\rm hw}$ as a free fitting parameter we find again an agreement between experiment and theory (dashed lines), for both, real and imaginary part. As one would expect from theory for the case of reflecting ends we see a significant imaginary part of the fidelity amplitude $f_I(t)$ (green and orange lines), whereas in the case of an absorbing end (black line) the imaginary part is nearly zero.
It is convenient to continue our discussion in terms of the fidelity (not its amplitude) introduced in Eq. (\[eq:F\_ab\]). In Fig. \[fig:03\] we present the experimental and theoretical fidelity results for the situations 50$\Omega$ load (black lines), open-end (green lines), and hard-wall (orange lines) as perturbation, for three frequency ranges. In case of the closed channels the total phase shift $\varphi(\nu)$ increases monotonically with frequency. Thus $\lambda_{\rm oe}$ and $\lambda_{\rm hw}$ are oscillating in counter phase, see Eq. (\[eq:s14\]). This induces a corresponding oscillation in the strength of the fidelity decay as seen in Fig. \[fig:03\]. For a more quantitative discussion we compare the experimentally determined fidelity decay (solid lines) to the theoretical curves (dotted and dashed lines). First of all, one sees that fitting (dashed lines) the experimental results again works well for all cases. Focusing on the 50$\Omega$ load case (black lines) also the theoretical results without free parameter (dotted lines) show good agreement with the experiment for the frequency ranges plotted in Figs. \[fig:03\] (ii) and (iii). Only the plot in Fig. \[fig:03\] (i) shows significant deviation between the theoretical result without free parameter and the experimental curve. We want to stress that the case shown here is the worst among all the investigated frequency ranges. Here the experimental fidelity amplitude $f_{50\Omega}$ shows a significant imaginary part. Thus the imaginary part of $\lambda_{50\Omega}$ is not zero, i.e. the 50$\Omega$ terminator does not correspond to perfect absorption, and Eq. (\[eq:s14\]) does not hold. This might be due to an antenna resonance, leading to an increased reflection from the channel $c$.
$\nu$/GHz $\lambda^{\rm exp}_{\rm 50\Omega}$ $\lambda^{\rm fit}_{\rm 50\Omega}$ $\lambda_{\rm oe}$ $\lambda_{\rm hw}$ $\lambda_W$ $T_a$
------- ----------- ------------------------------------ ------------------------------------ -------------------- -------------------- ------------- --------
(i) $7.2-7.7$ $0.19$ $0.37$ $0.65\,\imath$ $-0.04\,\imath$ $0.16$ $0.19$
(ii) $8.0-8.5$ $0.21$ $0.20$ $0.19\,\imath$ $-0.23\,\imath$ $0.21$ $0.22$
(iii) $8.7-9.2$ $0.24$ $0.21$ $0.05\,\imath$ $-0.83\,\imath$ $0.20$ $0.34$
: \[tab:01\] Coupling constants in the three different frequency ranges (i)-(iii). According to Eq. (\[eq:s15\]), $\lambda^{\rm exp}_{\rm 50\Omega}$ and $\lambda^{\rm fit}_{\rm 50\Omega}$ should be compared to $\lambda_W=\sqrt{\lambda_{\rm oe}\lambda_{\rm hw}}$, see main text for discussion.
Finally we perform a check on the coupling constants based on Eq. (\[eq:s15\]). Accordingly, the square root of the product of the coupling constants for open-end and hard-wall reflection should give $\lambda_{50\Omega}$. Table \[tab:01\] shows that for the frequency ranges (ii) and (iii) there is indeed good agreement between $\sqrt{\lambda_{\rm oe}\lambda_{\rm hw}}$, $\lambda^{\rm fit}_{\rm 50\Omega}$ and $\lambda^{\rm exp}_{\rm 50\Omega}$. In the case (i) $\sqrt{\lambda_{\rm oe}\lambda_{\rm hw}}$ agrees quite good with the experimental parameter $\lambda^{\rm exp}_{\rm 50\Omega}$, but the fitting parameter is much larger. This deviation reconfirms our arguments presented in the above discussion of the fidelity plot shown on Fig. \[fig:03\](i).
Conclusions {#sec:conclusions}
===========
In this work, we have studied the influence of the coupling to the continuum on the decay of fidelity. This complements previous experiments of our group, where the fidelity decay under the influence of various types of geometrical perturbations was studied [@sch05b; @hoeh08a; @sch05d; @bod09a] but for closed systems exclusively. To get rid of an overall absorption we used the concept of scattering fidelity introduced by us previously [@sch05b], defined as the parametric cross-correlation function of $S$-matrix elements normalized to the corresponding autocorrelation function.
On the theoretical side we have developed a model description of the fidelity decay in terms of a modified VWZ approach. The parametric cross-correlation function of $S$-matrix elements for two different $\lambda \ne \lambda'$ can be reduced to an autocorrelation function with a complex effective transmission coefficient \[Eq. (\[eq:Teff\])\], thus expressing coupling fidelity in terms of a modified VWZ integral. This theory holds for an arbitrary number of channels and describes the experimental coupling fidelity results well. It would be interesting to investigate, whether it is possible to relate coupling fidelity obtained via the VWZ ansatz to an approximation using the random plane wave conjecture, as used to explain the local fidelity [@hoeh08a].
We have found two additional important result. First, a smooth variation of the coupling, e.g. by varying the coupling to an attached wave guide will not easily yield the information about the effect of coupling to the continuum on the scattering fidelity. Each geometric variation will give rise to both a change of coupling and internal scattering properties, thus screening the purely external effect, as is also discussed in Appendix \[app:Exp\].
Second, we have included closed channels within the description of VWZ. The speed of the fidelity decay for the open-end and hard-wall reflection oscillates with frequency due to the corresponding variation of the phase with frequency. An important relation (\[eq:s15\]) between the coupling constants of the antenna terminations has been established, enabling us to connect the results found for the closed channel ($\lambda_{\rm oe}$, $\lambda_{\rm hw}$) to those for the open channel ($\lambda_{50\Omega}$). In all cases the fidelity decay for at least one of the reflecting antennas is faster than for the open channel, showing the strong influence of the imaginary part on the coupling constant $\lambda$.
One of us (D.V.S.) is grateful to I. Smolyarenko for useful discussions of the results. We would like to acknowledge the generous hospitality of CIC (Cuernavaca, Mexico) during our stay there at the program RMT-MEX09, where this work has been partly completed. The experiments have been founded by the Deutsche Forschungsgemeinschaft via the research group 760 “Scattering systems with complex dynamics”. T.H.S. thanks the DFG for support of a number of visits in Marburg. T.H.S. and T.G. have been supported by CONACyT under Grant No. 79988 and by PAPIIT, Universidad Nacional Autónoma de México under Grant No. IN-111607 and IN-114310.
Experiment with attached wave guide with variable coupling {#app:Exp}
==========================================================
In our first approach we used the setup shown in Fig. \[fig:04\] where the opening of the variable slit plays the role of the fidelity parameter. The setup is based on a quarter Sinai shaped billiard with length $l=\rm 342\,mm$, width $w=\rm 237\,mm$ and a quarter-circle of radius $r =\rm 70\, mm$, and an attached channel. The channel has a total length $l_c=\rm 243\,mm$ and a width $w_c=\rm16\,mm$. At position $a$ and $c$ two antennas were fixed and connected to the VNA. The complete $S$-matrix was measured in a frequency range from $\rm 9.5$ to $\rm 18.0\,GHz$ with a resolution of $\rm 0.1\,MHz$, where the wave guide only supports a single propagating mode, i.e. it acts as a single channel. The perturbation of the system was achieved by opening the channel from $d=0-16$mm in steps of 0.1mm using a slit diaphragm at the point of attachment. An ellipse insert with semiaxis $a=\rm 70\,mm$ and $b=40$mm was rotated to get an ensemble of 20 different systems for averaging. Additional elements were inserted into the billiard to avoid bouncing-ball resonances. The wave guide was terminated by a perfect absorber, which according to Eq. (\[eq:s13\]) should correspond to a purely imaginary coupling. As before the coupling constant $\lambda_W$ could be determined directly from a reflection measurement at antenna $c$. $\lambda_W$ could be varied from $\lambda_W=0$ (no coupling) to $\lambda_W=1$ (perfect coupling) by increasing the opening $d$ of the slit.
![\[fig:04\] Geometry of the billiard with attached wave guide. ](fig4){width=".95\columnwidth"}
 Experimental coupling fidelity $|f(t)|^2$ (solid lines) and theoretical results for the experimental parameter $\lambda_W$ (dotted line) and the fit parameter $\lambda_{\rm fit}$ (dashed line) for two openings $d=\rm 6.5\,mm$ with $\lambda_W=0.05$ and $\lambda_{\rm fit}=-0.18i$ (black), and $d=\rm 11.2\,mm$ with $\lambda_W=0.52$ and $\lambda_{\rm fit}=-0.55i$ (orange, light gray). The frequency window of the Fourier transform of the measured $S_{\rm aa}(\nu)$ was 13 to $\rm14\,GHz$ and the transmission coefficient for antenna $a$ was $T_a=0.95$.](fig5){width=".98\columnwidth"}
In Fig. \[fig:05\] the coupling fidelity decay is shown for two different perturbation strengths. The solid lines show the experimental results. With the formulas derived in Sec. \[sec:theory\], we calculated the expected theoretical fidelity decay assuming that the channel is totally open, i.e. the coupling is purely imaginary (dotted lines). There is obviously no agreement. This shows that something is wrong in the argumentation. For a further check we removed the absorbing end and the antenna in the channel and replaced it by a reflecting end thus closing the system. We did not find any noticeable difference to the case with the absorbing end and the antenna in the channel experimentally. So there is only one explanation: by far the major part of the wave is reflected directly at the slit, and only a minor part really penetrates into the channel. This means that the coupling is not imaginary but mainly real (up to perhaps a minor imaginary contribution), and we should use Eq. (\[eq:s14\]) instead of Eq. (\[eq:s13\]) for the interpretation of our results. The dashed lines in Fig. \[fig:05\] show the resulting theoretical curves with $\lambda_{\rm fit}$ as a free parameter according to the definition preceding Eq. (\[eq:s14\]). Now a perfect agreement between experiment and theory is found.
As a resume we can state that the variable slit works essentially as a scattering center leading to partial masking of the change of coupling by the change of scattering properties in the fidelity decay.
Derivation of EQ. (\[eq:Teff\]) and qualitative discussion {#app:theo}
==========================================================
The calculation of Eq. (\[eq:ss\_ft\]) proceeds along the same lines as in [@ver85a], hence we indicate below only the main steps and essential differences. First, we make use of the representation of resolvents and thus $S$-matrix elements \[Eq. (\[eq:s\_cc1\])\] in terms of Gaussian integrals over auxiliary “supervectors” consisting of both commuting and anticommuting (Grassmann) variables. This allows us to perform statistical averaging over GOE exactly. Then in the RMT limit $N\to\infty$, the remaining integration over the auxiliary field can be done in the saddle-point approximation. It turns out that there exists a nontrivial saddle-point manifold [@efe83] over which one has to integrate exactly. As a result, the two-point correlation function of the $S$-matrix elements in the energy domain acquires the form of a certain expectation value in field theory (nonlinear supersymmetric $\sigma$-model), $\langle(\cdots)\rangle=\int\mathrm{d}[\sigma_G]e^{\mathcal{L}(\varepsilon)}\mathcal{F}_{M}(\cdots)$, cf. Eq. (7.13) of Ref. [@ver85a]. In the notations of this paper, the effective Lagrangian reads $\mathcal{L}(\varepsilon)=\frac{1}{4}N\varepsilon\,\mathrm{trg}(\sigma_GL)$, with $\varepsilon$ being the energy difference. Definitions of the supertrace, $\mathrm{trg}$, as well as of the supermatrices $\sigma_G$ and $L$ can also be found there (see [@efe96] for a general reference). The pre-exponential terms omitted above depend on the coupling constants in the channels $a$ and $b$ ($a,b\neq c$), being thus the same as considered in [@ver85a]. They finally correspond to the expressions appearing explicitly in Eqs. (\[eq:J\]) and (\[eq:P\]). At last, the so-called channel factor $\mathcal{F}_M$ accounts for the coupling to all the channels. It is the term that requires modification due to both generally complex and varied coupling constants. In the \[1,2\] block notation (the “advanced-retarded” ordering of supermatrix elements), $\mathcal{F}_M$ reads $$\mathcal{F}_M = \prod_{k=1}^M \exp\left\{-\frac{1}{2} \mathrm{trg} \ln \left[ 1 + i
\left(\begin{array}{cc} \lambda_k & 0 \\ 0 & \lambda_k'^{*} \end{array}\right)
\sigma_G L \right]\right\},$$ where $\lambda_k=\lambda_k'$ for all channels save the varied one, $k\neq c$. By employing the “angular” parametrization of $\sigma_{G}$ in terms of the matrices $t_{12}$ and $t_{21}$, the subsequent evaluation of $\mathcal{F}_M$ goes in parallel with Sec. 7 of Ref. [@ver85a], with the final result being $$\label{eq:app1}
\mathcal{F}_M = \prod_{k=1}^M \exp \left[-\frac{1}{2} \mathrm{trg} \ln \left(
1 + T_k^{\mathrm{eff}} t_{12}t_{21} \right)\right]\,.$$ This is just a usual formula for the channel factor in the VWZ theory except for the effective transmission coefficient $T_c^{\mathrm{eff}}$ in the channel $c$ that is now given by expression (\[eq:Teff\]) (we note that $T_k^{\mathrm{eff}}=T_k$ if $k\neq c$) [@note2]. Performing finally the Fourier transform, the two-point correlation function in the time domain takes the form of Eqs. (\[eq:VWZ\])–(\[eq:I\]), with the above modification in the channel factor corresponding explicitly to the second line of Eq. (\[eq:I\]).
Although the subsequent evaluation cannot be made analytically and has to be done numerically, it is still useful to make some qualitative analysis. To this end we note that $P_{ab}(t)$ and $J_a(t)$ are quite similar in structure to the “norm leakage” decay function [@sav97] and the form factor of the Wigner time delays [@leh95b]. Following the analysis performed there (see also [@dit00]), one notices that the time dependence in question is mainly due to the channel factor \[Eq. (\[eq:app1\])\]. In the time domain, its typical behavior is $\sim\prod_{k=1}^M(1+\frac{2}{\beta}T_k t)^{-\beta/2}$, where $\beta=1$ is for the present case of time-invariant systems whereas $\beta=2$ is for the case of broken time invariance (GUE). The case of the coupling variation in the channel $c$ amounts then to replacing in this expression $T_c$ with $T_c^{\mathrm{eff}}$. This suggests the following heuristic formula for the coupling fidelity $$\label{eq:app2}
F_{\mathrm{surm}}(t) = \left[\frac{(1+2T_c t/\beta)(1+2T'_c t/\beta) }{
|1+2T^{\rm eff}_c t/\beta|^2}\right]^{\beta/2}\,.$$ For the parameters of $\lambda$ and $\lambda^\prime$ found in the experiment only deviations on the % level were found while making fit to surmise (\[eq:app2\]). We stress, however, that there is no control on approximations involved to derive this expression. One should generally expect that surmise (\[eq:app2\]) coincides closely with the exact result at small times (when it is given by an exponential dependence), while the exact asymptotic behavior at large times is reproduced up to a factor of the order of unity (as was indeed confirmed numerically). Therefore, we have used the exact supersymmetry result for all the figures and analysis of the main text.
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| ArXiv |
---
abstract: 'This paper investigates the dynamics of thin-shell in the presence of perfect fluid as well as scalar field. We formulate the equations of motion using Israel thin-shell formalism by taking the interior and exterior regions of Schwarzschild, Kerr as well as Kerr-Newmann black hole. We find numerical solutions of equations of motion and effective potential to analyze the scalar shell for collapse and expansion. It is found that the rate of collapse and expansion of scalar shell through shell’s radius depends on charge and rotation parameters. We conclude that the massive scalar shell leads to collapse of thin-shell, while massless scalar shell indicates both collapse as well as expansion.'
author:
- |
M. Sharif [^1] and Faisal Javed [^2]\
Department of Mathematics, University of the Punjab,\
Quaid-e-Azam Campus, Lahore-54590, Pakistan.
title: '**Collapse and Expansion of Scalar Thin-Shell for a Class of Black Holes**'
---
**Keywords:** Gravitational collapse; Scalar field; Israel thin-shell formalism.\
**PACS:** 04.20.-q; 04.40.Dg; 04.70.Bw; 75.78.-n
Introduction
============
Geon is a notion of electromagnetic-gravitational field that is held together in a confined region due its own gravity. This entity was introduced by Wheeler et al. [@1] who established particle-like solutions from classical electromagnetic field coupled to general relativity. They used the resulting solutions to analyze the scalar field. Bergmann and Leipnik [@2] studied solutions of the field equations in the presence of scalar field for Schwarzschild black hole (BH). In general relativity, scalar field appears in the low energy limit of string theory [@3]. There is no experimental evidence for the existence of such particles that are associated with the scalar field. Harada et al. [@4] studied that gravitational collapse of compact objects in scalar-tensor theories predict scalar field as a source of gravitational waves that can be identified by the advanced detectors.
The study of cosmological as well as astrophysical objects composed of scalar field has been the subject of great interest for many researchers. Kaup [@5] was the pioneer to investigate the geometrical configuration for complex massive scalar field. Ruffini and Bonazzola [@6] examined spherical geometries and found the equilibrium conditions for boson stars solutions. Seidel and Suen [@7] studied the dynamical evolution of compact stars associated with scalar field and found that boson stars have stable configurations for several values of the scalar fields. These results provide information corresponding to existence as well as formation of boson stars. Jetzer [@8] investigated the equilibrium configuration of boson stars and discussed their dynamical instability. Many people [@9] investigated the nature of spacetime singularity for massless scalar field with spherical geometry. Siebel et al. [@10] analyzed self-gravitating objects composed of massless scalar field and found that boson stars either oscillate or undergo collapse to form a BH. Bhattacharya et al. [@11] studied the collapse of spherical stars associated with massless scalar field and examined appropriate condition for the existence of a class of non-singular models.
The smooth matching of interior and exterior spacetimes helps to evaluate exact solutions of the field equations. This is also a key aspect to study the boundary of BHs, gravitational waves and contribution of matter at thin-shell. Israel [@12] developed thin-shell formalism to investigate the dynamics of fluid configuration at thin-shell. It is observed that the presence of thin layer of matter at thin-shell leads to jump discontinuity across the boundary of interior and exterior regions. In order to analyze thin-shell, Israel formalism has widely been used [@13].
De La Cruz and Israel [@14] generalized Israel thin-shell formalism for charged thin-shell in the absence of pressure. Kuchar [@15] examined the charged thin-shell problem by considering polytropic equation of state (EoS). Chase [@16] investigated the instability of spherically symmetric charged fluid shell. Boulware [@17] analyzed charged thin-shell and found that collapse produces a naked singularity when matter (with negative density) is located at thin-shell. N$\acute{u}\tilde{n}$ez [@18] studied spherically symmetric massive shell and found that the shell oscillates radially around a central compact object. N$\acute{u}\tilde{n}$ez et al. [@19] analyzed the effect of scalar field for Schwarzschild BH and found that massive scalar field leads to collapse of the shell when both pressure and gravitational mass are proportional. Pereira and Wang [@20] used thin-shell formalism to study the non-rotating shell constructed from two cylindrical regions and observed the behavior of collapsing shell. Sharif and his collaborators [@21] used this formalism to investigate spherical as well as planar collapse. Sharif and Abbas [@22] explored the dynamics of charged scalar thin-shell and concluded that for both (massless and massive scalar fields) shell can expand to infinity or collapse to zero size forming a curvature singularity. Sharif and Iftikhar [@23] explored a class of regular BHs and found that massless scalar shell leads to expansion, collapse and equilibrium structure while the massive case leads to collapse only.
This work is devoted to study the effect of scalar field on the dynamics of thin-shell using Israel formalism for a class of BHs. The paper is organized as follows. In section **2**, we construct the equations of motion by applying Israel thin-shell formalism for Schwarzschild, Kerr and Kerr-Newmann BHs. Section **3** explores massless and massive scalar fields to investigate these equations of motion. Finally, we summarize our results in the last section.
Equations of Motion
===================
In this section, we construct thin-shell for Schwarzschild, Kerr and Kerr-Newmann BHs. We consider a hypersurface ($\Sigma$) that divides a spherically symmetric spacetime into two four-dimensional manifolds $N^{+}$ and $N^{-}$ representing interior and exterior regions, respectively. The line element for interior and exterior regions of the Schwarzschild, Kerr and Kerr-Newmann BHs is given as $$\begin{aligned}
\nonumber
ds^2_\pm&=&-F_{\pm}(R)dt^2-\left(\frac{4m_{\pm}R}
{A}-\frac{2Q^2_{\pm}}{A}\right)a\sin^2\theta
dtd\phi+\left(\frac{A}{B}\right)dr^2+A d\theta^2\\\label{1}&+&
\left(R^2+a^2+\frac{2m_{\pm}a^2R\sin^2\theta}{A}-
\frac{a^2Q^2_{\pm}\sin^2\theta}{A}\right)\sin^2\theta d\phi^2,\end{aligned}$$ where $$\begin{aligned}
\nonumber
A=R^2+a^2cos^2\theta, \quad B=R^2-2m_{\pm}R+a^2+Q^2_{\pm},\end{aligned}$$ here $m_{\pm}$, $Q_{\pm}$ and $a$ denote the mass, charge and rotation parameters, respectively. Moreover, we assume that interior region contains more mass than the exterior region, i.e., $m_{+}\neq
m_-$, while the charge is uniformly distributed in both regions, i.e., $Q_+=Q_-=Q$. The explicit forms of the defining parameters corresponding to different BHs are given in Table **1**.\
\
**Table 1:** A Class of BHs.
**Name of BH** $F(R)$
------------------- ----------------------------------------------------------------------
**Schwarzschild** $F_{\pm1}(R)=\left(1-\frac{2m_{\pm}}{R}\right)$, $a=0$, $Q_{\pm}=0$
**Kerr** $F_{\pm2}(R)=\left(1-\frac{2m_{\pm}R}{A}\right)$, $Q_{\pm}=0$
**Kerr-Newmann** $F_{\pm3}(R)=\left(1-\frac{2m_{\pm}R}{A}+\frac{Q^2_{\pm}}{A}\right)$
We apply the intrinsic coordinates $\xi^{i}=(\tau,\theta,\phi)$ over $\Sigma$ at $R=R(\tau)$. Consequently, Eq.(\[1\]) yields $$\begin{aligned}
\nonumber
ds^2_\pm&=&\left[-F_{\pm}(R)+\frac{A}{B}\left(\frac{dR}{d\tau}
\right)^2\left(\frac{d\tau}{dt}\right)^2\right]dt^2-\left(\frac{4m_{\pm}R}
{A}-\frac{2Q^2_{\pm}}{A}\right)a\sin^2\theta d\phi
dt\\\label{2aa}&+&A d\theta^2+
\left(R^2+a^2+\frac{2m_{\pm}a^2R\sin^2\theta}{A}-
\frac{a^2Q^2_{\pm}\sin^2\theta}{A}\right)\sin^2\theta d\phi^2,\end{aligned}$$ where $\tau$ is the proper time. The corresponding induced metric for hypersurface is defined as $$\label{3aa}
ds^2=-d\tau^2+R^2(\tau)\left(d\theta^2+\sin^2\theta d\phi^2\right).$$ Comparing Eqs.(\[2aa\]) and (\[3aa\]), we obtain $$\begin{aligned}
\nonumber
\left[F_{\pm}(R)-\frac{A}{B}\left(\frac{dR}{d\tau}
\right)^2\left(\frac{d\tau}{dt}\right)^2\right]^\frac{1}{2}dt=(d\tau)_{\Sigma}.\end{aligned}$$ The outward unit normals $n_{\alpha}^{\pm}$ corresponding to interior as well as exterior region are $n_{\alpha}^{\pm}=(n_{0},n_{1},0,0)$, where $$\begin{aligned}
\nonumber
n_{0}&=&-\dot{R}\left(\frac{B + A \dot{R}^2}{B
F_{\pm}}\right)^{-\frac{1}{2}}\left[\frac{B}{A}+\left\{A B \dot{R}^2
F_{\pm} \left(-A \left(a^2 + R^2\right) +
a^2\sin^2\theta\right.\right.\right.
\\\nonumber &\times&\left.\left.\left.\left(Q^2 - 2 m_{\pm} R\right)
\right)\right\}\left\{\left(B + A \dot{R}^2\right) \left(A^2 F_{\pm}
\left(a^2 + R^2\right) - a^2 \left(Q^2 - 2 m_{\pm}
R\right)\right.\right. \right.\\\nonumber&\times&
\left.\left.\left.\left(A F_{\pm} - 4 Q^2 + 8 m_{\pm} R\right)
\sin^2\theta\right)\right\}^{-1}\right]^{-\frac{1}{2}},
\\\nonumber
n_{1}&=&\left[\frac{B}{A}+\left\{A B \dot{R}^2 F_{\pm} \left(-A
\left(a^2 + R^2\right) + a^2\sin^2\theta\left(Q^2 - 2 m_{\pm}
R\right)
\right)\right\}\right.\\\nonumber&\times&\left.\left\{\left(B + A
\dot{R}^2\right) \left(A^2 F_{\pm} \left(a^2 + R^2\right) - a^2
\left(Q^2 - 2 m_{\pm} R\right)\right.\right.
\right.\\\nonumber&\times& \left.\left.\left.\left(A F_{\pm} - 4 Q^2
+ 8 m_{\pm} R\right)
\sin^2\theta\right)\right\}^{-1}\right]^{-\frac{1}{2}}.\end{aligned}$$ Here, dot denotes derivative with respect to $\tau$. The extrinsic curvatures joining two sides of the shell are defined as $$\label{4}
K_{ij}^{\pm}=-n_{\beta}^{\pm}\left(\frac{d^2x_{\pm}^\beta}{d\xi^id\xi^j}
+\Gamma^\beta_{\mu\nu}\frac{dx^{\mu}_{\pm} dx^{\nu}_{\pm}} {d\xi^{i}
d\xi^{j}}\right),\quad \mu,\nu,\beta=0,1,2,3.$$
The discontinuity of extrinsic curvature appears due to the existence of thin layer of matter on $\Sigma$. The dynamics of this thin-shell is observed by using the field equations at $\Sigma$, i.e., by the Lanczos equations $$\label{5}
S_{ij}=\frac{1}{8\pi}\{g_{ij}K-[K_{ij}]\},\quad i,j=0,2,3,$$ where $[K_{ij}]=K^{+}_{ij}-K^{-}_{ij}$ and $K=tr[K_{ij}]=[K^{i}_{i}]$. The Lanczos equations for spherical thin-shell reduces to $$\begin{aligned}
\label{6}
\sigma=\frac{-1}{4\pi}[K_{\theta}^{\theta}], \quad
p=\frac{1}{8\pi}\left\{
[K_{\tau}^{\tau}]+[K_{\theta}^{\theta}]\right\}.\end{aligned}$$ In the absence of surface energy density ($\sigma$) and pressure ($p$), the connection between these geometries is referred as a boundary surface, otherwise it is known as thin-shell. The corresponding surface energy density can be expressed as $$\begin{aligned}
\nonumber
\sigma &=&\frac{B}{8\pi r A}\left[\frac{B}{A}+\left\{A B \dot{R}^2
F_{\pm} \left(-A \left(a^2 + R^2\right) + a^2\sin^2\theta\left(Q^2 -
2 m_{\pm} R\right) \right)\right\}\right.\\\nonumber &\times&\left.
\left\{\left(B + A \dot{R}^2\right) \left(A^2 F_{\pm} \left(a^2 +
R^2\right) - a^2 \left(Q^2 - 2 m_{\pm} R\right)\right.\right.
\right.\\\label{7}&\times& \left.\left.\left.\left(A F_{\pm}- 4 Q^2
+ 8 m_{\pm}
R\right)\sin^2\theta\right)\right\}^{-1}\right]^{-\frac{1}{2}}.\end{aligned}$$ The above equation can be rewritten as $$\begin{aligned}
\label{8}
\dot{R}^2 + V_{eff}(R)=0,\end{aligned}$$ where $$\begin{aligned}
\nonumber
V_{eff}(R)&=&\left\{B \left(B - 64 A \pi^2 r^2 \sigma^2\right)
\left(-A^2 F_{\pm} \left(a^2 + r^2\right) + a^2 \left(Q^2 - 2
m_{\pm} r\right) \right.\right.\\\nonumber&\times&
\left.\left.\left(A F_{\pm} - 4 Q^2 + 8 m_{\pm} r\right)
\sin^2\theta\right)\right\}\left\{A^3 B F_{\pm} \left(a^2 +
r^2\right) - a^2 A \right.\\\nonumber&\times&\left.\left(Q^2 - 2
m_{\pm} r\right) \left(B \left(A F_{\pm} - 4 Q^2 + 8 m_{\pm}
r\right) + 256 A \pi^2 r^2
\right.\right.\\\label{9}&\times&\left.\left.\left(Q^2 - 2 m_{\pm}
r\right) \sigma^2\right) \sin^2\theta\right\}^{-1}.\end{aligned}$$ It is observed that Eq.(\[8\]) satisfies the energy conservation law, i.e., the sum of the component of kinetic $(\dot{R}^2)$ and potential $(V_{eff}(R))$ energies vanishes at any time. The effective potential for Schwarzschild, Kerr and Kerr-Newmann BHs turns out to be $$\begin{aligned}
\label{10}
V_{eff1}(R) &=&-F_{\pm1} + 16\pi^2 R^2 \sigma^2,
\\\nonumber
V_{eff2}(R)&=&\left\{B \left(B - 16 A \pi^2 R^2 \sigma^2\right)
\left(A^2 F_{\pm2} \left(a^2 + R^2\right) + 2 a^2 m_{\pm} R \left(A
F_{\pm2} \right.\right.\right.\\\nonumber&+&\left.\left.\left. 8
m_{\pm} R\right) \sin^2\theta\right)\right\}\left\{A^3 B F_{\pm2}
\left(a^2 + R^2\right) + 2 a^2 A m_{\pm} R \left(A B F_{\pm2}
\right.\right.\\\label{11}&+&\left.\left.8 B m_{\pm} R - 128 A
m_{\pm} \pi^2 R^3 \sigma^2\right) \sin^2\theta\right\}^{-1},
\\\nonumber
V_{eff3}(R)&=&\{(B (B - 64 A \pi^2 R^2 \sigma^2) (-A^2 F_{\pm3} (a^2
+ R^2) + a^2(Q^2 - 2 m_{\pm} R)
\\\nonumber&\times&(A F_{\pm3}-4 Q^2 +
8 m_{\pm} R) \sin^2\theta))\} \{(A^3 B F_{\pm3} (a^2 + R^2) - a^2 A
\\\nonumber&\times&(Q^2 - 2 m_{\pm} R)(B (A F_{\pm3} - 4 Q^2 + 8
m_{\pm} R) + 256 A \pi^2 R^2\\\label{12}&\times&(Q^2 - 2 m_{\pm} R)
\sigma^2) \sin^2\theta\}^{-1}.\end{aligned}$$
Analysis of Equations of Motion
===============================
Here we investigate the scalar shell and its dynamical behavior for a class of BHs. We evaluate velocity of the scalar shell and its effective potential with respect to stationary observer. Also, we discuss the effect of charge as well as rotation parameter on the motion of scalar shell. For this purpose, we formulate a relation between surface energy density and the mass of thin-shell. The surface energy density and pressure follow the conservation equation $$\label{13}
p \frac{d\Delta}{d\tau}+\frac{d}{d\tau}(\sigma \Delta)=0,$$ where $\Delta=4\pi R^2$ represents area of the shell. Consequently, the conservation equation for $p=p(\sigma,R)$ takes the form $$\label{14}
\sigma'+\frac{2}{R}\left[\sigma+p(\sigma,R)\right]=0.$$ This equation can be solved by using EoS, $p=k\sigma$, yielding $$\begin{aligned}
\label{15}
\sigma=\sigma_{0}\left(\frac{R_{0}}{R}\right)^{2(k+1)},\end{aligned}$$ where $k$ is a constant and $R_{0}$ represents initial position of the shell at $\tau=\tau_{0}$ while $\sigma_0$ denotes surface density of the shell at $R_{0}$. Using the above equation, mass of the shell becomes $$\begin{aligned}
\label{16}
M=4\pi\sigma_{0}\left(\frac{R_{0}^{2(k+1)}}{R^{2k}}\right).\end{aligned}$$
In the following, we briefly discuss the dynamics of thin-shell using equations of motion.
The Schwarzschild Black Hole
----------------------------
The effective potential for Schwarzschild BH is given by $$\begin{aligned}
\label{17}
V_{eff1}(R) = -1+ \frac{M^2}{R^2} +\frac{2m_{\pm}}{R}.\end{aligned}$$ The corresponding equations of motion of the shell becomes $$\begin{aligned}
\label{18}
\dot{R} =\pm \left[1- \frac{M^2}{R^2}
-\frac{2m_{\pm}}{R}\right]^\frac{1}{2},\end{aligned}$$ here $\pm$ represents expansion and collapse of thin-shell. In Figure **1**, the left and right graphs describe the shell velocity for $\dot{R}>0$ and $\dot{R}<0$, respectively. The expansion (collapse) of the scalar shell is shown in left (right) graph, while blue and red curves correspond to interior and exterior regions. These graphs indicate that thin-shell velocity decreases positively (left plot) and increases negatively (right plot). Also, it is found that the velocity of interior region is greater than the exterior.
The Kerr Black Hole
-------------------
For Kerr BH, the effective potential takes the following form $$\begin{aligned}
\nonumber
V_{eff2}(R)&=&\left\{B \left(B-\frac{A M^2}{R^2}\right)
\left(A^2F_{\pm} \left(a^2 + R^2\right)+2 a^2 m_{\pm} R\sin^2\theta
\right.\right.\\\nonumber&\times&\left.\left. \left(A F_{\pm} +8
m_{\pm} R\right) \right)\right\}\left\{ \left(A^3 B F_{\pm}\left(a^2
+R^2 \right)+2AR a^2 m_{\pm}
\right.\right.\\\label{19}&\times&\left.\left(A B F_{\pm}-\frac{8 A
m_{\pm} M^2}{R} +8 B m R\right)\sin^2\theta\right\}^{-1}.\end{aligned}$$ In this case, the respective equations of motion becomes $$\begin{aligned}
\nonumber
\dot{R}&=&\pm\left[-\left\{B \left(B-\frac{A M^2}{R^2}\right)
\left(A^2F_{\pm} \left(a^2 + R^2\right)+2 a^2 m_{\pm} R\sin^2\theta
\right.\right.\right.\\\nonumber&\times&\left.\left.\left. \left(A
F_{\pm} +8 m_{\pm} R\right) \right)\right\}\left\{ \left(A^3 B
F_{\pm}\left(a^2 +R^2 \right)+2AR a^2 m_{\pm}
\right.\right.\right.\\\label{20}&\times&\left.\left.\left(A B
F_{\pm}-\frac{8 A m_{\pm} M^2}{R} +8 B m
R\right)\sin^2\theta\right\}^{-1}\right]^{\frac{1}{2}}.\end{aligned}$$ The plots for shell’s velocity is shown in Figure **2**. It is observed that the behavior of velocity in the presence of rotation parameter remains the same as for the Schwarzschild BH.
\[2\]
The Kerr-Newmann Black Hole
---------------------------
This BH is a generalization of the Kerr BH. The corresponding effective potential is $$\begin{aligned}
\nonumber
V_{eff3}(R)&=&-\left\{B \left(B-\frac{4A M^2}{R^2}\right)
\left(-A^2F_{\pm} \left(a^2 + R^2\right)+a^2\left(Q^2-2
m_{\pm}R\right) \right.\right.\\\nonumber&\times&\left.\left.
\left(A F_{\pm}-4Q^2 +8 m_{\pm} R\right)\sin^2\theta
\right)\right\}\left\{ \left(A^3 B F_{\pm}\left(a^2 +R^2
\right)-a^2A \right.\right.\\\nonumber&\times&\left.\left(\frac{16 A
\left(Q^2-2m_{\pm}R\right) M^2}{R^2} + B\left(A F_{\pm}-4Q^2+8
m_{\pm} R\right)\right)\right.\\\label{21}&
\times&\left.\left(Q^2-2m_{\pm}R\right)\sin^2\theta \right\}^{-1}.\end{aligned}$$ The equations of motion for this BH takes the form
\[3\]
$$\begin{aligned}
\nonumber
\dot{R}&=&\pm\left[\left\{B \left(B-\frac{4A M^2}{R^2}\right)
\left(-A^2F_{\pm} \left(a^2 + R^2\right)+a^2\left(Q^2-2
m_{\pm}R\right)
\right.\right.\right.\\\nonumber&\times&\left.\left.\left. \left(A
F_{\pm}-4Q^2 +8 m_{\pm} R\right)\sin^2\theta \right)\right\}\left\{
\left(A^3 B F_{\pm}\left(a^2 +R^2 \right)-a^2A
\right.\right.\right.\\\nonumber&\times&\left.\left.\left(\frac{16 A
\left(Q^2-2m_{\pm}R\right) M^2}{R^2} + B\left(A F_{\pm}-4Q^2+8
m_{\pm} R\right)\right)\right.\right.\\\label{22}&
\times&\left.\left.\left(Q^2-2m_{\pm}R\right)\sin^2\theta
\right\}^{-1}\right]^{\frac{1}{2}}.\end{aligned}$$
Figure **3** (upper panel is for exterior while lower is for interior region) indicates that the velocity of thin-shell depends on charge but there is no contribution of rotation parameter. When we increase charge, the velocity decreases positively for both spacetimes as shown in the left plot of upper as well as lower panel. It is observed that velocity increases negatively for these geometries as shown in the right plot of upper as well as lower panel.
Dynamics of Scalar Shell
------------------------
Here we examine the dynamical behavior of the scalar shell. For this purpose, we use a transformation $\left(u_{a}=\frac{\psi_{,a}}{\sqrt{\psi_{,b}\psi^{,b}}}\right)$ [@19], which relates surface energy density and pressure of a perfect fluid with potential function $V(\psi)$ and derivative of the scalar field. The corresponding surface energy density and pressure are obtained as follows $$\begin{aligned}
\label{23}
\sigma=-\frac{1}{2}\left[\psi_{,b}\psi^{,b}-2V(\psi)\right],\quad
p=-\frac{1}{2}\left[\psi_{,b}\psi^{,b}+2V(\psi)\right],\end{aligned}$$ where $V(\psi)=M^2\psi^2$. The stress-energy tensor in terms of scalar field is defined as $$\begin{aligned}
\nonumber
S_{ij}=\nabla_{i}\psi
\nabla_{j}\psi-\eta_{ij}\left[\frac{1}{2}(\nabla
\psi)^2-V(\psi)\right].\end{aligned}$$ Since, the induced metric is a function of proper time $\tau$, so $\psi$ depends on $\tau$. Consequently, Eq.(23) yields $$\begin{aligned}
\label{24}
\sigma=\frac{1}{2}\left[\dot{\psi}^2+2V(\psi)\right], \quad
p=\frac{1}{2}\left[\dot{\psi}^2-2V(\psi)\right].\end{aligned}$$ The total mass of the shell ($M=A \sigma$) in terms of scalar field is defined as $$\begin{aligned}
\label{25}
M=4\pi R^2 \sigma= 2\pi R^2[\dot{\psi}^2+2V(\psi)].\end{aligned}$$ Inserting Eqs.(\[24\]) and (\[25\]) in (\[14\]), we obtain $$\begin{aligned}
\label{26}
\ddot{\psi}+\frac{2\dot{R}}{R}\dot{\psi}+\frac{\partial V}{\partial
\psi}=0.\end{aligned}$$ This is a well-known Klein-Gordon (KG) equation and its representation in shell coordinate system is $\Box\psi+\partial
V/\partial \psi=0$.
In order to analyze the dynamical behavior of spherical scalar shell, we solve KG as well as energy conservation equation simultaneously for $\psi(\tau)$ and $R(\tau)$. The effective potential for the Schwarzschild, Kerr and Kerr-Newmann BHs in terms of scalar field are obtained as $$\begin{aligned}
\label{27}
V_{eff1}(R)&=&-1 + \frac{2 m_{\pm}}{R} +4 \pi^2
R^2\left[\dot{\psi}^2 + 2 V(\psi)\right]^2,
\\\nonumber
V_{eff2}(R)&=&\left\{B \left(B-4A \pi^2
R^2\left[\dot{\psi}^2+2V(\psi)\right]^2\right) \left(A^2F_{\pm}
\left(a^2 + R^2\right)+2 a^2
\right.\right.\\\nonumber&\times&\left.\left. \left(A F_{\pm} +8
m_{\pm} R\right)m_{\pm} R\sin^2\theta \right)\right\}\left\{
\left(A^3 B F_{\pm}\left(a^2 +R^2 \right)+2a^2
\right.\right.\\\nonumber&\times&\left.\left(A B F_{\pm}-32 A
m_{\pm}\pi^2 R^3[\dot{\psi}^2+2V(\psi)]^2 +8 B m
R\right)\right.\\\label{28}&\times&\left.
m_{\pm}AR\sin^2\theta\right\}^{-1},
\\\nonumber
V_{eff3}(R)&=&-\left\{B \left(B-16 \pi^2A R^2\left[\dot{\psi}^2 + 2
V(\psi)\right]^2\right) \left(-A^2F_{\pm} \left(a^2 + R^2\right)
\right.\right.\\\nonumber&+&\left.\left.a^2\left(Q^2-2
m_{\pm}R\right) \left(A F_{\pm}-4Q^2 +8 m_{\pm} R\right)\sin^2\theta
\right)\right\}\left\{\left(A^3 B F_{\pm}
\right.\right.\\\nonumber&\times&\left.\left(a^2 +R^2
\right)-a^2A\left(64 \pi^2A \left(Q^2-2m_{\pm}R\right)
R^2\left[\dot{\psi}^2 + 2 V(\psi)\right]^2
\right.\right.\\\label{29}& +&\left.\left. B\left(A F_{\pm}-4Q^2+8
m_{\pm} R\right)\right)\left(Q^2-2m_{\pm}R\right)\sin^2\theta
\right\}^{-1}.\end{aligned}$$ Equations (\[8\]) and (\[26\]) can be solved numerically for $m_-=0$ and $m_+=R_0=\sigma_0=k=M=1$. Figure **4** indicates the dynamical behavior of shell corresponding to Schwarzschild, Kerr and Kerr-Newmann BHs. Here, each graph describes the motion of scalar shell through upper and lower curves indicating the expanding and collapsing behavior, respectively. For Kerr BH, it is found that the increase in rotation parameter leads to enhancement of expansion as well as collapse rate. In case of Kerr-Newmann BH, we analyze that the presence of charge and rotation parameters decreases the expansion and collapse rate.
\[3\]
Now, we study the dynamical behavior of scalar shell for massless as well as massive scalar field.
### Massless Scalar Shell
Here we analyze the dynamical behavior of shell in the absence of scalar potential field $(V(\psi))$, i.e., massless scalar field. In this case, we do not need different EoS because the vanishing of $(V(\psi))$ leads to a direct relation between pressure and surface density ($p=\sigma$). Using the condition $V(\psi)=0$, the KG equation can be written as $\ddot{\psi}+\frac{2\dot{R}}{R}\dot{\psi}=0$ and its integration leads to $\dot{\psi}=\frac{\lambda}{R^2}$, where $\lambda$ is an integrating constant. The equations of motion for Schwarzschild, Kerr and Kerr-Newmann BHs become $$\begin{aligned}
\label{30}
\dot{R}^2&=&1 - \frac{2m_{\pm}}{R} - \frac{4 \pi^2 \lambda^4}{R^6},
\\\nonumber \dot{R}^2&=&-\left\{B \left(B-\frac{4A
\pi^2\lambda^4}{R^6}\right) \left(A^2F_{\pm} \left(a^2 +
R^2\right)+2 a^2 m_{\pm} R\sin^2\theta
\right.\right.\\\nonumber&\times&\left.\left. \left(A F_{\pm} +8
m_{\pm} R\right) \right)\right\}\left\{ \left(A^3 B F_{\pm}\left(a^2
+R^2 \right)+2AR a^2 m_{\pm}
\right.\right.\\\label{31}&\times&\left.\left(A B
F_{\pm}-\frac{32\pi^2 A m_{\pm} \lambda^4}{R^5} +8 B m
R\right)\sin^2\theta\right\}^{-1}, \\\nonumber \dot{R}^2&=&\left\{B
\left(B-\frac{16A \pi^2 \lambda^4}{R^6}\right) \left(-A^2F_{\pm}
\left(a^2 + R^2\right)+a^2\left(Q^2-2 m_{\pm}R\right)
\right.\right.\\\nonumber&\times&\left.\left. \left(A F_{\pm}-4Q^2
+8 m_{\pm} R\right)\sin^2\theta \right)\right\}\left\{ \left(A^3 B
F_{\pm}\left(a^2 +R^2 \right)-a^2A
\right.\right.\\\nonumber&\times&\left.\left(\frac{64\pi^2 A
\left(Q^2-2m_{\pm}R\right) \lambda^4}{R^6} + B\left(A F_{\pm}-4Q^2+8
m_{\pm} R\right)\right)\right.\\\label{32}&
\times&\left.\left(Q^2-2m_{\pm}R\right)\sin^2\theta \right\}^{-1}.\end{aligned}$$ The corresponding effective potential are $$\begin{aligned}
\label{33}
V_{eff1}(R)&=&-1 + \frac{2m_{\pm}}{R} + \frac{4 \pi^2
\lambda^4}{R^6}, \\\nonumber V_{eff2}(R)&=&\left\{B \left(B-\frac{4A
\pi^2\lambda^4}{R^6}\right) \left(A^2F_{\pm} \left(a^2 +
R^2\right)+2 a^2 m_{\pm} R\sin^2\theta
\right.\right.\\\nonumber&\times&\left.\left. \left(A F_{\pm} +8
m_{\pm} R\right) \right)\right\}\left\{ \left(A^3 B F_{\pm}\left(a^2
+R^2 \right)+2AR a^2 m_{\pm}
\right.\right.\\\label{34}&\times&\left.\left(A B
F_{\pm}-\frac{32\pi^2 A m_{\pm} \lambda^4}{R^5} +8 B m
R\right)\sin^2\theta\right\}^{-1},
\\\nonumber
V_{eff3}(R)&=&-\left\{B \left(B-\frac{16A \pi^2
\lambda^4}{R^6}\right) \left(-A^2F_{\pm} \left(a^2 +
R^2\right)+\left(Q^2-2 m_{\pm}R\right)
\right.\right.\\\nonumber&\times&\left.\left. \left(A F_{\pm}-4Q^2
+8 m_{\pm} R\right)a^2\sin^2\theta \right)\right\}\left\{ \left(A^3
B F_{\pm}\left(a^2 +R^2 \right)-a^2A
\right.\right.\\\nonumber&\times&\left.\left(\frac{64\pi^2 A
\left(Q^2-2m_{\pm}R\right) \lambda^4}{R^6} + B\left(A F_{\pm}-4Q^2+8
m_{\pm} R\right)\right)\right.\\\label{35}&
\times&\left.\left(Q^2-2m_{\pm}R\right)\sin^2\theta \right\}^{-1}.\end{aligned}$$
The radius of shell and effective potential for massless scalar field are plotted in Figures **5-7**. Figure **5** shows increasing and decreasing behavior of radius which leads to expansion and collapse, respectively. It is also observed that the rate of expansion as well as collapse increases by increasing $\lambda$. The right plot of lower panel shows that the contribution of charge and rotation parameters decreases the collapse as well as expansion rate. Figures **6** and **7** represent the dynamics of massless scalar shell through effective potential. These plots describe the expansion ($V_{eff}>0$), collapse ($V_{eff}<0$) and saddle points ($V_{eff}=0$) of massless scalar shell. Figures **6** and **7** show that the scalar shell has the same behavior of expansion, collapse as well as saddle points. It is shown that the dynamical behavior of shell decreases by enhancing charge and rotation parameters.
### Massive Scalar Shell
In this case, we discuss the dynamics of scalar shell through massive scalar field, i.e., $V(\psi)=M^2\psi^2$. From Eq.(\[24\]), we find the potential function and massive scalar field given as $$\begin{aligned}
\label{36}
\dot{\psi}^2=\sigma+p, \quad 2V(\psi)=p-\sigma .\end{aligned}$$ We consider surface pressure as explicit function of $R$, i.e., $p=p_{0}e^{-\gamma R}$, where $\gamma$ and $p_{0}$ are constants. Using Eq.(\[14\]) alongwith the value of $p$, we find $$\begin{aligned}
\label{37}
\sigma=\frac{\omega}{R^2}+\frac{2\left(1+\gamma
R\right)p_{0}e^{-\gamma R}}{\gamma^2 R^2},\end{aligned}$$ where $\omega$ is an integration constant. Using the values of energy density and surface pressure in Eq.(\[36\]), we obtain $$\begin{aligned}
\label{38}
V(\psi)&=&\frac{\omega}{2R^2}-\frac{p_{0}e^{-\gamma
R}}{2}\left(1-\frac{2(1+\gamma R)}{\gamma^2 R^2}\right),
\\\label{39}
\dot{\psi}^2&=&\frac{\omega}{R^2}-p_{0}e^{-\gamma
R}\left(1+\frac{2\left(1+\gamma R\right)}{\gamma^2 R^2}\right),\end{aligned}$$ which satisfy the KG equation. Using Eqs.(\[37\])-(\[39\]) in (\[27\])-(\[29\]), it follows that $$\begin{aligned}
\label{40}
V_{eff1}(R)&=&-1+ \frac{M^2}{R^2} +\frac{2m_{\pm}}{R}, \\\nonumber
V_{eff2}(R)&=&\left\{B \left(B-\frac{A M^2}{R^2}\right)
\left(A^2F_{\pm} \left(a^2 + R^2\right)+2 a^2 m_{\pm} R\sin^2\theta
\right.\right.\\\nonumber&\times&\left.\left. \left(A F_{\pm} +8
m_{\pm} R\right) \right)\right\}\left\{ \left(A^3 B F_{\pm}\left(a^2
+R^2 \right)+2AR a^2 m_{\pm}
\right.\right.\\\label{41}&\times&\left.\left(A B F_{\pm}-\frac{8 A
m_{\pm} M^2}{R} +8 B m R\right)\sin^2\theta\right\}^{-1},
\\\nonumber V_{eff3}(R)&=&-\left\{B \left(B-\frac{4A M^2}{R^2}\right)
\left(-A^2F_{\pm} \left(a^2 + R^2\right)+a^2\left(Q^2-2
m_{\pm}R\right) \right.\right.\\\nonumber&\times&\left.\left.
\left(A F_{\pm}-4Q^2 +8 m_{\pm} R\right)\sin^2\theta
\right)\right\}\left\{ \left(A^3 B F_{\pm}\left(a^2 +R^2
\right)-a^2A \right.\right.\\\nonumber&\times&\left.\left(\frac{16 A
\left(Q^2-2m_{\pm}R\right) M^2}{R^2} + B\left(A F_{\pm}-4Q^2+8
m_{\pm} R\right)\right)\right.\\\label{42}&
\times&\left.\left(Q^2-2m_{\pm}R\right)\sin^2\theta \right\}^{-1}.\end{aligned}$$ Also, the mass of shell can be expressed as $$\begin{aligned}
\label{43}
M=4\pi R^2\sigma=4\pi\omega+\frac{8\pi p_{0}e^{-\gamma
R}}{\gamma^2}\left(1+\gamma R\right).\end{aligned}$$
Figures **8-10** indicate the behavior of shell’s radius as well as effective potential for massive scalar field for $m_-=0$ and $m_+=R_0=\sigma_0=\omega=\gamma=Q=a=1$. In Figure **8**, the upper curve represents the expansion of scalar shell while the lower curve shows collapse. Figure **9** shows that the effective potential for massive scalar field leads to expansion, collapse as well as saddle points for $\omega\in(-2,2)$. It is also found that $-2\geq\omega\geq2$ leads the shell to collapse (negative effective potential) only (Figure **10**). For massive scalar field, we find that the effective potential exhibits similar behavior (expansion, collapse and saddle point) for all values of free parameters.
Final Remarks
=============
In this paper, we have studied the dynamics of scalar shell for a class of BHs using Israel thin-shell formalism. For this purpose, we have formulated the equations of motion which leads to the behavior of shell’s velocity. We have then investigated the dynamics of scalar shell for massless and massive scalar fields. The results can be summarized as follows.
- The motion of scalar shell shows expanding as well as collapsing behavior of the shell (Figures **1-4**). It is found that shell’s velocity has similar behavior for all considered BHs. For the Kerr-Newmann BH, it is analyzed that the velocity of shell decreases by enhancing charge parameter (Figure **3**). We have obtained that the rate of expansion and collapse for Schwarzschild BH is greater than the Kerr and Kerr-Newmann BHs (Figure **4**).
- For massless scalar field, it is observed that the presence of charge and rotation parameters affect the dynamical behavior of shell’s radius. For the Kerr and Kerr-Newmann BHs, expansion as well as collapse rate is small as compared to the Schwarzschild BH (Figure **5**). The behavior of effective potential indicates the expansion, collapse as well as the stable points (Figures **6-7**). For both interior and exterior spacetimes, expansion and collapse rates are decreased as $\lambda$ increases (Figures **6** and **7**).
- For massive scalar field, it is shown that the radius of shell either expands or undergoes collapse. We have found that Schwarzschild BH exhibits more expansion and collapse as compared to Kerr and Kerr-Newmann BHs. This describes that the contribution of charge and rotation parameters decreases these phenomena (Figure **8**). It is found that $V_{eff}$ shows expansion as well as collapse of thin-shell for $\omega\in(-2,2)$ while it indicates only collapse for other values of $\omega$ (Figures **9** and **10**). It is observed that the behavior of effective potential for three BHs overlaps for all values of charge as well as rotation parameter.
We conclude that the dynamical evolution of scalar shell can be expressed through continuous expansion, collapse and stable configuration.
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[^1]: [email protected]
[^2]: [email protected]
| ArXiv |
---
abstract: 'A wide range of stochastic processes that model the growth and decline of populations exhibit a curious dichotomy: with certainty either the population goes extinct or its size tends to infinity. There is a elegant and classical theorem that explains why this dichotomy must hold under certain assumptions concerning the process. In this note, I explore how these assumptions might be relaxed further in order to obtain the same, or a similar conclusion, and obtain both positive and negative results.'
address: 'Biomathematics Research Centre, University of Canterbury, Christchurch, New Zealand'
author:
- Mike Steel
title: 'Reflections on the extinction–explosion dichotomy'
---
Extinction, Borel–Cantelli lemma, population size, coupling, Markov chain
Introduction
============
The ‘merciless dichotomy’ (Section 5.2 of [@had]) concerning extinction refers to a very general property of stochastic processes that describes the long-term fate of populations. Roughly speaking, the result states that if there is always a strictly positive chance the population could become extinct in the future (depending, perhaps, on the current population size), then the population is guaranteed to either become extinct or to grow unboundedly large. More precisely, a formal version of this result, due to Jagers (Theorem 2 of [@jag]), applies to any sequence $X_1, X_2, \ldots, X_n \ldots $ of non-negative real-valued random variables that are defined on some probability space and which is absorbing at 0 (i.e. $X_n=0 \Rightarrow X_{n+1}=0$ for all $n$). It states that, provided: $$\label{strong}
{{\mathbb P}}(\exists r: X_r=0|X_1, X_2, \ldots, X_n) \geq \delta_x>0 \mbox{ whenever $X_n \leq x$}$$ holds for all positive integers $n$, then, with probability 1, either $X_n \rightarrow \infty$ or a value of $n$ exists for which $X_k=0$ for all $k\geq n$ (notice that $\delta_x$ can tend towards 0 at any rate as $x$ grows). This result applies to a wide variety of stochastic processes studied in evolutionary and population biology (e.g. Yule birth-death models, branching processes etc) and the proof in [@jag] involves an elegant and short application of the martingale convergence theorem.
Note that the processes in [@jag] (and here) need not be Markovian. Nevertheless, the lower-bound inequality condition in (\[strong\]) has a Markovian-like feature that it is required to hold for all values of $X_1, X_2, \ldots, X_{n-1}$ whenever $X_n$ is less than $x$. This raises the question of how much this uniform bounding across the previous history of the process might be relaxed without sacrificing the conclusion of certain extinction or explosion. In this short note, we consider possible extensions of Jagers’ theorem by weakening the assumption in (\[strong\]). Specifically, we will consider a lower bound that conditions just on the event that $0<X_n \leq x$, either alone or alongside another variable that is dependent on (but less complete than) the past history $X_1, \ldots, X_{n-1}$.
First, we consider what happens if the probability in the lower bound (\[strong\]) were to condition just on $0<X_n \leq x$. In this case, we describe a positive result that delivers a slightly weaker conclusion than the original theorem of Jagers. We then show that the full conclusion cannot be obtained by lower bounds that condition solely on $0<X_n\leq x$ by exhibiting a specific counterexample. However, in the final section, we show that the full conclusion of Jagers’ theorem can be obtained by conditioning on $0<X_n\leq x$, together with some partial information concerning the past history of the process.
A simple general lemma and its consequence for bounded populations
==================================================================
We first present an elementary but general limit result, stated within the usual notation of a probability space $(\Omega, \Sigma, {{\mathbb P}})$ consisting of a sigma-algebra $\Sigma$ of ‘events’ (subsets of the sample space $\Omega$) and a probability measure ${{\mathbb P}}$ (for background on probability theory, see [@borel]).
Suppose that $E_1, E_2,\ldots $ are [*increasing*]{} (i.e. $E_i \subseteq E_{i+1}$) and $E = \bigcup_{n=1}^{\infty}E_n$. For example, suppose that $E_n$ is the event that some particular ‘situation’ (e.g. extinction of the population) has arisen on or before a given time step $n$ (e.g. day, year). These events are increasing and their union $E$ is the event that the ‘situation’ eventually arises. We are interested in when ${{\mathbb P}}(E)=1$. A sufficient condition to guarantee this is to impose any non-zero lower bound on the probability that the ‘situation’ arises at time step $n$ given that it has not done so already; in other words, to require that the conditional probability ${{\mathbb P}}(E_n|\overline{E_{n-1}})$ is at least $\delta >0$ for all sufficiently large values of $n$ (throughout this paper an overline denotes the complementary event).
On the other hand, it is equally easy to check that if $p_n={{\mathbb P}}(E_n|\overline{E_{n-1}})$ is allowed to converge to zero sufficiently quickly (so the probability of the ‘situation’ first arising on day $n$ goes to zero sufficiently fast that $\sum_n p_n < \infty$), then it is possible for ${{\mathbb P}}(E)<1$. For example, if accidents occur independently and the probability of a particular accident is reduced each year by $1\%$ of its current value, then there is a positive probability that no accident will ever occur; but if the probability reduces at the rate $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \cdots,$ then an accident is guaranteed to eventually occur (by the second Borel–Cantelli lemma).
Rather than placing some lower bound on the probability that the situation arises at time step $n$, we can, following [@jag], make a weaker assumption that if the situation has not happened yet, there is always a non-vanishing chance that it will occur some time in the future (formally, requiring merely that ${{\mathbb P}}(E|\overline{E_n})$ is uniformly bounded away from $0$). For maximal generality, we also wish to avoid any Markovian or independence assumptions. The following lemma provides a sufficient condition for ${{\mathbb P}}(E)=1$ without any further assumptions, and uses an elementary argument that will be useful later.
\[figure7\]
\[mike\] Suppose $E_n$ is an increasing sequence with limit $E$ and suppose that for some $\epsilon > 0$, ${{\mathbb P}}(E|\overline{E_n}) \geq \epsilon$ holds for all $n \geq 1$. Then ${{\mathbb P}}(E)=1$.
[*Proof:*]{} Let $p_n = P(E_n)$. Then, by the law of total probability:
${{\mathbb P}}(E) = {{\mathbb P}}(E|\overline{E_n})(1 - p_n) + {{\mathbb P}}(E|E_n)p_n$.
Now, ${{\mathbb P}}(E|E_n) = 1$ and, by assumption, ${{\mathbb P}}(E|\overline{E_n}) \geq \epsilon$. Therefore:
${{\mathbb P}}(E) \geq \epsilon(1 - p_n) + p_n$.
Since the events $E_n$ are increasing, a well known and elementary result in probability theory ensures that ${{\mathbb P}}(E) = \lim_{n \to \infty} p_n$. So, letting $n \to \infty$ in the previous inequality gives:
${{\mathbb P}}(E) \geq \epsilon (1 - {{\mathbb P}}(E)) + {{\mathbb P}}(E)$,
which implies that ${{\mathbb P}}(E) = 1$, as claimed. $\Box$
Example 1
---------
Consider population of a species where $X_n$ denotes the size of the population at time step $n$. The event $E_n = \{X_n=0\}$ is the event that the population is extinct by time step $n$ and this increasing sequence has the limit $E$ equal to the event of eventual extinction. In this setting, Lemma \[mike\] provides the following special case of Jagers’ theorem.
\[coro1\] Suppose that $X_1, X_2, \ldots, X_n$ is a sequence of non-negative real-valued random variables that are absorbing at 0 and are constrained to lie between $0$ and $M$. Moreover, suppose that for some $\delta>0$ and all positive integers $n$ we have: ${{\mathbb P}}(\exists r: X_r=0|X_n \neq 0) \geq \delta.$ Then, with probability 1, a value $n$ exists for which $X_k=0$ for all $k \geq n$.
$\Box$
Remarks
-------
- One might view Lemma \[mike\] as a simple formulation of ‘Murphy’s Law’ – the idea that if something bad can happen, it will at some point (a popular claim often made in jest that has an interesting history [@murphy]). In that context, $E_n$ is simply the event that the ‘bad thing’ has happened on or before day $n$.
- The proof of Proposition \[mike\] shows that $\lim_{n \rightarrow \infty} {{\mathbb P}}(E|\overline{E_n})>0 \Longrightarrow {{\mathbb P}}(E)=1.$ The converse also holds, provided that ${{\mathbb P}}(E_n)<1$ for all $n$; indeed under that restriction, a sharper limit can be stated: ${{\mathbb P}}(E)=1 \Longrightarrow \lim_{n \rightarrow \infty} {{\mathbb P}}(E|\overline{E_n})=1.$ With a view towards Borel–Cantelli type results, note also that one can have: $\sum_{n \geq 1} {{\mathbb P}}(E|\overline{E_n}) = \infty$ and ${{\mathbb P}}(E) <1$, if, for example, ${{\mathbb P}}(E_n) = q-\frac{1}{n}$, where $q<1$.
- A general characterisation for when ${{\mathbb P}}(E)=1$ is the following result from [@bruss].
\[nice\] If $E_n$ is an increasing sequence of events with limit $E$, then ${{\mathbb P}}(E)=1$ if and only if either ${{\mathbb P}}(E_1)=1$ or ${{\mathbb P}}(E_i|\overline{E_{i-1}})=1$ for some $i$, or $\sum_{i=1}^\infty {{\mathbb P}}(E_{t_i}|\overline{E_{t_{i-1}}})= \infty$ for some strictly increasing sequence $t_i$.
A convergence in probability result for $X_n$
=============================================
We now consider what happens if the population size is not bounded above by some maximal value $M$ as in Corollary \[coro1\]. In this case, by weakening the conditioning in Inequality (\[strong\]) to just $X\in (0,m]$, one can still derive a result a result concerning convergence in probability (rather than almost sure convergence) of the population size to 0 or infinity, as we now show.
\[jag2\] Suppose that $X_1, X_2, \ldots, X_n$ is a sequence of non-negative real-valued random variables that are absorbing at 0, and that for each positive integer $m$, there is a value $\delta_m>0$ for which the following holds for all values of $n$: $$\label{boundful1}
{{\mathbb P}}(\exists r: X_r=0|X_n \in (0,m]) \geq \delta_m.$$ Then, for every $m \geq 1$, we have $\lim_{n \rightarrow \infty} {{\mathbb P}}(X_n =0 \cup X_n >m) = 1.$
[*Proof:*]{} Throughout this proof we will let $E$ denote the event $\{\exists r: X_r=0\}$. The proof of Proposition \[jag2\] relies on the following result.
> [**Claim:**]{} Both ${{\mathbb P}}(X_n=0|X_n \leq m)$ and ${{\mathbb P}}(E|X_n \leq m)$ converge to 1 as $n \rightarrow \infty$.
Proposition \[jag2\] follows directly from this claim, since, for any $m \geq 1$: $${{\mathbb P}}(X_n =0 \cup X_n >m) = {{\mathbb P}}(X_n=0) + {{\mathbb P}}(X_n > m)$$ $$\geq {{\mathbb P}}(X_n=0|X_n \leq m){{\mathbb P}}(X_n\leq m)+ {{\mathbb P}}(X_n >m).$$ By the claim, $ {{\mathbb P}}(X_n=0|X_n \leq m)$ converges to 1 as $n$ grows, and so the previous inequality ensures that $\lim_{n \rightarrow \infty} {{\mathbb P}}(X_n =0 \cup X_n >m) = 1,$ as required. Thus it suffices to establish the claim.
[*Proof of Claim:*]{} Consider any subsequence $n(k)$ of positive integers for which the bounded sequence ${{\mathbb P}}(X_{n(k)} \leq m)$ has a limit. Such subsequences exist (by the Bolzano–Weierstrass theorem), and since $\lim \inf_{n \rightarrow \infty} {{\mathbb P}}(X_n \leq m)>0$ (by (\[boundful1\])) for all $m\geq 1$, the limit of ${{\mathbb P}}(X_{n(k)} \leq m)$ for any such subsequence is strictly positive (this latter observation also ensures that some conditional probabilities below are well defined for large enough values of $k$). By the law of total probability: $${{\mathbb P}}(E|X_{n(k)} \leq m) = {{\mathbb P}}(E|X_{n(k)}=0) {{\mathbb P}}(X_{n(k)}=0|X_{n(k)} \leq m)$$ $$+ {{\mathbb P}}(E|X_{n(k)} \in (0, m]) {{\mathbb P}}(X_{n(k)}>0|X_{n(k)} \leq m).$$ Thus if we let $p_k = {{\mathbb P}}(X_{n(k)}=0|X_{n(k)} \leq m)$, then, by (\[boundful1\]): $$\label{pp1}
{{\mathbb P}}(E|X_{n(k)} \leq m) \geq 1 \cdot p_k +\delta_m(1-p_k).$$ Now, $p_{k} = {{\mathbb P}}(X_{n(k)}=0)/{{\mathbb P}}(X_{n(k)} \leq m)$ and so $\lim_{k \rightarrow \infty} p_{k} = \frac{\lim_{k\rightarrow \infty} {{\mathbb P}}(X_{n(k)}=0)}{\lim_{k\rightarrow \infty} {{\mathbb P}}(X_{n(k)} \leq m)}$, since the numerator and denominator limits are non-zero. Moreover, we have ${{\mathbb P}}(E)=\lim_{k\rightarrow \infty} {{\mathbb P}}(X_{n(k)}=0)$ and so: $$\label{pp2}
\lim_{k \rightarrow \infty} p_{k} = \lim_{k \rightarrow \infty} {{\mathbb P}}(E|X_{n(k)} \leq m).$$ Let $p$ denote the shared limit in Eqn. (\[pp2\]). Then, from Inequality (\[pp1\]) we have: $$p \geq p + \delta_m(1-p),$$ which implies that $p=1$. Thus, for [*all*]{} subsequences $n(k)$ of positive integers for which ${{\mathbb P}}(X_{n(k)} \leq m)$ has a limit, this limit takes the same value (namely 1). It follows from a well-known result in analysis (e.g. Theorem 11, p. 67 of [@mal92]) that the full sequence ${{\mathbb P}}(X_n \leq m)$ also converges to 1 as $n \rightarrow \infty$, and, therefore, so do the sequences ${{\mathbb P}}(X_n=0|X_n \leq m)$ and ${{\mathbb P}}(E|X_n \leq m)$. This establishes the two limit claims in the Claim, and so completes the proof of Proposition \[jag2\]. $\Box$
Notice that Proposition \[jag2\] also implies Corollary \[coro1\] by taking $m = M$ and $\delta_m = \delta$ in (\[boundful1\]).
The conclusion of Proposition \[jag2\] cannot be strengthened to almost sure convergence.
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Suppose $X_1, X_2, \ldots, X_n \ldots$ is a sequence of non-negative real-valued random variables that satisfy the conditions described in Proposition \[jag2\]. In this case, the proposition assures us that $X_n$ converges in probability either to 0 or to infinity. This is a weaker conclusion than the statement that, with probability $1$, either $X_n=0$ for all sufficiently large $n$, or $X_n \rightarrow \infty$. We now show, by an explicit example, that such a stronger conclusion (which holds under the stronger condition (\[strong\]) required for Jagers’ theorem) need not hold under just the conditions described in Proposition \[jag2\]. In other words, some additional conditioning on the past history of the process is required in order to secure the stronger conclusion (we describe this further in the next section).
Example 2
---------
Consider the following process. Let $X_n^1, n \geq 1$ be a sequence of independent random variables with: $${{\mathbb P}}(X^1_n) = \begin{cases}
1, & \mbox{ with probability } \frac{1}{n};\\
n, & \mbox{otherwise}.
\end{cases}$$ For each $k \geq 2$, let $X^k_n, n\geq 1$ be the (deterministic) random variables defined by: $${{\mathbb P}}(X^k_n) =
\begin{cases}
1, & \mbox{ with probability 1 for all $n \in [1, \ldots, 2^k)$};\\
0, & \mbox{with probability 1 for all $n \geq 2^k$}.
\end{cases}$$
Now, let $X_n$ be the stochastic process which selects $K=k$ with probability $\frac{1}{2^k}$ (for $k=1, 2,\ldots$) and then takes $X_n$ to be the process $X^K_n$ for all $n\geq 1$.
Firstly, note that this mixture process is well defined, since $\sum_{k \geq 1} {{\mathbb P}}(K=k) = 1$. Next, observe that since $X^1_n = 1$ infinitely often (with probability 1) by the Borel–Cantelli Lemma (for independent random variables) and since there is a probability of $\frac{1}{2}$ that $X_n = X^1_n$ for all $n$, then, with probability $\frac{1}{2}$, $X_n$ does not converge to infinity or hit zero (note that $X^1_n \neq 0$ for any $n$, and $X^1_n$ returns to 1 infinitely often and so does not tend to infinity).
Thus, to establish the claim regarding our example it suffices to show that Inequality (\[boundful1\]) applies. This can be verified, and the details are provided in the Appendix.
An extended extinction dichotomy theorem
========================================
The example in the previous section shows that in (\[boundful1\]) we need to supplement the condition $X_n \in ~(0, m]$ with some further information concerning the past history of the process, in order to guarantee eventual extinction or $X_n \rightarrow \infty$. Here, we provide a mild extension of Theorem 2 of [@jag] by conditioning on the number of times the process has dipped below each given value $m$ up to the present step of the process.
\[jag3\] Suppose $X_1, X_2, \ldots, X_n \ldots$ is a sequence of non-negative real-valued random variables that are absorbing at 0. For each positive integer $m \geq 1$, let $\kappa_m(X_1, \ldots, X_{n-1})$ count the number of $X_1,X_2, \ldots, X_{n-1}$ that are less than or equal to $m$. Suppose that for each positive integer $m$, there exists $\delta_m>0$ for which the following holds for all $n$.: $$\label{boundful3}
{{\mathbb P}}(\exists r: X_r=0| X_n \in (0, m], \kappa_m(X_1, \ldots, X_{n-1})) \geq \delta_m.$$ Then, with probability 1, either $X_n \rightarrow \infty$ or a value of $n$ exists for which $X_k=0$ for all $k\geq n$.
[*Proof:*]{} For any strictly positive integers $n$ and $m$, let $E_n$ be the event that $X_n=0$ and let $J_m$ be the event that $X_k \leq m$ for infinitely many values of $k$. Notice that $E_n$ and $J_m$ are both increasing sequences. Moreover, if we let $E= \bigcup_{n \geq 1}E_n$, $J = \bigcup_{m \geq 1} J_m$ and $\overline{J} =\bigcap_{m \geq 1} \overline{J_m}$, then $E$ is the event that some $k$ exists such that $X_k=0$ and $\overline{J}$ is the event that $X_n \rightarrow \infty$. We wish to show the following: $$\label{ejeq}
{{\mathbb P}}(E \cup \overline{J}) =1.$$ Notice that: $$\label{ej}
E \subseteq J_m \mbox{ for each $m\geq 1$}.$$ Furthermore, ${{\mathbb P}}(E) >0$ by Inequality (\[boundful3\]) applied to $n=1$, and any value of $m\geq 1$ for which ${{\mathbb P}}(X_1\leq m)>0$. Thus, from (\[ej\]), ${{\mathbb P}}(J_m)>0$ (and so ${{\mathbb P}}(J)>0$ also), so the conditional probabilities ${{\mathbb P}}(E|J)$ and ${{\mathbb P}}(E|J_m)$ are well defined, and for each $m \geq 1$, the inclusion (\[ej\]) gives: $$\label{EJ2}
{{\mathbb P}}(E) = {{\mathbb P}}(E|J_m){{\mathbb P}}(J_m).$$ We will show that: $$\label{basic}
{{\mathbb P}}(E|J_m) = 1 \mbox{ for each } m \geq 1,$$ which, combined with Eqn. (\[EJ2\]), gives ${{\mathbb P}}(E) = {{\mathbb P}}(J_m)$ for each $m \geq 1$. Thus, since ${{\mathbb P}}(J) = \lim_{m \rightarrow \infty} {{\mathbb P}}(J_m)$ (recall $J_m$ are increasing), we have ${{\mathbb P}}(E) = {{\mathbb P}}(J)$, and consequently ${{\mathbb P}}(E)+{{\mathbb P}}(\overline{J})=1,$ since $E$ and $\overline{J}$ are mutually exclusive. In this way we obtain the required identity (\[ejeq\]) that establishes the theorem.
Thus it suffices to establish Eqn. (\[basic\]). For this we employ a coupling-style argument. For each positive integer $m$, we will associate to $X_n$ a second sequence of random variables $Y_k, k\geq 1$ as follows. Let $O_m= \{n \geq 1: X_n \leq m\}$, and for each $k \leq |O_m|,$ let $Y_k = X_{\nu(k)}$ where the random variable $\nu(k)$ is the $k^{\rm th}$ element of $O_m$ under the natural ordering of the positive integers. If $O_m$ is finite, then set $Y_k =0$ for all $k>|O_m|$ (notice that this will not occur when we condition on $J_m$ below).
We may assume that the joint probability ${{\mathbb P}}(Y_k \neq 0, J_m)$ is strictly positive; otherwise ${{\mathbb P}}(Y_k=0|J_m)=1$ and so (\[basic\]) holds, since ${{\mathbb P}}(Y_k=0|J_m) \leq {{\mathbb P}}(E|J_m)$. Consequently, the conditional probabilities are well defined in the following equation: $$\label{big}
{{\mathbb P}}(E|Y_k \neq 0, J_m) = \sum_{n \geq 1} {{\mathbb P}}(E| X_n \in (0, m], \nu(k)=n, J_m)\cdot {{\mathbb P}}(\nu(k)=n|Y_k \neq 0, J_m).$$ From (\[ej\]) and (\[boundful3\]), we obtain the following equality and inequality, respectively: $$\label{one}
{{\mathbb P}}(E| X_n \in (0, m], \nu(k) = n, J_m) \geq {{\mathbb P}}(E| X_n \in (0, m], \nu(k)=n) \geq \delta_m,$$ where the first inequality is from (\[ej\]) and the second inequality is from (\[boundful3\]), since conditioning on the conjunction $X_n \in (0, m], \nu(k) = n$ is equivalent to conditioning on the conjunction of $X_n \in (0,m]$ and $\kappa_m(X_1, \ldots, X_{n-1}) = k-1$. Substituting (\[one\]) into the right-hand side of (\[big\]) gives ${{\mathbb P}}(E|Y_k \neq 0, J_m) \geq \delta_m$. Thus, we have: $$\label{good}
{{\mathbb P}}(E|J_m) ={{\mathbb P}}(E|Y_k \neq 0, J_m){{\mathbb P}}(Y_k \neq 0|J_m) + 1 \cdot {{\mathbb P}}(Y_k=0|J_m)$$ $$\geq \delta_m(1-p_k) + 1\cdot p_k,$$ where $p_k = {{\mathbb P}}(Y_k=0|J_m),$ and where the factor $1$ is because, conditional on $J_m$, the event $E$ occurs whenever $Y_k=0$. Now, $\{Y_k=0\}$ is an increasing sequence in $k$, so if we let $\mathcal{Y}:= \bigcup_{k\geq 1}\{Y_k=0\}$, then: $$\label{helps1}
p:= \lim_{k \rightarrow \infty} p_k={{\mathbb P}}({\mathcal Y}|J_m) .$$ Moreover: $$\label{helps2}
{{\mathbb P}}(E|J_m) = {{\mathbb P}}({\mathcal Y}|J_m).$$ Applying (\[helps1\]) and (\[helps2\]) into (\[good\]) gives: $p \geq \delta_m(1-p)+p,$ which, in turn, implies that $p=1$ (since $\delta_m>0$). Thus, ${{\mathbb P}}(E|J_m)=p=1$, which establishes (\[basic\]) and so completes the proof.
$\Box$
Concluding remarks
==================
Notice that Theorem \[jag3\] implies Theorem 2 of [@jag], since the lower bound (\[boundful3\]) involves conditioning on aggregates of values for $X_1, \ldots, X_{n}$, so it holds automatically under the lower bound (\[strong\]). Notice also that the proof of Theorem \[jag3\], though longer than the elegant martingale argument for Theorem 2 of [@jag], requires merely elementary notions in probability.
It turns out that the collection of random variables $\kappa_m(X_1, \ldots, X_k)$ across all (real) values of $m$ and all integer values of $k$ between 1 and $n$ suffices to determine the sequence of random variables $X_1, \ldots, X_n$ (by induction on $k$), so it is not immediately clear that Theorem \[jag3\] really allows greater generality than Theorem 2 of [@jag]. Therefore we provide an example to show that this is indeed the case. Informally, the extra generality in Theorem \[jag3\], arises from imposing fewer inequalities: in (\[boundful3\]) there are $n$ inequalities corresponding to the $n$ possible values that $\kappa_m(X_1, \ldots, X_{n-1})$ can take, while in (\[strong\]), there are potentially infinitely many, corresponding to all possible values for $X_1, \ldots, X_{n-1}$ (and for $X_n \leq x$).
Example 3
---------
Roughly speaking, the stochastic process we will construct becomes extinct unless it oscillates regularly within a fixed range for an initial period, and the longer that it oscillates the greater the chance that it will escape to infinity rather than become extinct. We show that such a process satisfies (\[boundful3\]) but not (\[strong\]).
First, consider a simple Markov chain $Y_n$ on the three states $0,1,2$ that starts in state 2 (i.e. $Y_1=2$ with probability 1) and with transition probabilities described as follows:
- 0 is an absorbing state;
- from state 1 or state 2, the next state is chosen with equal probability ($\frac{1}{3}$) from 0,1,2.
Thus, with probability 1, a value $n$ exists for which $Y_k =0$ for all $k\geq n$.
We will say that a sequence of values $y_1, y_2, y_3, \ldots, y_k$ from $\{1,2\}$ is a [*terminated flip sequence*]{} (of length $k$) if $y_1 = 2$ and $y_{i} = y_{i-1}$ only for $i=k$. For example $(2,1,2,1,2,1,2,1,1)$ and $(2,1,2,1,2,2)$ are terminated flip sequences of lengths nine and six respectively.
We use $Y_n$ to define our process $X_n$ which takes non-negative integer values as follows. If there is no value $N \geq 4$ for which $Y_1, Y_2, \ldots Y_N$ is a terminated flip sequence, then set $X_n=Y_n$ for all $n$; in which case $X_n$ absorbs at 0 with probability 1. On the other hand, if a value $N \geq 4$ exists for which $Y_1, Y_2, \ldots Y_N$ is a terminated flip sequence, then, conditional on this value of $N$, $X_n = Y_n$ for all $n \leq N$, and for $n> N$, $X_n = Z^N_{n-N}$, where $Z^N_1, Z^N_2, \ldots$ is a second Markov chain on the state space $\{0\} \cup \{ N-1, N, N+1, N+2,\ldots\}$. This second chain has $Z_1= N-1$ (with probability 1), and has transitions from each state $i \geq N-1$ to $0$ and to $i+1$ with probabilities of $2^{-i}$ and $1- 2^{-i}$, respectively.
Notice that, although the process $X_n$ is absorbing at $0$, it fails to satisfy (\[strong\]) since, for an terminated flip sequence $(x_1,x_2, \ldots, x_n)$, of length 4 or more, we have $x_n \leq 2$ and yet: $${{\mathbb P}}(\exists r: X_r=0| \wedge_{i=1}^n \{X_i =x_i\}) = \sum_{j=n-1}^\infty \frac{1}{2^j} \rightarrow 0, \mbox{ as } n \rightarrow \infty.$$
To show that $X_n$ satisfies (\[boundful3\]), we consider the cases $m=1$, $m=2$ and $m>2$ separately. For $m=1$, (\[boundful3\]) is equivalent to the following inequality holding for all $n \geq 1$: $$\label{m1}
{{\mathbb P}}(\exists r: X_r=0| X_n =1, \kappa_1(X_1, \ldots, X_{n-1})) \geq \delta_1>0.$$ Now, if $\kappa_1(X_1, \ldots, X_{n-1}) \neq \lfloor (n-1)/2\rfloor$ then $X_1, \ldots, X_n$ cannot be a terminated flip sequence, and so, with probability at least $\frac{1}{3}$, we have $X_{n+1} = 0$. On the other hand, if $\kappa_1(X_1, \ldots, X_{n-1}) = \lfloor (n-1)/2\rfloor$ then the probability that $Y_1, Y_2, \ldots, Y_n$ is a terminated flip sequence of length 4 or more is bounded away from 1 as $n$ grows, and so the event $\{\exists r: X_r=0\}$ has a probability that is bounded away from $0$ for all $n$ when we condition on $\kappa_1(X_1, \ldots, X_{n-1})$ and $X_n = 1$. Thus a value $\delta_1>0$ can be chosen to satisfy (\[m1\]) for all $n \geq 1$.
For $m=2$, (\[boundful3\]) is equivalent to the following inequality holding for all $n \geq 1$: $$\label{m2}
{{\mathbb P}}(\exists r: X_r=0| X_n \in (0,2])\geq \delta_2>0.$$ Notice that $\kappa_2$ has vanished, since $X_n \in (0,2]$ implies that $\kappa_2(X_1, \ldots, X_{n-1}) = n-1$ with probability 1. Now, conditional on $X_n \in (0,2]$, the probability that $Y_1, Y_2, \ldots, Y_n$ is a terminated flip sequence of length 4 or more is bounded away from 1 as $n$ grows, and so the event $\{\exists r: X_r=0\}$ has a probability that is bounded away from $0$ for all $n$ when we condition on$X_n \in \{1,2\}$. Thus a value $\delta_2>0$ can be chosen to satisfy (\[m2\]) for all $n \geq 1$.
Finally, for each $m>2$, for all $n \geq 1$: $${{\mathbb P}}(\exists r: X_r=0| X_n \in (0,m], \kappa_m(X_1, \ldots, X_{n-1})) \geq 2^{-m}>0,$$ so we can set $\delta_m = 2^{-m}$ for all $m>2$. In summary, for all values of $m$, $X_n$ satisfies (\[boundful3\]) for all $n$, as claimed.
Acknowledgments
===============
I thank Elchanan Mossel for several helpful comments concerning an earlier version of this manuscript, and Elliott Sober for some motivating discussion. I also thank the Allan Wilson Centre for funding support for this work.
References
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Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes, 3rd ed. Oxford University Press, New York, USA.
Haccou, P., Jagers, P., and Vatutin, V.A., (2005). Branching processes: variation, growth, and extinction of populations. Cambridge Studies in Adaptive Dynamics, Cambridge University Press, Cambridge UK.
Jagers, P. (1992). Stabilities and instabilities in population dynamics. J. Appl. Probab. 29(4): 770–780.
Malik, S.C. and Arora, S. (1992). Mathematical Analysis. New Age International. New Delhi, India.
Wikipedia (2014) http://en.wikipedia.org/wiki/Murphy’s law (accessed 15 September 2014).
Appendix: Proof that Example 2 satisfies Inequality (\[boundful1\])
===================================================================
Firstly, if $m<1$, then conditioning on $X_n\leq m$ is equivalent to conditioning on $K>1$ and so we can take any positive value for $\delta_m$ (even $= 1$) and satisfy Inequality (\[boundful1\]).
Next, suppose that $m \geq 1$, and, for any $n \geq 1$, write: $$\label{eq0}
n = 2^q +r, \mbox{ where } 0\leq r < 2^q, q\geq 0.$$ SInce ${{\mathbb P}}(\exists r: X_r=0|X_n \in (0,m]) = {{\mathbb P}}(K>1|X_n\in (0,m])$ we have: $$\label{eq1}
{{\mathbb P}}(\exists r: X_r=0|X_n \in (0,m]) = \sum_{k \geq 2} {{\mathbb P}}(K=k|X_n\in (0,m]),$$ and, from Bayes’ identity: $$\label{eq2}
{{\mathbb P}}(K=k|X_n\in (0,m]) = \frac{{{\mathbb P}}(X_n \in (0,m]|K=k){{\mathbb P}}(K=k)}{{{\mathbb P}}(X_n \in (0,m])}.$$
Now, for any $k \geq 2$ (and still with $m>1$): $$\label{eqx}
{{\mathbb P}}(X_n \in (0,m]|K=k) =
\begin{cases}
1, & \mbox{ provided $k \geq q+1$};\\
0, & \mbox{otherwise};
\end{cases}$$ since $X_n^k=1$ with probability 1 for all $n \in [1, \ldots, 2^k)$. Consequently, the numerator of (\[eq2\]) equals $\frac{1}{2^k}$ ( $={{\mathbb P}}(K=k)$) when $k \geq q+1$ and is zero otherwise.
Now, the denominator of (\[eq2\]), namely ${{\mathbb P}}(X_n \in (0,m])$, can be written as: $$\label{eq3}
\left[\sum_{k \geq 2} {{\mathbb P}}(X_n \in (0,m]|K=k){{\mathbb P}}(K=k)\right] + {{\mathbb P}}(X_n\in (0,m]|K=1){{\mathbb P}}(K=1).$$ From (\[eqx\]), the first term in (\[eq3\]) is: $$\label{eq4}
\sum_{k \geq 2} {{\mathbb P}}(X_n \in (0,m]|K=k){{\mathbb P}}(K=k)= \sum_{k \geq q+1} \frac{1}{2^k} = \frac{1}{2^q}.$$ Regarding the second term in (\[eq3\]), observe that: $${{\mathbb P}}(X_n\in (0,m]|K=1) =
\begin{cases}
\frac{1}{n}, & \mbox{ provided $n > m$};\\
1 , & \mbox{if $n \leq m$. }
\end{cases}$$ Therefore, recalling (\[eq0\]), the second term in (\[eq3\]) is: $$\label{eq5}
\begin{cases}
\frac{1}{n} \times \frac{1}{2} = \frac{1}{2^{q+1}+2r} \leq \frac{1}{2^{q+1}} , & \mbox{ provided $n >m$};\\
1 \times \frac{1}{2}, & \mbox{when $n \leq m$}.
\end{cases}$$ Consequently, by combining (\[eq3\]), (\[eq4\]) and (\[eq5\]) into (\[eq2\]) (and noting again that $\sum_{k \geq q+1} \frac{1}{2^k} = \frac{1}{2^q}$) we have that if $n >m$, then $\sum_{k \geq 2} {{\mathbb P}}(K=k|X_n\in (0,m]) \geq
\frac{\frac{1}{2^q}}{\frac{1}{2^q} + \frac{1}{2^{q+1}}} \geq \frac{1}{2},$ while if $n \leq m$, then $\sum_{k \geq 2} {{\mathbb P}}(K=k|X_n\in (0,m]) \geq \frac{\frac{1}{2^q}}{\frac{1}{2^q} + \frac{1}{2}} \geq \frac{\frac{1}{2^q}}{1+ \frac{1}{2}} \geq \frac{2}{3}\cdot \frac{1}{2^q} \geq \frac{2}{3m},$ where the last inequality is from $ m \geq n \geq 2^q$. Thus, if we take $\delta_1 = \frac{1}{2}$ and $\delta_m = \frac{2}{3m}$ for each $m \geq 2$, then, from (\[eq1\]), $ {{\mathbb P}}(\exists r: X_r=0|X_n \in (0,m])\geq \delta_m$ for all $n, m$, as claimed.
| ArXiv |
---
abstract: 'High-frequency (up to $\omega = 6 \,10^4 \un{rad/s}$) rheological measurements combined with light scattering investigations show that an isotropic and multiconnected phase of surfactant micelles exhibits a terminal relaxation time of a few $\un{\mu s}$, much smaller than in solutions of entangled wormlike micelles. This result is explained in terms of the local hexagonal order of the microscopic structure and we discuss its relevance for the understanding of dynamic behaviour in related systems, such as wormlike micelles and sponge phases.'
author:
- 'D. Constantin[^1]'
- 'J.-F. Palierne'
- 'É. Freyssingeas'
- 'P. Oswald'
title: 'High-frequency rheological behaviour of a multiconnected lyotropic phase'
---
In recent years, experimental evidence was presented as to the existence of isotropic phases consisting of connected surfactant micelles [@danino1; @kato1; @kato2]. It has been proposed that they provide an intermediate structure between entangled wormlike micelles and sponge phases [@porte1; @drye1]. Indeed, experimental results [@porte1; @appell1; @khatory1] show that, in some ionic wormlike micellar systems, a dramatic decrease in both viscosity and relaxation time is induced by increasing the counterion concentration, feature that could be explained by the appearance of connections in the micellar network. On the theoretical side, models for the flow behaviour of these connected phases have been developed [@drye1; @lequeux1], and rheology data has been interpreted according to these models in order to characterize the appearance of connections, qualitatively [@narayanan1; @hassan1; @aitali1] or quantitatively [@in1]. Throughout this body of work, however, only the relaxation modes specific to polymer systems have been considered. This approach is certainly valid in dilute phases with not too many connections, but it must fail when the density of connections becomes important and in concentrated systems, where the micelles begin to interact (sterically or otherwise). How does the system behave then and which are the relevant concepts ?
In this Letter, we try to answer these questions by investigating a concentrated and highly connected isotropic phase of a nonionic surfactant/water mixture. We argue that, in the absence of reptation (suppressed by the connections), it can be short-range order (for a concentrated system) that dominates the rheological behaviour.
We employ high-frequency rheology and dynamical light scattering (DLS) to study the isotropic phase in the [ ]{}lyotropic mixture, where [ ]{}is the non-ionic surfactant hexa-ethylene glycol mono-n-dodecyl-ether, or (for the phase diagram see [@mitchell]). Its dynamic behaviour has already been investigated by measuring the shear viscosity [@strey1; @darrigo1], sound velocity and ultrasonic absorption [@darrigo1] as well as NMR relaxation rates [@burnell1], all pointing to the presence of wormlike micelles (at least above 10 % surfactant concentration by weight [@darrigo1]). In previous experiments [@sallen1; @constantin] we have shown that, for 50 % wt surfactant concentration, above the hexagonal mesophase, the isotropic phase has a structure consisting of surfactant cylinders that locally preserve the hexagonal order over a distance $d$ that varies from about $40 \, \un{nm}$ at $40 { {\,}^{\circ} \mbox{C}}$ to $25 \, \un{nm}$ at $60 { {\,}^{\circ} \mbox{C}}$. Between the cylinders there is a large number of thermally activated connections (with an estimated density $n \sim 10^{6} \, \un{\mu m^{-3}}$) [@constantin].
We prepared the [ ]{}mixture with 50.0 % [ ]{}weight concentration. The surfactant was purchased from Nikko Chemicals Ltd. and used without further purification. We used ultrapure water from Fluka Chemie AG. The mixture was carefully homogenized by repeatedly heating, stirring and centrifuging and then allowed to equilibrate at room temperature over a few days.
Rheology measurements were performed in a piezorheometer, the principle of which has been described in reference [@cagnon] : the liquid sample of thickness $60 \un{\mu m}$ is contained between two glass plates mounted on piezoelectric ceramics. One of the plates is made to oscillate vertically with an amplitude of about 1 nm by applying a sine wave to the ceramic. This movement induces a squeezing flow in the sample and the stress transmitted to the second plate is measured by the other piezoelectric element. The shear is extremely small : $\gamma \leq 10^{-4}$, so the sample structure is not altered by the flow. The setup allows us to measure the storage ($G'$) and loss ($G''$) shear moduli for frequencies ranging from $1$ to $6 \, 10^{4} \un{rad/s}$ with five points per frequency decade. The entire setup is temperature regulated within $0.05 { {\,}^{\circ} \mbox{C}}$ and hermetically sealed to avoid evaporation.
Ten temperature points in the isotropic phase have been investigated, from $38.85 { {\,}^{\circ} \mbox{C}}$ (transition temperature from the hexagonal phase) up to $48 { {\,}^{\circ} \mbox{C}}$. The results are displayed in figure \[fig1\]. For clarity, only curves corresponding to 40, 42, 44, 46, and $48 { {\,}^{\circ} \mbox{C}}$ are plotted. Values below $1 \un{Pa}$ (solid horizontal line) are not reliable, as the signal/noise ratio becomes poor. At low frequencies, the response is purely viscous; it is only above $\omega = 10^{3} \un{rad/s}$ that there is a noticeable increase in the value of the storage modulus $G'$.
On general grounds, the low-frequency behaviour of the storage and loss moduli in a fluid is [@ferry] : $G' \propto \omega ^2$ and $G'' \propto \omega$. The slope of $G'$ vs. $\omega$ yields the “zero-shear viscosity” $\eta _0$ and the two curves cross at a frequency $\omega=1/\tau$, where $\tau$ is the terminal relaxation time. The ratio $\eta _0 / \tau$ defines a shear modulus. If $\tau$ is the only relevant time scale in the system, the complex modulus $G^*(\omega) = G' + i G''$ has a simple analytical expression, known as the Maxwell model [@ferry] : $$\label{maxwell}
G^*(\omega) = \frac{i \omega \eta _0}{1+i \omega \tau} \, .$$ The relaxation time $\tau$ separates two regimes : for $\omega \tau \ll 1$, the system can be considered as a viscous fluid with viscosity $\eta _0$, while for $\omega \tau \gg 1$ it exhibits elasticity, with a shear modulus $G_{\infty} = \eta _0 / \tau$.
As shown in \[fig2\], we obtain robust results for the static viscosity $\eta _0$ and for the relaxation time $\tau$ (plotted vs. temperature in figure \[fig3\]).
The temperature variation of the parameters $\eta _0$ and $\tau$ can be described by Arrhenius laws; for the viscosity : $$\label{arrhenius}
\eta _0 (T) = \eta _0 (T^*) \exp \left [ \frac{E_{\eta}}{k_B} \left (\frac{1}{T}-\frac{1}{T^*}\right ) \right ] \, ,$$ yielding an activation energy $E_{\eta} = 35 \pm 1 \, k_B T$ (solid curve in figure \[fig3\]). For comparison, continuous shear measurements in a Couette rheometer (Haake, model RS100), give an activation energy $E_{\eta} = 31 \, k_B T$ [@sallen3]. The relaxation time has an activation energy $E_{\tau} = 38 \pm 6 \, k_B T$ (solid curve in figure \[fig3\]). Within experimental precision, $E_{\eta} = E_{\tau}$. The high-frequency elastic modulus is therefore constant in temperature : $$\label{eq:ginf}
G_{\infty} = \eta _0 / \tau = 44 \pm 6 \, 10^3 \, \un{Pa} \, .$$
The DLS setup uses an Ar laser ($\lambda = 514 \, \un{nm}$), delivering up to $1.5 \un{W}$, a thermostated bath of an index matching liquid (decahydronaphthalene, $n = 1.48$), a photomultiplier and a PC-controlled 256 channel Malvern correlator with sample times as fast as $0.1 \un{\mu s}$. The scattering vector $q$ varies in the range $4 \, 10^{6}$ – $3 \, 10^{7} \un{m^{-1}}$. The signal is monoexponential over the whole range. In figure \[fig5\] we show the relaxation rate $\Omega (q)$ vs. $q^2$ for temperatures between $40$ and $49 { {\,}^{\circ} \mbox{C}}$ . The data fit well to a diffusion law (although there is a slight indication of super-diffusive behaviour). Since the scattered intensity is related to the variations in refractive index produced by concentration fluctuations, we obtain the collective diffusion constant for the concentration field; its temperature variation can be described by an Arrhenius fit (solid curve in figure \[fig5\] – inset) with an activation energy $E_D \simeq 4 \, k_B T$. The average value : $$D = 1.65 \, 10^{-10} \un{m^2/s}
\label{eq:diff}$$ is in good agreement with the one previously obtained from directional-growth experiments [@sallen2] : $D = 1.2 \, 10^{-10} \un{m^2/s}$ at the transition temperature ($38.85 { {\,}^{\circ} \mbox{C}}$).
In unconnected wormlike micellar systems [@drye1; @cates1; @cates2], the relevant relaxation process is reptation, the micelle gradually disengaging from its initial deformed environment and adopting a stress-free configuration. The typical reptation time is given by : $\tau _{\rm{rep}} \simeq L_{\rm{m}} ^2 / D_{\rm{c}}$, with $L_{\rm{m}}$ the average length of a micelle and $D_{\rm{c}}$ the curvilinear diffusion constant. However, if the micelles can break up (with a lifetime $\tau _{\rm{br}}$) this provides an additional pathway for disengagement, the two resulting ends being free to recombine in a different environment. For $ \tau _{\rm{br}} \ll \tau _{\rm{rep}}$, the terminal relaxation time is given by : $\tau = (\tau _{\rm{br}} \tau _{\rm{rep}})^{1/2}$ [@cates1]. As an illustration, in the CTAB/H$_2$0/KBr system the typical micelle length is $L_{\rm{m}} \simeq 1 \mu\rm{m}$, while $\tau$ varies between 0.1 and 1 s depending on the surfactant concentration [@candau1].
Let us now consider the effect of connections; following Drye and Cates [@drye1], we will introduce a typical micelle length between cross-links $L_{\rm{c}}$. The effect of the connections is that reptation occurs on distances of the order of $L_{\rm{c}}$, instead of the much larger $L_{\rm{m}}$ [@cates2]. This explains the fact (counterintuitive at first sight) that connecting the network does in fact reduce the viscosity. If $L_{\rm{c}}$ is small enough, the network is saturated, and the concept of entanglement is no longer applicable; neither is the reptation mechanism. The system we investigate is well in the saturated case, since the typical distance between connections on a micelle is only four times the mean distance between micelles [@constantin].
What is then the origin of viscoelasticity ? We begin the discussion of our results with the very general observation that, when a system is dynamically correlated over a typical distance $L$, one can only observe elastic behaviour by probing the system on scales smaller than the correlation distance [@dimension]. The time $\tau$ needed to relax the stress can then be estimated as : $$\label{tau}
\tau \sim L^2 / (2 \delta D) \, ,$$ where $\delta$ is the space dimension and $D$ is the diffusion constant associated to the relaxation process (a classical example is provided by the Nabarro–Herring creep in solids [@quere]). The system under investigation is very concentrated so, in contrast with the semi-dilute wormlike micellar solutions usually studied, the micelle-micelle interaction plays an important role in the dynamics of the phase. This interaction locally induces hexagonal order as mentioned above; the relevant correlation length is the distance $d$ over which the micelles preserve local order. A pictorial representation is given in figure \[fig6\] : consider a material with short-range order confined between two plates. The system can be seen as consisting of elasticity-endowed units of typical size $d$, the correlation distance. After applying an instantaneous shear $\gamma$ by moving the upper plate to the left, one such unit (represented in thick line) has been advected from point 1 to point 2. At time $t=0^+$ after the deformation, the stress on the upper plate is $\sigma = G_{\infty} \gamma$. Since there is no long-range restoring force, once the particles equilibrate their internal configuration (over a distance $d$), the elastic stress is completely relaxed; thus, after a time $\tau$ given by eq. \[tau\], $\sigma = 0$.
Does this mechanism account for the observed behaviour ? In light of the previous discussion, let us estimate the relaxation time for our system. With the value of $d$ obtained from X-ray scattering and the DLS collective diffusion coefficient (eq. \[eq:diff\]), one has : $$\tau \simeq d^2 / (6D) \sim 10^{-6} \un{s} \, ,
\label{tau2}$$ in good agreement with the experimental results (figure \[fig3\]).
A rough estimate of $G_{\infty}$ can be obtained by noticing that at short range (less than $d$), the structure of the phase resembles that of the hexagonal one, so it should exhibit a similar shear modulus when probed on very short scales. The shear modulus of the hexagonal phase can be estimated as $G_{\rm{hex}} = k_B T / a^3 \simeq 2 \, 10^4 \un{Pa}$ (with $a=6 \un{nm}$ the lattice parameter), in agreement with our result (eq. \[eq:ginf\]). This value can also be compared with preliminary measurements of the shear modulus in the hexagonal phase of [ ]{}[@pieranski1] yielding : $$G_{\rm{hex}} \simeq 2 \, 10^{5} \un{Pa}$$ at room temperature, of the same order of magnitude as our result. The shear modulus of the hexagonal phase should vary very little with temperature, in agreement with our experimental findings.
However, our very simple model does not accurately describe the temperature variation of the physical parameters in equation \[tau2\]. An Arrhenius fit of $d(T)$ (from the X-ray data of reference [@constantin]) yields an activation energy $E_d = 7 \pm 1 \, k_B T$. From equation (7) we would expect that : $$E_{\tau} = 38 \pm 6 \, k_B T \sim 2 E_d + E_D = 18 \pm 2 \, k_B T$$ which is clearly off by a factor of two.
A tentative explanation involves the possible anisotropy of the correlated domains; in this case, the value obtained from the X-ray diffractogram is an average between a transverse correlation length $d=d_{\bot}$ (which is the one relevant for the relaxation) and a $d_{\|}$ (which need not exhibit the same temperature variation). The same observation applies for $D$ : we measure an average value, but at small scale the structure is anisotropic.
A more detailed comparison with theory requires additional data on unsaturated structures. We are currently investigating the same isotropic phase at lower surfactant concentration, where preliminary experiments show rather complicated rheological behaviour.
Finally, we suggest that this approach can also be applied to sponge phases, the characteristic distance being $\xi$, the correlation length. These phases are equally very fluid and, at low frequency (up to at least $10^2 \un{s^{-1}}$), display pure Newtonian behaviour [@snabre1; @vinches1]. Within the framework of the same highly simplified model (equation \[tau\]), we predict a relaxation time of order $\tau \sim \xi ^2 / (6D)$. For instance, in the /hexanol/water system at 5.3 % volume fraction of membrane, where $\xi \simeq 0.1 \un{\mu m}$ and $D \simeq 2 \, 10^{-12} \un{m^2/s}$ [@freyssingeas1], we expect $\tau \sim 10^{-3} \un{s}$.
In conclusion, we study the dynamics of the isotropic (micellar) phase in the [ ]{}mixture at high concentration, where it is highly connected. We show that the observed viscoelastic behaviour can be related to the local hexagonal order of the system.
We would like to thank P. Pieranski for communicating experimental results prior to publication and R. Strey for providing a reprint.
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. From the softening of the structure the authors infer that, even at 50 % surfactant concentration, the isotropic phase is still composed of wormlike micelles (they assume that a connected structure would be more rigid). In fact, as discussed in references [@porte1; @drye1; @appell1; @khatory1; @lequeux1] and in the present work, connecting the micelles can render the structure more fluid.
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Provided that the local order is 2D or 3D. If the order is lamellar (1D), the associated shear modulus is zero.
experiments in progress.
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[^1]: E-mail address :
| ArXiv |
---
abstract: 'In global seismology Earth’s properties of fractal nature occur. Zygmund classes appear as the most appropriate and systematic way to measure this local fractality. For the purpose of seismic wave propagation, we model the Earth’s properties as Colombeau generalized functions. In one spatial dimension, we have a precise characterization of Zygmund regularity in Colombeau algebras. This is made possible via a relation between mollifiers and wavelets.'
author:
- |
Günther Hörmann and Maarten V. de Hoop\
*Department of Mathematical and Computer Sciences*,\
*Colorado School of Mines, Golden CO 80401*
title: 'Geophysical modelling with Colombeau functions: Microlocal properties and Zygmund regularity'
---
Introduction
============
*Wave propagation in highly irregular media*. In global seismology, (hyperbolic) partial differential equations the coefficients of which have to be considered generalized functions; in addition, the source mechanisms in such application are highly singular in nature. The coefficients model the (elastic) properties of the Earth, and their singularity structure arises from geological and physical processes. These processes are believed to reflect themselves in a multi-fractal behavior of the Earth’s properties. Zygmund classes appear as the most appropriate and systematic way to measure this local fractality (cf. [@Holschneider:95 Chap.4]).
*The modelling process and Colombeau algebras*. In the seismic transmission problem, the diagonalization of the first order system of partial differential equations and the transformation to the second order wave equation requires differentiation of the coefficients. Therefore, highly discontinuous coefficients will appear naturally although the original model medium varies continuously. However, embedding the fractal coefficient first into the Colombeau algebra ensures the equivalence after transformation and yields unique solvability if the regularization scaling $\ga$ is chosen appropriately (cf. [@LO:91; @O:89; @HdH:01]). We use the framework and notation (in particular, $\G$ for the algebra and $\A_N$ for the mollifier sets) of Colombeau algebras as presented in [@O:92].
An interesting aspect of the use of Colombeau theory in wave propagation is that it leads to a natural control over and understanding of ‘scale’. In this paper, we focus on this modelling process.
Basic definitions and constructions
===================================
Review of Zygmund spaces
------------------------
We briefly review homogeneous and inhomogeneous Zygmund spaces, ${\ensuremath{\dot{C}_*}}^s(\R^m)$ and ${\ensuremath{C_*}}^s(\R^m)$, via a characterization in pseudodifferential operator style which follows essentially the presentation in [@Hoermander:97], Sect. 8.6. Alternatively, for practical and implementation issues one may prefer the characterization via growth properties of the discrete wavelet transform using orthonormal wavelets (cf. [@Meyer:92]).
Classically, the Zygmund spaces were defined as extension of Hölder spaces by boundedness properties of difference quotients. Within the systematic and unified approach of Triebel (cf. [@Triebel:I; @Triebel:II]) we can simply identify the Zygmund spaces in a scale of inhomogeneous and homogeneous (quasi) Banach spaces, $B^s_{p q}$ and $\dot{B}^s_{p q}$ ($s\in\R$, $0 < p, q \leq \infty$), by ${\ensuremath{C_*}}^s(\R^m) = B^s_{\infty \infty}(\R^m)$ and ${\ensuremath{\dot{C}_*}}^s(\R^m) = \dot{B}^s_{\infty \infty}$. Both ${\ensuremath{C_*}}^s(\R^m)$ and ${\ensuremath{\dot{C}_*}}^s(\R^m)$ are Banach spaces.
To emphasize the close relation with mollifiers we describe a characterization of Zygmund spaces in pseudodifferential operator style in more detail.
Let $0 < a < b$ and choose $\vphi_0\in\D(\R)$, $\vphi_0$ symmetric and positive, $\vphi_0(t) = 1$ if $|t| < a$, $\vphi_0(t) = 0$ if $|t| > b$, and $\vphi_0$ strictly decreasing in the interval $(a,b)$. Putting $\vphi(\xi) = \vphi_0(|\xi|)$ for $\xi\in\R^m$ then defines a function $\vphi\in\D(\R^m)$. Finally we set $$\psi(\xi) = - \inp{\xi}{\grad\vphi(\xi)}$$ and note that if $a < |\xi| < b$ then $\psi(\xi) = - \vphi_0'(|\xi|) |\xi| > 0$. We denote by ${\cal M}(\R^m)$ the set of all pairs $(\vphi,\psi)\in\D(\R^m)^2$ that are constructed as above (we usually suppress the dependence of ${\cal
M}$ on $a$ and $b$ in the notation).
We are now in aposition to state the characterization theorem for the inhomogeneous Zygmund spaces as subspaces of $\S'(\R^m)$. It follows from [@Triebel:88], Sec. 2.3, Thm. 3 or, alternatively, from [@Hoermander:97], Sec. 8.6. Note that all appearing pseudodifferential operators in the following have $x$-independent symbols and are thus given simply by convolutions.
\[inh\_Z\] Assume that $a \leq 1/4$ and $b \geq 4$ and choose $(\vphi,\psi)\in {\cal M}(\R^m)$ arbitrary. Let $s\in\R$ then $u\in\S'(\R^m)$ belongs to the inhomogeneous Zygmund space of order $s$ ${\ensuremath{C_*}}^s(\R^m)$ if and only if $${\ensuremath{|u|_{{\ensuremath{C_*}}^{s}}}} := \linf{\vphi(D)u} +
\sup\limits_{0 < t < 1}\Big( t^{-s} \linf{\psi(tD)u}\Big) <
\infty .$$
(Note that we made use of the modification for $q=\infty$ in [@Triebel:88], equ. (82).)
\[Z\_rem\]
1. ${\ensuremath{|u|_{{\ensuremath{C_*}}^{s}}}}$ defines an equivalent norm on ${\ensuremath{C_*}}^s$. In fact that all norms defined as above by some $(\vphi,\psi)\in{\cal M}(\R^m)$ are equivalent can be seen as in [@Hoermander:97], Lemma 8.6.5.
2. If $s\in \R_+ \setminus \N$ then $C^s_*(\R^m)$ is the classical Hölder space of regularity $s$. Denoting by ${\ensuremath{\lfloor s \rfloor}}$ the greatest integer less than $s$ it consists of all ${\ensuremath{\lfloor s \rfloor}}$ times continuously differentiable functions $f$ such that $\d^\al f$ is bounded when $|\al|
\leq {\ensuremath{\lfloor s \rfloor}}$ and globally Hölder continuous with exponent $s-{\ensuremath{\lfloor s \rfloor}}$ if $|\al| = {\ensuremath{\lfloor s \rfloor}}$.
3. Due to the term $\linf{\vphi(D)u}$ the norm ${\ensuremath{|u|_{{\ensuremath{C_*}}^{s}}}}$ is not homogeneous with respect to a scale change in the argument of $u$.
4. If $u\in\L^\infty(\R^m)$ then (cf. [@Hoermander:97], Sect. 8.6) $$u(x) = \vphi(D)u(x) + \int\limits_1^\infty \psi(D/t)u(x) \frac{dt}{t}
\qquad \text{ for almost all } x .$$ Using $\vphi(\xi) = \int_0^1 \psi(\xi/t)/t \,dt$ this can be rewritten in the form $u(x) = \int_0^\infty \psi(D/t)u(x)/t \,dt$ and resembles Calderon’s classical identity in terms of a continuous wavelet transform (cf. [@Meyer:92], Ch. 1, (5.9) and (5.10)).
5. In a similar way one can characterize the homogeneous Zygmund spaces as subspaces of $\S'(\R^m)$ modulo the polynomials ${\cal P}$. A proof can be found in [@Triebel:82], Sec. 3.1, Thm. 1. We may identify $\S'/{\cal P}$ with the dual space $\S_0'(\R^m)$ of $\S_0(\R^m) = \{ f\in\S(\R^m) \mid \d^\al \FT{f}(0) = 0 \,
\forall \al\in\N_0^m \}$, the Schwartz functions with vanishing moments, by mapping the class $u+{\cal P}$ with representative $u\in\S'$ to $u\mid_{\S_0}$. Assume that $a \leq 1/4$ and $b \geq 4$ and choose $\psi\in\D(\R^m)$ as constructed above and let $s\in\R$ and $u\in\S'(\R^m)$. Then $u\!\!\mid_{\S_0}$ belongs to the homogeneous Zygmund space ${\ensuremath{\dot{C}_*}}^s(\R^m)$ of order $s$ if and only if $${\ensuremath{|u|_{{\ensuremath{\dot{C}_*}}^{s}}}} :=
\sup\limits_{0 < t < \infty}\Big( t^{-s} \linf{\psi(tD)u}\Big) <
\infty .$$ (Note that we use the modification for $q=\infty$ in [@Triebel:82], equ. (16).)
The continuous wavelet transform
--------------------------------
Following [@Holschneider:95] we call a function $g\in\L^1(\R^m)\cap\L^\infty(\R^m)$ with $\int g = 0$ a *wavelet*. We shall say that it is a *wavelet of order $k$* ($k\in\N_0$) if the moments up to order $k$ vanish, i.e., $\int x^\al g(x) dx = 0$ for $|\al|\leq k$.
The (continuous) wavelet transform is defined for $f\in\L^p(\R^m)$ ($1\leq p \leq \infty$) by ($\eps > 0$) $$\label{wf_trafo}
W_g f(x,\eps) = \int\limits_{\R^m} f(y) \frac{1}{\eps^m}
\bar{g}(\frac{y-x}{\eps}) \, dy = f * (\bar{\check{g}})_\eps (x)$$ where we have used the notation $\check{g}(y) = g(-y)$ and $g_\eps(y) =
g(y/\eps)/\eps^n$. By Young’s inequality $W_g f(.,\eps)$ is in $\L^p$ for all $\eps > 0$ and $W_g$ defines a continuous operator on this space for each $\eps$.
If $g\in C_c(\R^m)$ we can define $W_g f$ for $f\in\L^1_{\text{loc}}(\R^m)$ directly by the same formula (\[wf\_trafo\]). If $g\in\S_0(\R^m)$ then $W_g$ can be extended to $\S'(\R^m)$ as the adjoint of the wavelet synthesis (cf.[@Holschneider:95], Ch. 1, Sects. 24, 25, and 30) or directly by $\S'$-$\S$-convolution in formula (\[wf\_trafo\]).
If $f$ is a polynomial and $g\in\S_0$ it is easy to see that $W_g f = 0$. In fact, $f$, $g$, and $W_g f$ are in $\S'$ and $\FT{(W_g f(.,\eps))} =
\FT{f} \bar{\FT{g}}(\eps .)$. Since $g$ is in $\S_0$ the Fourier transform $\FT{g}(\eps .)$ is smooth and vanishes of infinite order at $0$. But $\FT{f}$ has to be a linear combination of derivatives of $\de_0$ implying $\FT{f} \bar{\FT{g}}(\eps .) = 0$. Therefore the wavelet transform ‘is blind to polynomial parts’ of the analyzed function (or distribution) $f$. In terms of geophysical modelling this means that a polynomially varying background medium is filtered out automatically.
Wavelets from mollifiers
------------------------
The Zygmund class characterization in Theorem \[inh\_Z\] (and remark \[Z\_rem\],(v)) used asymptotic estimates of scaled smoothings of the distribution which resembles typical mollifier constructions in Colombeau theory. In this subsection we relate this in turn directly to the wavelet transform obtaining the well-known wavelet characterization of Zygmund spaces.
Let $\chi\in\S(\R^m)$ with $\int \chi = 1$ and define the function $\mu$ by $$\label{mo_to_wv}
\ovl{\check{\mu}(x)} := m \chi(x) + \inp{x}{\grad\chi(x)} \, .$$ Then $\mu$ is in $\S(\R^m)$ and is a wavelet since a simple integration by parts shows that $$\begin{gathered}
(-1)^{|\al|} \ovl{\int \mu(x) x^\al \, dx} =
\int \bar{\check{\mu}}(x) (-x)^\al \, dx \\
= (-1)^{|\al|+1} \int x^\al \chi(x)\, dx \sum_{j=1}^m \al_j
= (-1)^{|\al|+1} |\al| \int x^\al \chi(x)\, dx \; .\end{gathered}$$ $\int \mu = 0$ and if $|\al| > 0$ we have $\int x^\al \mu(x)\,dx = 0$ if and only if $\int x^\al \chi(x) \,dx = 0$. Therefore $\mu$ defined by (\[mo\_to\_wv\]) is a wavelet of order $N$ if and only if the mollifier $\chi$ has vanishing moments of order $1 \leq |\al|
\leq N$.
Furthermore, by straightforward computation, we have $$\label{mo_to_wv_2}
(\bar{\check{\mu}})_\eps(x) =
-\eps \diff{\eps} \big(\chi_\eps (x)\big)$$ yielding an alternative of (\[mo\_to\_wv\]) in the form $\bar{\check{\mu}}(x) = - \diff{\eps}\big(\chi_\eps(x)\big)\mid_{\eps=1}$.
If $(\vphi,\psi)\in{\cal M}(\R^m)$ arbitrary and $\chi$, $\mu$ are the unique Schwartz functions such that $\FT{\chi} = \vphi$ and $\FT{\mu} = \psi$, then straightforward computation shows that $\mu$ and $\chi$ satisfy the relation (\[mo\_to\_wv\]). Therefore since $\mu$ is then a real valued and even wavelet we have for $u\in\S'$ $$\psi(tD)u(x) = t^m \bar{\check{\mu}}(\frac{.}{t})*u(x) = W_\mu u(x,t) \;
.$$ Hence the distributions $u$ in the Zygmund class ${\ensuremath{C_*}}^s(\R^m)$ can be characterized in terms of a wavelet transform and a smoothing pseudodifferential operator by $ \linf{\vphi(D)u} < \infty$ and $ \sup_{0 < t < 1} \Big( t^{-s} \linf{W_\mu u(.,t)}\Big) < \infty$. We have shown
Let $(\FT{\chi},\FT{\mu})\in{\cal M}(\R^m)$. A distribution $u\in\S'(\R^m)$ belongs to the Zygmund class ${\ensuremath{C_*}}^s(\R^m)$ if and only if $$\label{Z_W_char}
\linf{u * \chi} < \infty \quad \text{and} \quad
\linf{W_\mu u(.,r)} = O(r^s) \;\; (r \to 0) .$$
\[W\_rem\]
1. Observe that the condition on $\FT{\chi}$ implies that $\chi$ and hence $\mu$ can never have compact support. If this characterization is to be used in a theory of Zygmund regularity detection within Colombeau algebras one has to allow for mollifiers of this kind in the corresponding embedding procedures. This is the issue of the following subsection. Nevertheless we note here that according to remarks in [@Jaffard:97a (2.2) and (3.1)] and, more precisely, in [@Meyer:98 Ch.3] the restrictions on the wavelet itself in a characterization of type (\[Z\_W\_char\]) may be considerably relaxed — depending on the generality one wishes to allow for the analyzed distribution $u$. However, in case $m=1$ and $u$ a function a flexible and direct characterization (due to Holschneider and Tchamitchian) can be found in [@Daubechies:92], Sect. 2.9, or [@Holschneider:95], Sect. 4.2.
2. There are more refined results in the spirit of the above theorem describing local Hölder (Zygmund) regularity by growth properties of the wavelet transform (cf. in particular [@Holschneider:95], Sect. 4.2, [@JM:96], and [@Jaffard:97a]).
3. The counterpart of (\[Z\_W\_char\]) for $\L^1_{\text{loc}}$-functions in terms of (discrete) multiresolution approximations is [@Meyer:92], Sect. 6.4, Thm. 5.
Colombeau modelling and wavelet transform
=========================================
Embedding of temperate distributions
------------------------------------
We consider a variant of the Colombeau embedding $\iota_\chi^\ga : \D'(\R^m) \to \G(\R^m)$ that was discussed in [@HdH:01], subsect. 3.2. As indicated in remark \[W\_rem\],(i) we need to allow for mollifiers with noncompact support in order to gain the flexibility of using wavelet-type arguments for the extraction of regularity properties from asymptotic estimates. On the side of the embedded distributions this forces us to restrict to $\S'$, a space still large enough for the geophysically motivated coefficients in model PDEs.
Recall ([@HdH:01], Def. 11) that an admissible scaling is defined to be a continuous function $\ga : (0,1) \to \R_+$ such that $\ga(r) = O(1/r)$, $\ga(r) \to\infty$, and $\ga(sr) = O(\ga(r))$ if $0<s<1$ (fixed) as $r\to 0$.
Let $\ga$ be an admissible scaling, $\chi\in\S(\R^m)$ with $\int \chi = 1$, then we define $\iota_\chi^\ga : \S'(\R^m) \to \G(\R^m)$ by $$\iota_\chi^\ga(u) = \cl{(u * \chi^\ga(\phi,.))_{\phi\in\A_0(\R^m)}} \qquad
u\in\S'(\R^m)$$ where $$\chi^\ga(\phi,x) = \ga(l(\phi_0))^m \chi(\ga(l(\phi_0)) x)
\quad \text{ if } \phi = \phi_0\otimes \cdots \otimes\phi_0
\quad \text{ with } \phi_0\in\A_0 .$$
$\iota_\chi^\ga$ is well-defined since $(\phi,x) \to u*\chi^\ga(\phi,x)$ is clearly moderate and negligibility is preserved under this scaled convolution. By abuse of notation we will write $\iota_\chi^\ga(u)(\phi,x)$ for the standard representative of $\iota_\chi(u)$.
The following statements describe properties of such a modelling procedure resembling the original properties used by M. Oberguggenberger in [@O:89], Prop.1.5, to ensure unique solvability of symmetric hyperbolic systems of PDEs (cf. [@O:89; @LO:91]). The definition of Colombeau functions of logarithmic and bounded type is given in [@O:92], Def. 19.2, the variation used below is an obvious extension.
1. $\iota_\chi^\ga : \S'(\R^m) \to \G(\R^m)$ is linear, injective, and commutes with partial derivatives.
2. $\forall u\in\S'(\R^m)$: $\iota_\chi^\ga(u) \approx u$.
3. If $u\in W^{-1}_\infty(\R^m)$ then $\iota_\chi^\ga(u)$ is of *$\ga$-type*, i.e., there is $N\in\N_0$ such that for all $\phi\in\A_N(\R^m)$ there exist $C > 0$ and $1 > \eta > 0$: $$\sup\limits_{y\in\R^m} | \iota_\chi^\ga(u)(\phi_\eps,y)| \leq
N \ga(C\eps) \quad 0 < \eps < \eta .$$
4. If $u\in\L^\infty(\R^m)$ then $\iota_\chi^\ga(u)$ is of bounded type and its first order derivatives are of $\ga$-type.
*ad (i),(ii):* Is clear from $\chi_\eps :=
\chi^\ga(\phi_\eps,.) \to \de$ in $\S'$ as $\eps\to 0$ and the convolution formula.
*ad (iii):* Although this involves only marginal changes in the proof of [@O:89], Prop. 1.5(i), we recall it here to make the presentation more self-contained.
Let $u = u_0 + \sum_{j=1}^m \d_j u_j$ with $u_j\in\L^\infty$ ($j=0,\ldots,m$) then with $\ga_\eps := \ga(\eps l(\phi_0))$ $$\begin{gathered}
|u*\chi_\eps(x)| \leq \linf{u_0 * \chi_\eps} +
\sum_{j=1}^m \linf{u_j * \d_j(\chi_\eps)} \\
\leq \linf{u_0} \lone{\chi} +
\ga_\eps \sum_{j=1}^m \linf{u_j} \lone{\d_j\chi} \\
= \ga_\eps \big( \frac{\linf{u_0} \lone{\chi}}{\ga_\eps} +
\sum_{j=1}^m \linf{u_j} \lone{\d_j \chi} \big)\end{gathered}$$ where the expression within brackets on the r.h.s. is bounded by some constant $M$, dependent on $u$ and $\chi$ only but independent of $\phi$, as soon as $\eps < \eta$ with $\eta$ chosen appropriately (and dependent on $M$, $u$, $\chi$, and $\phi$). Therefore the assertion is proved by putting $N \geq M$ and $C = l(\phi_0)$.
*ad (iv):* is proved by similar reasoning
In particular, we can model a fairly large class of distributions as Colombeau functions of logarithmic growth (or log-type) thereby ensuring unique solvability of hyperbolic PDEs incorporating such as coefficients.
1. If $\ga(\eps) = \log(1/\eps)$ then $\iota_\chi^\ga(W^{-1,\infty}) \subseteq \{ U\in\G \mid U
\text{ is of log-type } \}$ and $$\iota_\chi^\ga(\L^\infty) \subseteq
\{ U\in\G \mid U \text{ of bounded type and }
\d^\alpha U \text{ of log-type for } |\al| = 1 \} .$$
2. If $u\in W^{-k,\infty}(\R^m)$ for $k\in\N_0$ then $\iota_\chi^\ga(u)$ is of $\ga^k$-type. In particular, there is an admissible scaling $\ga$ such that $\iota_\chi^\ga(u)$ and all first order derivatives $\d_j \iota_\chi^\ga(u)$ ($j = 1,\ldots,m$) are of log-type.
Wave front sets under the embedding
-----------------------------------
One of the most important properties of the embedding procedure introduced in [@HdH:01] was its faithfulness with respect to the microlocal properties if ‘appropriately measured’ in terms of the set of $\ga$-regular Colombeau functions $\G_\ga^\infty(\R^m)$ ([@HdH:01], Def. 11). But there the proof of this microlocal invariance property heavily used the compact support property of the standard mollifier $\chi$ which is no longer true in the current situation. In this subsection we show how to extend the invariance result to the new embedding procedure defined above.
Let $w\in\S'(\R^m)$, $\ga$ an admissible scaling, and $\chi\in\S(\R^m)$ with $\int \chi = 1$ then $$WF_g^\ga(\iota_\chi^\ga(w)) = WF(w) .$$
The necessary changes in the proof of [@HdH:01], Thm. 15, are minimal once we established the following
If $\vphi\in\D(\R^m)$ and $v\in\S'(\R^m)$ with $\supp(\vphi)
\cap \supp(v) = \emptyset$ then $\vphi\cdot\iota_\chi^\ga(v)\in\G_\ga^\infty$.
Using the short-hand notation $\chi_\eps = \chi^\ga(\phi_\eps,.)$ and $\ga_\eps = \ga(\eps l(\phi_0))$ we have $$\d^\be\big(\vphi (v*\chi_\eps)\big)(x) =
\ga_\eps^m \sum_{\al\leq\be} \binom{\be}{\al} \d^{\be-\al}\vphi(x) \,
\ga_\eps^{|\al|}\, \dis{v}{\d^{\al}\chi(\ga_\eps(x-.))} \; .$$ Hence we need to estimate terms of the form $\ga_\eps^{|\al|}\, \dis{v}{\d^{\al}\chi(\ga_\eps(x-.))}$ when $x\in\supp(\vphi) =: K$. Let $S$ be a closed set satisfying $\supp(v)
\subset S \subset \R^m \setminus K$ and put $d = \text{dist}(S,K)>0$. Since $v$ is a temperate distribution there is $N\in\N$ and $C > 0$ such that $$\ga_\eps^{|\al|} |\dis{v}{\d^{\al}\chi(\ga_\eps(x-.))}| \leq
C \ga_\eps^{|\al|} \sum_{|\sig|\leq N} \sup\limits_{y\in S}
|\d^\sig\big( \d^\al \chi(\ga_\eps(x-y)) \big)| \, .$$ $\chi\in\S$ implies that each term in the sum on the right-hand side can be estimated for arbitrary $k\in\N$ by $$\sup\limits_{y\in S} |\d^{\sig+\al}\chi(\ga_\eps(x-y)) \big)|
\ga_\eps^{|\sig|} \leq
\ga_\eps^{|\sig|} \sup\limits_{y\in S} C_k (1+\ga_\eps|x-y|)^{-k}
\leq C'_k \ga_\eps^{|\sig|-k}/ d^k$$ if $x$ varies in $K$. Since $|\al|+|\sig| \leq |\be|+ N$ we obtain $$\linf{\d^\be\big(\vphi (v*\chi_\eps)\big)} \leq
C' \ga_\eps^{m+N +|\be|-k}$$ with a constant $C'$ depending on $k$, $v$, $\vphi$, $d$, and $\chi$ but $k$ still arbitrary. Choosing $k = |\be|$, for example, we conclude that $\vphi\cdot\iota_\chi^\ga(v)$ has a uniform $\ga_\eps$-growth over all orders of derivatives. Hence it is a $\ga_\eps$-regular Colombeau function.
Referring to the proof (and the notation) of [@HdH:01], Thm. 15, we may now finish the proof of the theorem simply by carrying out the following slight changes in the two steps of that proof.
*Ad step 1:* Choose $\psi\in\D$ such that $\psi = 1$ in a neighborhood of $\supp(\vphi)$ and write $$\vphi (w*\chi_\eps) = \vphi ((\psi w)*\chi_\eps) + \vphi
(((1-\psi)w)*\chi_\eps \; .$$ The first term on the right can be estimated by the same methods as in [@HdH:01] and the second term is $\ga$-regular by the lemma above.
*Ad step 2:* Rewrite $$\vphi w = \vphi \psi w = \vphi (\psi w - (\psi w)*\chi_\eps) +
\vphi((\psi w)*\chi_\eps)$$ and observe that the reasoning of [@HdH:01] is applicable since $\Sigma_g^\ga(\vphi\, \iota_\chi^\ga(\psi w)) \subseteq
\Sigma_g^\ga(\vphi\, \iota_\chi^\ga(w))$ by the above lemma.
The modelling procedure and wavelet transforms
----------------------------------------------
Simple wavelet-mollifier correspondences as in subsection 2.3 allow us to rewrite the Colombeau modelling procedure and hence prepare for the detection of original Zygmund regularity in terms of growth properties in the scaling parameters.
A first version describes directly $\iota_\chi^\ga$ but involves an additional nonhomogeneous term.
\[inhom\_lemma\] If $\chi\in\S(\R^m)$ has the properties $\int\chi = 1$ and $\int x^\al \chi(x) dx = 0$ ($0 < |\al| \leq N$) then $\bar{\check{\mu}} =
-\diff{\eps}(\chi_\eps)\mid_{\eps=1}$ defines a wavelet of order $N$ and we have for any $f\in\S'(\R^m)$ $$\iota_\chi^\ga(f)(\phi,x) = f * \chi (x) + \!\!\!\!
\int\limits_{1/\ga(l(\phi))}^{1}\!\!\!\! W_\mu f(x,r)\, \frac{dr}{r} \;.$$
Let $\eps > 0$ then eq. (\[mo\_to\_wv\_2\]) implies $W_\mu f(x,\eps) =
f * (\bar{\check{\mu}})_\eps (x) =
-\eps \diff{\eps}\big( f * \chi_\eps(x) \big)$ and integration with respect to $\eps$ from $1/\ga(l(\phi))$ to $1$ yields $$- \!\!\!\! \int\limits_{1/\ga(l(\phi))}^{1}
\!\!\!\! W_\mu f(x,\eps) \frac{d\eps}{\eps} = f * \chi (x) -
\iota_\chi^\ga(f)(\phi,x) \; .$$
A more direct mollifier wavelet correspondence is possible via derivatives of $\iota_\chi^\ga$ instead.
\[hom\_lemma\] If $\chi\in\S(\R^m)$ with $\int \chi =1$ then for any $\al\in\N_0^n$ with $|\al| > 0$ $$\label{chi_al}
\chi_\al(x) = \ovl{(\d^\al\chi)\check{\ }(x)}$$ is a wavelet of order $|\al|-1$ and for any $f\in\S'(\R^m)$ we have $$\d^\al \iota_\chi^\ga(f)(\phi,x) =
\ga(l(\phi))^{|\al|}\, W_{\chi_\al} f (x,\frac{1}{\ga(l(\phi))}) \; .$$
Let $|\be| < |\al|$ then $\int x^\be D^\al\chi(x) \, dx =
(-D)^\be(\xi^\al \FT{\chi}(\xi))\mid_{\xi=0} = 0$ which proves the first assertion. The second assertion follows from $$\d^\al \iota_\chi^\ga(f)(\phi) = \ga^{|\al|+m} f *
\d^\al\chi(\ga .) =
\ga^{|\al|} f *
\ovl{\big(\ga^m (\d^\al\bar{\chi})\check{\ }(\ga .)\big)\check{\ }}$$ with the short-hand notation $\ga = \ga(l(\phi))$.
Both lemmas \[inhom\_lemma\] and \[hom\_lemma\] may be used to translate (global) Zygmund regularity of the modeled (embedded) distribution $f$ via Thm. \[Z\_W\_char\] into asymptotic growth properties with respect to the regularization parameter. To what extent this can be utilized to develop a faithful and completely intrinsic Zygmund regularity theory of Colombeau functions may be subject of future research.
Zygmund regularity of Colombeau functions: the one-dimensional case
===================================================================
If we combine the basic ideas of the Zygmund class characterization in 2.3 with the simple observations in 3.3 we are naturally lead to define a corresponding regularity notion intrinsically in Colombeau algebras as follows.
\[Z\_C\_def\] Let $\ga$ be an admissible scaling function and $s$ be a real number. A Colombeau function $U\in\G(\R^m)$ is said to be *globally of $\ga$-Zygmund regularity $s$* if for all $\alpha\in\N_0^m$ there is $M\in\N_0$ such that for all $\phi\in\A_M(\R^m)$ we can find positive constants $C$ and $\eta$ such that $$\label{ZC_def}
|\d^\al U(\phi_\eps,x)| \leq
\begin{cases}
C & \text{if $|\al| < s$}\\
C \;\ga_\eps^{|\al|-s} & \text{if $|\al| \geq s$}
\end{cases}
\qquad
x\in\R^m, 0<\eps<\eta .$$ The set of all (globally) $\ga$-Zygmund regular Colombeau functions of order $s$ will be denoted by $\G_{*,\ga}^s(\R^m)$.
A detailed analysis of $\G_{*,\ga}^s$ in arbitrary space dimensions and not necessarily positive regularity $s$ will appear elsewhere. Here, as an illustration, we briefly study the case $m=1$ and $s>0$ in some detail. Concerning applications to PDEs this would mean that we are allowing for media of typical fractal nature varying continuously in one space dimension. For example one may think of a coefficient function $f$ in ${\ensuremath{C_*}}^s(\R)$ to appear in the following ways.
1. Let $f$ be constant outside some interval $(-K,K)$ and equal to a typical trajectory of Brownian motion in $[-K,K]$; it is well-known that with probability $1$ those trajectories are in ${\ensuremath{\dot{C}_*}}^s(\R)$ whenever $s < 1/2$. This is proved, e.g., in [@Holschneider:95], Sect. 4.4, elegantly by wavelet transform methods.
2. We refer to [@Zygmund:68], Sect. V.3, for notions and notation in this example. Then similarly to the above one can set $f=0$ in $(-\infty,0]$, $f=1$ in $[2\pi,\infty)$ and in $[0,2\pi]$ let $f$ be Lebesgue’s singular function associated with a Cantor-type set of order $d\in\N$ with (constant) dissection ratio $0 < \xi < 1/2$. Then $f$ belongs to ${\ensuremath{C_*}}^s(\R)$ with $s = \log(d+1)/|\log(\xi)|$. (The classical triadic Cantor set corresponds to the case $d=2$ and $\xi = 1/3$.)
We have already seen that the Colombeau embedding does not change the microlocal structure (i.e., the $\ga$-wave front set) of the original distribution. We will show now that also the refined Zygmund regularity information is accurately preserved. If $n\in\N_0$ we denote by $C^n_b(\R)$ the set of all $n$ times continuously differentiable functions with the derivatives up to order $n$ bounded. Note that $C^n_b(\R)$ is a strict superset of ${\ensuremath{C_*}}^{n+1}(\R)$.
Let $\chi\in\A_0(\R)$ and $s>0$. Define $n\in\N_0$ such that $n < s \leq n+1$ then we have $$\iota_\chi^\ga(C^n_b(\R)) \cap \G_{*,\ga}^s(\R) =
\iota_\chi^\ga({\ensuremath{C_*}}^s(\R)) .$$ In other words, in case $0< s <1$ we can precisely identify those Colombeau functions that arise from the Zygmund class of order $s$ within all embedded bounded continuous functions.
We use the characterizations in [@Daubechies:92], Thms. 2.9.1 and 2.9.2 and the remarks on p. 48 following those; choosing a smooth compactly supported wavelet $g$ of order $n$ we may therefore state the following[^1]: $f\in C^n_b(\R)$ belongs to ${\ensuremath{C_*}}^s(\R)$ if and only if there is $C>0$ such that $$\label{Daub}
|W_g f(x,r)| \leq C r^s \quad \text{ for all } x .$$
Now the proof is straightforward. First let $f\in{\ensuremath{C_*}}^s(\R)$. If $|\al| - 1 < n$ then $|\d^\al \iota_\chi^\ga(f)| =
|\iota_\chi^\ga(\d^\al f)| \leq \linf{\d^\al f} \lone{\chi}$ by Young’s inequality. If $|\al| > n$ we use lemma \[hom\_lemma\] and set $\ga_\eps =
\ga(\eps l(\phi))$ to obtain $$\d^\al \iota_\chi^\ga(f)(\phi_\eps,x) = \ga_\eps^{|\al|} W_{\chi_\al}
f(x,\ga_\eps^{-1})$$ where $\chi_\al$ is a wavelet of order at least $n$. Hence (\[Daub\]) gives an upper bound $C \ga_\eps^{|\al| - s}$ uniformly in $x$. Hence (\[ZC\_def\]) follows.
Finally, if we know that $f\in C^n_b(\R)$ and $\iota_\chi^\ga(f) \in \G_{*,\ga}^s(\R)$ then combination of (\[ZC\_def\]) and lemma \[hom\_lemma\] gives if $|\al| \geq s$ $$|\ga_\eps^{|\al|} W_{\chi_\al} f(x,\ga_\eps^{-1})| =
|\d^\al \iota_\chi^\ga(f)(\phi_\eps,x)| \leq C \ga_\eps^{|\al| - s}$$ uniformly in $x$. Hence another application of (\[Daub\]) proves the assertion.
[10]{}
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[^1]: Note that we do not use the wavelet scaling convention adapted to $\L^2$-spaces here
| ArXiv |
---
abstract: 'Very high energy gamma-rays ($E_\gamma >$ 20 GeV) from blazars traversing cosmological distances through the metagalactic radiation field can convert to electron-positron pairs in photon-photon collisions. The converted gamma rays initiate electromagnetic cascades driven by inverse-Compton scattering off the microwave background photons. The cascades shift the injected gamma ray spectrum to MeV-GeV energies. Randomly oriented magnetic fields rapidly isotropize the secondary electron-positron beams resulting from the beamed blazar gamma ray emission, leading to faint gamma-ray halos. Using a model for the time-dependent metagalactic radiation field consistent with all currently available far-infrared-to-optical data, we compute (i) the expected gamma-ray attenuation in blazar spectra, and (ii) the cascade contribution from faint, unresolved blazar to the extragalactic gamma-ray background as measured by EGRET, assuming a generic emitted spectrum extending to an energy of 10 TeV. The latter cascade contribution to the EGRET background is fed by the assumed >20 GeV emission from the hitherto undiscovered sources, and we estimate their dN-dz distribution taking into account that the nearby (z<0.2) fraction of these sources must be consistent with the known (low) numbers of sources above 300 GeV.'
author:
- 'Tanja M. Kneiske'
- Karl Mannheim
title: 'BL Lac Contribution to the Extragalactic Gamma-Ray Background'
---
[ address=[Universitaet Wuerzburg, Am Hubland, 97074 Wuerzburg, Germany]{} ]{}
[ address=[Universitaet Wuerzburg, Am Hubland, 97074 Wuerzburg, Germany]{} ]{}
Introduction
============
An isotropic, diffuse background radiation presumably due to faint, unresolved extragalactic sources has been observed in nearly all energy bands. The confirmation of an extragalactic gamma-ray background by EGRET (Energetic Gamma-Ray Experiment Telescope) on board the Compton Gamma Ray Observatory has extended the spectrum up to an energy of $\sim$50 GeV. A first analysis of the data resulted in a total flux of $(1.45\pm 0.05) \cdot 10^{-5}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$ above 100 MeV and a spectrum which could be fitted by a power law with an spectra index of $-2.1\pm 0.03$ [@lit:sreekumar]. These values are strongly dependent on the foreground emission model which is subtracted from the observed intensity to obtain the extragalactic residual [@lit:hunter]. Since using the old foreground model, a residual GeV halo remained after subtraction (in addition to the isotropic extragalactic background), the foreground model had to be improved. This lead to a new analysis of the EGRET data, and a new result for the extragalactic background spectrum, now showing a dip a GeV energies and an overall weaker intensity of $(1.14\pm 0.12) \cdot 10^{-5}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$ [@lit:strong]. This new result can help us to understand the origin of the extragalactic background radiation.
Since EGRET detected a large number of extragalactic gamma-ray sources belonging to the blazar class of AGN, a reasonable assumption is that the gamma background is produced by unresolved AGN. Using a gamma-ray luminosity function from EGRET data [@lit:chiang]Chiang & Mukherjee (1998) came to the result that only 25% to 50% of the gamma background could be explained by blazars. [@lit_stecker96] were able to explain 100% of the background, but were facing a problem with the deficit of observed faint, nearby blazars.
The new idea which will be presented in this paper is to extend the existing models by assuming a population of BL Lacs with a spectral energy distribution such that their flux at EGRET energies is too low to be generally detected, while their very high energy gamma ray flux is strong. Since most of these sources are at redshifts high enough for pair attenuation to take place, a significant part of their VHE emission is reprocessed by cascades contributing to the diffuse background, but not to the single source counts.
During the paper we use a Hubble constant of $H_0=71$km s$^{-1}$ Mpc$^{-1}$ and a flat universe with the cosmological parameters $\Omega$=0.3 and $\Omega_\Lambda$=0.7.
![Pair attenuation optical depth for various redshifts and MRF models. The labeling of the line styles are explained in [@lit:kneiske2]. The crossing point with the line $\tau=1$ defines the exponential cutoff energy.[]{data-label="fig:Tau"}](Tau_bis06_201004){height=".3\textheight"}
Gamma-Ray Background
====================
If the gamma-ray background is produced by unresolved sources, it can be described by
$$F_{E_\gamma} = \frac{1}{\Omega} \int^{z_m}_0 dz \frac{dV}{dz}
\int^{\infty}_{L_{\rm min}} \frac{dN}{dV dL}
F_{E_\gamma}(z, L) dL, \ \ \ \
\label{eq:gammaback}$$
with $\Omega$ the solid angle coverage of the survey ($\Omega_{\rm EGRET}=10.4$), $\frac{dV}{dz}$ the volume element, $L_{min}$ is the luminosity of the weakest source, $\frac{dN}{dV dL}$ the luminosity function and $F_{E_\gamma}(z_q, L)$ the flux of the gamma sources, depending on their luminosity and redshift. The luminosity function of resolved EGRET sources, extended to the faint end has been computed by [@lit:chiang]. We used their model changing only the spectral index from $\alpha=2.1$ to $\alpha=2.3$. The new spectral index was determined by fitting the reanalyzed EGRET data at $ <2.0$ GeV.
The remaining excess of the measured gamma-ray background we ascribe to high-energy peaked blazars belonging to the HBL and ExBL classes (defined by [@lit:Ghisellini]. We calculate the flux from this sources using equation \[eq:gammaback\]. The spectral energy distribution of these sources between 100 MeV and 10 TeV and their luminosity function (LF) is poorly known, and we have to make some theoretical assumptions for them which are described the next sections.
![Spectrum of the extragalactic gamma-ray background (open circles: Sreekumar (1998), filled diamonds: Strong et al. (2004)). The contribution of the HBL component (thick solid line) is compared with the spectrum without any absorption and reemission (thin solid line) and the contribution of the secondary photons only (dashed line)[]{data-label="fig:gammabackHBL"}](Gammaback_HBL){height=".3\textheight"}
Template Spectra
================
A number of extragalactic gamma-ray sources have been detected with imaging air-Cherenkov telescopes (Table 6, [@lit:horan]). Four of them (with redshifts $z=0.03, 0.03, 0.129, 0.048$) were bright enough to resolve their spectra in the TeV energy band. The observed spectra are presumably modified by gamma ray attenuation, i.e. $$F_{\rm obs}(E)=F_{\rm int}(E)\exp[-\tau_{\gamma\gamma}(E,z)]$$ where $\tau_{\gamma\gamma}(E,z)$ is the optical depth for gamma-rays (Fig. \[fig:Tau\]). We used various model parameters for the metagalactic radiation field (MRF) to bracket the range of the un-absorbed (intrinsic) spectra.
Depending on the model of the MRF the intrinsic spectra show turnovers or broad maxima around a few TeV. The intrinsic spectrum of H1426+428 which has a larger redshift (z=0.129) could have a maximum at 10 TeV or higher. We use a mean of the spectra of Mkn501, Mkn421 and ES1959+650 as a template for the HBL-type sources and a spectrum like H1426+428 as the template spectrum for the ExBL-types. Each template spectrum is modeled using two power laws. The parameters are two spectral indices and the location of the maximum. For HBL the spectral index at low energies is $\alpha=1.7$ and at high energies $\alpha=2.3$ with a maximum at 4 TeV. The spectral index of the ExBLs is $\alpha=1.2$ with a maximum at 10 TeV.
![Spectrum of the extragalactic gamma-ray background. The contribution of the ExBL component (thick solid line) is compared with the spectrum without any absorption and reemission (thin solid line) and the contribution of the secondary photons only (dashed line)[]{data-label="fig:gammabackExBL"}](Gammaback_ExBL){height=".3\textheight"}
The absorption of the primary photons is calculated using the MRF model presented in [@lit:kneiske1], and the reemission is calculated using the radiative transfer equation employing an inverse-Compton emission term due to scattering off the microwave background. The flux from a population of gamma-ray sources contributing to the gamma-ray background is the sum of the primary and the secondary flux $F_{E_\gamma}(z_q, L)=F^{\rm p}_{E_\gamma}(z_q, L)+F^{\rm s}_{E_\gamma}(z_q, L)$.
TeV-Luminosity Function
=======================
Although the secondary cascade contribution is likely to be isotropic, and therefore much fainter than the beamed primary gamma rays, we ignore this effect which is compensated by the correspondingly larger number of hosts which contribute to the extragalactic background light. By doing so, we must use the LF of the parent population of the secondary emission, e.g. the LF of XBLs and ExBLs. HBL and ExBL are x-ray selected BL Lacs showing indications of correlated x-ray/gamma ray emission. We will use x-ray observations to develop a TeV-luminosity function. Using the ROSAT-All-Sky-Survey, [@lit:Bade] and [@lit:laurent] could develop a luminosity function with a maximum in the number of sources at z=0.2. [@lit:rector] and [@lit:caccianiga] found a maximum of sources around z=0.3 using a sample of the Einstein-Medium-Sensitivity Survey and the Radio-Emitting-X-Ray-Sources catalog, respectively. We will instead use the results of [@lit:Beckmann] who combined all the available data. To obtain a relation between the gamma-ray flux and x-ray flux we used the calculations from [@lit:costaghiss]. They presented for 33 BL Lacs multi-wavelength spectra using a theoretical SSC-model and various observations. From their results we fitted a relation between the luminosity at 1 keV $L_{(1~\rm keV)}$ and the gamma-ray luminosity above 0.3 TeV $L_{(\rm TeV)}$
$$\label{eq:LTeVLx}
L_{\rm TeV}=2.6535 \cdot 10^{-4} L_{(1~\rm keV)}^{0.15781} \ \ \
[10^{48}\mathrm{erg \ s}^{-1} ].$$
The TeV-luminosity function can be written as broken power law $\frac{dN}{dV}(dL_{0, \rm TeV}) \propto \left(L_{0, \rm TeV}\right)^{\alpha_{\rm LF}}$ with $\alpha_{\rm LF}=-0.9$ for $L_{0, \rm TeV} \le L_{\rm B}$ and $\alpha_{\rm LF}=-1.4$ for $L_{0, \rm TeV} > L_B$ with $L_B$ break luminosity. The evolution can be described by $$\rho(z) \propto (1+z)^{\alpha_\rho} .
\label{eq:dNdz_bllac}$$ with a steep rise of $\alpha_{\rho}=10$ for $z \le 0.15$ and $\alpha_{\rho}=-3$ for $z > 0.15$ (values fitted from the distribution shown in Beckmann et al. 2003).
The maximum and the minimum of the luminosity function has been calculated from the maximum and minimum of the x-ray LF. The absolute value of the luminosity function has been chosen to match the EGRET gamma-ray background data. The bright ($L_{(\rm TeV)}>2.7\cdot 10^{-4}$ $10^{-48}$erg s$^{-1}$) sources are defined as ExBLs while the faint end of the LF is assumed to represent the HBLs. The number density is comparable with the observed EGRET blazar number density.
Discussion
==========
![Spectrum of the extragalactic gamma-ray background. The total spectrum is produced by three components: EGRET sources with a spectral index of 1.3 (thin solid line), HBL (dotted-dashed line) and ExBL (dashed line). For all three component the effect of extragalactic absorption and reemission via inverse Compton scattering is taken into account.[]{data-label="fig:gammaback"}](Back_Blazar_LF_191004){height=".3\textheight"}
The results for the various contributions to the extragalactic gamma-ray background are shown in Figure \[fig:gammaback\]. The thin solid line denotes the EGRET blazar contribution. Due to the new spectral index and the reanalyzed EGRET data the unresolved EGRET blazars now produce about 75-80% of the background flux. The dashed line represents the background flux made due to the HBL population, while the dot-dashed line describes the flux corresponding to the ExBL contribution. Comparing the total flux as a sum of the three contributions and the EGRET data, the agreement is acceptable. As can be seen in Fig. \[fig:gammabackHBL\] and Fig. \[fig:gammabackExBL\] the primary flux of the BL Lac population produces only a small contribution in the EGRET energy range. The secondary photons can contribute about 20% to the gamma-ray background. Although the number of ExBL is less then 10 % of the total number of BL Lac objects the flux is of the same order of magnitude as the HBL flux. The adopted values of the spectral index and the maximum at 10 TeV have a large influence on the secondary photon contribution.
The number density of BL Lacs in our model is comparable with that of the EGRET blazars. Nevertheless, only 7 HBLs have been observed with Imaging Air Cherenkov Telescopes above 300 GeV (in the published records). The reason for this comparatively small number is the effect of pair attenuation and the limited sensitivity of the Cherenkov telescopes. E.g., for AGN at a redshift of 0.2 photons with energies $>300$ GeV are undergoing the pair production process. With Cherenkov Telescopes of the Whipple tpye ($E_{\rm thr}\approx 300$, $F_{\rm lim} \approx 10^{-11}$) still most sources within this redshift range are too faint to be detected in a typical 10-50 h observation campaign. The observational constraint of small zenith angles further reduces the number of observable sources number by a factor of roughly $\sim 1/4$. Assuming the number density to fit the gamma-ray background data a telescope of the Whipple type would only be able to observe $\approx$ 18 BL Lacs from this population.
Our assumptions can be tested by the next generation of Cherenkov Telescopes. Fig. \[fig:dNdz\] shows the number of BL Lacs depending on the flux limit and the threshold energy of a telescope. A telescope with a threshold energy of 50 GeV and a flux limit of $F_{\mathrm lim} \approx 10^{-11}$ should in principle be able to observe about 25% of 1500 sources.
![Number of observable BL Lacs as a function of energy threshold and flux limit of Cherenkov Telescopes.[]{data-label="fig:dNdz"}](dNdz_AnEthr_Ges){height=".3\textheight"}
This research was gratefully supported by the BMB+f under grant 05AM9MGA.
[99]{} Sreekumar, P. et al. 1998, ApJ, 494 523 Hunter, S. D. et al. 1997, ApJ, 481, 205 Strong, A.W., Moskalenko I.V. & O. Reimer, O. 2004, acc. bei A&A, astro-ph/0405441 Chiang, J. und Mukherjee, R. 1998, ApJ, 496, 752 Stecker, F. W. und Salamon, M.H. 1996, ApJ, 464, 600 Ghisellini, G. et al. 1998, MNRAS, 301, 451 Horan, D. et al. 2002, ApJ, 571, 753 Kneiske, T.M., Mannheim, K. & Hartmann, D. 2002, ApJ, 386, 1 Bade, N. et al. 1998, A&A, 334, 459 Laurent-M"uhleisen, S.A. et al. 1999, ApJ, 525, 127 Rector, T.A. et al. 2000, AJ, 120, 1626 Caccianiga, A. 2002, ApJ, 566, 181 Beckmann, V., Engels, D., Bade, N. und Wucknitz, O. 2003, A&A, 401, 927 Costamante, L. und Ghisellini, G. 2002, A&A, 384, 56
Kneiske, T.M., Bretz, T., Mannheim, K. & Hartmann, D. 2004, ApJ, 413, 807
| ArXiv |
---
author:
- 'Quentin De Mourgues\'
title: |
\
\
of the KZB Classification Theorem
---
Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action have been classified by Kontsevich and Zorich, and by Boissy through methods involving both combinatorics algebraic geometry, topology and dynamical systems. Our precedent paper [@DS17] as well as the one of Fickenscher [@Fic16] proposed an ad hoc combinatorial proof of this classification.
However, unlike those two previous combinatorial proofs, we develop in this paper a general method, called the labelling method, which allows one to classify Rauzy-type dynamics in a much more systematic way. We apply the method to the Rauzy dynamics and obtain a third combinatorial proof of the classification. The method is versatile and will be used to classify three other Rauzy-type dynamics in follow-up articles.
Another feature of this paper is to introduce an algorithmic method to work with the sign invariant of the Rauzy dynamics. With this method we can prove most of the identities appearing in the literature so far ([@KZ03],[@Del13] [@Boi13] [@DS17]...) in an automatic way.
The sign invariant {#sec.signinv}
==================
Arf functions for permutations {#ssec.arf_inv}
------------------------------
For $\s$ a permutation in $\kS_n$, let () = \# { 1i<j n | (i)<(j) } i.e. $\chi(\s)$ is the number of pairs of non-crossing edges in the diagram representation of $\s$.
Let $E=E(\s)$ be the subset of $n$ edges in $\cK_{n,n}$ described by $\s$. For any $I \subseteq E$ of cardinality $k$, the permutation $\s|_{I} \in \kS_k$ is defined in the obvious way, as the one associated to the subgraph of $\cK_{n,n}$ with edge-set $I$, with singletons dropped out, and the inherited total ordering of the two vertex-sets.
Define the two functions $$\begin{aligned}
A(\s)
&:=
\sum_{I \subseteq E(\s)} (-1)^{\chi(\s|_I)}
\ef;
&
\Abar(\s)
&:=
\sum_{I \subseteq E(\s)} (-1)^{|I|+\chi(\s|_I)}
\ef.\end{aligned}$$ When $\s$ is understood, we will just write $\chi_I$ for $\chi(\s|_I)$. The quantity $A$ is accessory in the forthcoming analysis, while the crucial fact for our purpose is that the quantity $\Abar$ is invariant in the $\perms_n$ dynamics.
In the following section, we define a technique to demonstrate identities of the arf invariant involving differents configurations.
Automatic proofs of Arf identitites {#ssec.arfcalcseasy}
-----------------------------------
We will *not* try to evaluate Arf functions of large configurations starting from scratch. We will rather compare the Arf functions of two (or more) configurations, which differ by a finite number of edges, and establish linear relations among their Arf functions. The method we develop here, gives an algorithm to find and check Arf identities.
In order to have the appropriate terminology for expressing this strategy, let us define the following: Given a permutation $\s$ define $\s_{k,\ell}$ to be a permutation with $k$ marks on its bottom line and $\ell$ marks on its top line. The marks are all at distinct positions and do not touch the corners of the permutation. These marks break the bottom (respectively top) line into $k+1$ open interval $P_{-,1},\ldots,P_{-,k+1}$ (respectively $\ell+1$ open interval $P_{+,1},\ldots,P_{+,\ell+1}$).
For example if $k=1,\ell=3$: $$\put(50,-30){$P_{-,1}$}\put(100,-30){$P_{-,2}$}\put(26,25){$P_{+,1}$}\put(60,25){$P_{+,2}$}\put(90,25){$P_{+,3}$}\put(125,25){$P_{+,4}$}
\s_{k,\ell}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex1.pdf}}$$
Let $\s_{k,\ell,E'}$ be the permutation obtained by adding a set of edges $E'$ on the marks of permutation $\s_{k,\ell}$ with the following convention: an edge $e\in E'$ is a pair $(i.x,j.y)$. The edge connects the $i$th bottom mark and the $j$th top mark, and it is ordered as the $x$th edge within the bottom mark and the $y$th edge within the top mark. Note that if $i=0$ of $i=k+1$ (likewise of $j$) this implies that the edge is connected to a corner of the permutation.
For example if $k=1,\ell=3$ and $E'=\{(0.1,2.2),(1.1,3.1),(1.2,1.1),(1.3,2.1),(2,1.2)\}$: $$\put(60,-30){$P_{-,1}$}\put(110,-30){$P_{-,2}$}\put(36,25){$P_{+,1}$}\put(70,25){$P_{+,2}$}\put(105,25){$P_{+,3}$}\put(135,25){$P_{+,4}$}
\s_{k,\ell,E'}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex2.pdf}}$$ We will define an algorithm that allows one to check if, for all $\s$, we have $\sum^n_{i=1} K_i\Abar(\s_{k,\ell,E^i})=0$ or $\sum^n_{i=1} K_iA(\s_{k,\ell,E^i})=0$ for some $k,\ell,(E^i)_i,(K_i)_i,n.$
\[def.637647\] Let $\s_{k,\ell,E'}$, $P_{-,1},\ldots,P_{-,k+1}$ and $P_{+,1},\ldots,P_{+,\ell+1}$ be as defined above. Then define the $m \times (k+1)(\ell+1)$ matrix valued in $\gf_2$ Q\_[e,ij]{} := {
[ll]{} 1 &\
0 &
. For $v \in \gf_2^{(k+1)(\ell+1)}$, let $|v|$ be the number of entries equal to $1$. Similarly, identify $v$ with the corresponding subset of $[(k+1)(\ell+1)]$. Given such a construction, introduce the following functions on $(\gf_2)^{(k+1)(\ell+1)}$ $$\begin{aligned}
A_{k,\ell,E'}(v)
&:=
\sum_{u \in (\gf_2)^{E'}}
(-1)^{\chi_u + (u,Qv)}
\ef;
&
\Abar_{k,\ell,E'}(v)
&:=
\sum_{u \in (\gf_2)^{E'}}
(-1)^{|u|+\chi_u + (u,Qv)}
\ef.\end{aligned}$$
The construction is illustrated in Figure \[fig.arf\_ex\_def\].
![\[fig.arf\_ex\_def\]The permutation $\s_{1,4,\{(0.1,0.1),(0.2,3.1),(0.3,1.1),(1.1,4.1),(1.2,2.1)\}}$. We cannot show the full matrix $Q$ for such a big example, but we can give one row, for the edge which has the label $e$ in the drawing. The row $Q_e$ reads $(Q_e)_{11, 12, \ldots, 15, 21, \ldots, 25}=(1,1,0,0,0,\,0,0,1,1,1)$.](FigFol/Figure4_fig_arf_ex_def.pdf)
Let us comment on the reasons for introducing such a definition. The quantities $A_{k,\ell,E'}(v)$ (respectively $\Abar_{k,\ell,E'}(v)$) do not depend on $\s$ and allows to sum together many contributions to the function $A(\s_{k,\ell,E'})$. Our goal is to have $E'$ of fixed size, while $E$ (the edge set of $\s$) is arbitrary and of unbounded size, so that the verification of our properties, as it is confined to the matrix $Q$, involves a finite data structure. Thus the algorithm will be exponential in $|E'|$ which will not be a problem for small sizes.
Indeed, let us split in the natural way the sum over subsets $I$ of $E\bigcup E'$ the edge set of $\s_{k,\ell,E'}$ namely $$\sum_{I_0 \subseteq E\bigcup E'} f(I_0)
=
\sum_{I \subseteq E}
\sum_{I' \subseteq E'}
f(I \cup I')$$ For $I$ and $J$ two disjoint sets of edges, call $\chi_{I,J}$ the number of pairs $(i,j) \in I \times J$ which do not cross. Then clearly $$\chi_{I \cup J} = \chi_{I} + \chi_{J} + \chi_{I,J}$$ Now let $u(I') \in \{0,1\}^{E'}$ be the vector with entries $u_e=1$ if $e \in I'$ and $0$ otherwise. Let $m(I)=\{m_{ij}(I)\}$ be the $(k+1) \times (\ell+1)$ matrix describing the number of edges connecting the intervals $P_{-,i}$ to $P_{+,j}$ in $\s$, and let $v(I)=\{v_{ij}(I)\}$, $v_{ij} \in \{0,1\}$ be the parities of the $m_{ij}$’s. Call $I_{ij}$ the restriction of $I$ to edges connecting $P_{-,i}$ and $P_{+,j}$. Clearly, $$\chi_{I',I}=\sum_{ij} \chi_{I',I_{ij}}
= \sum_{e,ij} u_e Q_{e,ij} m_{ij}=(u(I'),Qm(I)),$$ which has the same parity as the analogous expression with $v$’s instead of $m$’s. I.e. we have $$(-1)^{\chi_{I',I}}=(-1)^{(u(I'),Qv(I))}.$$ Now, while the $m$’s are in $\bN$, the vector $v$ is in a linear space of finite cardinality, which is crucial for allowing a finite analysis of our expressions.
As a consequence, $$\begin{aligned}
\label{eq.arf_explain_def}
A(\s_{k,\ell,E^i})
&=
\sum_{I \subseteq E}(-1)^{\chi_I}A_{k,\ell,E'}(v(I))
\ef;
\\\label{eq.arf_explain_def_1}
\Abar(\s_{k,\ell,E^i})
&=\sum_{I \subseteq E}(-1)^{|I|+\chi_I}\Abar_{k,\ell,E'}(v(I))
\ef.\end{aligned}$$
Thus we have the following proposition:
\[prop.ArfProp\]\[prop.Arf0\] Let $k,\ell \in \N$ and let $(E^i)_{1\leq i\leq n}$ be a family of edge set. Then the two following statements are equivalent:
1. For all $v \in GF_2^{(k+1)(\ell+1)},$ we have $\sum_{i=1}^n K_i A_{k,\ell,E^i}(v)=0$.
2. for all $\s,$ we have $\sum_{i=1}^n K_i A(\s_{k,\ell,E^i})=0.$
The same statement holds for $\Abar$.
Statement 1 implies 2 due to equation (\[eq.arf\_explain\_def\]).\
Let us show the converse: If $\s$ has no edge then $\sum_{i=1}^n K_i A(\s_{k,\ell,E^i})=0$ is equivalent to $\sum_{i=1}^n K_i A_{k,\ell,E^i}(v)=0$ with $v$ being the zero vector of $(GF_2)^{(k+1)(\ell+1)}$.
Then we choose the family of permutations $(\s^{a,b})_{a,b}$ with exactly one edge connecting $P_{-,a}$ to $P_{+,b}$. Then if we define $v_{a,b}$ to be the vector $(GF_2)^{(k+1)(\ell+1)}$ with exactly one 1 at position $ab$ we have
&\_[i=1]{}\^n K\_i A(\^[a,b]{}\_[k,,E\^i]{})=0\
& \_[i=1]{}\^n K\_i ( A\_[k,,E\^i]{}(0) + A\_[k,,E\^i]{}(v\_[a,b]{})) =0\
& \_[i=1]{}\^n K\_i A\_[k,,E\^i]{}(v\_[a,b]{}) =0\
in the last line we have used that $\sum_{i=1}^n K_i A_{k,\ell,E^i}(0)=0$.
Thus inductively we show that $\sum_{i=1}^n K_i A_{k,\ell,E^i}(v)=0$ for any $v\in (GF_2)^{(k+1)(\ell+1)}$ with at most $p\leq n$ ones.
This theorem is very important since it reduces the problem of calculating Arf identities for permutations of any size to a check of an Arf identity for $2^{(k+1)(\ell+1)}$ values. Thus in exponential time in $k\ell$ we can calculate $A_{k,\ell,E'}(v)$ for every $v \in GF_2^{(k+1)(\ell+1)}$ and decide if a given Arf identity is correct.
We can even do better:
Let $k,\ell \in \N$ and let $(E^i)_{1\leq i\leq n}$ be a family of edge set. We can decide (in exponential time in $k\ell$) if there exists $x_1,\ldots,x_n$ such that $\sum_{i=1}^n x_i A(\s_{k,\ell,E'})=0$.
For every $v$ we have an equation $\sum_{i=1}^n x_i A_{k,\ell,E^i}(v) =0$ with the $n$ unknown variables. So there are $2^{(k+1)(\ell+1)}$ equations. We can find the subspace of solution in time exponential in $k\ell$.
The previous proposition can be used when we suspect a relation between a few configurations without knowing the coefficients. The algorithm demands little more than the previous one for the verification so it remains usable for small $|E'|$.
Finally we can actually enumerate all the possible Arf identities:
There is an algorithm that enumerate all the arf identities with at most $n$ terms and on an edge set $E'$ of size at most $h$.
This algorithm is really not praticable. However it can be used in the following case: we have two terms and we want to find an arf identity relating them to one another but the previous algorithm failed (i.e there are no identity containing only those two terms). Then we use this algorithm to find a third term (or a fourth etc...) for which an identity exists.
We can even propose a generalisation of this framework: let us choose two permutations $\pi_{-}$ and $\pi_{+}$ of size $k+1$ and $\ell+1$ respectively then $\s_{k,\ell,E',\pi_{-},\pi_{+}}$ is obtained from $\s_{k,\ell,E'}$ by permuting the $P_{-,i}$ with $\pi_{-}$ and $P_{+,i}$ with $\pi_{+}$.
For example if $k=1,\ell=3$ and $E'=\{(0.1,2.2),(1.1,3.1),(1.2,1.1),(1.3,2.1),(2,1.2)\}$: $$\put(60,-30){$P_{-,1}$}\put(110,-30){$P_{-,2}$}\put(36,25){$P_{+,1}$}\put(70,25){$P_{+,2}$}\put(105,25){$P_{+,3}$}\put(135,25){$P_{+,4}$}
\s_{k,\ell,E'}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex2.pdf}}
\put(130,-30){$P_{-,\pi_{-}1}$}\put(180,-30){$P_{-,\pi_{-}2}$}\put(100,25){$P_{+,\pi_{+}1}$}\put(140,25){$P_{+,\pi_{+}2}$}\put(175,25){$P_{+,\pi_{+}3}$}\put(210,25){$P_{+,\pi_{+}4}$}
\qquad \qquad\s_{k,\ell,E',\pi_{-},\pi_{+}}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arf_ex2.pdf}}.$$ For an example with $\pi_{-}=id_2$ and $\pi_{+}=(2,1)$ (the reversing permutation $\omega$ at size 2) we have: $$\put(55,-30){$P_{-,1}$}\put(95,-30){$P_{-,2}$}\put(55,25){$P_{+,1}$}\put(85,25){$P_{+,2}$}
\s_{1,1,E'}=\raisebox{-20pt}{\includegraphics[scale=2.5]{P05_Arf/figure/fig_arfproof_s11-eps-converted-to.pdf}}
\put(125,-30){$P_{-,1}$}\put(165,-30){$P_{-,2}$}\put(120,50){$P_{+,2}$}\put(160,50){$P_{+,1}$}
\qquad \qquad\s_{1,1,E',\pi_{-},\pi_{+}}=\raisebox{-20pt}{\includegraphics[scale=2.5]{FigFol/fig_arfproof_general_model_ex.pdf}}$$ It is easily checked that the previous theorems continue to hold for this generalisation once we introduce for $v \in \gf_2^{(k+1)(\ell+1)}$ the following function on $(\gf_2)^{(k+1)(\ell+1)}$ (similar definition for $\Abar_{k,\ell,E',\pi_{-},\pi_{+}}$) $$\begin{aligned}
A_{k,\ell,E',\pi_{-},\pi_{+}}(v)
&:=
\sum_{u \in (\gf_2)^{E'}}
(-1)^{\chi_u + (u,Q(P^{-1}_{\pi_{-}}vP_{\pi_{+}}))}
\ef;\end{aligned}$$ Where in the expression $(P^{-1}_{\pi_{-}}vP_{\pi_{+}})$, $v$ is identified to the matrix of size $(k+1)\times (\ell+1)$ and $P_{\pi_{-}}$ and $P_{\pi_{+}}$ are the permutation matrices associated to $\pi_{-}$ and $\pi_{+}$.
The framework of automatic proofs of Arf identities we have developped is rather general. Most of the identities found in the litterature (see [@KZ03], [@Boi13], [@DS17], [@Del13], [@Gut17]) can be obtained in this setting.
Let us now apply the algorithm to find Arf identities.
It is convenient to introduce the notation $\vec{A}(\s)=\begin{pmatrixsm} \Abar(\s) \\ A(\s) \end{pmatrixsm}$. We have
\[prop.fingerred\] $$\begin{aligned}
\label{prop.signinvdyn}
\Abar\bigg(\,\tau=\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_t1.pdf}}\,\bigg)
&=
\Abar\bigg(\,\s=\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_s1.pdf}}\,\bigg)
\\
\label{eq.546455a}
\vec{A}\bigg(\,
%\tau=
\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_t3.pdf}}\,\bigg)
&=
\begin{pmatrix}
1 & -1 \\ 1 & 1
\end{pmatrix}
\vec{A}\bigg(\,
%\s=
\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_s3.pdf}}\,\bigg)
\ef;
\\
\label{eq.546455b}
\vec{A}\bigg(\,
%\tau=
\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_t4.pdf}}\,\bigg)
&=
\begin{pmatrix}
0 & 0 \\ 0 & 2
\end{pmatrix}
\vec{A}\bigg(\,
%\s=
\raisebox{-12pt}{\includegraphics[scale=1.8]{FigFol/Figure4_fig_arfproof_s3.pdf}}\,\bigg)
\ef;\end{aligned}$$
Clearly equation (\[prop.signinvdyn\]) prove the invariance of the arf invariant for the dynamics since $\s=L(\tau)$ and the case $R$ is deduced by symmetry.
Exceptional classes
===================
In this appendix, when using a matrix representation of configurations, it is useful to adopt the following notation: The symbol $\epsilon$ denotes the $0 \times 0$ empty matrix. The symbol denotes a square block in a matrix (of any size $\geq 0$), filled with an identity matrix. A diagram, containing these special symbols and the ordinary bullets used through the rest of the appendix, describes the set of all configurations that could be obtained by specifying the sizes of the identity blocks. In such a syntax, we can write equations of the like $$\begin{aligned}
\id
&:=
\raisebox{-1mm}{\includegraphics[width=4.8mm]{FigFol/FigureA2_fig_matr_id.pdf}}
=
\epsilon \cup
\raisebox{-3mm}{\includegraphics[width=8.8mm]{FigFol/FigureA2_fig_matr_Bid.pdf}}
=
\epsilon \cup
\raisebox{5.8mm}{\includegraphics[width=8.8mm,
angle=180]{FigFol/FigureA2_fig_matr_Bid.pdf}}
\ef;
% \\
% \intertext{and}
&
\id'
&:=
\raisebox{-7mm}{\includegraphics[width=16.8mm]{FigFol/fig_matr_idp.pdf}}
\ef.\end{aligned}$$ The sets $\id$ and $\id'$ contain one element per size, $\id_n$ and $\id'_n$, for $n \geq 0$ and $n \geq 3$ respectively.
The two exceptional classes $\Id_n$ and $\tree_n$ contain the configurations $\id_n$ and $\id'_n$, respectively.
We have the following proposition:
\[pro\_excep\_std\] The permutation $\s=\id_n$ (respectively $\s=\id'_n)$ is the only permutation of $\Id_n$ (respectively $\Id'_n$) with $\s(1)=1$ and $\s(2)=2$.
The structure of the classes $\Id_n$ is summarised by the following relation: := \_n \_n = ( \_[k 1]{} (X\_[RL]{}\^[k]{} X\_[LR]{}\^[k]{} X\_[LL]{}\^[k]{} X\_[RR]{}\^[k]{}) ) where the configurations $X_{\cdot \cdot}^{k}$ are defined as in figure \[fig.struct\_id\] (discard colours for the moment).\
$$\begin{aligned}
X_{LL}^{(k)}
&=
\quad
\makebox[0pt][l]{\raisebox{-16mm}{\includegraphics[width=33mm]{FigFol/FigureA2_fig_matr_CidGen1.pdf}}}
\hspace{-16pt}
\raisebox{-30pt}{$\rotatebox{45}{$k \left\{\rule{0pt}{50pt}\right. $}$}
&
X_{RR}^{(k)}
&=
\makebox[0pt][l]{\raisebox{-16mm}{\includegraphics[width=33mm]{FigFol/FigureA2_fig_matr_CidGen2.pdf}}}
\hspace{28pt}\raisebox{14pt}{$\rotatebox{45}{$\left. \rule{0pt}{50pt}\right\} k $}$}
\\
X_{RL}^{(k)}
&=
\makebox[0pt][l]{\raisebox{-16mm}{\includegraphics[width=38mm]{FigFol/FigureA2_fig_matr_CidGen3.pdf}}}
\hspace{-2pt}
\raisebox{-30pt}{$\rotatebox{45}{$k \left\{\rule{0pt}{50pt}\right. $}$}
&
X_{LR}^{(k)}
&=
\makebox[0pt][l]{\raisebox{-16mm}{\includegraphics[width=38mm]{FigFol/FigureA2_fig_matr_CidGen4.pdf}}}
\hspace{42pt}\raisebox{14pt}{$\rotatebox{45}{$\left. \rule{0pt}{50pt}\right\} k $}$}\end{aligned}$$
We can now prove the lemma \[lem.StFamNotmanyId\_tobeproveninAppC\] that we introduced in section 10.4 !!!!!!!!!.\
This is equivalent to say that there are no pairs of permutations $\s_1, \s_2 \in \Id_n$ which allow for a block decomposition $$\begin{aligned}
\s_1 &= \begin{array}{|c|}
\hline
\rule{5pt}{0pt}
A
\rule{5pt}{0pt}
\\
\hline
B \\
\hline
\end{array}
&
\s_2 &= \begin{array}{|c|}
\hline
\rule{5pt}{0pt}
B
\rule{5pt}{0pt}
\\
\hline
A \\
\hline
\end{array}\end{aligned}$$ If the block $A$ has $\ell$ rows, we say that $\s_2$ is the result of shifting $\s_1$ by $\ell$.
Clearly, at the light of the structure of configurations that we have presented (refer in particular to Figure \[fig.struct\_id\]), this pattern is incompatible with $\s_1$ or $\s_2$ being $\id_n$ (as a non-trivial shift produces a configuration which is not even irreducible), so we have excluded the cases in which, still with reference to the figure, we have only one violet block, and the number of black points is at least 3, for $X^{(k)}_{LL}$ and $X^{(k)}_{RR}$, and at least 4, for $X^{(k)}_{LR}$ and $X^{(k)}_{RL}$. Note that the black points are the positions in the grid which are south-west or north-east extremal (i.e., positions $(i,j) \in \s$ such that there is no $(i',j') \in \s$ with $i'<i$ and $j'<j$, or the analogous statement with $i'>i$ and $j'>j$). Let us call *number of records*, $\rho(\s)$, this parameter. Thus we have that configurations in $X^{(k)}_{LL}$ and $X^{(k)}_{RR}$ have $\rho=2k+1$, and configurations in $X^{(k)}_{LR}$ and $X^{(k)}_{RL}$ have $\rho=2k+2$.
Now, if we perform a shift within one block of consecutive ascents, it is easily seen, by investigation of the sub-configuration at the right of the entry of the new configuration in the bottom-most row, or the one at the left of the entry of the new configuration in the top-most row, that the resulting structure is incompatible with the structure of $\Id$. The same reasoning apply if we perform the shift at the beginning/end of a non-empty diagonal block, which is not the one at the bottom-right/top-left. On the other side, if we perform a shift in any other configuration, we have a new configuration in which $\rho$ has strictly decreased. As $\s_2$ is a non-trivial shift of $\s_1$, and $\s_1$ is a non-trivial shift of $\s_2$, we can thus conclude.
| ArXiv |
---
abstract: 'This is a survey of the historical development of the Spectral Standard Model and beyond, starting with the ground breaking paper of Alain Connes in 1988 where he observed that there is a link between Higgs fields and finite noncommutative spaces. We present the important contributions that helped in the search and identification of the noncommutative space that characterizes the fine structure of space-time. The nature and properties of the noncommutative space are arrived at by independent routes and show the uniqueness of the Spectral Standard Model at low energies and the Pati–Salam unification model at high energies.'
title: |
A survey of spectral models of gravity\
coupled to matter
---
Dedicated to Alain Connes
Introduction
============
In 1988, at the height of the string revolution, there appeared an alternative way to think about the structure of space-time, based on the breathtaking progress in the new field of noncommutative geometry. Despite the success of string theory in incorporating gravity, consistency of the theory depended on the existence of supersymmetry as well as six or seven extra dimensions. Enormous amount of research was carried out to obtain the Standard Model from string compactification, which even up to day did not materialize. Most compactifications start in ten dimensions with the Yang–Mills gauge group $E_{8}\times E_{8}$ requiring a very large number of fields to become massive at high energies. In a remarkable paper, Alain Connes laid down the blue print of a new innovative approach to uncover the origin of the Standard Model and its symmetries [@C90]. The foundation of this approach was based on noncommutative geometry, a field he founded few years before [@C85] (see also [@C94]). Alain realized that by making space slightly noncommutative by tensoring the four dimensional space with a space of two points, one gets a parallel universe where the distance between the two sheets is of the order of $10^{-16}$ cm, with the unexpected bonus of having the Higgs scalar field mediating between them. Although this looked similar to the idea of Kaluza–Klein, there were essential differences, mainly in avoiding the huge number of the massive tower of states as well as obtaining the Higgs field in a representation which is not the adjoint. Soon after this work inspired similar approaches also based on extending the four-dimensional space to become noncommutative [@Dub88; @DKM89; @DKM89b; @DKM90; @CFF92].
In this survey we will review the key developments that allowed noncommutative geometry to deepen our understanding of the structure of space-time and explain from first principles why and how nature dictates the existence of the elementary particles and their fundamental interactions. In Section 2 we will start by reviewing the pioneering work of Alain Connes [@C90] introducing the basic mathematical definitions and structures needed to define a noncommutative space. We summarize the characteristic ingredients in the construction of the Connes–Lott model and later generalizations by others. We then consider how to develop the analogue of Riemannian geometry for non-commutative spaces, and to incorporate the gravitational field in the Connes–Lott model. In Section 3 we present a breakthrough in the development of noncommutative geometry with the introduction of the reality operator which led to the definition of KO dimension of a noncommutative space. With this it became possible to present the reconstruction theorem of Riemannian geometry from noncommutative geometry. Section 4 covers the formulation and applications of the spectral action principle where the spectrum of the Dirac operator plays a dominant role in the study of noncommutative spaces. This key development allowed to obtain the dynamics of the Standard Model coupled to gravity in a non-ambiguous way, and to study geometric invariants of noncommutative spaces. We then show that incorporating right-handed neutrinos with the fundamental fermions forces a change in the algebra of the noncommutative space and the use of real structures to impose simultaneously the reality and chirality conditions on fundamental states, singling out the KO dimension to be 6. We show in detail how the few requirements about KO dimension, Majorana masses for right-handed neutrinos and the first order condition on the Dirac operator, singles out the geometry of the Standard Model. In Section 5 we present a classification of finite noncommutative spaces of KO dimension 6 showing the almost uniqueness of the Standard spectral model. In Section 6 we give a prescription of constructing spectral models from first principles and show that the spectral Standard Model agrees with the available experimental limits, provided that the scale giving mass to the right-handed neutrinos is promoted to a singlet scalar field. We then show that there exists a more general case where the first order condition on the Dirac operator is removed, the singlet scalar fields become part of a larger representation of the Pati–Salam model. The Standard Model becomes a special point in the spontaneous breaking of the Pati–Salam symmetries. In Section 6 we show that a different starting point where a Heisenberg like quantization condition in terms of the Dirac operator considered as momenta and two possible Clifford-algebra valued maps from the four-dimensional manifold to two four-spheres $S^{4}$ result in noncommutative spaces with quantized volumes. The Pati–Salam model and its various truncations are uniquely determined as the symmetries of the spaces solving the constraint. Section 7 contains the conclusions and a discussion of possible directions of future research.
### Acknowledgements {#acknowledgements .unnumbered}
The work of A. H. C. is supported in part by the National Science Foundation Grant No. Phys-1518371. He also thanks the Radboud Excellence Initiative for hosting him at Radboud University where this research was carried out. We would like to thank Alain Connes for sharing with us his insights and ideas.
Early days of the spectral Standard Model
=========================================
The first serious attempt to utilize the ideas of noncommutative geometry in particle physic was made by Alain Connes in 1988 in his paper “Essay on physics and noncommutative geometry” [@C90]. He observed that it is possible to change the structure of the (Euclidean) space-time so that the action functional gives the Weinberg-Salam model. The main emphasis was on the conceptual understanding of the Higgs field, which he calls, the black box of the standard model. The qualitative picture was taken to be of a two-sheeted Euclidean space-time separated by a distance of the order of $10^{-16}$ cm. In order to simplify the presentation, and to easily follow the historical development, we will use a uniform notation, representing old results in a new format. It is therefore more efficient to start with the basic definitions.
Noncommutative spaces and differential calculus
-----------------------------------------------
A noncommutative space is determined from the spectral data $\left(
\mathcal{A},\mathcal{H},D,\gamma,J\right) $ where $\mathcal{A}$ is an associative algebra with unit element $1$ and involution \*, $\mathcal{H}$ a Hilbert space carrying a faithful representation $\pi$ of the algebra, $D$ is a self-adjoint operator on $\mathcal{H}$ with $\left( D^{2}+1\right) ^{-1}$ compact, $\gamma$ is the unitary chirality operator and $J$ an anti-unitary operator on $\mathcal{H}$, the reality structure. The operator $J$ was introduced later in 1994 [@C95].
In the model proposed in 1988, there were ambiguities in defining the algebra and the action on the Hilbert space. These were rectified in the 1990 paper [@CL91] with John Lott, in what became known as the Connes–Lott model. In order to appreciate the enormous progress made over the years, we will summarize this model in a simplified presentation. A more detailed account can be found in the early reviews [@VG93; @MGV98; @Kas93; @Kas96; @KasS96; @KasS97]. Note that at around the same time a derivation based differential calculus was introduced by others in [@Dub88; @DKM89; @DKM89b; @DKM90] with many similarities to the model proposed by Connes in 1988.
We first need to first introduce new ingredients. Given a unital involutive algebra $\mathcal{A}$, the universal differential algebra over $\mathcal{A}$ is defined as $$\Omega^{\ast}\left( \mathcal{A}\right) ={\displaystyle\bigoplus\limits_{n=0}^{\infty}}
\Omega^{n}\left( \mathcal{A}\right)$$ where we set $\Omega^{0}\left( \mathcal{A}\right) =\mathcal{A}$, and take$$\Omega^{n}\left( \mathcal{A}\right) =\left\{
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}da_{2}^{i}\cdots da_{n}^{i},\qquad a_{j}^{i}\in
\mathcal{A},\quad\forall i,j\right\} ,\quad n=1,2,\cdots$$ Here $da$ denotes an equivalence class of $\mathcal{A}$, modulo the following relations $$d\left( a\cdot b\right) =da\cdot b+a\cdot db,\qquad d1=0,\qquad d^{2}=0$$ An element of $\Omega^{n}\left( \mathcal{A}\right) $ is called a form of degree $n.$ One forms can be considered as connections on a line bundle whose space of sections is given by the algebra $\mathcal{A}$. A one form $\rho
\in\Omega^{1}\left( \mathcal{A}\right) $ is expressed in the form$$\rho={\displaystyle\sum\limits_{i}}
a^{i}db^{i},\qquad a^{i},b^{i}\in\mathcal{A}$$ and since $d1=0,$ we may impose the condition ${\displaystyle\sum\limits_{i}}
a^{i}b^{i}=1,$ without any loss in generality. We say that $\left(
\mathcal{H},D\right) $ is a Dirac K-cycle for $\mathcal{A}$ if and only if there exists an involutive representation $\pi$ of $\mathcal{A}$ on $\mathcal{H}$ satisfying $\pi\left( a\right) ^{\ast}=\pi\left( a^{\ast}\right) $ with the properties that $\pi\left( a\right) $ and $\left[ D,\pi\left(
a\right) \right] $ are bounded operators on $\mathcal{H}$ for all $a\in\mathcal{A}$. The K-cycle is called even if there exists a chirality operator $\gamma$ such that $\gamma D=-D\gamma,$ $\gamma=\gamma^{-1}=\gamma^{\ast}$ and $\left[ \gamma,\pi\left( a\right) \right] =0,$ otherwise it is odd. The action of $\pi$ on $\Omega^{\ast}\left(
\mathcal{A}\right) $ is defined as $$\pi\left(
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\right) ={\displaystyle\sum\limits_{i}}
\pi\left( a_{0}^{i}\right) [D,\pi\left( a_{1}^{i}\right) ]\cdots\lbrack
D,\pi\left( a_{n}^{i}\right) ]$$ The space of auxiliary fields is defined by $$\mathrm{Aux}=\ker\pi+d\,\ker\pi$$ where $$\ker\pi={\displaystyle\bigoplus\limits_{n=0}^{\infty}}
\left\{
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\,:\pi\left(
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\right) =0\right\}$$ and $$d\ker\pi={\displaystyle\bigoplus\limits_{n=0}^{\infty}}
\left\{
{\displaystyle\sum\limits_{i}}
da_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\,:\pi\left(
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\right) =0\right\}$$ The integral of a form $\alpha\in\Omega^{\ast}\left( \mathcal{A}\right) $ over a noncommutative space of metric dimension $d$ is defined by setting$${\displaystyle\int}
\alpha=\mathrm{Tr}_{w}\left( \pi\left( \alpha\right) D^{-d}\right)$$ where $\mathrm{Tr}_{w}$ is the Dixmier trace.
Two-sheeted spacetime
---------------------
A simple extension of space-time is taken as a product of continuous four-dimensional manifold times a discrete set of two points. The algebra is $\mathcal{A}=\mathcal{A}_{1}\otimes\mathcal{A}_{2}$ acting on the Hilbert space $\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ where $\mathcal{A}_{1}=C^{\infty}\left( M\right) $ and $\mathcal{A}_{2}=M_{2}\left(
\mathbb{C}\right) \oplus M_{1}\left( \mathbb{C}\right) ,$ the algebra of $2\times2$ and $1\times1$ matrices. The Hilbert space is that of spinors of the form $$L=\left(
\begin{array}
[c]{c}l\\
e
\end{array}
\right)$$ where $l$ is a doublet and $e$ is a singlet. The spinor $L$ satisfies the chirality condition $\gamma_{5}\otimes\Gamma_{1}L=L$ where $\Gamma
_{1}=\mathrm{diag}\left( 1_{2},-1\right) $ is a grading operator. From this we deduce that $l$ is a left-handed spinor and $e$ is right handed, and we thus write $l=\left(
\begin{array}
[c]{c}\nu_{L}\\
e_{L}\end{array}
\right) $ and $e=e_{R}.$ The Dirac operator is given by $D=D_{1}\otimes1+\gamma_{5}\otimes D_{2}$ where $D_{1}=\gamma^{\mu}\partial_{\mu}$ and $D_{2}$ is the Dirac operator on $\mathcal{A}_{2}$ such that $$D_{l}=\left(
\begin{array}
[c]{cc}\gamma^{\mu}\partial_{\mu}\otimes1_{2}\otimes1_{3} & \gamma_{5}\otimes
M_{12}\otimes k\\
\gamma_{5}\otimes M_{21}\otimes k^{\ast} & \gamma^{\mu}\partial_{\mu}\otimes1\otimes1_{3}\end{array}
\right)$$ where $M_{21}=M_{12}^{\ast}$ and $k$ is a $3\times3$ family mixing matrix representing Yukawa couplings for the leptons. The $1\times2$ matrix $M_{12}$ turns out to be the vev of the Higgs field and is taken as $M_{12}=\mu\left(
\begin{array}
[c]{c}0\\
1
\end{array}
\right) =H_{0}.$ The elements $a\in\mathcal{A}$ have the representation $a=\left(
\begin{array}
[c]{cc}a_{1} & 0\\
0 & a_{2}\end{array}
\right) $ where $a_{1},$ $a_{2}$ are $2\times2$ and $1\times1$ unitary valued functions. A quick calculation shows that the self-adjoint one-form $\rho$ has the representation$$\pi_{1}\left( \rho\right) =\left(
\begin{array}
[c]{cc}A_{1}\otimes1_{3} & \gamma_{5}\otimes H\otimes k\\
\gamma_{5}\otimes H^{\ast}\otimes k^{\ast} & A_{2}\otimes1_{3}\end{array}
\right)$$ where $$\begin{aligned}
A_{1} & =\gamma^{\mu}{\displaystyle\sum\limits_{i}}
a_{1}^{i}\partial_{\mu}b_{1}^{i},\qquad A_{2}=\gamma^{\mu}{\displaystyle\sum\limits_{i}}
a_{2}^{i}\partial_{\mu}b_{2}^{i},\\
H & =H_{0}+{\displaystyle\sum\limits_{i}}
a_{1}^{i}H_{0}b_{2}^{i}.\end{aligned}$$ The quarks are introduced by taking for the finite space a bimodule structure relating two algebras $\mathcal{A}$ and $\mathcal{B}$ where the algebra $\mathcal{B}$ is taken to be $M_{1}\left( \mathbb{C}\right) \oplus
M_{3}\left( \mathbb{C}\right) $ commuting with the action of $\mathcal{A}.$ In addition, the mass matrices in the Dirac operator are taken to be zero when acting on elements of $\mathcal{B}.$ The one-form $\eta\in\Omega^{1}\left(
\mathcal{B}\right) $ has the simple form $B_{1}\mathrm{diag}\left(
1_{2},1\right) $ where $B_{1}$ is a gauge field associated with $M_{1}\left(
\mathbb{C}\right) .$ The Hilbert space for the quarks is $$Q=\left(
\begin{array}
[c]{c}q_{L}\\
u_{R}\\
d_{R}\end{array}
\right) ,\qquad q_{L}=\left(
\begin{array}
[c]{c}u_{L}\\
d_{L}\end{array}
\right)$$ The representation of $a\in\mathcal{A}$ is $a\rightarrow\left( a_{1},a_{2},\overline{a}_{2}\right) $ where $a_{1}$ and $a_{2}$ are a $2\times2$ and $1\times1$ complex valued functions. The Dirac operator acting on the quark Hilbert space is $$D_{q}=\left(
\begin{array}
[c]{ccc}\gamma^{\mu}\left( \partial_{\mu}+\cdots\right) \otimes1_{2}\otimes1_{3} &
\gamma_{5}\otimes M_{12}\otimes k^{\prime} & \gamma_{5}\otimes\widetilde{M}_{12}\otimes k^{^{\prime\prime}}\\
\gamma_{5}\otimes M_{12}^{\ast}\otimes k^{\prime\ast} & \gamma^{\mu}\left(
\partial_{\mu}+\cdots\right) \otimes1_{3} & 0\\
\gamma_{5}\otimes\widetilde{M}_{12}^{\ast}\otimes k^{^{\prime\prime}\ast} &
0 & \gamma^{\mu}\left( \partial_{\mu}+\cdots\right) \otimes1_{3}\end{array}
\right)$$ where $k^{\prime}$ and $k^{\prime\prime}$ are $3\times3$ family mixing matrices and $\widetilde{M}_{12}=\mu\left(
\begin{array}
[c]{c}1\\
0
\end{array}
\right) .$ The one form in $\Omega^{1}\left( \mathcal{A}\right) $ has then the representation $$\pi_{q}\left( \rho\right) =\left(
\begin{array}
[c]{ccc}A_{1}\otimes1_{3} & \gamma_{5}\otimes H\otimes k^{\prime} & \gamma_{5}\otimes\widetilde{H}\otimes k^{^{\prime\prime}}\\
\gamma_{5}\otimes H^{\ast}\otimes k^{\prime\ast} & A_{2}\otimes1_{3} & 0\\
\gamma_{5}\otimes\widetilde{H}^{\ast}\otimes k^{^{\prime\prime}\ast} & 0 &
\overline{A}_{2}\otimes1_{3}\end{array}
\right)$$ where $\widetilde{H}_{a}=\epsilon_{ab}H^{b}.$ When acting on the algebra $\mathcal{B}$ the Dirac operator has zero mass matrices and the one-form $\eta$ in $\Omega^{1}\left( \mathcal{B}\right) $ has the representation $\pi_{q}\left( \eta\right) =B_{2}\mathrm{diag}\left( 1_{2},1\right) $ where $B_{2}$ is the gauge field associated with $M_{3}\left( \mathbb{C}\right) .$ Imposing the unimodularity condition on the algebras $\mathcal{A}$ and $\mathcal{B}$ would then relate the $U\left( 1\right) $ factors in both algebras so that $\mathrm{tr}\left( A_{1}\right) =0,$ $A_{2}=B_{1}=-\mathrm{tr}\left( B_{2}\right) \equiv\frac{i}{2}g_{1}B$. With these we can then write $$\begin{aligned}
A_{1} & =-\frac{i}{2}g_{2}\sigma^{a}A_{a}\\
B_{2} & =-\frac{i}{6}g_{1}B-\frac{i}{2}g_{3}V^{i}\lambda_{i}$$ where $g_{1},$ $g_{2}$ and $g_{3}$ are the $U\left( 1\right) ,$ $SU\left(
2\right) $ and $SU\left( 3\right) $ gauge coupling constants, $\sigma^{a}$ and $\lambda^{i}$ are the Pauli and Gell-Mann matrices respectively. The fermionic actions for the leptons and quarks are then given by $$\begin{aligned}
\left\langle L,\left( D+\rho+\eta\right) L\right\rangle & ={\displaystyle\int}
d^{4}x\sqrt{g}\left( \overline{L}\left( D_{l}+\pi_{l}\left( \rho\right)
+\pi_{l}\left( \eta\right) \right) L\right) \\
\left\langle Q,\left( D+\rho+\eta\right) Q\right\rangle & ={\displaystyle\int}
d^{4}x\sqrt{g}\left( \overline{Q}\left( D_{q}+\pi_{q}\left( \rho\right)
+\pi_{q}\left( \eta\right) \right) Q\right)\end{aligned}$$ These terms can be easily checked to reproduced all the fermionic terms of the Standard Model.
The bosonic action is the sum of the square of curvatures in both the lepton and quark sectors. These are given by $$\begin{aligned}
I_{l} & =\mathrm{Tr}\left( C_{l}\left( \theta_{\rho}+\theta_{\eta}\right)
^{2}D_{l}^{-4}\right) \\
I_{q} & =\mathrm{Tr}\left( C_{q}\left( \theta_{\rho}+\theta_{\eta}\right)
^{2}D_{q}^{-4}\right)\end{aligned}$$ where $$\theta_{\rho}\equiv d\rho+\rho^{2}$$ is the curvature of $\rho,$ and $C_{l}$ and $C_{q}$ are constant elements of the algebra. Since the representation $\pi$ has a kernel, the auxiliary fields must be projected out. This step mainly affects the potential. After some algebra one can show that the bosonic action given above reproduces all the bosonic interactions of the Standard Model with the same number of parameters. If one assumes that $C_{l}$ and $C_{q}$ belong to the center of the algebra, then one can get fixed values for the top quark mass and Higgs mass. The main advantage of the noncommutative construction of the Standard Model is that one gets a geometrical understanding of the origin of the Higgs field and a unification of the gauge and Higgs sectors. One sees that the Higgs fields are the components of the one form along discrete directions.
Constructions beyond the Standard Model
---------------------------------------
The early constructions of the Standard Model provided encouragements to look further into noncommutative spaces. The construction was also complicated with some ambiguities such as the independence of the lepton and quark sectors, the construction of the Higgs potential and projecting out the auxiliary fields. It was then natural to ask whether it is possible to go beyond the Standard Model. In particle physics the route taken was to consider larger groups such as $SU\left( 5\right) $ or $SO(10)$ which contains $U\left( 1\right)
\times SU\left( 2\right) \times SU\left( 3\right) $ as a subgroup. The main advantage of GUT is that the fermionic fields are unified in one or two representations, the most attractive possibility being $SO(10)$ where the spinor representation $16_{s}$ contains all the known fermions in addition to the right-handed neutrino. The simplicity in the fermionic sector did not make the theory more predictive because of the arbitrariness of the Higgs sector. There are many possible Higgs representations that can break the symmetry spontaneously from $SO(10)$ to $SU\left( 3\right) \times U\left( 1\right)
.$ In the noncommutative construction the Higgs sector is more constrained which was taken as an encouragement to explore the possibility of considering larger matrix algebras. As an example if one arranges the leptons in the form $L=\left(
\begin{array}
[c]{c}l_{L}\\
l_{R}\end{array}
\right) $ where $l=\left(
\begin{array}
[c]{c}\nu\\
e
\end{array}
\right) $ then the corresponding algebra will be $M_{2}\left( \mathbb{C}\right) \oplus M_{2}\left( \mathbb{C}\right) .$ A natural possibility is then to consider a discrete space of four points and where the fermions are arranged in the format $\psi=\left(
\begin{array}
[c]{c}l_{L}\\
l_{R}\\
l_{L}^{c}\\
l_{R}^{c}\end{array}
\right) $ and the representation $\pi$ acting on $\mathcal{A}$ is given by $\pi\left( a\right) =\mathrm{diag}\left( a_{1},a_{2},\overline{a}_{1},\overline{a}_{2}\right) $ where $a_{1},$ $a_{2}$ are $2\times2$ complex matrices. The resulting model has $SU\left( 2\right) _{L}\times SU\left(
2\right) _{R}\times U\left( 1\right) _{B-L}$ with the Higgs fields in the representations $\left( 2,2\right) ,$ $\left( 3,1\right) +\left(
1,3\right) $ of $SU\left( 2\right) _{L}\times SU\left( 2\right) _{R}.$ We can summarize the steps needed to construct noncommutative particle physics models. First we specify the fermion representations then we choose the number of discrete points and the symmetry between them. From this we deduce the appropriate algebra and the map $\pi$ acting on the Hilbert space of spinors. Finally we write down the Dirac operator acting on elements of the algebra and choose the mass matrices to correspond to the desired vacuum of the Higgs fields.
To illustrate these steps consider the chiral space-time spinors $P_{+}\psi$ to be in the $16_{s}$ representation of $SO(10),$ where $P_{+}$ is the $SO(10)$ chirality operator, and the number of discrete points to be four. The Hilbert space is taken to be $\Psi=\left(
\begin{array}
[c]{c}P_{+}\psi\\
P_{+}\psi\\
P_{-}\psi^{c}\\
P_{-}\psi^{c}\end{array}
\right) $ where $\psi^{c}=BC\overline{\psi}^{T},$ $C$ being the charge conjugation matrix while $B$ is the $SO\left( 10\right) $ conjugation matrix. The finite algebra is taken to be $\mathcal{A}_{2}=P_{+}\left(
\mathsf{Cliff\,SO}\left( 10\right) \right) P_{+},$ and the finite Hilbert space $\mathcal{H}_{2}=\mathbb{C}^{32}.$ Let $\pi_{0}$ denote the representation of the algebra $\mathcal{A}$ on the Hilbert space $\mathcal{H}$ and let $\overline{\pi}_{0}$ denote the anti representation defined by $\overline{\pi}_{0}\left( a\right) =B\overline{\pi_{0}\left( a\right)
}B^{-1}.$ We then define $\pi\left( a\right) =\pi_{0}\left( a\right)
\oplus\pi_{0}\left( a\right) \oplus\overline{\pi}_{0}\left( a\right)
\oplus\overline{\pi}_{0}\left( a\right) .$ The Dirac operator is taken to be $$\left(
\begin{array}
[c]{cccc}\gamma^{\mu}\partial_{\mu}\otimes1_{32}\otimes1_{3} & \gamma_{5}\otimes
M_{12}\otimes K_{12} & \gamma_{5}\otimes M_{13}\otimes K_{13} & \gamma
_{5}\otimes M_{14}\otimes K_{14}\\
\gamma_{5}\otimes M_{12}^{\ast}\otimes K_{12}^{\ast} & \gamma^{\mu}\partial_{\mu}\otimes1_{32}\otimes1_{3} & \gamma_{5}\otimes M_{23}\otimes
K_{23} & \gamma_{5}\otimes M_{24}\otimes K_{24}\\
\gamma_{5}\otimes M_{13}^{\ast}\otimes K_{13}^{\ast} & \gamma_{5}\otimes
M_{23}^{\ast}\otimes K_{23}^{\ast} & \gamma^{\mu}\partial_{\mu}\otimes
1_{32}\otimes1_{3} & \gamma_{5}\otimes M_{34}\otimes K_{34}\\
\gamma_{5}\otimes M_{14}^{\ast}\otimes K_{14}^{\ast} & \gamma_{5}\otimes
M_{24}^{\ast}\otimes K_{24}^{\ast} & \gamma_{5}\otimes M_{34}^{\ast}\otimes
K_{34}^{\ast} & \gamma^{\mu}\partial_{\mu}\otimes1_{32}\otimes1_{3}\end{array}
\right)$$ where the $K_{mn}$ are $3\times3$ family mixing matrices commuting with $\pi\left( a\right) .$ We may impose the exchange symmetries $1\leftrightarrow2$ and $3\leftrightarrow4$ so that $M_{12}=M_{12}^{\ast
}=\mathcal{M}_{0},$ $M_{13}=M_{14}=M_{23}=M_{24}=\mathcal{N}_{0},$ $M_{34}=M_{34}^{\ast}=B\overline{\mathcal{M}}_{0}B^{-1}.$ Computing $\pi\left( \rho\right) $ we get$$\pi\left( \rho\right) =\left(
\begin{array}
[c]{cccc}A & \gamma_{5}\mathcal{M}K_{12} & \gamma_{5}\mathcal{N}K_{13} & \gamma
_{5}\mathcal{N}K_{14}\\
\gamma_{5}\mathcal{M}K_{12}^{\ast} & A & \gamma_{5}\mathcal{N}K_{23} &
\gamma_{5}\mathcal{N}K_{24}\\
\gamma_{5}\mathcal{N}^{\ast}K_{13}^{\ast} & \gamma_{5}\mathcal{N}^{\ast}K_{23}^{\ast} & B\overline{A}B^{-1} & \gamma_{5}B\overline{\mathcal{M}}B^{-1}K_{34}\\
\gamma_{5}\mathcal{N}^{\ast}K_{14}^{\ast} & \gamma_{5}\mathcal{N}^{\ast}K_{24}^{\ast} & \gamma_{5}B\overline{\mathcal{M}}B^{-1}K_{34}^{\ast} &
B\overline{A}B^{-1}\end{array}
\right)$$ where $$\begin{aligned}
A & =P_{+}{\displaystyle\sum\limits_{i}}
a^{i}\gamma^{\mu}\partial_{\mu}b^{i}P_{+}\\
\mathcal{M}+\mathcal{M}_{0} & =P_{+}{\displaystyle\sum\limits_{i}}
a^{i}\mathcal{M}_{0}b^{i}P_{+}\\
\mathcal{N}+\mathcal{N}_{0} & =P_{+}{\displaystyle\sum\limits_{i}}
a^{i}\mathcal{N}_{0}B\overline{b}^{i}B^{-1}P_{-}$$ One sees immediately that the Higgs fields $\mathcal{M}$ and $\mathcal{N}$ are in the $16_{s}\times16_{s}$ and $16_{s}\times\overline{16}_{s}$ representations. Equating the action of $A$ on $\psi$ and $\psi^{c}$ will reduce it to an $SO\left( 10\right) $ gauge field. Specifying $\mathcal{M}_{0}$ and $\mathcal{N}_{0}$ determines the breaking pattern of $SO\left(
10\right) .$ One can then proceed to construct the bosonic sector and project out the auxiliary fields to determine the potential. There are very limited number of models one can construct. These models, however, will suffer the same problems encountered in the GUT construction, mainly that of low unification scale of $10^{14}$ Gev implying fast rate of proton decay which is ruled out experimentally.
Coupling matter to gravity
--------------------------
The dynamics of the gravitational force is based on Riemannian geometry. It is therefore natural to study the nature of the gravitational field in noncommutative geometry. The original attempt [@CFF93; @CFG95] was based on generalizing the basic notions of Riemannian geometry, notably the theory of linear connections on differential forms. (Note that an alternative route that takes vector fields as a starting point ends with a derivation based differential calculus as in [@Dub88] ([*cf.*]{} [@Mad95]). In line with the Connes–Lott model, we will instead take differential forms as our starting point. For more details we also refer to the exposition in [@Lnd97 Sect. 10.3]).
First one defines the metric as an inner product on a cotangent space. Then one shows that every cycle over $\mathcal{A}$ yields a notion of cotangent bundle associated with $\mathcal{A}$ and a Riemannian metric on the cotangent bundle $\Omega_{D}^{1}\left( \mathcal{A}\right) .$ With the connection $\nabla$ the Riemann curvature of $\nabla$ on $\Omega_{D}^{1}\left( \mathcal{A}\right) $ is defined by $R\left( \nabla\right) :=-\nabla^{2}$ and the torsion by $T=d-m\circ\nabla$ where $m$ is the tensor product. Requiring $\nabla$ to be unitary and the torsion to vanish we obtain the Levi–Civita connection. If $\Omega_{D}^{1}\left( \mathcal{A}\right) $ is a finitely generated module, then it admits a basis $e^{A},$ $A=1,2,\cdots,N,$ and the connection $\omega_{B}^{A}\in\Omega_{D}^{1}\left( \mathcal{A}\right) $ is defined by $\nabla e^{A}=-\omega_{B}^{A}\otimes e^{B}.$ The components of the torsion $T\left( \nabla\right) $ are defined by $T^{A}=T\left( \nabla\right)
e^{A}$ then $T^{A}\in\Omega_{D}^{2}\left( \mathcal{A}\right) $ is given by $$T^{A}=de^{A}+\omega_{B}^{A}e^{B}$$ Similarly, components of the curvature $R_{B}^{A}\in\Omega_{D}^{2}\left(
\mathcal{A}\right) $ satisfy the defining property that $R\left( \nabla\right) e^{A}=R_{B}^{A}\otimes e^{B}$ so that $$R_{B}^{A}=d\omega_{B}^{A}+\omega_{C}^{A}\omega_{B}^{C}.$$ The analogue of the Einstein–Hilbert action is then $$I\left( \nabla\right) :=\kappa^{-2}\left\langle R_{B}^{A}e^{B},e_{A}\right\rangle$$ where $\kappa^{-1}$ is the Planck scale. Computing this action for the product space $M_{4}\times Z_{2}$ one finds that $$I\left( \nabla\right) =2{\displaystyle\int\limits_{M}}
d^{4}x\sqrt{g}\left( \kappa^{-2}r-2\partial_{\mu}\sigma\partial^{\mu}\sigma\right)$$ where $r$ is the scalar curvature of the Levi–Civita connection of the Riemannian manifold $M_{4}$ coupled to a scalar field $\sigma.$ Applying this construction to the Connes–Lott model is rather involved because the two sheets are not treated symmetrically, being associated with two different algebras. The complication arise because the projective module is not free and the basis $e^{A}$ is constrained. The Einstein–Hilbert action in this case is given by $$I\left( \nabla\right) =2{\displaystyle\int\limits_{M}}
d^{4}x\sqrt{g}\left( \kappa^{-2}\frac{3}{2}r-2\left( 3+\lambda\right)
\partial_{\mu}\sigma\partial^{\mu}\sigma+c\left( \lambda\right) e^{-2\sigma
}\right)$$ where $\lambda=\mathrm{Tr}\left( kk^{\ast}\right) ^{2}-1.$ To understand the significance of the field $\sigma$, we note that by examining the Dirac operator one finds that the field $\phi=e^{-\kappa\sigma}$ now replaces the weak scale. Thus quantum corrections to the classical potential will depend on $\sigma,$ thus the vev of $\sigma$ could be determined from the minimization equations.
The spectral action principle
=============================
Despite the success of the Connes–Lott model and the generalizations that followed in giving a geometrical meaning to the Higgs field and unifying it with the gauge fields, it was felt that the construction is not satisfactory. The first unpleasant feature was the use of the bimodule structure to introduce the $SU\left( 3\right) $ symmetry and the second is the use of unimodularity condition to get the correct hypercharge assignments to the particles. Another major problem was the existence of mirror fermions as a consequence of the fact that the conjugation operator on fermions gives independent fields. In addition, there was arbitrariness in the construction of the potential in the bosonic sector associated with the step of eliminating the auxiliary fields.
Real structures on spectral triples {#sect:st}
-----------------------------------
The first breakthrough came in 1995 with the publication of Alain Connes’ paper “Noncommutative geometry and reality” [@C95]. In this paper, the notion of real structure is introduced, motivated by Atiyah’s KR theory and Tomita’s involution operator $J.$ A hint for the necessity of the reality operator can be taken from physics. We have seen that space-time spinors, which are elements of the Hilbert space satisfy a chirality condition. The charge conjugation operator, when acting on these spinors, produces a conjugate element, which in general is independent. It is possible to replace the chirality condition, with a reality one, known as the Majorana condition which equates the two. Imposing both conditions, chirality and reality, simultaneously can only occur in certain dimensions. The action of the anti-linear isometry $J$ on the algebra $\mathcal{A}$ satisfies the commutation relation $\left[ a,b^{\mathrm{o}}\right] =0,$ $\forall
a,b\in\mathcal{A}$ where$$b^{\mathrm{o}}=Jb^{\ast}J^{-1},\qquad\forall b\in\mathcal{A}$$ so that $b^{\mathrm{o}}\in$ $\mathcal{A}^{\mathrm{o}}.$ This gives a bimodule, using the representation of $\mathcal{A}\otimes\mathcal{A}^{\mathrm{o}}$, given by $$a\otimes b^{\mathrm{o}}\rightarrow aJb^{\ast}J^{-1},\qquad\forall
a,b\in\mathcal{A}$$ We define the fundamental class $\mu$ of the noncommutative space as a class in the $KR$-homology of the algebra $\mathcal{A}\otimes\mathcal{A}^{\mathrm{o}}$ having the involution $$\tau\left( a\otimes b^{\mathrm{o}}\right) =b^{\ast}\otimes\left( a^{\ast
}\right) ^{\mathrm{o}},\qquad\forall a,b\in\mathcal{A}$$ The $KR$-homology cycle implements the involution $\tau$ given by $$\tau\left( w\right) =JwJ^{-1},\qquad\forall w\in\mathcal{A}\otimes
\mathcal{A}^{\mathrm{o}}$$ These imply that the $KR$-homology is periodic with period $8$ and the dimension $n$ modulo $8$ is determined from the commutation rules $$J^{2}=\varepsilon,\qquad JD=\varepsilon^{\prime}DJ,\qquad J\gamma
=\varepsilon^{\prime\prime}\gamma J$$ where $\varepsilon,$ $\varepsilon^{\prime},$ $\varepsilon^{\prime\prime}\in\left\{ -1,1\right\} $ are given as function of $n$ modulo $8$ according to the table$$\begin{tabular}
[c]{l|llllllll}$n$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$\\
\hline
$\varepsilon$ & $1$ & $1$ & $-1$ & $-1$ & $-1$ & $-1$ & $1$ & $1$\\
$\varepsilon^{\prime}$ & $1$ & $-1$ & $1$ & $1$ & $1$ & $-1$ & $1$ & $1$\\
$\varepsilon^{\prime\prime}$ & $1$ & & $-1$ & & $1$ & & $-1$ &
\end{tabular}$$ It is not surprising that this table agrees with the one obtained by classifying in which dimensions a spinor obey the Majorana and Weyl conditions. The intersection form $K_{\ast}\left( \mathcal{A}\right) \times
K_{\ast}\left( \mathcal{A}\right) \rightarrow\mathbb{Z}$ is obtained from the Fredholm index of $D$ in $K_{\ast}\left( \mathcal{A}\otimes
\mathcal{A}^{\mathrm{o}}\right) .$ Using the Kasparov intersection product, Poincare duality is formulated in terms of the invertibility of $\mu$ and that $D$ is an operator of order one implies the condition$$\left[ \left[ D,a\right] ,b^{\mathrm{o}}\right] =0,\qquad\forall
a,b\in\mathcal{A}$$ Next we consider automorphisms of the algebra $\mathcal{A}$ denoted by $\mathrm{Aut}\left( \mathcal{A}\right) .$ This comprises both of inner and outer automorphisms. Inner automorphisms $\mathrm{Int}\left( \mathcal{A}\right) $ is a normal subgroup of $\mathrm{Aut}\left( \mathcal{A}\right) $ defined by $$\alpha\left( f\right) =ufu^{\ast},\qquad\forall f\in\mathcal{A},\qquad
u\,u^{\ast}=u^{\ast}u=1$$ The group $\mathrm{Aut}^{+}\left( \mathcal{A}\right) $ of automorphisms of the involutive algebra $\mathcal{A}$ are implemented by a unitary operator $U$ in $\mathcal{H}$ commuting with $J$ satisfying $$\alpha\left( x\right) =UxU^{-1}\,\,\qquad\forall x\in\mathcal{A}$$ For Riemannian manifolds $M$, this plays the role of the group of diffeomorphisms $\mathrm{Diff}^{+}\left( M\right) ,$ which preserves the $K$-homology fundamental class of $M.$ Let $\mathcal{E}$ be a finite projective, hermitian right $\mathcal{A}$-module, and define the algebra $\mathcal{B}=\mathrm{End}\left( \mathcal{A}\right) $ as the Morita equivalence of the algebra $\mathcal{A}$ with a hermitian connection $\nabla$ on $\mathcal{E}$ defined as the linear map $\nabla:\mathcal{E\rightarrow
E\otimes}_{\mathcal{A}}\Omega_{D}^{1}$ satisfying $$\begin{aligned}
\nabla\left( \zeta a\right) & =\left( \nabla\zeta\right) a+\zeta\otimes
da,\qquad\forall\zeta\in\mathcal{E},\,a\in\mathcal{A}\\
d\left( \zeta,\eta\right) & =\left( \zeta,\nabla\eta\right) -\left(
\nabla\zeta,\eta\right) ,\qquad\forall\zeta,\,\eta\in\mathcal{E}$$ where $da=\left[ D,a\right] $ and $\Omega_{D}^{1}$ is the bimodule of operators of the form $$A={\displaystyle\sum\limits_{i}}
a_{i}\left[ D,b_{i}\right] ,\qquad a_{i},b_{i}\in\mathcal{A}$$ Since any algebra is Morita equivalent to itself with $\mathcal{E}=\mathcal{A},$ applying the construction given above yields the inner deformation of the spectral geometry. The unitary equivalence is implemented by the representation $u\rightarrow\widetilde{U}=u\left( Ju\,J^{-1}\right)
=u\left( u^{\mathrm{o}}\right) ^{\ast}$ so that the Dirac operator that includes inner fluctuations$$D_{A}=D+A+JAJ^{-1}$$ where $A=A^{\ast}$ transforms as $D_{A}\rightarrow\widetilde{U}D_{A}\widetilde{U}^{-1}$ provided that $$A\rightarrow u\,Au^{\ast}+u\left[ D,u^{\ast}\right]$$ This will ensure that the inner product $$\left( \Psi,D_{A}\Psi\right)$$ is invariant under the transformation $\Psi\rightarrow\widetilde{U}\Psi.$ This expression will then take care of all fermionic interactions which, as will be seen in the next section, removes the arbitrariness in specifying the action of the connection on the Hilbert space.
The spectral action principle
-----------------------------
The next breakthrough came a year later in 1996 in the work of Chamseddine and Connes entitled “The spectral action principle” [@CC96]. The basic observation is that for a noncommutative space defined by spectral data, the emphasis is shifted from the coordinates $x$ of a geometric space to the spectrum $\Sigma
\sqsubset\mathbb{R}$ of the operator $D$. We postulate the following hypothesis$$\text{The physical action depends only on }\Sigma$$ The existence of Riemannian manifolds which are isospectral but not isometric shows that the spectral action principle is stronger than the usual diffeomorphism invariance. In the usual Riemannian case the group $\mathrm{Diff}\left( M\right) $ of diffeomorphisms of $M$ is canonically isomorphic to the group $\mathrm{Aut}\left( \mathcal{A}\right) $ of automorphisms of the algebra $\mathcal{A}=C^{\infty}\left( M\right) .$ To each $\varphi\in\mathrm{Diff}\left( M\right) $ one associates the algebra preserving map $\alpha_{\varphi}:\mathcal{A}\rightarrow\mathcal{A}$ given by $$\alpha_{\varphi}\left( f\right) =f\circ\varphi^{-1}\qquad\forall f\in
C^{\infty}\left( M\right) =\mathcal{A}$$ The prescription to determine the bosonic action with some cutoff energy scale $\Lambda$ is to first replace the Hilbert space $\mathcal{H}$ by the subspace $\mathcal{H}_{\Lambda}$ defined by $$\mathcal{H}_{\Lambda}=\mathrm{range\,}\chi\left( \frac{D}{\Lambda}\right)$$ where $\chi$ is a suitable smooth positive function, restricting both $D$ and $\mathcal{A}$ to this subspace maintaining the commutation relations for the algebra. This procedure is superior to the lattice approximation because it does respect the geometric symmetry group. The [*spectal action functional*]{} is then given by the $$\operatorname{Tr}\chi\left( \frac D \Lambda \right).$$ For a noncommutative space which is a tensor product of a continuous manifold times a discrete space, the functional $\operatorname{Tr}\chi\left( \frac{D}{\Lambda}\right) $ can be expanded in an asymptotic series in $\Lambda$, rendering the computation amenable to a heat kernel expansion. This procedure will be illustrated in the next section. More general methods to analyze the spectral action have also been developed, see [@FGLV98] for an early result and also the recent book [@EI18]. An interpretation of the spectral action as the von Neumann entropy of a second-quantized spectral triple has been found recently in [@CCS18] ([*cf.*]{} [@DK19]).
To summarize, the breakthroughs carried out in the short period 1995-1996, defining the reality operator $J$ and developing the spectral action principle will allow to remove the ambiguities encountered before in the construction of the noncommutative spectral Standard Model.
The spectral Standard Model
===========================
At the time that the spectral action was formulated, it was clear that this principle forms a unifying framework for gravity and particle physics of the Standard Model. As said, this led to much activity ([*cf.*]{} [@SUW02]) in the years that followed. Also shortcomings of the approach were pointed out quite quickly, such as the notorious fermion-doubling problem [@LMMS97; @GIS98]. This doubling —or actually, quadrupling— was due to the incorporation of left-right, particle-anti-particle degrees of freedom both in the continuum spinor space and in the finite noncommutative space. At the technical level this was a crucial starting point, allowing for a product geometry to describe gravity coupled to the Standard Model.
Nevertheless, it was a somewhat disturbing feature which, together with the apparent arbitrariness of the choice of a finite geometry and the abscence of neutrino mixing in the model, led Connes to eventually resolve these problems in [@C06]. At the same time John Barrett [@Bar06] arrived at the same conclusion (see also the recent uniqueness result [@Bes19]), even though his motivation came from the desire to have noncommutative geometry with a Lorentzian signature.
The crucial insight in both of these works is that one should allow for a KO-dimension for the finite space $F$ which is different from the metric dimension (which is zero). More specifically, the KO-dimension of the finite space should be 6 (modulo 8), so that the product of the continuum $M$ with $F$ is 10 modulo 8. The precise structure of the spectral Standard Model (see Section \[sect:spectr-SM\]) is then best understood using the classification of all irreducible finite noncommutative geometries of KO-dimension 6 which we now briefly recall.
Classification of irreducible geometries {#sect:irr}
----------------------------------------
In [@CC07b] Chamseddine and Connes classified [*irreducible*]{} finite real spectral triples of KO-dimension 6. This lead to a remarkably concise list of spectral triples, based on the matrix algebras $M_N(\C) \oplus M_N(\C)$ for some $N$. We remark that earlier classification results were obtained [@Kra97; @PS98] which were also exploited in a search Beyond the Standard Model (see Remark \[rem:beyond-sm\] below).
A finite real spectral triple $(A,H,D;J, \gamma)$ is called [*irreducible*]{} if the triple $(A,H,J)$ is irreducible. More precisely, we demand that:
1. The representations of $A$ and $J$ in $H$ are irreducible;
2. The action of $A$ on $H$ has a separating vector.
We will prove the main result of [@CC07b] using an alternative approach which is based on [@Sui14 Sect. 3.4].
\[thm:irr-geom\] Let $(A,H,D;J,\gamma)$ be an irreducible finite real spectral triple of KO-dimension 6. Then there exists a positive integer $N$ such that $A \simeq M_N(\C) \oplus M_N(\C)$.
Let $(A,H,D;J,\gamma)$ be an arbitrary finite real spectral triple. We may then decompose $$A= \bigoplus_{i=1}^N M_{n_i}(\C), \qquad H = \bigoplus_{i,j=1}^N \C^{n_i} \otimes( \C^{n_j})^\circ \otimes V_{ij},$$ with $V_{ij}$ corresponding to the multiplicities as before. Now each $\C^{n_i} \otimes \C^{n_j}$ is an irreducible representation of $A$, but in order for $H$ to support a real structure $J:H \to H$ we need both $\C^{n_i} \otimes (\C^{n_j})^\circ$ and $\C^{n_j} \otimes (\C^{n_i})^\circ$ to be present in $H$. Moreover, an old result of Wigner [@Wig60] for an anti-unitary operator with $J^2 =1$ assures that already with multiplicities $\dim V_{ij}=1$ there exists such a real structure. Hence, the irreducibility condition (1) above yields $$H = \C^{n_i} \otimes (\C^{n_j})^\circ \oplus \C^{n_j} \otimes (\C^{n_i})^\circ,$$ for some $i,j \in \{ 1,\ldots, N\}$. Then, let us consider condition (2) on the existence of a separating vector. Note first that the representation of $A$ in $H$ is faithful only if $A= M_{n_i}(\C) \oplus M_{n_j}(\C)$. Second, the stronger condition of a separating vector $\xi$ then implies $n_i = n_j$, as it is equivalent to $A' \xi = H$ for the commutant $A'$ of $A$ in $H$. Namely, since $A' = M_{n_j}(\C) \oplus M_{n_i}(\C)$ with $\dim A' = n_i^2 + n_j^2$, and $\dim H = 2n_i n_j$ we find the desired equality $n_i=n_j$.
With the complex finite-dimensional algebras $A$ given as a direct sum $M_N(\C) \oplus M_N(\C)$,[^1] the additional demand that $H$ carries a symplectic structure $I^2=-1$ yields real algebras of which $A$ is the complexification. We see that this requires $N=2k$ so that one naturally considers triples $(A,H,J)$ for which $$\label{eq:classif}
A= M_{k}(\bH) \oplus M_{2k}(\C); \qquad H= \C^{2(2k)^2}.$$
Noncommutative geometry of the Standard Model {#sect:spectr-SM}
---------------------------------------------
The above classification of irreducible finite geometries of KO-dimension 6 forms the starting point for the derivation of the Standard Model from a noncommutative manifold [@CCM07]. Hence, it is based on the matrix algebra $M_N(\C) \oplus M_N(\C)$ for $N \geq 1 $. Let us make the following two additional requirements on the irreducible finite geometry $(A,H_F,D_F;J_F,\gamma_F)$:
1. The finite-dimensional Hilbert space $H_F$ carries a symplectic structure $I^2 = -1$;
2. the grading $\gamma_F$ induces a non-trivial grading on $A$, by mapping $$a \mapsto \gamma_F a \gamma_F,$$ and selects an even subalgebra $A^\ev \subset A$ consisting of elements that commute with $\gamma_F$.
But the first demand sets $A=M_k(\bH) \oplus M_{2k}(\C)$, represented on the Hilbert space $\C^{2(2k)^2}$. The second requirement sets $k \geq 2$; we will take the simplest $k=2$ so that $H_F = \C^{32}$. [^2] Indeed, this allows for a $\gamma_F$ such that $$\begin{aligned}
A^\ev &= \bH_R \oplus \bH_L \oplus M_4(\C), \nn
\intertext{where $\bH_R$ and $\bH_L$ are two copies (referred to as {\em right} and {\em left}) of the quaternions; they are the diagonal of $M_2(\bH) \subset A$. The Hilbert space can then be decomposed according to the defining representations of $A^\ev$, }
\label{eq:HF-PS}
H_F &= (\C^2_R \oplus \C^2_L) \otimes (\C^4)^{\circ} \oplus \C^4 \otimes ( (\C^2_R)^{\circ} \oplus (\C^2_L)^{\circ}).
\intertext{According to this direct sum decomposition, we write }
\label{eq:dirac-sm}
D_F&= \begin{pmatrix} {S}&{T^*}\\ {T}&{\bar S} \end{pmatrix}
$$ Moreover, $J_F$ is the anti-unitary operator that flips the two 16-dimensional components in Equation .
The key result is that if we assume that $T$ is non-trivial, then the first-order condition selects the maximal subalgebra of the Standard Model, that is to say, $A_F= \C \oplus \bH \oplus M_3(\C)$.
[[@CCM07 Prop. 2.11]]{} \[prop:subalg-sm\] Up to $*$-automorphisms of $A^\ev$, there is a unique $*$-subalgebra $A_F\subset A^\ev$ of maximal dimension that allows $T \neq 0$ in . It is given by $$A_F = \left\{ \left( q_\lambda, q , \begin{pmatrix} q &0 \\ 0&m \end{pmatrix} \right): \lambda \in \C, q \in \bH_L, m \in M_3(\C) \right\} \subset \bH_R \oplus \bH_L \oplus M_4(\C),$$ where $\lambda \mapsto q_\lambda$ is the embedding of $\C$ into $\bH$, with $$q_\lambda = \begin{pmatrix} \lambda & 0 \\ 0 & \bar\lambda \end{pmatrix}.$$ Consequently, $A_F \simeq \C \oplus \bH \oplus M_3(\C)$.
The restriction of the representation of $A$ on $H_F$ to the subalgebra $A_F$ gives a decomposition of $H_F$ into irreducible (left and right) representations of $\C$, $\H_L$ and $M_3(\C)$. For instance, $$\label{eq:decomp-subalg}
(\C^2_R \oplus \C^2_L) \otimes (\C^4)^{\circ} \leadsto (\C \oplus \overline \C \oplus \C^2_L) \otimes \left ((\C)^\circ \oplus (\C^3)^\circ \right).$$ and similarly for $\C^4 \otimes ( (\C^2_R)^{\circ} \oplus (\C^2_L)^{\circ})$. In order to connect to the physics of the Standard Model, let us introduce an orthonormal basis for $H_F$ that can be recognized as the fermionic particle content of the Standard Model, and subsequently write the representation of $A_F$ in terms of this basis.
We let the subspace of $H_F$ displayed in Equation be represented by basis vectors $\{ \nu_R ,e_R, (\nu_L,e_L)\}$ of the so-called [*lepton space*]{} $H_l$ and basis vectors $\{ u_R,d_R,(u_L,d_L)\}$ of the [*quark space*]{} $H_q$. Their reflections with respect to $J_F$ are the [*anti-lepton space*]{} $H_{\bar l}$ and the [*anti-quark space*]{} $H_{\bar q}$, spanned by $\{ \bar{\nu_R} ,\bar{e_R}, (\bar{\nu_L},\bar{e_L})\}$ and $\{\bar{u_R}, \bar{d_R},( \bar{u_L},\bar{d_L})\}$, respectively. The three colors of the quarks are given by a tensor factor $\C^3$ and when we take into account [*three generations*]{} of fermions and anti-fermions by tripling the above finite-dimensional Hilbert space we obtain $$\begin{aligned}
H_F := \left( H_l \oplus H_{\bar l} \oplus H_q \oplus H_{\bar q} \right)^{\oplus3} .\end{aligned}$$ Note that $H_l = \C^4$, $H_q=\C^4 \otimes \C^3$, $H_{\bar l} = \C^4$, and $H_{\bar q} = \C^4 \otimes \C^3$.
An element $a=(\lambda,q,m)\in A_F$ acts on the space of leptons $H_l$ as $q_\lambda \oplus q$, and acts on the space of quarks $H_q$ as $(q_\lambda \oplus q) \otimes 1_3$. For the action of $a$ on an anti-lepton $\bar l\in H_{\bar l}$ we have $a\bar l = \lambda 1_4\bar l$, and on an anti-quark $\bar q\in H_{\bar q}$ we have $a\bar q = (1_4 \otimes m) \bar q$.
The $\Z_2$-grading $\gamma_F$ is such that left-handed particles have eigenvalue $+1$ and right-handed particles have eigenvalue $-1$. The anti-linear operator $J_F$ interchanges particles with their anti-particles, so $J_F f = \bar f$ and $J_F \bar f = f$, with $f$ a lepton or quark.
The first indication that the subalgebra $A_F$ is relevant for the Standard Model —to say the least— comes from the fact that the Standard Model gauge group can be derived from the unitaries in $A_F$. We restrict to the [*unimodular gauge group*]{}, $$\mathrm{SU}(A_F) = \left\{ u \in A_F: u^* u = uu^* = 1, \det(u) = 1\right\}$$ where $\det$ is the determinant of the action of $u$ in $H_F$. It then follows that, up to a finite abelian group we have $$\mathrm{SU}(A_F) \sim U(1) \times SU(2) \times SU(3)$$ and the hypercharges are derived from the unimodularity condition to be the usual ones: $$\begin{aligned}
\begin{array}{l|cccccccc}
\text{Particle} & \nu_R & e_R & \nu_L & e_L & u_R & d_R & u_L & d_L \\
\hline
\text{Hypercharge} & 0 & -2 & -1 & -1 & \frac43 & -\frac23 & \frac13 & \frac13 \\
\end{array}\end{aligned}$$
Let us now turn to the form of the finite Dirac operator, and see what we can say about the components of the matrix $D_F$ as displayed in . Recall that we are looking for a self-adjoint operator $D_F$ in $H_F$ that commutes with $J_F$, anti-commutes with $\gamma_F$, and fulfills the first-order conditions with resepct to $A_F$: $$[[D,a],J b J^{-1}]=0; \qquad (a,b \in A_F).$$ We also require that $D_F$ commutes with the subalgebra $\C_F = \{ (\lambda,\lambda,0) \} \subset A_F$ which physically speaking corresponds to the fact that the photon remains massless. Then it turns out [@C06 Theorem 1] (see also [@CCM07 Theorem 2.21]\[page:moduli-dirac\]) that any $D_F$ that satisfies these assumptions is of the following form: in terms of the decomposition of $H_F$ in particle ($H_l^{\oplus 3} \oplus H_q^{\oplus 3}$) and anti-particles ($H_{\bar l}^{\oplus 3} \oplus H_{\bar q}^{\oplus 3}$) the operator $S$ is $$\begin{aligned}
S_l &:= \left.S\right|_{H_l^{\oplus 3}} = \begin{pmatrix} 0&0&Y_\nu^*&0 \\0&0&0&Y_e^*\\ Y_\nu&0&0&0\\ 0&Y_e&0&0\end{pmatrix} ,
\label{eq:yukawa-l} \\
S_q \otimes 1_3 &:= \left.S\right|_{H_q^{\oplus 3}} =\begin{pmatrix} 0&0&Y_u^*&0 \\0&0&0&Y_d^* \\Y_u&0&0&0 \\0&Y_d&0&0\end{pmatrix} \otimes1_3 ,\label{eq:yukawa-q}\end{aligned}$$ where $Y_\nu$, $Y_e$, $Y_u$ and $Y_d$ are some $3\times3$ matrices acting on the three generations, and $1_3$ acting on the three colors of the quarks. The symmetric operator $T$ only acts on the right-handed (anti)neutrinos, so it is given by $T\nu_R = Y_R\bar{\nu_R}$, for a certain $3\times3$ symmetric matrix $Y_R$, and $Tf=0$ for all other fermions $f\neq\nu_R$. Note that $\nu_R$ here stands for a vector with $3$ components for the number of generations.
The above classification result shows that the Dirac operators $D_F$ give all the required features, such as mixing matrices for quarks and leptons, unbroken color and the see-saw mechanism for right-handed neutrinos. Let us illustrate the latter in some more detail. The mass matrix restricted to the subspace of $H_F$ with basis $\{\nu_L, \nu_R, \bar{\nu_L}, \bar{\nu_R}\}$ is given by $$\begin{aligned}
\begin{pmatrix} 0&Y_\nu^*&Y_R^*&0 \\ Y_\nu&0&0&0\\ Y_R&0&0&\bar Y_\nu^* \\ 0&0&\bar Y_\nu&0\end{pmatrix} .\end{aligned}$$ Suppose we consider only one generation, so that $Y_\mu = m_\nu$ and $Y_R = m_R$ are just scalars. The eigenvalues of the above mass matrix are then given by $$\begin{aligned}
\pm \frac12 m_R \pm \frac12 \sqrt{{m_R}^2 + 4{m_\nu}^2} .\end{aligned}$$ If we assume that $m_\nu \ll m_R$, then these eigenvalues are approximated by $\pm m_R$ and $\pm\frac{{m_\nu}^2}{m_R}$. This means that there is a heavy neutrino, for which the Dirac mass $m_\nu$ may be neglected, so that its mass is given by the Majorana mass $m_R$. However, there is also a light neutrino, for which the Dirac and Majorana terms conspire to yield a mass $\frac{{m_\nu}^2}{m_R}$, which is in fact much smaller than the Dirac mass $m_\nu$. This is called the *seesaw mechanism*. Thus, even though the observed masses for these neutrinos may be very small, they might still have large Dirac masses (or Yukawa couplings).
Of course, in the physical applications one chooses $Y_\nu, Y_e$ to be the [*Yukawa mass matrices*]{} and $Y_R$ is the [*Majorana mass matrix*]{}. There has been searches for additional conditions to be satisfied by the spectral triple $(A_F,H_F,D_F)$ to further constrain the form of $D_F$, see for instance [@BBB15; @BF18; @KL18; @DAS18; @DS18].
The gauge and scalar fields as inner fluctuations
-------------------------------------------------
We here derive the precise form of internal fluctuations $A_\mu$ for the above spectral triple of the Standard Model (following [@CCM07 Sect. 3.5] or [@Sui14 Sect. 11.5]).
Take two elements $a=(\lambda,q,m)$ and $b=(\lambda',q',m')$ of the algebra $\A = C^\infty(\C\oplus\bH \oplus M_3(\C))$. According to the representation of $A_F$ on $H_F$, the inner fluctuations $A_\mu = -ia\partial_\mu b$ decompose as $$\begin{aligned}
\Lambda_\mu &:= -i\lambda\partial_\mu\lambda'; \qquad
\Lambda_\mu' := -i\bar\lambda\partial_\mu\bar\lambda'
\intertext{on $\nu_R$ and $e_R$, respectively, and as }
Q_\mu &:= -iq\partial_\mu q';\qquad
V_\mu' := -im\partial_\mu m'\end{aligned}$$ acting on $(\nu_l,e_L)$ and $H_{\bar q}$, respectively. On all other components of $H_F$ the gauge field $A_\mu$ acts as zero. Imposing the hermiticity $\Lambda_\mu=\Lambda_\mu^*$ implies $\Lambda_\mu\in\R$, and also automatically yields $\Lambda_\mu' = -\Lambda_\mu$. Furthermore, $Q_\mu = Q_\mu^*$ implies that $Q_\mu$ is a real-linear combination of the Pauli matrices, which span $i\,su(2)$. Finally, the condition that $V_\mu'$ be hermitian yields $V_\mu' \in i\,u(3)$, so $V_\mu'$ is a $U(3)$ gauge field. As mentioned above, we need to impose the unimodularity condition to obtain an $SU(3)$ gauge field. Hence, we require that the trace of the gauge field $A_\mu$ over $H_F$ vanishes, and we obtain $$\begin{aligned}
\left.\operatorname{Tr}\right|_{H_{\bar l}}\big( \Lambda_\mu 1_4 \big) + \left.\operatorname{Tr}\right|_{H_{\bar q}}\big( 1_4\otimes V_\mu' \big) = 0 \quad\Longrightarrow\quad \operatorname{Tr}(V_\mu') = - \Lambda_\mu .\end{aligned}$$ Therefore, we can define a traceless $SU(3)$ gauge field $V_\mu$ by $\bar V_\mu := - V_\mu' - \frac13 \Lambda_\mu$. The action of the gauge field $B_\mu = A_\mu - J_FA_\mu J_F^{-1}$ on the fermions is then given by $$\begin{aligned}
\left.B_\mu\right|_{H_l} &= \begin{pmatrix} 0&0& \\ 0&-2\Lambda_\mu& \\ &&Q_\mu-\Lambda_\mu1_2\end{pmatrix} , \notag\\
\label{eq:Gauge_field_SM}
\left.B_\mu\right|_{H_q} &= \begin{pmatrix} \frac43\Lambda_\mu1_3+V_\mu&0& \\ 0&-\frac23\Lambda_\mu1_3+V_\mu& \\ &&(Q_\mu+\frac13\Lambda_\mu1_2)\otimes1_3+1_2\otimes V_\mu\end{pmatrix} .\end{aligned}$$ for some $U(1)$ gauge field $\Lambda_\mu$, an $SU(2)$ gauge field $Q_\mu$ and an $SU(3)$ gauge field $V_\mu$.
Note that the coefficients in front of $\Lambda_\mu$ in the above formulas are precisely the aforementioned (and correct!) hypercharges of the corresponding particles.
Next, let us turn to the scalar field $\phi$, which is given by $$\begin{aligned}
\label{eq:higgs_field_SM}
\left.\phi\right|_{H_l} &= {
\left(\!\!\!\begin{array}{c@{~}c}0&Y^*\\#3&0\\\end{array}\!\!\!\right)
} , & \left.\phi\right|_{H_q} &= {
\left(\!\!\!\begin{array}{c@{~}c}0&X^*\\#3&0\\\end{array}\!\!\!\right)
} \otimes1_3 , & \left.\phi\right|_{H_{\bar l}} &= 0 , & \left.\phi\right|_{H_{\bar q}} &= 0 ,\end{aligned}$$ where we now have, for complex fields $\phi_1,\phi_2$, $$\begin{aligned}
Y &= {
\left(\!\!\!\begin{array}{c@{~}c}Y_\nu\phi_1&-Y_e\bar\phi_2\\#3&Y_e\bar\phi_1\\\end{array}\!\!\!\right)
} , & X &= {
\left(\!\!\!\begin{array}{c@{~}c}Y_u\phi_1&-Y_d\bar\phi_2\\#3&Y_d\bar\phi_1\\\end{array}\!\!\!\right)
} . \end{aligned}$$ The scalar field $\Phi$ is then given by $$\begin{aligned}
\label{eq:Higgs_field_SM}
\Phi = D_F + {
\left(\!\!\!\begin{array}{c@{~}c}\phi&0\\#3&0\\\end{array}\!\!\!\right)
} + J_F{
\left(\!\!\!\begin{array}{c@{~}c}\phi&0\\#3&0\\\end{array}\!\!\!\right)
}J_F^* = {
\left(\!\!\!\begin{array}{c@{~}c}S+\phi&T^*\\#3&\bar{(S+\phi)}\\\end{array}\!\!\!\right)
} .\end{aligned}$$ Finally, one can compute that the action of the gauge group $\mathrm{SU}(A_F)$ by conjugation on the fluctuated Dirac operator $$\begin{aligned}
D_\omega = \dirac\otimes 1 + \gamma^\mu\otimes B_\mu + \gamma_M\otimes\Phi\end{aligned}$$ is implemented by $$\begin{gathered}
\Lambda_\mu \mapsto \Lambda_\mu - i \lambda\partial_\mu\bar\lambda , \quad
Q_\mu \mapsto qQ_\mu q^* - iq\partial_\mu q^* , \quad
\bar V_\mu \mapsto m\bar V_\mu m^* - im\partial_\mu m^* , \\
H \mapsto \bar\lambda\,q H ,\end{gathered}$$ for $\lambda\in C^\infty\big(M,U(1)\big)$, $q\in C^\infty\big(M,SU(2)\big)$ and $m\in C^\infty\big(M,SU(3)\big)$ and we have written the [*Higgs doublet*]{} as $$H:= {
\left(\!\!\!\begin{array}{c}\phi_1+1\\#2\\\end{array}\!\!\!\right)
}$$ For the detailed computation we refer to [@CCM07 Sect. 3.5] or [@Sui14 Prop. 11.5].
Summarizing, the gauge fields derived take values in the Lie algebra $u(1) \oplus su(2) \oplus su(3)$ and transform according to the usual Standard Model gauge transformations. The scalar field $\phi$ transforms as the Standard Model Higgs field in the defining representation of $SU(2)$, with hypercharge $-1$.
Spectral action
---------------
The spectral action for the above spectral Standard Model has been computed in full detail in [@CCM07 Section 4.2] and confirmed in [*e.g.*]{} [@Sui14 Theorem 11.10]. Since it would lie beyond the scope of the present review, we refrain from repeating this computation. Instead, we summarize the main result, which is that the Lagrangian derived from the spectral action is $$\begin{aligned}
S_B= \int &\Bigg( \frac{48\chi_4\Lambda^4}{\pi^2} - \frac{c\chi_2\Lambda^2}{\pi^2} + \frac{d\chi(0)}{4\pi^2} + \left(\frac{c\chi(0)}{24\pi^2} - \frac{4\chi_2\Lambda^2}{\pi^2} \right) s - \frac{3\chi(0)}{10\pi^2} (C_{\mu\nu\rho\sigma})^2 \notag\\
&\quad+ \frac14 Y_{\mu\nu} Y^{\mu\nu} + \frac14 W_{\mu\nu}^a W^{\mu\nu,a} + \frac14 G_{\mu\nu}^i G^{\mu\nu,i} + \frac{b\pi^2}{2a^2\chi(0)} |H|^4 \notag\\
&\quad- \frac{2a\chi_2\Lambda^2 - e\chi(0)}{a\chi(0)} |H|^2 + \frac{1}{12} s |H|^2 + \frac12 |D_\mu H|^2 \Bigg) \sqrt{g} d^4x ,\end{aligned}$$ where $\chi_j = \int_0^\infty \chi(v) v^{j-1} dv$ are the moments of the function $\chi$, $j>0$, $s=-R$ is the scalar curvature, $Y_{\mu\nu}, W_{\mu\nu}$ and $G_{\mu\nu}$ are the field strengths of $Y_\mu, Q_\mu$ and $V_\mu$, respectively and the covariant derivative $D_\mu H$ is given by $$\begin{aligned}
\label{eq:Higgs_kin_gauge}
D_\mu H = \partial_\mu H + \frac12 i g_2 W_\mu^a \sigma^a H - \frac12 i g_1 Y_\mu H .\end{aligned}$$ Moreover, we have defined the following constants $$\begin{aligned}
\label{eq:abcde_SM}
a &= \operatorname{Tr}\big(Y_\nu^*Y_\nu + Y_e^*Y_e + 3Y_u^*Y_u + 3Y_d^*Y_d\big) , \notag\\
b &= \operatorname{Tr}\big((Y_\nu^*Y_\nu)^2 + (Y_e^*Y_e)^2 + 3(Y_u^*Y_u)^2 + 3(Y_d^*Y_d)^2\big) , \notag\\
c &= \operatorname{Tr}\big(Y_R^*Y_R\big) , \\
d &= \operatorname{Tr}\big((Y_R^*Y_R)^2\big) , \notag\\
e &= \operatorname{Tr}\big(Y_R^*Y_R Y_\nu^*Y_\nu\big) . \notag\end{aligned}$$ The normalization of the kinetic terms imposes a relation between the coupling constants $g_1,g_2,g_3$ and the coefficients $\chi_0$, of the form $$\begin{aligned}
\label{eq:couplings_norm}
\frac{\chi(0)}{2\pi^2} {g_3}^2 = \frac{\chi(0)}{2\pi^2} {g_2}^2 = \frac{5\chi(0)}{6\pi^2} {g_1}^2 = \frac14 .\end{aligned}$$ The coupling constants are then related by $$\begin{aligned}
{g_3}^2 = {g_2}^2 = \frac53 {g_1}^2 ,\end{aligned}$$ which is precisely the relation between the coupling constants at unification, common to grand unified theories (GUT). We shall further discuss this in Section \[sect:pheno\].
Fermionic action in KO-dimension 6
----------------------------------
As already announced above, the shift to KO-dimension 6 for the finite space solved the fermion doubling problem of [@LMMS97]. Let us briefly explain how this works, following [@C06].
The crucial observation is that in KO-dimension $2 \equiv 4+6 \mod 8$ the following pairing $$( \psi, \psi') \mapsto (J \psi, D_\omega \psi')$$ is a skew-symmetric form on the $+1$-eigenspace of $\gamma$ in $\H$. This skew-symmetry is in concordance with the Grassmann nature of fermionic fields $\psi$, guaranteeing that the following action functional is in fact non-zero: $$S_F = \frac 12 \langle J \xi , D_A \xi \rangle$$ for $\xi$ a Grassmann variable in the $+1$-eigenspace of $\gamma$.
This then solves the fermion doubling, or actually quadrupling as follows. First, the restriction to the chiral subspace of $\gamma$ takes care of a factor of two. Then, the functional integral involving anti-commuting Grassman variables delivers a Pfaffian, which takes care of a square root. That this indeed works has been worked out in full detail for the case of the Standard Model in [@CCM07 Section 4.4.1] or [@Sui14 Section 11.4].
Phenomenological consequences {#sect:pheno}
-----------------------------
The first phenomenological consequence one can derive from the spectral Standard Model is an upper bound on the mass of the top quark. In fact, the appearance of the constant $a$ in both the fermionic and the bosonic action allows to derive $$\begin{aligned}
\label{eq:masses_ferm_W}
\operatorname{Tr}\big(m_\nu^*m_\nu + m_e^*m_e + 3m_u^*m_u + 3m_d^*m_d\big) = 2{g_2}^2{v}^2 = 8 {M_W}^2 .\end{aligned}$$ It is natural to assume that the mass $m_{\text{top}}$ of the top quark is much larger than all other fermion masses, except possibly a Dirac mass that arises from the seesaw mechanism as was described above. If we write $m_\nu = \rho m_{\text{top}}$ then the above relation would yield the constraint $$\begin{aligned}
\label{eq:top_mass}
m_{\text{top}} \lesssim \sqrt\frac8{3+\rho^2} M_W .\end{aligned}$$ The relations between the coupling constants and $\chi(0)$ suggests that we have grand unification of the coupling constants. Moreover, from the action functional we see that the quartic Higgs coupling constant $\lambda$ is related to $\chi(0)$ as well via $$\lambda = 24 \frac{b}{a^2} g_2^2.$$ Thus, the spectral Standard Model imposes relations between the coupling constants and bounds on the fermion masses. These relations were used in [@CCM07] as input at (or around) grand unification scale $\Lambda_\GUT$, and then run down using one-loop renormalization group equations to ’low energies’ where falsifiable predictions were obtained.
![Observed and expected exclusion limits for a Standard Model Higgs boson at the 95-percent confidence level for the combined CDF and DZero analyses. (Fermilab) []{data-label="fig:fermilab"}](fermilab.jpg)
In fact, the mass of the top quark can indeed be found to get an acceptable value, however, for the Higgs mass it was found that $$167 \operatorname{GeV}\leq m_h \leq 176 \operatorname{GeV}.$$ Given that there were not much models in particle physics around that could produce falsifiable predictions, it is somewhat ironical that the first exclusion results on the mass of the Higgs that appeared in 2009 from Fermilab hit exactly this region. See Figure \[fig:fermilab\]. And, of course, with the discovery of the Higgs at $m_h \approx 125.5 \operatorname{GeV}$ in [@ATLAS12; @CMS12] one could say that the spectral Standard Model was not in a particularly good shape at that time.
Beyond the Standard Model with noncommutative geometry
======================================================
Even though the incompatibility between the spectral Standard Model and the experimental discovery of the Higgs with a relatively low mass was not an easy stroke at the time, it also led to a period of reflection and reconsideration of the premises of the noncommutative geometric approach. In fact, it was the beginning of yet another exciting chapter in our story on the spectral model of gravity coupled with matter. As we will see in this and the next chapter, once again the input from experiment is taken as a guiding principle in our search for the spectral model that goes Beyond the Standard Model.
\[rem:beyond-sm\] We do not pretend to give a complete overview of the literature here, but only indicate some of the highlights and actively ongoing research areas.
Other searches beyond the Standard Model with noncommutative geometry include [@ISS04; @Ste06; @Ste07; @Ste09; @Ste09b; @Ste13], adopting a slightly different approach to almost-commutative manifolds as we do.
There is another aspect that was studied is the connection between supersymmetry and almost-commutative manifolds. It turned out to be very hard —if not impossible— to combine the two. A first approach is [@Cha94] and more recently the intersection was studied in [@BroS10; @BroS11; @BBS16].
Resilience of the spectral Standard Model
-----------------------------------------
In 2012 it was realized how a small correction of the spectral Standard Model gives an intriguing possibility to go beyond the Standard Model, solving at the same time a problem with the stability of the Higgs vacuum given the measured low mass $m_h$. This is based on [@CC12], but for which some of the crucial ingredients surprisingly enough were already present in the 2010 paper [@CC10].
Namely, in the definition of the finite Dirac operator $D_F$ of Equation \[eq:dirac-sm\], we can replace $Y_R$ by $Y_R \sigma$, where $\sigma$ is a real scalar field on $M$. Strictly speaking, this brings us out of the class of almost-commutative manifolds $M \times F$, since part of $D_F$ now varies over $M$ and this was the main reason why it was disregarded before. However, since from a physical viewpoint there was no reason to assume $Y_R$ to be constant, it was treated as a scalar field already in [@CC10]. This was only fully justified in subsequent papers (as we will see in the next subsections) where the scalar field $\sigma$ arises as the relic of a spontaneous symmetry breaking mechanism, similar to the Higgs field $h$ in the electroweak sector of the Standard Model. We will discuss a few of the existing approaches in the literature in the next few sections. For now, let us simply focus on the phenomenological consequences of this extra scalar field.
Thus we replace $Y_R$ by $Y_R \sigma$ and analyze the additional terms in the spectral action. The scalar sector becomes $$\begin{gathered}
S_H' := \int_M \bigg( \frac{bf(0)}{2\pi^2} |H|^4 - \frac{2af_2\Lambda^2}{\pi^2} |H|^2 +\frac{ef(0)}{\pi^2} \sigma^2|H|^2\\
-\frac{cf_2\Lambda^2}{\pi^2} \sigma^2 + \frac{df(0)}{4\pi^2} \sigma^4 + \frac{af(0)}{2\pi^2} |D_\mu H|^2 + \frac{1}{4 \pi^2} f(0) c ( \partial_\mu \sigma)^2 \bigg) \sqrt{g} dx,\end{gathered}$$ where we ignored the coupling to the scalar curvature.
We exploit the approximation that $m{_{\scriptscriptstyletop}}$, $m_\nu$ and $m_R$ are the dominant mass terms. Moreover, as before we write $m_\nu = \rho m{_{\scriptscriptstyletop}}$. That is, the expressions for $a,b,c,d$ and $e$ in now become $$\begin{aligned}
a &\approx m{_{\scriptscriptstyletop}}^2 (\rho^2 +3),\\
b &\approx m{_{\scriptscriptstyletop}}^4 (\rho^4 +3),\\
c & \approx m_R^2,\\
d&\approx m_R^4,\\
e&\approx \rho^2 m_R^2 m{_{\scriptscriptstyletop}}^2. \end{aligned}$$ In a unitary gauge, where $H = \begin{pmatrix} h \\ 0 \end{pmatrix}$, we arrive at the following potential: $$\L{_{\scriptscriptstylepot}}(h,\sigma) = \frac{1}{24} \lambda_h h^4 + \frac12 \lambda_{h \sigma} h^2 \sigma^2 + \frac14 \lambda_\sigma \sigma^4 - \frac{4 g_2^2}{\pi^2} f_2 \Lambda^2 (h^2 + \sigma^2),$$ where we have defined coupling constants $$\begin{aligned}
\label{eq:scalar-couplings}
\lambda_h &= 24 \frac{\rho^4 + 3}{(\rho^2+3)^2} g_2^2,&
\lambda_{h \sigma} &= \frac{8 \rho^2}{\rho^2 +3} g_2^2,&
\lambda_\sigma &= 8 g_2^2.\end{aligned}$$ This potential can be minimized, and if we replace $h$ by $v+h$ and $\sigma$ by $w+ \sigma$, respectively, expanding around a minimum for the terms quadratic in the fields, we obtain: $$\begin{aligned}
\L{_{\scriptscriptstylepot}}(v+h,w+\sigma)|{_{\scriptscriptstyle\text{quadratic}}} &= \frac16 v^2 \lambda_h v^2 + 2 vw \lambda_{h \sigma} \sigma h + w^2 \lambda_\sigma \sigma^2 \\
&= \frac12 \begin{pmatrix} h & \sigma \end{pmatrix} M^2 \begin{pmatrix} h \\ \sigma \end{pmatrix},\end{aligned}$$ where we have defined the mass matrix $M$ by $$M^2 = 2 \begin{pmatrix} \frac16 \lambda_h v^2 & \lambda_{h \sigma} vw \\ \lambda_{h \sigma} vw & \lambda_\sigma w^2 \end{pmatrix}.$$ This mass matrix can be easily diagonalized, and if we make the natural assumption that $w$ is of the order of $m_R$, while $v$ is of the order of $M_W$, so that $v \ll w$, we find that the two eigenvalues are $$\begin{aligned}
m_+^2 &\sim 2 \lambda_\sigma w^2 + 2 \frac{\lambda_{h \sigma}^2}{\lambda_\sigma} v^2,\\
m_-^2 &\sim 2 \lambda_h v^2 \left( \frac16 - \frac{\lambda_{h \sigma}^2}{\lambda_h \lambda_\sigma}\right).\end{aligned}$$ We can now determine the value of these two masses by running the scalar coupling constants $\lambda_h, \lambda_{h \sigma}$ and $\lambda_\sigma$ down to ordinary energy scalar using the renormalization group equations for these couplings that were derived in [@GLPR10], referring to [@CC12; @Sui14] for full details.
![A contour plot of the Higgs mass $m_h$ as a function of $\rho^2$ and $t = \log (\Lambda{_{\scriptscriptstyleGUT}}/M_Z)$. The red line corresponds to $m_h = 125.5~\operatorname{GeV}$.[]{data-label="fig:higgsmass"}](higgsmass.png)
The result varies with the chosen value for $\Lambda_\GUT$ and the parameter $\rho$. The mass of $\sigma$ is essentially given by the largest eigenvalue $m_+$ which is of the order $10^{12}~ \operatorname{GeV}$ for all values of $\Lambda_\GUT$ and the parameter $\rho$. The allowed mass range for the Higgs, [*i.e.*]{} for $m_-$, is depicted in Figure \[fig:higgsmass\]. The expected value $m_h=125.5 ~\operatorname{GeV}$ is therefore compatible with the above noncommutative model. Moreover, without the $\sigma$ the $\lambda_h$ turns negative at energies around $10^{12} \operatorname{GeV}$. Furthermore, this calculation implies that there is a relation (given by the red line in the Figure) between the ratio $m_\nu/m{_{\scriptscriptstyletop}}$ and the unification scale $\Lambda_\GUT$.
Pati–Salam unification and first-order condition {#sect:patisalam}
------------------------------------------------
In order to see how we one can use the noncommutative geometric approach to go beyond the Standard Model it is important to trace our steps that led to the spectral Standard Model in the previous Section. The route started with the classification of the algebras of the finite space ([*cf.*]{} Equation ). The results show that the only algebras which solve the fermion doubling problem are of the form $M_{2a}(\mathbb{C})\oplus M_{2a}(\mathbb{C})$ where $a$ is an even integer. An arbitrary symplectic constraint is imposed on the first algebra restricting it from $M_{2a}(\mathbb{C})$ to $M_{a}(\mathbb{H}).$ The first non-trivial algebra one can consider is for $a=2$ with the algebra $$M_{2}(\mathbb{H})\oplus M_{4}(\mathbb{C}).$$ Coincidentally, and as explained in the introduction, the above algebra comes out as a solution of the two-sided Heisenberg quantization relation between the Dirac operator $D$ and the two maps from the four spin-manifold and the two four spheres $S^{4}\times S^{4}$ [@CCM14; @CCM15]. This removes the arbitrary symplectic constraint and replaces it with a relation that quantize the four-volume in terms of two quanta of geometry and have far reaching consequences on the structure of space-time. We will come back to this in the last Section.
The existence of the chirality operator $\gamma$ that commutes with the algebra breaks the quaternionic matrices $M_{2}(\mathbb{H})$ to the diagonal subalgebra and leads us to consider the finite algebra $$\mathcal{A}_{F}=\mathbb{H}_{R}\oplus\mathbb{H}_{L}\oplus M_{4}(\mathbb{C}).$$ This algebras is the simplest candidate to search for new physics beyond the Standard Model. In fact, the inner automorphism group of $\mathcal{A=C}^{\infty}\left(
M\right) \otimes\mathcal{A}_{F}$ is recognized as the Pati–Salam gauge group $SU(2)_{R}\times SU(2)_{L}\times SU(4)$, and the corresponding gauge bosons appear as inner perturbations of the (spacetime) Dirac operator [@CCS13b]. Thus, we are considering a spectral Pati–Salam model as a candidate beyond the Standard Model. Let us further analyze this model and its phenomenological consequences.
An element of the Hilbert space $\Psi\in\mathcal{H}$ is represented by $$\Psi_{M}=\left(
\begin{array}
[c]{c}\psi_{A}\\
\psi_{A^{^{\prime}}}\end{array}
\right) ,\quad\psi_{A^{\prime}}=\psi_{A}^{c}$$ where $\psi_{A}^{c}$ is the conjugate spinor to $\psi_{A}.$ Thus all primed indices $A^{\prime}$ correspond to the Hilbert space of conjugate spinors. It is acted on by both the left algebra $M_{2}\left( \mathbb{H}\right) $ and the right algebra $M_{4}\left( \mathbb{C}\right) $. Therefore the index $A$ can take $16$ values and is represented by $$A=\alpha I$$ where the index $\alpha$ is acted on by quaternionic matrices and the index $I$ by $M_{4}\left( \mathbb{C}\right) $ matrices. Moreover, when the grading breaks $M_{2}\left( \mathbb{H}\right) $ into $\mathbb{H}_{R}\oplus\mathbb{H}_{L}$ the index $\alpha$ is decomposed to $\alpha
=\overset{.}{a},a$ where $\overset{.}{a}=\overset{.}{1},\overset{.}{2}$ (dotted index) is acted on by the first quaternionic algebra $\ \mathbb{H}_{R}$ and $a=1,2$ is acted on by the second quaternionic algebra $\ \mathbb{H}_{L}$. When $M_{4}\left( \mathbb{C}\right) $ breaks into $\mathbb{C}\oplus M_{3}\left( \mathbb{C}\right) $ (due to symmetry breaking or through the use of the order one condition as in [@CC07b]) the index $I$ is decomposed into $I=1,i$ and thus distinguishing leptons and quarks, where the $1$ is acted on by the $\mathbb{C}$ and the $i$ by $M_{3}\left(
\mathbb{C}\right) .$ Therefore the various components of the spinor $\psi
_{A}$ are $$\begin{aligned}
\psi_{\alpha I} & =\left(
\begin{array}
[c]{cccc}\nu_{R} & u_{iR} & \nu_{L} & u_{iL}\\
e_{R} & d_{iR} & e_{L} & d_{iL}\end{array}
\right) ,\qquad i=1,2,3\\
& =\left( \psi_{\overset{.}{a}1},\psi_{\overset{.}{a}i},\psi_{a1},\psi
_{ai}\right) ,\qquad a=1,2,\quad\overset{.}{a}=\overset{.}{1},\overset{.}{2}\nonumber\end{aligned}$$ This is a general prediction of the spectral construction that there is $16$ fundamental Weyl fermions per family, $4$ leptons and $12$ quarks.
The (finite) Dirac operator can be written in matrix form$$D_{F}=\left(
\begin{array}
[c]{cc}D_{A}^{B} & D_{A}^{B^{^{\prime}}}\\
D_{A^{^{\prime}}}^{B} & D_{A^{^{\prime}}}^{B^{^{\prime}}}\end{array}
\right) ,\label{eq:dirac}$$ and must satisfy the properties $$\gamma_{F}D_{F}=-D_{F}\gamma_{F}\qquad J_{F}D_{F}=D_{F}J_{F}$$ where $J_{F}^{2}=1.$ A matrix realization of $\gamma_{F}$ and $J_{F}$ are given by $$\gamma_{F}=\left(
\begin{array}
[c]{cc}G_{F} & 0\\
0 & -\overline{G}_{F}\end{array}
\right) ,\qquad G_{F}=\left(
\begin{array}
[c]{cc}1_{2} & 0\\
0 & -1_{2}\end{array}
\right) ,\qquad J_{F}=\left(
\begin{array}
[c]{cc}0_{4} & 1_{4}\\
1_{4} & 0_{4}\end{array}
\right) \circ\mathrm{cc}$$ where $\mathrm{cc}$ stands for complex conjugation. These relations, together with the hermiticity of $D$ imply the relations $$\left( D_{F}\right) _{A^{^{\prime}}}^{B^{^{\prime}}}=\left( \overline
{D}_{F}\right) _{A}^{B}\,\qquad\left( D_{F}\right) _{A^{^{\prime}}}^{B}=\left( \overline{D}_{F}\right) _{B}^{A^{\prime}}$$ and have the following zero components [@CC10] $$\begin{aligned}
\left( D_{F}\right) _{aI}^{bJ} & =0=\left( D_{F}\right) _{\overset{.}{a}I}^{\overset{.}{b}J}\\
\left( D_{F}\right) _{aI}^{\overset{.}{b}^{\prime}J^{\prime}} & =0=\left(
D_{F}\right) _{\overset{.}{a}I}^{b^{\prime}J\prime}$$ leaving the components $\left( D_{F}\right) _{aI}^{\overset{.}{b}J}$, $\left( D_{F}\right) _{aI}^{b^{\prime}J^{\prime}}$ and $\left(
D_{F}\right) _{\overset{.}{a}I}^{\overset{.}{b}^{\prime}J^{\prime}}$ arbitrary. These restrictions lead to important constraints on the structure of the connection that appears in the inner fluctuations of the Dirac operator. In particular the operator $D$ of the full noncommutative space given by $$D=D_{M}\otimes1+\gamma_{5}\otimes D_{F}$$ gets modified to $$D_{A}=D+A_{\left( 1\right) }+JA_{\left( 1\right) }J^{-1}+A_{\left(
2\right) }$$ where $$A_{\left( 1\right) }={\displaystyle\sum}
a\left[ D,b\right] ,\,\qquad A_{2}={\displaystyle\sum}
\widehat{a}\left[ A_{\left( 1\right) },\widehat{b}\right] ,\qquad
\widehat{a}=JaJ^{-1}$$
We have shown in [@CCS13b] that components of the connection $A$ which are tensored with the Clifford gamma matrices $\gamma^{\mu}$ are the gauge fields of the Pati–Salam model with the symmetry of $SU\left( 2\right) _{R}\times
SU\left( 2\right) _{L}\times SU\left( 4\right) .$ On the other hand, the non-vanishing components of the connection which are tensored with the gamma matrix $\gamma_{5}$ are given by $$\left( A\right) _{aI}^{\overset{.}{b}J}\equiv\gamma_{5}
\Sigma _{aI}^{\overset{.}{b}J},\qquad\left( A\right) _{aI}^{b^{\prime}J^{\prime}}=\gamma_{5}H_{aIbJ},\qquad\left( A\right)
_{\overset{.}{a}I}^{\overset{.}{b}^{\prime}J^{\prime}}\equiv\gamma
_{5}H_{\overset{.}{a}I\overset{.}{b}J}$$ where $H_{aIbJ}=H_{bJaI}$ and $H_{\overset{.}{a}I\overset{.}{b}J}=H_{\overset{.}{b}J\overset{.}{a}I}$, which is the most general Higgs structure possible. These correspond to the representations with respect to $SU\left( 2\right) _{R}\times SU\left( 2\right) _{L}\times SU\left(
4\right) :$$$\begin{aligned}
\Sigma_{aI}^{\overset{.}{b}J} & =\left( 2_{R},2_{L},1\right) +\left(
2_{R},2_{L},15\right) \\
H_{aIbJ} & =\left( 1_{R},1_{L},6\right) +\left( 1_{R},3_{L},10\right) \\
H_{\overset{.}{a}I\overset{.}{b}J} & =\left( 1_{R},1_{L},6\right) +\left(
3_{R},1_{L},10\right)\end{aligned}$$ We note, however, that the inner fluctuations form a semi-group and if a component $\left( D_{F}\right) _{aI}^{\overset{.}{b}J}$ or $\left(
D_{F}\right) _{aI}^{b^{\prime}J^{\prime}}$ or $\left( D_{F}\right)
_{\overset{.}{a}I}^{\overset{.}{b}^{\prime}J^{\prime}}$ vanish, then the corresponding $A$ field will also vanish. We can distinguish three cases: 1) Left-right symmetric Pati–Salam model with fundamental Higgs fields $\Sigma_{aI}^{\overset{.}{b}J},$ $H_{aIbJ}$ and $H_{\overset{.}{a}I\overset{.}{b}J}.$ In this model the field $H_{aIbJ}$ should have a zero vev. 2) A Pati–Salam model where the Higgs field $H_{aIbJ}$ that couples to the left sector is set to zero which is desirable because there is no symmetry between the left and right sectors at low energies. 3) If one starts with $\left( D_{F}\right) _{aI}^{\overset{.}{b}J}$ or $\left( D_{F}\right)
_{aI}^{b^{\prime}J^{\prime}}$ or $\left( D_{F}\right) _{\overset{.}{a}I}^{\overset{.}{b}^{\prime}J^{\prime}}$ whose values are given by those that were derived for the Standard Model, then the Higgs fields $\Sigma
_{aI}^{\overset{.}{b}J},$ $H_{aIbJ}$ and $H_{\overset{.}{a}I\overset{.}{b}J}$ will become composite and expressible in terms of more fundamental fields $\Sigma_{I}^{J},$ $\Delta_{\overset{.}{a}J}$ and $\phi_{\overset{.}{a}}^{b}$ . We refer to this as the composite model. It has the scalar field $\sigma$ discussed in the previous section as a remnant after spontaneous symmetry breaking [@CCS13b]. In fact, contrary to some claims in the literature it is possible to perform the potential analysis in this case in unitarity gauge and arrive at the conclusion that the field content contains the scalar field $\sigma$ ([*cf.*]{} Appendix \[app:potential\]).
Depending on the precise particle content we may determine the renormalization group equations of the Pati–Salam gauge couplings $g_{R},g_{L},g$. In [@CCS15] we have run them to look for unification of the coupling $g_{R}=g_{L}=g$. The boundary conditions are taken at the intermediate mass scale $\mu=m_{R}$ to be the usual (e.g. [@Moh86 Eq. (5.8.3)]) $$\frac{1}{g_{1}^{2}}=\frac{2}{3}\frac{1}{g^{2}}+\frac{1}{g_{R}^{2}},\qquad
\frac{1}{g_{2}^{2}}=\frac{1}{g_{L}^{2}},\qquad\frac{1}{g_{3}^{2}}=\frac
{1}{g^{2}},\label{eq:couplings-relations}$$ in terms of the Standard Model gauge couplings $g_{1},g_{2},g_{3}$. At the mass scale $m_{R}$ the Pati–Salam symmetry is broken to that of the Standard Model, and we take it to be the same scale that is present in the see-saw mechanism. It should thus be of the order $10^{11}-10^{13}$GeV. What we have found in [@CCS15] (and this was confirmed by others in [@AMST15]) is that in all three cases it is possible to achieve grand unification of the couplings, while connecting to Standard Model physics in the broken, low-energy phase. An example of a running of the gauge coupling is illustrated in Figure \[fig:ps-running\].
![Running of the gauge couplings of the Standard Model gauge couplings (below scale $m_R \approx 10^{11} \operatorname{GeV}$) and the Pati–Salam gauge coupling (above scale $m_R$) in case 2.[]{data-label="fig:ps-running"}](PSrunningNoOrder1.png)
Grand symmetry and twisted spectral triples
-------------------------------------------
In [@DLM14] the next-to-next case[^3] in the list of irreducible geometries in Equation was considered: $k=4$. Thus, one considers $$A_G = M_4(\bH) \oplus M_8(\C); \qquad H_F := \C^{128}.$$ where $128$ is exactly the number of spinor and internal degrees of freedom combined (including the aforementioned fermion quadruplication). The geometry is then $$\left( C^\infty(M,A_G), L^2(M) \otimes H_F, D_M + \gamma_M D_F \right)$$ where one has to assume that the spinor bundle on $M$ has been trivialized to gather the spinor and internal fermionic degrees of freedom in a single Hilbert space $H_F$.
Note that the above geometry is not a direct product of the continuum with a discrete space. In fact, both the algebra and the Dirac operator $D_M$ contain spinor indices. As a consequence the commutator $[D_M, a]$ can become unbounded, thus challenging one of the basic axioms of spectral triples. Instead, it is possible to guarantee that [*twisted*]{} commutators are bounded so that this example fits in the general framework of twisted spectral triples developed in [@CM08]. In [@DM14] the authors identify an inner automorphism $\rho = R (\cdot )R$ of $A_G$ such that $$[D,a]_\rho = D a - \rho (a) D$$ is bounded.
An interesting question that arises at this point is how to generate inner fluctuations of twisted spectral triples. This was analyzed in full detail from a mathematical viewpoint in [@LM16; @LM17]. One of the intriguing aspects is the self-adjointness of the Dirac operator under fluctuations (even gauge transformations): for this to be respected one has to impose a compatibility between the twist and the fluctuation.
An alternative route was suggested in [@DFLM17]. Namely, one may drop the above condition of self-adjointness and instead look for operators that are Krein-self-adjoint, using the Krein structure on the Hilbert space that is induced by the operator $R$ (defining the twist $\rho$). This will have an intriguing appearance of the Lorentzian structure (given by the Krein inner product) from a purely algebraic and Euclidean starting point. Here we also refer to the nice overview given in [@Liz18].
Algebraic constraints on the finite geometry
--------------------------------------------
An interesting question to consider —in particular in light of theories that go Beyond the Standard Model— is whether one can [*derive*]{} the restricted form of the Dirac operator $D_F$ in . We highlight a few approaches to this question that are present in the literature.
First of all, as mentioned already on page , the form of the $D_F$ in terms of the matrices $Y_\nu, Y_e, Y_u, Y_d$ and $Y_R$ as in Equations and appears naturally in the study of moduli of finite Dirac operators. The only constraint (in addition to the usual conditions layed out in Section \[sect:st\]) there was that the photon remained massless.
An attempt was made to make the latter condition less [*ad hoc*]{} is [@BF13; @BF14; @BF18]. They proposed to generalize noncommutative geometry to non-associative noncommutative geometry, thus allowing for non-associative algebras. The crucial idea —which goes back to Eilenberg— is to combine the (differential) algebra and (Hilbert space) bimodule into a single algebra, and understand the conditions such as commutant property and first-order conditions as consequences of associativity of the pertinent algebra $B$. However, this associativity is a strong constraint and accordingly further restrict the geometry described by $D_F$. Note that non-associative algebras have also been used in the context of noncommutative geometry and particle physics to predict the number of families (to be three) [@TD18]
Another approach to analyzing the form of the Dirac operator $D_F$ by imposing algebraic conditions is taken by [@Dab17; @DAS18]. Here the authors adopt the principle that, similar to differential forms in the continuum, the finite Hilbert space should be a Morita equivalence between $A$ and the Clifford algebra generated by $A_F$ and $D_F$. One finds that the aforementioned form of $D_F$ does not satisfy this condition but additional entries in $D_F$ should be non-zero. This gives rise to a model Beyond the Standard Model: an analysis of the phenomenological consequences is performed in [@KL18; @DS18]. In [@Ayd19] it was then found that this model does not exhibit grand unification of the Standard Model couplings.
Volume quantization and uniqueness of SM
========================================
In the classification of finite noncommutative spaces we arrived at the result that the algebra $\mathcal{A}_{F}=\left( \mathbb{H}_{R}\mathbb{\oplus H}_{L}\right) \oplus M_{4}\left( \mathbb{C}\right) $ was the first possibility out of many of the form $\mathcal{A}_{F}=\left( M_{n}\left(
\mathbb{H}\right) _{R}\mathbb{\oplus}M_{n}\left( \mathbb{H}\right)
_{L}\right) \oplus M_{4n}\left( \mathbb{C}\right) $. in addition we made an assumption, that seemed arbitrary, of the existence of antilinear isometry that reduced the algebra $M_{4n}\left( \mathbb{C}\right) $ to $\left(
M_{n}\left( \mathbb{H}\right) _{R}\mathbb{\oplus}M_{n}\left( \mathbb{H}\right) _{L}\right) $. It is necessary to have a stronger evidence of the uniqueness of our conclusions that helps us to avoid making the above mentioned assumptions. Surprisingly, the new evidence came in the process of solving a seemingly completely independent problem, encoding low dimensional geometries, and in particular dimension four.
Higher form of Heisenberg’s commutation relations
-------------------------------------------------
Starting with the simple example of one dimensional geometries, consider the equation $$U^{\ast}\left[ D,U\right] =1,\qquad U^{\ast}U=1$$ where $D$ is self-adjoint operator. Assuming that the one dimensional space is a closed curve parameterized by coordinate $x$ and the Dirac operator to be $D=-i\frac{d}{dx}+\alpha$ the above equation simplifies to $$-iU^{\ast}dU=dx$$ Writing $U=e^{in\theta}$ we obtain $dx=nd\theta.$ Integrating both sides implies that the length of the one dimensional curve is an integer multiple of $2\pi$, the length of $S^1$$${\displaystyle\oint\limits_{C}}
dx=n\left( 2\pi\right)$$ To adopt this construction to higher dimensions, we note that we can characterize the circle $S^{1}$ by the equation $Y^{A}Y^{A}=1,$ $A=1,2$, $Y^{A\ast}=Y^{A}.$ Assembling the two coordinates $Y^{1},$ $Y^{2}$ in one matrix, define $Y=Y^{A}\Gamma_{A},$ where $\Gamma_{A},$ $A=1,2$ are taken to be $2\times2.$ In addition we identify $\Gamma_{1}=\sigma_{1},$ $\Gamma
_{2}=\sigma_{2},$ the Pauli matrices, and define $\Gamma=-i\Gamma_{1}\Gamma_{2}=\sigma_{3}$ so that $\Gamma_{+}=\frac{1}{2}\left( 1+\Gamma\right)
$ is a projection operator. We notice that we can write $$Y=\left(
\begin{array}
[c]{cc}0 & Y^{1}-iY^{2}\\
Y^{1}+iY^{2} & 0
\end{array}
\right) =\left(
\begin{array}
[c]{cc}0 & U^{\ast}\\
U & 0
\end{array}
\right)$$ where $U=Y^{1}-iY^{2}$ and $U^{\ast}U=1.$ The expression $$\left\langle \Gamma_{+}Y\left[ D,Y\right] \right\rangle =1\label{oned}$$ where $\left\langle {}\right\rangle $ is defined to be the trace over the Clifford algebra defined by $\Gamma_{A},$ gives back the equation $U^{\ast
}\left[ D,U\right] =1.$
For higher dimensional geometries we consider a Riemannian manifold with dimension $n$ and where the algebra $\mathcal{A}$ is taken to be $C^{\infty
}\left( M\right) ,$ the algebra of continuously differentiable functions, while the operator $D$ is identified with the Dirac operator given by $$D_{M}=\gamma^{\mu}\left( \frac{\partial}{\partial x^{\mu}}+\omega_{\mu
}\right) ,$$ where $\gamma^{\mu}=e_{a}^{\mu}\gamma^{a}$ and $\omega_{\mu}=\frac{1}{4}\omega_{\mu bc}\gamma^{bc}$ is the $SO(n)$ Lie-algebra valued spin-connection with the (inverse) vielbein $e_{a}^{\mu}$ being the square root of the (inverse) metric $g^{\mu\nu}=e_{a}^{\mu}\delta^{ab}e_{b}^{\nu}.$ The gamma matrices $\gamma^{a}$ are anti-hermitian $\left( \gamma^{a}\right) ^{\ast
}=-\gamma^{a}$ that define the Clifford algebra $\left\{ \gamma^{a},\gamma
^{b}\right\} =-2\delta^{ab}.$ The Hilbert space $\mathcal{H}$ is the space of square integrable spinors $L^{2}\left( M,S\right) .$ The chirality operator $\gamma$ in even dimensions is then given by $$\gamma=\left( i\right) ^{\frac{n}{2}}\gamma^{1}\gamma^{2}\cdots\gamma^{n}$$ Starting with manifolds of dimension $2$ we first define the two sphere by the equation $Y^{A}Y^{A}=1,$ $A=1,2,3$, $Y^{A\ast}=Y^{A}.$ Assembling the three coordinates $Y^{1},$ $Y^{2},$ $Y^{3}$ in one matrix, defining $Y=Y^{A}\Gamma_{A},$ where $\Gamma_{A},$ $A=1,2,3$ are taken to be $2\times2$ Pauli matrices. Notice that in this case $\Gamma\equiv-i\Gamma_{1}\Gamma_{2}\Gamma_{3}=1$ and to generalize equation (\[oned\]) to two dimensions the factor $\Gamma$ can be dropped, and we write instead$$\frac{1}{2!}\left\langle Y\left[ D,Y\right] ^{2}\right\rangle =\gamma
\label{heisenberg2}$$ The reason we have to include the chirality operator $\gamma$ on the two dimensional manifold $M$ is that the Dirac operator $D$ appears twice yielding a product of the form $\gamma_{1}\gamma_{2}=-i\gamma.$ A simple calculation shows that the above equation in component form is given by $$\frac{1}{2!}\epsilon^{\mu\nu}\epsilon_{ABC}Y^{A}\partial_{\mu}Y^{B}\partial_{\nu}Y^{C}=\det\left( e_{\mu}^{a}\right)$$ which is a constraint on the volume form of $M_{2}.$ This implies that the volume of $M_{2}$ will be an integer multiple of the area of the unit $2$-sphere$$\begin{aligned}
{\displaystyle\int\limits_{M_{2}}}
d^{2}x\sqrt{g} & ={\displaystyle\int}
\epsilon_{ABC}Y^{A}dY^{B}dY^{C}\\
& =n(4\pi)\end{aligned}$$ where $n$ is the winding number. An example of a map $Y$ with winding number $n$ is $$Y\equiv Y^{1}+iY^{2}=\frac{2z^{n}}{\left\vert z\right\vert ^{2n}+1},\qquad
Y^{3}=\frac{\left\vert z\right\vert ^{2n}-1}{\left\vert z\right\vert ^{2n}+1},\qquad z=x^{1}+ix^{2}$$ From this we deduce that the pullback $Y^{\ast}\left( w_{n}\right) $ is a differential form that does not vanish anywhere. This in turn implies that the Jacobian of the map $Y$ does not vanish anywhere, and that $Y$ is a covering of the sphere. The sphere is simply connected, and on each connected component $M_{j}\subset M_{n}$, the restriction of the map $Y$ to $M_{j}$ is a diffeomorphism, implying that the manifold must be disconnected, with each piece having the topology of a sphere. To allow for two dimensional manifolds with arbitrary topology, our first observation is that condition (\[heisenberg2\]) involves the commutator of the Dirac operator $D$ and the coordinates $Y.$ In momentum space $D$ is the Feynman-slashed $\gamma^{\mu
}p_{\mu}$ momentum and $Y$ are the Feynman-slashed coordinates. This suggests that the quantization condition is a higher form of Heisenberg commutation relation quantizing the phase space formed by coordinates and momenta. We first notice that although the quantization condition is given in terms of the noncommutative data, the operator $J$ is the only one missing. We therefore modify the condition to take $J$ into account. The operator $J$ transforms $Y$ into its commutant $Y^{\prime}=iJYJ^{-1}$ so that $\left[ Y,Y^{\prime
}\right] =0$. Thus let $Y=Y^{A}\Gamma_{A}$ and $Y^{\prime}=iJYJ^{-1}$ and $\Gamma_{A}^{\prime}=iJ\Gamma_{A}J^{-1}$ so that we can write $$Y=Y^{A}\Gamma_{A},\qquad Y^{\prime}=Y^{\prime A}\Gamma_{A}^{\prime},$$ satisfying $Y^{2}=1$ and $Y^{\prime2}=1$ with the Clifford algebras $C_{\pm}$$$\begin{aligned}
\left\{ \Gamma_{A},\Gamma_{B}\right\} & =2\,\delta_{AB},\ \quad(\Gamma
_{A})^{\ast}=\Gamma_{A}\label{Cplus}\\
\left\{ \Gamma_{A}^{\prime},\Gamma_{B}^{\prime}\right\} & =-2\,\delta
_{AB},\ \quad(\Gamma_{A}^{\prime})^{\ast}=-\Gamma_{A}^{\prime}\label{Cminus}$$ We immediately see that the Clifford algebra $C_{+}=M_{2}\left(
\mathbb{C}\right) $ and $C_{-}=\mathbb{H}.$ We then define the projection operator $e=\frac{1}{2}\left( 1+Y\right) $ satisfying $e^{2}=e$ and similarly $e^{\prime}=\frac{1}{2}\left( 1+Y^{\prime}\right) $ satisfying $e^{\prime2}=e^{\prime}.$ From the tensor product of $E=ee^{\prime}$ satisfying $E^{2}=E,$ we construct $Z=2E-1$ satisfying $Z^{2}=1$ and allowing us to write $$\frac{1}{2}\left\langle Z\left[ D,Z\right] ^{2}\ \right\rangle =\gamma$$ A straightforward calculation reveals that this relation splits as the sum of two non-interfering parts$$\frac{1}{2}\left\langle Y\left[ D,Y\right] ^{2}\right\rangle +\frac{1}{2}\left\langle Y^{\prime}\left[ D,Y^{\prime}\right] ^{2}\right\rangle
=\gamma$$ which in component form reads$$\frac{1}{2!}\epsilon^{\mu\nu}\epsilon_{ABC}\left( Y^{A}\partial_{\mu}Y^{B}\partial_{\nu}Y^{C}+Y^{^{\prime}A}\partial_{\mu}Y^{^{\prime}B}\partial_{\nu}Y^{^{\prime}C}\right) =\det\left( e_{\mu}^{a}\right)$$ We will show later, when considering the four dimensional case that this modification allows to reconstruct two dimensional manifolds of arbitrary topology from the pullbacks of the maps $Y,$ $Y$’.
For three dimensional manifolds $\gamma=1$ and in analogy with the one-dimensional case we write $$\frac{1}{3!}\left\langle \Gamma_{+}Y\left[ D,Y\right] ^{3}\right\rangle
=1\label{universal}$$ where $Y=Y^{A}\Gamma_{A},$ $A=1,\ldots4,$ $Y^{2}=1,$ $Y=Y^{\ast},$ $\Gamma
_{A}$ are $4\times4$ Clifford algebra matrices $C_{+}$ where $\left\{
\Gamma_{A},\Gamma_{B}\right\} =2\,\delta_{AB}$. In this representation of the $\Gamma$ matrices we have $\Gamma=\Gamma_{5}=\Gamma_{1}\Gamma_{2}\Gamma
_{3}\Gamma_{4}=\left(
\begin{array}
[c]{cc}1_{2} & 0\\
0 & -1_{2}\end{array}
\right) $ so that $\Gamma_{+}=\frac{1}{2}\left( 1+\Gamma\right) $ is a projection operator. In $d=3,$ we can write $$Y=Y^{A}\Gamma_{A}=\left(
\begin{array}
[c]{cc}0 & U^{\ast}\\
U & 0
\end{array}
\right)$$ where $U$ is a unitary $2\times2$ matrix such that it could be written in the form $U=\exp\left( i\left( \alpha_{0}1+\alpha_{a}\sigma^{a}\right) \right)
$ so that $U^{\ast}U=1$. It is easy to check that $\left\langle Y\left[
D,Y\right] ^{3}\right\rangle =0$ and that the component form of the above relation is $$\det\left( e_{\mu}^{a}\right) =\frac{1}{3!}\epsilon^{\mu\nu\rho}\mathrm{Tr}\left( U^{\ast}\partial_{\mu}UU^{\ast}\partial_{\nu}UU^{\ast
}\partial_{\rho}U\right)$$ whose integral is the winding number of the $SU(2)$ group manifold. Again, using the reality operator $J$ we act on the Clifford algebra $Y^{\prime}=iJYJ^{-1}$ so that $\left[ Y,Y^{\prime}\right] =0$, then $\Gamma_{A}^{\prime}=iJ\Gamma_{A}J^{-1}$ satisfies $\left\{ \Gamma_{A}^{\prime
},\Gamma_{B}^{\prime}\right\} =-2\,\delta_{AB},$ $(\Gamma_{A}^{\prime})^{\ast}=-\Gamma_{A}^{\prime}$. Forming the projection operators $e=\frac
{1}{2}\left( 1+Y\right) ,$ $e^{\prime}=\frac{1}{2}\left( 1+Y^{\prime
}\right) $, we form the tensor product $E=ee^{\prime}$ we define the field $Z=2E-1,$ and thus the two sided relation becomes $$\frac{1}{3!}\left\langle \Gamma_{+}\Gamma_{+}^{\prime}Z\left[ D,Z\right]
^{3}\right\rangle =1$$ A lengthy calculation shows that the component form of this relation separates into two parts without interference terms$$\begin{aligned}
\det\left ( e_{\mu}^{a}\right) & =\frac{1}{3!}\epsilon^{\mu\nu\rho}\bigg(
\mathrm{Tr}\left( U^{\ast}\partial_{\mu}UU^{\ast}\partial_{\nu}UU^{\ast
}\partial_{\rho}U\right)\\
&\qquad \qquad +\mathrm{Tr}\left( U^{^{\prime}\ast}\partial_{\mu
}U^{\prime}U^{^{\prime}\ast}\partial_{\nu}U^{\prime}U^{^{\prime}\ast}\partial_{\rho}U^{\prime}\right) \bigg)\end{aligned}$$ Finally, for four dimensional manifolds the Clifford algebras $C_{+}$ and $C_{-}$ defined as in (\[Cplus\]) (\[Cminus\]) with $\Gamma_{A},$ $\Gamma_{A}^{\prime}$, $A=1,\cdots,5$ are known to be given by $C_{+}=M_{2}\left( \mathbb{H}\right) $ and $C_{-}=M_{4}\left(
\mathbb{C}\right) .$ The quantization condition takes the same form as the two dimensional case$$\frac{1}{4!}\left\langle Z\left[ D,Z\right] ^{4}\ \right\rangle
=\gamma\label{Heisenberg}$$ This relation separates into two non-interfering terms $$\frac{1}{4!}\left\langle Y\left[ D,Y\right] ^{4}\ \right\rangle +\frac
{1}{4!}\left\langle Y^{\prime}\left[ D,Y^{\prime}\right] ^{4}\ \right\rangle
=\gamma$$ the component form of which is given by $$\begin{aligned}
\det\left( e_{\mu}^{a}\right) &=\frac{1}{4!}\epsilon^{\mu\nu\kappa\lambda
}\epsilon_{ABCDE}\bigg( Y^{A}\partial_{\mu}Y^{B}\partial_{\nu}Y^{C}\partial_{\kappa}Y^{D}\partial_{\lambda}Y^{E}\\
&\qquad \qquad \qquad +Y^{^{\prime}A}\partial_{\mu
}Y^{^{\prime}B}\partial_{\nu}Y^{^{\prime}C}\partial_{\kappa}Y^{^{\prime}D}\partial_{\lambda}Y^{^{\prime}E}\bigg)\end{aligned}$$ One can verify that similar considerations fail when the dimension of the manifold $n>4$ as there are interference terms between the $Y$ and $Y^{\prime
}.$ Integrating both sides imply$${\displaystyle\int\limits_{M_{4}}}
d^{4}x\sqrt{g}=\frac{8}{3}\pi^{2}\left( N+N^{\prime}\right)$$ where $N$, $N^{\prime}$ are the winding numbers of the two maps $Y,$ $Y^{\prime}.$ An example of a map $Y$ with winding number $n$ is given by $$\begin{aligned}
Y & \equiv Y^{4}1+Y^{i}e_{i}=\frac{2x^{n}}{x^{n}\overline{x}^{n}+1},\\
Y^{5} & =\frac{x^{n}\overline{x}^{n}-1}{x^{n}\overline{x}^{n}+1},\end{aligned}$$ where $x=x^{4}1+x^{i}e_{i}$ and $e_{i},$ $i=1,2,3$ are the quaternionic complex structures $e_{i}^{2}=-1,$ $e_{i}e_{j}=\epsilon_{ijk}e_{k},$ $i\neq
j.$
Volume quantization
-------------------
Consider the smooth maps $\phi_{\pm}:M_{n}\rightarrow S^{n}$ then their pullbacks $\phi_{\pm}^{\ast}$ would satisfy $$\phi_{+}^{\ast}\left( \alpha\right) +\phi_{-}^{\ast}\left( \alpha\right)
=\omega, \label{integer}$$ where $\alpha$ is the volume form on the unit sphere $S^{n}$ and $\omega\left( x\right) $ is an $n-$form that does not vanish anywhere on $M_{n}.$ We have shown that for a compact connected smooth oriented manifold with $n<4$ one can find two maps $\phi_{+}^{\ast}\left( \alpha\right) $ and $\phi_{-}^{\ast}\left( \alpha\right) $ whose sum does not vanish anywhere, satisfying equation (\[integer\]) such that $\int \omega\in\mathbb{Z}.$ The proof for $n=4$ is more difficult and there is an obstruction unless the second Stieffel–Whitney class $w_{2}$ vanishes, which is satisfied if $M$ is required to be a spin-manifold and the volume to be larger than or equal to five units. The key idea in the proof is to note that the kernel of the Jacobian of the map $Y$ is a hypersurface $\Sigma$ of co-dimension $2$ and therefore $$\dim\Sigma=n-2.$$ We can then construct a map $Y^{\prime}=Y\circ\psi$ where $\psi$ is a diffeomorphism on $M$ such that the sum of the pullbacks of $Y$ and $Y^{\prime}$ does not vanish anywhere. The coordinates $Y$ are defined over a Clifford algebra $C_{+}$ spanned by $\left\{ \Gamma_{A},\Gamma_{B}\right\}
=2\delta_{AB}.$ For $n=2$, $C_{+}=M_{2}\left( \mathbb{C}\right) $ while for $n=4$, $C_{+}=M_{2}\left( \mathbb{H}\right) \oplus M_{2}\left(
\mathbb{H}\right) $ where $\mathbb{H}$ is the field of quaternions. However, for $n=4,$ since we will be dealing with irreducible representations we take $C_{+}=M_{2}\left( \mathbb{H}\right) .$ Similarly the coordinates $Y^{\prime}$ are defined over the Clifford algebra $C_{-}$ spanned by $\left\{ \Gamma_{A}^{\prime},\Gamma_{B}^{\prime}\right\} =-2\delta_{AB}$ and for $n=2$, $C_{-}=\mathbb{H\oplus H}$ and for $n=4$, $C_{-}=M_{4}\left(
\mathbb{C}\right) .$ The operator $J$ acts on the two algebras $C_{+}\oplus
C_{-}$ in the form $J\left( x,y\right) =\left( y^{\ast},x^{\ast}\right) $ (i.e. it exchanges the two algebras and takes the Hermitian conjugate). The coordinates $Z=\frac{1}{2}\left( Y+1\right) \left( Y^{\prime}+1\right)
-1,$ then define the matrix algebras [@CCM14] $$\begin{aligned}
\mathcal{A}_{F} & =M_{2}\left( \mathbb{C}\right) \oplus\mathbb{H},\qquad
n=2\\
\mathcal{A}_{F} & =M_{2}\left( \mathbb{H}\right) \oplus M_{4}\left(
\mathbb{C}\right) ,\qquad n=4.\end{aligned}$$ One, however, must remember that the maps $Y$ and $Y^{\prime}$ are functions of the coordinates of the manifold $M$ and therefore the algebra associated with this space must be $$\begin{aligned}
\mathcal{A} & =C^{\infty}\left( M,\mathcal{A}_{F}\right) \\
& =C^{\infty}\left( M\right) \otimes\mathcal{A}_{F}.\end{aligned}$$ To see this consider, for simplicity, the $n=2$ case with only the map $Y.$ The Clifford algebra $C_{-}=\mathbb{H}$ is spanned by the set $\left\{
1,\Gamma^{A}\right\} ,$ $A=1,2,3,$ where $\left\{ \Gamma^{A},\Gamma
^{B}\right\} =-2\delta^{AB}.$ We then consider functions which are made out of words of the variable $Y$ formed with the use of constant elements of the algebra [@C00] $${\displaystyle\sum\limits_{i=1}^{\infty}}
a_{1}Ya_{2}Y\cdots a_{i}Y,\qquad a_{i}\in\mathbb{H},$$ which will generate arbitrary functions over the manifold which is the most general form since $Y^{2}=1$. One can easily see that these combinations generate all the spherical harmonics. This result could be easily generalized by considering functions of the fields $$Z=\frac{1}{2}\left( Y+1\right) \left( Y^{\prime}+1\right) -1,\qquad
Y\in\mathbb{H},\quad Y^{\prime}\in M_{2}\left( \mathbb{C}\right) ,$$ showing that the noncommutative algebra generated by the constant matrices and the Feynman slash coordinates $Z$ is given by [@C00] $$\mathcal{A}=C^{\infty}\left( M_{2}\right) \otimes\left( \mathbb{H+}M_{2}\left( \mathbb{C}\right) \right) .$$ We now restrict ourselves to the physical case of $n=4.$ Here the algebra is given by $$\mathcal{A}=C^{\infty}\left( M_{4}\right) \otimes\left( M_{2}(\mathbb{H})\mathbb{+}M_{4}\left( \mathbb{C}\right) \right) .$$ The associated Hilbert space is $$\mathcal{H}=L^{2}\left( M_{4},S\right) \otimes\mathcal{H}_{F}.$$ The Dirac operator mixes the finite space and the continuous manifold non-trivially$$D=D_{M}\otimes1+\gamma_{5}\otimes D_{F},$$ where $D_{F\text{ }}$ is a self adjoint operator in the finite space. The chirality operator is $$\gamma=\gamma_{5}\otimes\gamma_{F},$$ and the anti-unitary operator $J$ is given by $$J=J_{M}\gamma_{5}\otimes J_{F},$$ where $J_{M}$ is the charge-conjugation operator $C$ on $M$ and $J_{F}$ the anti-unitary operator for the finite space. Thus an element $\Psi
\in\mathcal{H}$ is of the form $\Psi=\left(
\begin{array}
[c]{c}\psi_{A}\\
\psi_{A^{\prime}}\end{array}
\right) $ where $\psi_{A}$ is a $16$ component $L^{2}\left( M,S\right) $ spinor in the fundamental representation of $\mathcal{A}_{F}$ of the form $\psi_{A}=\psi_{\alpha I}$ where $\alpha=1,\cdots,4$ with respect to $M_{2}\left( \mathbb{H}\right) $ and $I=1,\cdots,4$ with respect to $M_{4}\left( \mathbb{C}\right) $ and where $\psi_{A^{\prime}}=C\psi
_{A}^{\ast}$ is the charge conjugate spinor to $\psi_{A}$ [@CC10]. The chirality operator $\gamma$ must commute with elements of $\mathcal{A}$ which implies that $\gamma_{F}$ must commute with elements in $\mathcal{A}_{F}.$ Commutativity of the chirality operator $\gamma_{F}$ with the algebra $\mathcal{A}_{F}$ and that this $\mathbb{Z}/2$ grading acts non-trivially reduces the algebra $M_{2}\left( \mathbb{H}\right) $ to $\mathbb{H}_{R}\oplus\mathbb{H}_{L}$ [@CCM14]. Thus the $\gamma_{F}$ is identified with $\gamma_{F}=\Gamma^{5}=\Gamma^{1}\Gamma^{2}\Gamma^{3}\Gamma^{4}$ and the finite space algebra reduces to $$\mathcal{A}_{F}=\mathbb{H}_{R}\oplus\mathbb{H}_{L}\oplus M_{4}\left(
\mathbb{C}\right) .$$ This can be easily seen by noting that an element of $M_{2}\left(
\mathbb{H}\right) $ takes the form $\left(
\begin{array}
[c]{cc}q_{1} & q_{2}\\
q_{3} & q_{4}\end{array}
\right) $ where each $q_{i},$ $i=1,\cdots,4,$ is a $2\times2$ matrix representing a quaternion. Taking the representation of $\Gamma^{5}=\left(
\begin{array}
[c]{cc}1_{2} & 0\\
0 & -1_{2}\end{array}
\right) $ to commute with $M_{2}\left( \mathbb{H}\right) $ implies that $q_{2}=0=q_{3},$ thus reducing the algebra to $\mathbb{H}_{R}\oplus
\mathbb{H}_{L}.$ Therefore the index $\alpha=1,\cdots,4$ splits into two parts, $\overset{.}{a}=\overset{.}{1},\overset{.}{2}$ which is a doublet under $\mathbb{H}_{R}$ and $a=1,2$ which is a doublet under $\mathbb{H}_{L}$. The spinor $\Psi$ further satisfies the chirality condition $\gamma\Psi=\Psi$ which implies that the spinors $\psi_{\overset{.}{a}I}$ are in the $\left(
2_{R},1_{L},4\right) $ with respect to the algebra $\mathbb{H}_{R}\mathbb{\oplus H}_{L}\oplus M_{4}\left( \mathbb{C}\right) $ while $\psi
_{aI}$ are in the $\left( 1_{R},2_{L},4\right) $ representation. The finite space Dirac operator $D_{F}$ is then a $32\times32$ Hermitian matrix acting on the $32$ component spinors $\Psi.$ In addition we take three copies of each spinor to account for the three families, but will omit writing an index for the families. At present we have no explanation for why the number of generations should be three. The Dirac operator for the finite space is then a $96\times96$ Hermitian matrix. The Dirac action is then given by [@CCM07] $$\left( J\Psi,D\Psi\right) .$$ We note that we are considering compact spaces with Euclidean signature and thus the condition $J\Psi=\Psi$ could not be imposed. It could, however, be imposed if the four dimensional space is Lorentzian [@Bar06].The reason is that the $KO$ dimension of the finite space is $6$ because the operators $D_{F},$ $\gamma_{F}$ and $J_{F}$ satisfy$$J_{F}^{2}=1,\qquad J_{F}D_{F}=D_{F}J_{F},\qquad J_{F}\gamma_{F}=-\gamma
_{F}J_{F}.$$ The operators $D_{M},$ $\gamma_{M}=\gamma_{5},$ and $J_{M}=C$ for a compact manifold of dimension $4$ satisfy $$J_{M}^{2}=-1,\qquad J_{M}D_{M}=D_{M}J_{M},\qquad J_{M}\gamma_{5}=\gamma
_{5}J_{M}. \label{Euclidean}$$ Thus the $KO$ dimension of the full noncommutative space $\left(
\mathcal{A},\mathcal{H},D\right) $ with the decorations $J$ and $\gamma$ included is $10$ and satisfies $$J^{2}=-1,\qquad JD=DJ,\qquad J\gamma=-\gamma J.$$ We have shown in [@CCM07] that the path integral of the Dirac action, thanks to the relations $J^{2}=-1$ and $J\gamma=-\gamma J$, yields a Pfaffian of the operator $D$ instead of its determinant and thus eliminates half the degrees of freedom of $\Psi$ and have the same effect as imposing the condition $J\Psi=\Psi.$
We have also seen that the operator $J$ sends the algebra $\mathcal{A}$ to its commutant, and thus the full algebra acting on the Hilbert space $\mathcal{H}$ is $\mathcal{A\otimes A}^{o}.$ Under automorphisms of the algebra $$\Psi\rightarrow U\Psi,$$ where $U=u\widehat{u}$ with $u\in\mathcal{A},$ $\widehat{u}\in\mathcal{A}^{o}$ with $\left[ u,\widehat{u}\right] =0$, it is clear that Dirac action is not invariant.
At this point it is clear that we have retrieved all our conclusions we have before arriving at a unique possibility, which is to have a noncommutative space corresponding to the Pati–Salam Model we considered before, and in the special case where the Dirac operator and algebra satisfy the order one condition, the result is the noncommutative space of the Standard Model. We have thus succeeded in obtaining the Pati–Salam Model and Standard Model as unique possibilities starting with the two sided Heisenberg like equation (\[Heisenberg\]) thus eliminating all other possibilities obtained in classifying finite noncommutative spaces of KO dimension $6.$ There is no need to assume the existence of an isometry that reduces the first algebra from $M_{4}\left( \mathbb{C}\right) $ to $M_{2}\left( \mathbb{H}\right) $, and no need to assume that the KO dimension of the finite space to be $6.$ These results are very satisfactory and serve to enhance our confidence of the fine structure of space time as given by the above derived noncommutative space.
Outlook: towards quantization
=============================
Starting with the simple observation that the Higgs field could be interpreted as the link between two parallel sheets separated by a distance of the order of $10^{-16}$ cm it took enormous effort to identify a noncommutative space where the spectrum of the Standard Model could fit. Small deviations from the model, such as the need for a real structure and a KO dimension $6$, were taken as input to fine tune and determine precisely the noncommutative space. The spectral action principle proved to be very efficient way in evaluating the bosonic sector of the theory. Having identified the noncommutative space, the next target was to understand why nature would chose the Standard Model and not any other possibility. A classification of finite spaces revealed the special nature of the the finite part of the noncommutative space identified. Work on encoding manifolds with dimensions equal to four satisfying a higher form of Heisenberg type equation showed that the most general solution of this equation is that of a noncommutative space which is a product of a four-dimensional Riemannian spin-manifold times the finite space corresponding to a Pati–Salam unification model. The Standard Model is a special case of this space where a first order differential condition is satisfied. After a long journey the reasons why nature chose the Standard Model is now reduced to determining solutions of a higher form of Heisenberg equation. With such little input, it is quite satisfying to learn that it is possible to answer many of the questions which puzzled theorists for a long time. We now know why there are 16 fermions per generation, why the gauge group is $SU\left(
3\right) \times SU\left( 2\right) \times U\left( 1\right) ,$ an explanation of the Higgs field and origin of spontaneous symmetry breaking. The Spectral model also predicts a Majorana mass for the right-handed neutrinos and explains the see-saw mechanism. We thus understand unification of all fundamental forces as a geometrical theory based on the spectral action principle of a noncommutative space.
Naturally, there are many questions that are still unanswered, and this motivates the need for further research to address these problems using noncommutative geometry considerations. To conclude, we mention few of the possible directions of future research. One important aspect to consider is the renormalizability properties of the spectral model. Another problem is to study the quantum properties of the Dirac operator and whether it could be related to the pullbacks of the maps used in determining the quanta of geometry. The future of noncommutative geometry in the program of unification of all fundamental interactions looks now to be very promising.
Pati–Salam model: potential analysis {#app:potential}
====================================
We here include the scalar potential analysis for the composite Pati–Salam model, as described in Section \[sect:patisalam\] above.
If there is unification of lepton and quark couplings, then $\rho=1$ so that the $\Sigma^I_J$-field decouples. In that case we have $$\begin{gathered}
\L_{pot} (\phi_{\overset{.}{a}}^{b},\Delta_{\overset{.}{a}I})=
-\mu^2 \phi_{\overset{.}{a}}^{c} \phi_{c}^{\overset{.}{a}}
- \nu^2 \left( \Delta_{\overset{.}{a}K}\overline{\Delta}^{\overset{.}{a}K}\right) ^{2}+ \lambda_{\Sigma}
\phi_{a}^{\overset{.}{c}}
\phi_{\overset{.}{c}}^{b}
\phi_{b}^{\overset{.}{d}}
\phi_{\overset{.}{d}}^{a}
\\
+ \lambda_H \left(\Delta_{\overset{.}{a}K}\overline{\Delta}^{\overset{.}{a}L}\Delta_{\overset{.}{b}L}\overline{\Delta}^{\overset{.}{b}K} \right)^2
+\lambda_{H \Sigma}
\left( \Delta_{\overset{.}{a}J}\overline{\Delta}^{\overset{.}{a}J}\Delta_{\overset{.}{c}I}\overline{\Delta
}^{\overset{.}{d}I}\right) \phi_{b}^{\overset{.}{c}} \phi_{\overset{.}{d}}^{b}\end{gathered}$$ where we have absorbed some constant factors by redefining the couplings $\lambda_H, \lambda_{H \Sigma}$ and $\lambda_\Sigma$.
We choose unitarity gauge for the $\Delta$ and $\phi$-fields, in the following precise sense.
\[lma:unitary-gauge1\] For each value of the fields $\{ \phi_{\dot a}^b, \Delta_{\dot a I}\}$ there is an element $(u_R,u_L,u) \in SU(2)_R \times SU(2)_L \times SU(4)$ such that $$\begin{aligned}
u_R \begin{pmatrix} \phi_{\dot 1}^1 & \phi_{\dot 1}^2 \\[1mm] \phi_{\dot 2}^1 & \phi_{\dot 2}^2
\end{pmatrix} u_L^* & = \begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix}
\intertext{and}
u_R \begin{pmatrix} \Delta_{\dot 1 1}& \Delta_{\dot 1 2}& \Delta_{\dot 1 3}& \Delta_{\dot 1 4}\\ \Delta_{\dot 2 1}& \Delta_{\dot 2 2}& \Delta_{\dot 2 3}& \Delta_{\dot 2 4}\end{pmatrix} u^t &= \begin{pmatrix} 1+\delta_0 & 0& 0 & 0 \\ \delta_1 & \eta_1& 0 &0 \end{pmatrix}\end{aligned}$$ where $h ,\delta_0, \delta_1, \eta_1$ are real fields and $\chi$ is a complex field.
Consider the singular value decomposition of the $2 \times 2$ matrix $(\phi_{\dot a}^b)$: $$\begin{pmatrix} \phi_{\dot 1}^1 & \phi_{\dot 1}^2 \\[1mm] \phi_{\dot 2}^1 & \phi_{\dot 2}^2
\end{pmatrix} = U \begin{pmatrix} h & 0 \\ 0 & k \end{pmatrix} V^*$$ for unitary $2 \times 2$ matrices $U,V$ and real coefficients $h,k$. If we define $$\begin{aligned}
u_R &= \begin{pmatrix} 1 & 0 \\ 0 & \det U\end{pmatrix}U^* \in SU(2)_R\\
u_L &= \begin{pmatrix} 1 & 0 \\ 0 & \det V\end{pmatrix} V^*\in SU(2)_L\end{aligned}$$ it follows that $$\begin{aligned}
u_R \begin{pmatrix} \phi_{\dot 1}^1 & \phi_{\dot 1}^2 \\[1mm] \phi_{\dot 2}^1 & \phi_{\dot 2}^2
\end{pmatrix} u_L^* & = \begin{pmatrix} h & 0 \\ 0 & k \det UV^* \end{pmatrix}
=: \begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix}.\end{aligned}$$ Next, we consider $\Delta_{\dot aI}$ and write $$\left(\Delta_{\dot aI}\right) = \begin{pmatrix} u_1^* \\ u_2^* \end{pmatrix},\qquad \text{with } u_a^* = \begin{pmatrix} \Delta_{\dot a 1}& \Delta_{\dot a 2}& \Delta_{\dot a 3}& \Delta_{\dot a 4} \end{pmatrix}$$ for $a=1,2$. We may suppose that the vectors $u_1,u_2$ are such that their inner product $u_1^* u_2$ is a real number. Indeed, if this is not the case, then multiply $\Delta_{\dot aI}$ by a matrix in $SU(2)_R$ as follows: $$\begin{pmatrix} u_1^* \\ u_2^* \end{pmatrix}\mapsto
\begin{pmatrix} \alpha & 0 \\ 0 & \alpha^* \end{pmatrix}\begin{pmatrix} u_1^* \\ u_2^* \end{pmatrix} = \begin{pmatrix} \alpha u_1^* \\ \alpha^* u_2^* \end{pmatrix}.$$ Now the inner product is $(\alpha^* u_1)^* \alpha u_2 = (\alpha)^2 u_1^* u_2$ and we may choose $\alpha$ so as to cancel the phase of $u_1^* u_2$. Moreover, this transformation respects the above form of $\phi_{\dot a}^b$ after a $SU(2)_L$-transformation of exactly the same form: $$\begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix} \mapsto
\begin{pmatrix} \alpha & 0 \\ 0 & \alpha^* \end{pmatrix} \begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix} \begin{pmatrix} \alpha & 0 \\ 0 & \alpha^* \end{pmatrix} ^* = \begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix} .$$ Thus let us continue with the vectors $u_1,u_2$ satisfying $u_1^* u_2 \in \R$. We apply Gramm-Schmidt orthonormalization to $u_1$ and $u_2$, to arrive at the following orthonormal set of vectors $\{e_1,e_2\}$ in $\C^4$: $$e_1 = \frac{u_1 }{\| u_1\|}; \qquad e_2 = \frac{u_2 - \frac{u_1^* u_2}{\|u_1\|} u_1}{\|u_2 - \frac{u_1^* u_2}{\|u_1\|} u_1\|}.$$ We complete this set by choosing two additional orthonormal vectors $e_3$ and $e_4$ and write a unitary $4 \times 4$ matrix: $$U = \begin{pmatrix} e_1 & e_2 & e_3 & e_4 \end{pmatrix}$$ The sought-for matrix $u \in SU(4)$ is determined by $$u^t = U \begin{pmatrix} 1_3 & 0 \\ 0 & \det U^* \end{pmatrix}$$ so as to give $$\left(\Delta_{\dot aI}\right) u^t = \begin{pmatrix} u_1^* e_1 & 0 & 0 & 0 \\ u_2^* e_1 & u_2^* e_2 & 0 & 0 \end{pmatrix} =: \begin{pmatrix} 1+\delta_0 & 0 & 0 & 0 \\ \delta_1 & \eta_1 & 0 & 0 \end{pmatrix}\qedhere$$
Note that this is compatible with the dimension of the quotient of the space of field values by the group. Indeed, the fields $\phi_{\dot a}^b$ and $\Delta_{\dot a I}$ span a real 24-dimensional space (at each manifold point). The dimension of the orbit space is then $24- \dim P$ with $P$ a principal orbit of the action of $SU(2)_R \times SU(2)_L \times SU(4)$ on the space of field values. This dimension $ \dim P$ is determined by the dimension of the group and of a principal isotropy group.
First, we see that up to conjugation there is always a $SU(2)$-subgroup of $SU(4)$ leaving $\Delta_{\dot aI}$ invariant: it corresponds to $SU(2)$-transformations in the space orthogonal to the vectors $\Delta_{\dot 1 I}$ and $\Delta_{\dot 2I}$ in $\C^4$. Moreover, one can compute that the isotropy subgroup of the field values $$\begin{pmatrix} \phi_{\dot a}^b \end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}; \qquad \begin{pmatrix} \Delta_{\dot a I} \end{pmatrix} = \begin{pmatrix}1 & 0& 0 & 0 \\ 1 & 1 & 0 & 0 \end{pmatrix}$$ is given by $\mathbb Z_2 \times SU(2)$. Hence, the dimension of the principal orbit is $21 - 3 = 18$ so that the orbit space is 6-dimensional. This corresponds to the 4 real fields $h,\delta_0,\delta_1,\eta_1$ and the complex field $\chi$.
We allow for the colour $SU(3)$-symmetry not to be broken spontaneously, hence we only choose unitarity gauge in the $SU(2)_R \times SU(2)_L \times U(1)$-representations. That is, we retain the row vector $\Delta_{\dot 2 I}$ for $I=1,\ldots, 4$ as a variable and write $$\begin{pmatrix} \Delta_{\dot a I} \end{pmatrix} = \begin{pmatrix}\sqrt{w}+\delta_0/\sqrt w & 0& 0 & 0 \\ \delta_1/\sqrt{w} & \eta_1/\sqrt{w} & \eta_2/\sqrt{w} & \eta_3/\sqrt{w} \end{pmatrix}$$ so that $(\eta_i)$ forms a scalar $SU(3)$-triplet field (so-called [*scalar leptoquarks*]{}). The reason for the rescaling with $\sqrt{w}$ is that it yields the right kinetic terms for $\delta_0,\delta_1$ and $\eta$. Indeed, from the spectral action we then have $$\begin{aligned}
\frac{1}{2}\partial_{\mu}H_{\overset{.}{a}I\overset{.}{b}J}\partial^{\mu}H^{\overset{.}{a}I\overset{.}{b}J}
&=\frac{1}{2} \partial_\mu \left(\Delta_{\overset{.}{a}J}\Delta_{\overset{.}{b}I}\right) \partial^\mu \left(\Delta^{\overset{.}{a}J}\Delta^{\overset{.}{b}J}\right)
\\
&\sim \sum_{a=0}^1 \partial_\mu \delta_a \partial^\mu \delta^a + \partial_\mu \eta \partial^\mu \eta^* + \text{ higher order}\end{aligned}$$ The scalar potential becomes in terms of the fields $h,\chi,\delta_0,\delta_1, \eta_i$: $$\begin{aligned}
&\L_{pot}(h,\chi, \delta_0,\delta_1,\eta) = - \mu^2 ( h^2 + |\chi|^2) -\nu^2 \left( (w+ \delta_0)^2 + \delta_1^2 + |\eta|^2 \right)^2 /w^2
\\
\nn &
\quad + \lambda_{H\Sigma} \left( (w+\delta_0)^2 h^2 + (\delta_1^2 + |\eta|^2) |\chi|^2 \right)\left( (w+ \delta_0)^2 + \delta_1^2 + |\eta|^2 \right) /w^2 \\
\nn &
\quad + \lambda_H \left( (w+\delta_0)^4 + 2 (w+ \delta_0)^2 \delta_1^2+ (\delta_1^2 +|\eta|^2)^2 \right)^2/w^4 + \lambda_\Sigma (h^4+ |\chi|^4) \end{aligned}$$ As we are interested in the truncation to the Standard Model, we look for extrema with $\langle \delta_1\rangle =\langle \eta_i\rangle=0$, whilst setting $\langle h \rangle =v, \langle \delta_0 \rangle = 0, \langle \chi \rangle = x$. Note that the symmetry of these vevs is $$\begin{gathered}
\left\{
\left(\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^* \end{pmatrix},
\begin{pmatrix} \lambda^* & 0 \\ 0 & \lambda \end{pmatrix},
\begin{pmatrix} \lambda^* & 0 \\ 0 & m \end{pmatrix}\right): \lambda \in U(1) , m \in SU(3) \right\}
\\\subset SU(2)_R \times SU(2)_L \times SU(4)\end{gathered}$$ In other words, $SU(2)_R \times SU(2)_L \times SU(4)$ is broken by the above vevs to $U(1) \times SU(3)$.
The first derivative of $V$ vanishes for these vevs precisely if $$\begin{aligned}
2v (w^2 \lambda_{H \Sigma} +2v^2 \lambda_\Sigma - \mu^2 ) = 0,\\
4 x^3 \lambda_\Sigma - 2x \mu^2 = 0,\\
4w (2w^2 \lambda_H + v^2 \lambda_{H \Sigma} - \nu^2) =0.\end{aligned}$$ This gives rise to the fine-tuning of $v,w$ as in [@CC12]: $$w^2 \lambda_{H \Sigma} +2v^2 \lambda_\Sigma - \mu^2 , \qquad 2w^2 \lambda_H + v^2 \lambda_{H \Sigma} - \nu^2$$ choosing $\mu$ and $\nu$ such that the solutions $v,w$ are of the desired orders. Moreover, we find that the vev for $\chi$ either vanishes or is equal to $x=\sqrt{\mu^2/2\lambda_\Sigma}$. Note that this latter vev appears precisely at the entry $k^d h$ (or $k^e h$) of the finite Dirac operator, which we have disregarded by setting $\rho=1$.
If $\langle \chi \rangle = x =0$ then the Hessian is (derivatives with respect to $h,\chi,\delta_0,\delta_1, \eta$): $$\left(
\begin{smallmatrix}
8 v^2 \lambda_\Sigma & 0 & 8 v w \lambda_{H \Sigma} & 0 & 0 \\
0 & -2 w^2\lambda_{H\Sigma} -4 v^2 \lambda \Sigma & 0 & 0 & 0 \\
8 v w \lambda_{H\Sigma} & 0 & 32 w^2 \lambda_H & 0 & 0 \\
0 & 0 & 0 & -2 v^2 \lambda_{H\Sigma} & 0 \\
0 & 0 & 0 & 0 & -8 \lambda_H w^2-2 v^2 \lambda_{H\Sigma} w^2 {\bf 1}_3 \\
\end{smallmatrix}
\right)$$ where the ${\bf 1}_3$ is the identity matrix in colour space, corresponding to the $\eta$-field. This Hessian is not positive definite so we disregard the possibility that $\langle \chi \rangle =0$.
If $x=\sqrt{\mu^2/2\lambda_\Sigma}$ then the Hessian is $$\left(\begin{smallmatrix}
8v^2 \lambda_\Sigma & 0 & 8vw \lambda_{H\Sigma} & 0 & 0 \\
0 & 4w^2 \lambda_{H \Sigma}+8v^2 \lambda_\Sigma & 0 & 0 & 0\\
8vw \lambda_{H\Sigma} & 0 & 32 w^2 \lambda_H & 0 & 0 \\
0 & 0 & 0 & w^2 \frac{\lambda_{H \Sigma}^2}{\lambda_\Sigma} & 0 \\
0 & 0 & 0 & 0 & w^2 \frac{ \lambda_{H\Sigma}^2 - 8 \lambda_H \lambda_\Sigma}{\lambda_\Sigma}{\bf 1}_3
\end{smallmatrix}\right)$$ which is positive-definite if $$\label{eq:positive-mass}
\lambda_{H\Sigma}^2 \geq 8 \lambda_H \lambda_\Sigma.$$ Note that this relation may hold only at high-energies. The masses for $\chi$, $\delta_1$ and $\eta$ are then readily found to be: $$\begin{aligned}
m_\chi^2 &= 4w^2 \lambda_{H \Sigma}+8v^2 \lambda_\Sigma,\\
m_{\delta_1}^2 &= w^2 \frac{\lambda_{H \Sigma}^2}{\lambda_\Sigma} ,\\
m_{\eta}^2 &=w^2 \frac{ \lambda_{H\Sigma}^2 - 8 \lambda_H \lambda_\Sigma}{\lambda_\Sigma} .\end{aligned}$$ Under the assumption that $v^2 \approx 10^2 \operatorname{GeV}, w^2 \approx 10^{11} \operatorname{GeV}$ we have $m_\chi^2 \approx 10^{11} \operatorname{GeV}$ and $m_{\delta_1}^2, m_{\eta} \approx 10^{11}\operatorname{GeV}$.
The (non-diagonal) $h$ and $\delta_0$ sector has mass eigenstates as in [@CC12]: $$\begin{gathered}
m_\pm^2 = 16 w^2\lambda_H +4 v^2 \lambda_\Sigma \\\pm 4\sqrt{16 w^{4} \lambda_H^2 + v^4 \lambda_\Sigma^2 + 4v^2 w^2 \left(\lambda_{H \Sigma}^2 - 2 \lambda_H \lambda_\Sigma \right)}\end{gathered}$$ Under the assumption that $v^2 \ll w^2$ we can expand the square root: $$\begin{aligned}
&4 \sqrt{ 16 \lambda_H^2 w^{4} \left(1 + \frac{\lambda_\Sigma^2}{\lambda_H^2}\frac{v^4}{w^{4}} + \frac{\lambda_{H \Sigma}^2 - 2 \lambda_H \lambda_\Sigma}{4\lambda_H^2} \frac{v^2}{w^2}\right)}\\ \nn
&\qquad\approx 16 \lambda_H w^2 \left(1 +\frac{\lambda_{H \Sigma}^2 - 2 \lambda_H \lambda_\Sigma}{8\lambda_H^2} \frac{v^2}{w^2}\right)
\\\nn
&\qquad= 16\lambda_H w^2 +\frac{2\lambda_{H \Sigma}^2}{\lambda_H} v^2
- 4\lambda_\Sigma v^2.\end{aligned}$$ Consequently, $$\begin{aligned}
m_+ &\approx 32 \lambda_H w^2 + 2 \frac{2\lambda_{H \Sigma}^2}{\lambda_H} v^2, \\
m_- &\approx 8 \lambda_\Sigma v^2 \left(1- \frac{\lambda_{H \Sigma}^2}{4\lambda_H \lambda_\Sigma} \right).\end{aligned}$$ which are of the order of $10^{11}$ and $10^2 \operatorname{GeV}$, respectively. This requires that we have at low energies $$\label{eq:positive-higgs-mass}
4\lambda_H \lambda_\Sigma \geq \lambda_{H \Sigma}^2,$$ which fully agrees with [@CC12] when we identify $\delta_0 \equiv \sigma$ and with the couplings related via $$\lambda_H = \frac14 \lambda_\sigma , \qquad \lambda_{H \Sigma} = \frac12 \lambda_{h\sigma}, \qquad \lambda_\Sigma = \frac14 \lambda_h$$ Note the tension between Equations and , calling for a careful study of the running of the couplings in order to guarantee positive mass eigenstates at their respective energies.
We have summarized the scalar particle content of the above model in Table \[table:part-cont\].
$$\begin{array}{c||ccc}
& U(1)_Y & SU(2)_L & SU(3) \\
\hline\hline
\begin{pmatrix}\phi_1^0 \\ \phi_1^+ \end{pmatrix} = \begin{pmatrix} \phi_{\dot 1}^1\\ \phi_{\dot 1}^2 \end{pmatrix} & 1 & 2 & 1 \\[4mm]
\begin{pmatrix}\phi_2^- \\ \phi_2^0 \end{pmatrix} = \begin{pmatrix} \phi_{\dot 2}^1\\ \phi_{\dot 2}^2 \end{pmatrix}& -1 & 2 & 1 \\[4mm]
\delta_0 & 0 & 1 & 1 \\
\delta_1 & -2 & 1 & 1 \\
\eta & -\frac 23& 1 & 3
\end{array}$$
In terms of the original scalar fields $\phi_{\dot a}^b$ and $\Delta_{\dot aI}$ the vevs are of the following form: $$\begin{aligned}
\begin{pmatrix} \phi_{\dot a}^b \end{pmatrix} &= \begin{pmatrix} v & 0 \\ 0 & \sqrt{\mu^2/2 \Lambda_\Sigma}\end{pmatrix}\\
\begin{pmatrix} \Delta_{\dot a I} \end{pmatrix} &= \begin{pmatrix} w & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} .\end{aligned}$$ This shows that there are two scales of spontaneous symmetry breaking: at $10^{11}-10^{12} \operatorname{GeV}$ we have $$SU(2)_R \times SU(2)_L \times SU(4) \to U(1)_Y \times SU(2)_L \times SU(3)$$ and then at electroweak scale (both $v$ and $\mu$) we have $$U(1)_Y \times SU(2)_L \times SU(3) \to U(1)_Q \times SU(3)$$
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[^1]: The case $N=1$ was exploited successfully in [@DS11] for a noncommutative description of abelian gauge theories.
[^2]: Also other algebras that appear in the classification of irreducible geometries of KO-dimension have been considered in the literature: besides the case $N=4$ that we consider here the simplest case $N=1$ is relevant for the noncommutative geometric description of quantum electrodynamics [@DS11] and the case $N = 8$ leads to the ‘grand algebra’ of [@DLM14; @DLM14b].
[^3]: The case $k=3$ was ruled out by physical considerations [@DLM14].
| ArXiv |
---
author:
- 'S. Ole Warnaar'
- Wadim Zudilin
date: September 2019
title: '$q$-rious and $q$-riouser'
---
Dick Askey is known not just for his beautiful mathematics and his many amazing theorems, but also for posing numerous interesting and important open problems. Dick being Dick, these problems are hardly ever isolated, and often intended to demonstrate the unity of analysis, number theory and combinatorics. On this ocassion we wish to take the reader down the rabbit hole created by one such problem, published as *Advanced Problem 6514* by the American Mathematical Monthly in April 1986 [@Askey86]. Dick’s inspiration for the problem was derived from the Macdonald–Morris constant term conjecture for the root system $\mathrm{G}_2$ [@Macdonald82; @Morris82] as well as much earlier work of P. Chebyshev [@Tchebichef82] and E. Landau [@Landau85] on the integrality of factorial ratios. Problem 6514 asks for a proof of the integrality of $$A(m,n)=
\frac{(3m+3n)!\,(3n)!\,(2m)!\,(2n)!}{(2m+3n)!\,(m+2n)!\,(m+n)!\,m!\,n!\,n!}$$ for all non-negative integers $m$ and $n$.
There are multiple reasons — some of them very deep, see e.g., [@Bober09; @Rodriguez-Villegas05; @Soundararajan19a; @Soundararajan19b] — for wanting to classify integer-valued factorial ratios such as Chebyshev’s $$C(n)=\frac{(30n)!\,n!}{(15n)!\,(10n)!\,(6n)!}\,.$$ Given a particular such ratio, integrality can always be verified by computing the $p$-adic order of the factorials entering the quotient. This is exactly what all eight solvers of Problem 6514 did. Such a verification, however, provides little insight into which ratios are integral and which ones are not, and from the editorial comments to the problem it is clear that Dick would have liked to see other types of solutions too. Indeed, it is remarked that
> \[\] the editor \[read: Dick Askey\] feels there is still room for other methods, involving perhaps combinatorial interpretations or manipulation of generating functions. In this particular case, the proposer remarks that $A(m,n)$ should be the constant term of the Laurent polynomial $$\begin{gathered}
> \quad\qquad \big((1-x)(1-1/x)(1-y)(1-1/y)(1-y/x)(1-x/y)\big)^m \\[1mm]
> \times \big((1-xy)(1-1/xy)(1-y/x^2)(1-x^2/y)(1-y^2/x)(1-x/y^2)\big)^n.\quad\end{gathered}$$
Incidentally, L. Habsieger [@Habsieger86] and D. Zeilberger [@Zeilberger87] both proved the $\mathrm{G}_2$ Macdonald–Morris constant term conjecture shortly after Dick Askey posed his problem. The submission dates of their respective papers (the 12th of May and the 2nd of June 1986) were well within the deadline of the 31st of August for submitting solutions to Problem 6514 to the Monthly. In fact, in the acknowledgement of his paper Zeilberger thanks Dick Askey for “rekindling his interest in the Macdonald conjecture”, so maybe he should belatedly be considered the 9th solver of Askey’s problem.
The height of a factorial ratio is the number of factorials in the denominator minus the number of factorials in the numerator, so that the height of $A(m,n)$ is two whereas the height of $C(n)$ is one. A one-parameter family of height-$k$ factorial ratios $$F(n)=\frac{(a_1\,n)!\cdots (a_{\ell}\, n)!}
{(b_1\, n)!\cdots (b_{k+\ell}\, n)!}$$ is balanced if $a_1+\dots+a_{\ell}=b_1+\dots+b_{k+\ell}$. All balanced, integral, height-one factorial ratios $F(n)$ were classified in 2009 by J. Bober [@Bober09]. In relation to this classification we should mention F. Rodriguez-Villegas’ observation [@Rodriguez-Villegas05] that if $F(n)$ is a balanced, height-one factorial ratio then the hypergeometric function $\sum_{n{\geqslant}0} F(n) z^n$ is algebraic if and only if $F(n)$ is integral. This observation was key to Bober’s proof, allowing him to use the Beukers–Heckman classification [@BH89] of $_nF_{n-1}$ hypergeometric functions with finite monodromy group. A proof not reliant on the Beukers–Heckman theory was recently found by K. Soundararajan [@Soundararajan19a]. By extending his method he also obtained a partial classification in the height-two case [@Soundararajan19b].
Despite the availability of the number-theoretic, $p$-adic approach to factorial ratios, the question of integrality is very interesting from a purely combinatorial point of view. The simplest example is of course provided by the height-one binomial coefficients $$\frac{(m+n)!}{m!\,n!},$$ whose integrality can be established combinatorially (as well as probabilistically, algebraically, etc.) with little effort. However, to the best of our knowledge, no combinatorial proof is known of the integrality of Chebyshev’s $C(n)$.
A related open problem arises from our joint work [@WZ11] from 2011. In [@WZ11] we observed that if each factorial $m!$ in an integral factorial ratio is replaced by a $q$-factorial $$[m]!=[m]_q!=\prod_{i=1}^m\frac{1-q^i}{1-q},$$ then the resulting $q$-factorial ratio is a polynomial with non-negative integer coefficients. The polynomiality and integrality parts are trivial but the positivity — which was referred to in [@WZ11] as ‘$q$-rious positivity’ — is completely open. The only (irreducible) cases that are proven are the three two-parameter families of height one given by $$\frac{[m+n]!}{[m]!\,[n]!},\qquad
\frac{[2m]!\,[2n]!}{[m]!\,[n]!\,[m+n]},\qquad
\frac{[m]!\,[2n]!}{[2m]!\,[n]!\,[n-m]!}\quad (m{\geqslant}n),$$ where the first family corresponds to the $q$-binomial coefficients and the second family to the $q$-super Catalan numbers. In the $q$-case no arithmetic approach is available, and given the lack of combinatorial methods to deal with integrality, a combinatorial approach to $q$-rious positivity seems hopeless.[^1] Perhaps the most tractable problem is to analytically prove, along the lines of [@WZ11], the positivity of the known two-parameter families of height two, such as $$A_q(m,n)=
\frac{[3m+3n]!\,[3n]!\,[2m]!\,[2n]!}
{[2m+3n]!\,[m+2n]!\,[m+n]!\,[m]!\,[n]!\,[n]!}\in\mathbb Z[q]$$ and $$C_q(m,n)=\frac{[6m+30n]!\,[n]!}
{[3m+15n]!\,[2m+10n]!\,[m]!\,[6n]!}\in\mathbb Z[q].$$ For the first family, which is the $q$-analogue of $A(m,n)$, it is known that [@Cherednik95; @Habsieger86; @Zeilberger87] $$\begin{gathered}
A_q(m,n) \\=\operatorname*{CT}\limits_{x,y}\Big[
\big(x,q/x,y,q/y,y/x,qx/y;q\big)_m
\big(xy,q/xy,y/x^2,qx^2/y,y^2/x,qx/y^2;q\big)_n\Big],\end{gathered}$$ where $(a_1,\dots,a_k;q)_n:=\prod_{i=1}^k \prod_{j=1}^n (1-a_i q^{j-1})$. This interpretation as a $\mathrm{G}_2$ constant term gives little insight into the positivity of the coefficients. It would appear that the second two-parameter family has not occurred before. For $q=1$ it arose earlier this year in the (partial) classification of height-two factorial ratios by Soundararajan [@Soundararajan19b] mentioned above. It should be noted that if one were to prove the $q$-rious positivity of $C_q(m,n)$ then this immediately would imply the positivity of the $q$-analogue of Chebyshev’s factorial ratio since $$C_q(n)=\frac{[30n]!\,[n]!}{[15n]!\,[10n]!\,[6n]!}=C_q(0,n).$$
[99]{}
<span style="font-variant:small-caps;">R. Askey</span>, Advanced Problem 6514, *Amer. Math. Monthly* **93**:4 (1986) 304–305; Solution of Advanced Problem 6514, *Amer. Math. Monthly* **94**:10 (1987) 1012–1014.
<span style="font-variant:small-caps;">F. Beukers</span> and <span style="font-variant:small-caps;">G. Heckman</span>, Monodromy for the hypergeometric function $_nF_{n-1}$, *Invent. Math.* **95**:2 (1989) 325–354.
<span style="font-variant:small-caps;">J.W. Bober</span>, Factorial ratios, hypergeometric series, and a family of step functions, *J. London Math. Soc.* (2) **79**:2 (2009) 422–444.
<span style="font-variant:small-caps;">I. Cherednik</span>, Double affine Hecke algebras and Macdonald’s conjectures, *Ann. of Math.* (2) **141**:1 (1995) 191–216.
<span style="font-variant:small-caps;">L. Habsieger</span>, La $q$-conjecture de Macdonald–Morris pour $G_2$, *C. R. Acad. Sci. Paris Sér. I Math.* **303**:6 (1986) 211–213.
<span style="font-variant:small-caps;">E. Landau</span>, Sur les conditions de divisibilité d’un produit de factorielles par un autre, in *Collected Works*, Vol. I (Thales-Verlag, Essen, 1985), p. 116.
<span style="font-variant:small-caps;">I.G. Macdonald</span>, Some conjectures for root systems, *SIAM J. Math. Anal.* **13**:6 (1982) 988–1007.
<span style="font-variant:small-caps;">W.G. Morris</span>, *Constant term identities for finite and affine root systems: conjectures and theorems*, PhD thesis (Univ. Wisconsin-Madison, 1982).
<span style="font-variant:small-caps;">F. Rodriguez-Villegas</span>, Integral ratios of factorials and algebraic hypergeometric functions, *Oberwolfach Rep.* **2** (2005) 1814–1816.
<span style="font-variant:small-caps;">K. Soundararajan</span>, Integral factorial ratios, *Preprint* [`arXiv:1901.05133 [math.NT]`](http://arxiv.org/abs/1901.05133) (2019).
<span style="font-variant:small-caps;">K. Soundararajan</span>, Integral factorial ratios: irreducible examples with height larger than 1, *Preprint* [`arXiv:1906.06413 [math.NT]`](http://arxiv.org/abs/1906.06413) (2019).
<span style="font-variant:small-caps;">P.L. Tchebichef</span>, Mémoire sur les nombres premiers, *J. Math. Pures Appl.* **17** (1852) 366–390.
<span style="font-variant:small-caps;">S.O. Warnaar</span> and <span style="font-variant:small-caps;">W. Zudilin</span>, A $q$-rious positivity, *Aequat. Math.* **81**:1-2 (2011) 177–183.
<span style="font-variant:small-caps;">D. Zeilberger</span>, A proof of the $\mathrm{G}_2$ case of Macdonald’s root system-Dyson conjecture, *SIAM J. Math. Anal.* **18**:3 (1987) 880–883.
[^1]: There are of course countless methods to show that the $q$-binomial coefficients have non-negative integer coefficients, but no combinatorial interpretation of the $q$-super Catalan numbers is known. In fact, not even a combinatorial interpretation of the ordinary super Catalan numbers is known.
| ArXiv |
---
abstract: 'We report long-slit spectroscopic observations of the quasar SDSS J082303.22+052907.6 ($z_{\rm CIV}$$\sim$3.1875), whose Broad Line Region (BLR) is partly eclipsed by a strong damped Lyman-$\alpha$ (DLA; log$N$(HI)=21.7) cloud. This allows us to study the Narrow Line Region (NLR) of the quasar and the Lyman-$\alpha$ emission from the host galaxy. Using [cloudy]{} models that explain the presence of strong NV and PV absorption together with the detection of SiII$^*$ and OI$^{**}$ absorption in the DLA, we show that the density and the distance of the cloud to the quasar are in the ranges 180 $<$ $n_{\rm H}$ $<$ 710 cm$^{-3}$ and 580 $>$ $r_0$ $>$230 pc, respectively. Sizes of the neutral($\sim$2-9pc) and highly ionized phases ($\sim$3-80pc) are consistent with the partial coverage of the CIV broad line region by the CIV absorption from the DLA (covering factor of $\sim$0.85). We show that the residuals are consistent with emission from the NLR with CIV/Lyman-$\alpha$ ratios varying from 0 to 0.29 through the profile. Remarkably, we detect extended Lyman-$\alpha$ emission up to 25kpc to the North and West directions and 15kpc to the South and East. We interpret the emission as the superposition of strong emission in the plane of the galaxy up to 10kpc with emission in a wind of projected velocity $\sim$500km s$^{-1}$ which is seen up to 25kpc. The low metallicity of the DLA (0.27 solar) argues for at least part of this gas being in-falling towards the AGN and possibly being located where accretion from cold streams ends up.'
date: 'Accepted ....... Received ....... '
title: ' A coronagraphic absorbing cloud reveals the narrow-line region and extended Lyman-$\alpha$ emission of QSO J0823$+$0529 [^1]'
---
\[firstpage\]
galaxies: evolution – intergalactic medium – quasars: absorption lines – quasars: individual: SDSS J082303.22$+$052907.6
Introduction
============
Luminous high-redshift quasars consist of supermassive black holes residing at the center of massive galaxies and growing through mass accretion of gas in an accretion disk. Bright quasars play an important role in shaping their host galaxies through the emission of ionizing flux but also through launching powerful and high-velocity outflows of gas. These outflows inject energy and material to the disk of the galaxy and may be influencing the physical state up to larger distances. It has remained unclear however what are the mechanisms that drive energy from the very center of the active galactic nuclei (AGN) to the outskirts of the galaxy.
Observational evidence for outflows and winds in AGNs is seen as prominent radio-jets in radio-loud sources, broad absorption lines observed in broad absorption line (BAL) quasars, or through the photoionized warm absorber which is frequently observed in the soft X-rays (e.g. Crenshaw et al. 2003). Gravitational micro-lensing studies have shown that the primary X-ray emission region in AGN is of the order of a few tens of gravitational radii in size (e.g. Dai et al. 2010) and X-ray spectroscopy shows that highly ionized outflows launched from this region are seen in high-$z$ quasars (Chartas et al. 2009) and in at least 40% of them with velocities up to 0.1 c (Gofford et al. 2013).
Outflows are observed also on large scales. Mullaney et al. (2013) used the SDSS spectroscopic data base to study the one-dimensional kinematic properties of \[OIII\]$\lambda$5007 by performing multicomponent fitting to the optical emission-line profiles of about 24000, $z<0.4$ optically selected AGNs. They showed that approximately 17 percent of the AGNs have emission-line profiles that indicate their ionized gas kinematics are dominated by outflows and a considerably larger fraction are likely to host ionized outflows at lower levels. Harrison et al. (2014) find high-velocity ionized gas (velocity widths of about 600-1500 km s$^{-1}$) with observed spatial extents of (6-16) kpc in a representative sample of $z<0.2$, luminous (i.e. $L$\[O [iii]{}\]$>$10$^{41.7}$ erg s$^{-1}$) type 2 AGNs. Therefore galaxy-wide energetic outflows are not confined to the most extreme star-forming galaxies or radio-luminous AGNs.
If outflows are observed both on small and large scales, how the small scale outflows are transported at larger distances remains unclear (Faucher-Giguère et al. 2012, Ishibashi & Fabian 2015, King & Pounds 2015). This is however a crucial question as these outflows are massive and energetic enough to significantly influence star formation in the host galaxy and provide significant metal enrichment to the interstellar and intergalactic media (e.g. Dubois et al. 2013). At high redshift where quasars are more luminous, the consequences of such outflows are of first importance for galaxy formation. One way to study the interplay between the quasar and its surrounding is to search for Lyman-$\alpha$ emission around quasars (e.g. Hu & Cowie 1987, Hu et al. 1996, Petitjean et al. 1996, Bunker et al. 2003, Christensen et al. 2006). These observations reveal gas infalling onto the galaxy (Weidinger et al. 2005), positive feed-back from the AGN (Rauch et al. 2013) or a correlation between the luminosity of the extended emission and the ionizing flux from the quasar (North et al. 2012).
Very recently, we searched quasar spectra from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2014) for the rare occurrences where a strong damped Lyman-$\alpha$ absorber (DLA) blocks the Broad Line Region emission (BLR) from the quasar and acts as a natural coronagraph to reveal narrow Lyman-$\alpha$ emission from the host galaxy (Finley et al. 2013; see also Hennawi et al. 2009). This constitutes a new way to have direct access to the quasar host galaxy and possibly, when the size of the DLA is small enough, to the very center of the AGN. Out of a total of more than 70,000 spectra of $z>2$ quasars (Pâris et al. 2012), we gathered a sample of 57 such quasars and followed-up six of them with the slit spectrograph Magellan-MagE to search for the very special cases where the DLA coronagraph reveals the very center of the host galaxy and extended Lyman-$\alpha$ emission. In the course of this follow-up program, we found one object SDSS J0823+0529 where the DLA does not cover the Lyman-$\alpha$ broad line region entirely and reveals the emission of the Lyman-$\alpha$ and C [iv]{} narrow line regions. We show here that this is a unique opportunity to study the link between the properties of the central regions of the AGN to that of the gas in the halo of the quasar.
The paper is organized as follows. In Section 2 we describe the observations and data reduction. We derive properties of the gas associated with the DLA (metallicity, ionization state, density, distance to the quasar, typical size) in Section 3. We discuss the properties of the quasar narrow line region and of the extended Lyman-$\alpha$ emission in Sections 4 and 5, respectively, We then finally, present our conclusions in Section 6. In this work, we use a standard CDM cosmology with $\Omega_{\Lambda}$ = 0.73, $\Omega_{m}$ = 0.27, and H$_0$ =70 km s$^{-1}$ Mpc$^{-1}$ (Komatsu et al. 2011). Therefore 1 arcsec corresponds to about 7.1 kpc at the redshift of the quasar ($z$ = 3.1875 see below). In the following we will use solar metallicities from Asplund et al. (2009).
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Observations and data reduction
===============================
The spectrum of the quasar SDSS J0823$+$0529 was observed with the Magellan Echellete spectrograph (MagE; Marshall et al. 2008) mounted on the 6.5 m Clay telescope located at Las Campanas Observatory. MagE is a medium-resolution long-slit spectrograph that covers the full wavelength range of the visible spectrum (3200 $\textup{\AA}$ $-$ 1 $\mu$ m). Its 10 arcsec long slit and 0.30$^{"}$ per pixel sampling in the spatial direction, are ideal for observing high-redshift extended astrophysical objects. The spectrograph was designed to have high throughput in the blue, with the peak throughput reaching 22 $\%$ at 5600 $\textup{\AA}$ including the telescope.
The quasar was observed in December 2012 with an one arcsec width slit aligned along two different position angles (PAs) for 1 hour each. One position was South-North (PA = 0) and the other position was East-West (PA = 90). Another 1-hour exposure with PA = 90 was taken in February 2013 but the resultant spectrum has a lower SNR. Following each exposure, the spectrum of a standard star was also recorded allowing us to precisely flux-calibrate the quasar spectra. The seeing, measured on the extracted trace of the quasar is 1.06 arcsec for PA = 0 and 0.93 arcsec for PA = 90.
We reduced the spectrum using the Mage\_Reduce pipeline written in the Interactive Data Language (IDL) by George Becker[^2]. In addition to the 1-dimensional (1D) spectrum, the pipeline provides 2-dimensional (2D) sky-subtracted science frames as well as the corresponding 2D wavelength solution corrected for the vacuum heliocentric velocity shifts. These 2D images will later be used to rectify the curved spectral orders (see below). Here, we note that since there is an extended Lyman-$\alpha$ emission in order 9 of our 2D spectra, we preferentially used a wider extraction window to extract this order.
Two-dimensional spectra when imaged onto a detector are often curved with respect to the natural (x,y)-coordinate system of the detector defined by the CCD columns (Kelson 2003). To rectify the curvature of the orders we first rebin the wavelength area (given by the 2D wavelength solution) and position area (given by the 2D slitgrid array provided by the pipeline) increasing the number of pixel by a factor of hundred. We define a new 2D array, with one dimension representing the wavelength ($\lambda$) and the other the position on the slit ($s$). For each pixel on the 2D spectrum of the quasar, we take its corresponding wavelength and its position on the slit directly in the rebinned 2D areas. Therefore, the (x,y) coordinates of each pixel (defined by the CCD columns) can now be transformed to an ($s$, $\lambda$) coordinate defined by the new 2D array introduced above. In this new ($s$, $\lambda$)-coordinate system, the curvature of the orders and the tilting of the spectral lines are all rectified. In this study, we use these rectified 2D images when we discuss the spatial extension of the Lyman-$\alpha$ emission line in the quasar spectrum. Note that the rebinning process has very little effect on the Lyman-$\alpha$ emission because it is conveniently placed in the middle of the corresponding order where curvature is at minimum.
Finally, the extracted 1D spectrum of each individual order is corrected for the relative spectral response of the instrument and flux-calibrated using the spectrum of a standard star (HR 1544) observed during the same night. The spectrum of the standard star can be found in the ESO standard star catalogue webpage[^3]. We emphasize that our standard star spectrum was obtained immediately after observing the quasar spectrum. We also flux calibrated in the same way the 2D spectra in the Lyman-$\alpha$ range. These flux-calibrated spectra are then combined weighting each pixel by the inverse of its variance. The resulting spectrum (after binning with a 3 $\times$ 3 box) has $\sim$ 27 km s$^{-1}$ per pixel and 3 pixels per resolution element, and therefore FWHM $\sim$ 80 km s$^{-1}$.
We have fitted the quasar C [iv]{} emission line with two Gaussian functions (to mimic the two lines of the doublet) to estimate the redshift of the quasar. We derive $z_{\rm em}$ = 3.1875. This is $\sim$330 km s$^{-1}$ smaller than the redshift of the DLA ($z_{\rm
DLA}$ = 3.1910 derived from the fit of Si [ii]{} and Fe [ii]{} absorptions). However, it is well known that redshifts from C [iv]{} are smaller by up to 600 km s$^{-1}$ compared to redshifts from narrow forbidden lines (e.g. Hewett & Wild 2010). Therefore we cannot be certain that the DLA is redshifted compared to the quasar. We would need to detect \[O[iii]{}\] lines redshifted to the infra-red to have a better estimate of $z_{\rm em}$.
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A DLA acting as a coronagraph
=============================
Figure \[QSO\_whole\_spect\] shows the co-added spectrum of SDSS J0823+0529 using all the exposures taken at different PAs. It is apparent that a strong DLA is located close to the redshift of the quasar. However, strong Lyman-$\alpha$ emission is present in the center of the DLA trough. This is possible if the absorbing cloud has a transverse extension which is smaller than the central narrow line region (NLR) of the quasar. Note that the strength of the Lyman-$\alpha$ emission together with the size of the cloud derived below makes it very improbable that the Lyman-$\alpha$ emission could be a consequence of star formation in the DLA itself.
The redshift of the DLA ($z_{\rm DLA}$ = 3.1910) was determined by conducting a simultaneous Voigt profile fit of the Fe [ii]{}, Si [ii]{}$^{*}$, and O [i]{}$^{**}$ absorption profiles.
Elemental abundances
--------------------
A Voigt profile fit was conducted on the damped Lyman-$\alpha$ absorption line of this system, resulting in a neutral hydrogen column density of log $N_{\rm HI}$ = 21.70 $\pm$ 0.10. Note that the placement of the quasar continuum is very uncertain as the DLA covers the Lyman-$\alpha$ and the N [v]{} broad emission lines. However, the core and especially the red wing of the DLA profile allowed us to satisfactorily constrain the H [i]{} column density. We will come back to this in Section 3.4.
We detect absorption lines from O [i]{}, O [i]{}$^{**}$, Si [ii]{}, Si [ii]{}$^*$, Fe [ii]{}, Al [ii]{}, Al [iii]{}, Ar [i]{}, C [ii]{}, C [ii]{}$^{*}$, C [iv]{}, Si [iv]{}, P [v]{} and N [v]{}. Absorption profiles are shown in Fig. \[j08231D\] and results from fitting these lines are given in Tables \[lowion\] $\&$ \[hiion\]. Techniques used here are similar to those in Fathivavsari et al (2013). It must be noted that most of the lines are saturated, preventing us from deriving accurate column densities especially at the resolution of our data (R$\sim$4000).
The profiles are dominated by a main strong component clearly seen in particular in Fe [ii]{}$\lambda$1608 and Si [ii]{}$^*$$\lambda$1533. To constrain the Doppler parameter of this component, we take advantage of the fact that Fe [ii]{}$\lambda$1608 is not saturated and well defined while Fe [ii]{}$\lambda$1611 is not detected (see Fig \[dopplerb\]). We start by fitting together Si [ii]{}$\lambda$1808 and 1526, imposing the presence of the main component with a fixed Doppler parameter. We then use the resulting decomposition to fit the Fe [ii]{} lines. Results are shown in Fig \[dopplerb\] for $b$ = 10 and 20 km/s. It is apparent that components narrower than $b$ $\sim$ 20 km/s are not favored and $b$ = 10 km/s is definitely rejected. Therefore, we are confident that imposing $b$ $\sim$ 20 km/s for the main component in the DLA will give us a good estimate of column densities.
For some of the species, several lines with very different oscillator strengths (either in doublets or multiplets) are present so that we can derive robust estimates of the column densities. This is the case for Si [ii]{}, Si [ii]{}$^{*}$, and Fe [ii]{}. From this, we derive metallicities relative to solar, \[Si/H\] = $-$0.79 and \[Fe/H\] = $-$1.87. We also derive an upper limit on $N$(Fe [iii]{}).
We also fit the high-ionization species. We did not try to tie the components in different profiles because this high-ionization phase could be highly perturbed. In Table \[hiion\], except for the first two components of the C [iv]{} absorption profiles, all reported column densities are upper limits because the profiles are strongly saturated. It can be seen however that the decompositions in sub-components are fairly consistent between the different species.
We detect absorptions from O [i]{}$^{**}$ and Si [ii]{}$^{*}$ which are rarely detected in DLAs (see Noterdaeme et al. 2015, Neeleman et al. 2015) and are more commonly seen in DLAs associated to GRB afterglows (Vreeswijk et al. 2004; Chen et al. 2005; Fynbo et al. 2006). Absorption from the fine structure state of Si [ii]{} will be used to constrain the density of the absorbing cloud (see below).
The high measured depletion of iron relative to silicon (i.e. \[Si/Fe\] = +1.08) in this DLA suggests the presence of dust. Consequently, extinction due to dust might be significant along this line of sight. Indeed, the median g $-$ r color for a sample of 697 non-BAL quasars with redshift within $\Delta$$z$ = $\pm$0.02 around $z_{\rm em}$ = 3.1910 is ${\left< g-r \right>}$ = 0.30 when $g-r$ = 1.1 for QSO J0823+0529. The reddening for this line of sight is E(B-V) = 0.09, measured with an SMC reddening law template, which places it among the most reddened of the sight lines in the Finley et al. (2013) statistical sample. We will take this reddening into account in the following while discussing the properties of the quasar.
Physical conditions in the DLA gas
----------------------------------
In this section we use the photo-ionization code [cloudy]{} to constrain the ionization state of the absorber and its distance to the quasar central engine. Observed ionic ratios of Si [ii]{}, Si [iv]{}, Al [iii]{}, and Ar [i]{} are used to constrain the plane parallel models constructed for a range of ionization parameters. The calculations were stopped when a neutral hydrogen column density of log $N$(H [i]{}) = 21.70 is reached. Solar relative abundances are assumed and the metallicity is taken to be Z = 0.16Z$_{\odot}$ from the Silicon abundance of the DLA derived in Section 3.1.
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Redshift & Ion & log(N)$^a$ & log(N)\
$ $ & $ $ & $\rm Observed$ & $\rm Model$\
3.190974 & Si [ii]{} & 16.42$\pm$0.10 & 16.53\
& Si [ii]{}$^{*}$ & 15.49$\pm$0.30 & 15.49\
& Fe [ii]{} & 15.33$\pm$0.20 & 16.50\
& Fe [iii]{} & $\le$ 15.00 & 15.09\
& Ar [i]{} & 14.97$\pm$0.20 & 15.04\
& Al [iii]{} & 14.80$\pm$0.10 & 14.64\
& O [i]{}$^{**}$ & 16.45$\pm$0.45 & 15.70\
& O [i]{} & 17.08$\pm$0.50 & 17.62\
\
\[lowion\]
[c c c c c]{} Redshift & Ion & b & log(N) & log(N)\
$ $ & $ $ & \[km s$^{-1}$\] & $\rm Observed$ & $\rm Model$$^a$\
3.185037 & C [iv]{} & 23 & 13.60$\pm$0.10 & $ $\
3.186553 & C [iv]{} & 32 & 13.90$\pm$0.10 & $ $\
3.189333 & C [iv]{} & 86$\pm$10 & $\ge$15.40$\pm$0.10 & $ $\
3.190953 & C [iv]{} & 38$\pm$14 & $\ge$14.65$\pm$0.30 & $ $\
3.192634 & C [iv]{} & 39$\pm$6 & $\ge$15.40$\pm$0.50 & $ $\
Total(C [iv]{}) & & & $\ge$15.75 & 18.43\
3.189333 & Si [iv]{} & 56$\pm$10 & $\ge$14.30$\pm$0.10 & $ $\
3.190953 & Si [iv]{} & 25$\pm$14 & $\ge$14.60$\pm$0.60 & $ $\
3.192634 & Si [iv]{} & 25$\pm$6 & $\ge$15.72$\pm$0.90 & $ $\
Total(Si [iv]{}) & & & $\ge$15.75 & 16.14\
3.189821 & N [v]{} & 99$\pm$39 & $\ge$14.80$\pm$0.30 & $ $\
3.191157 & N [v]{} & 34$\pm$29 & $\ge$14.89$\pm$0.20 & $ $\
3.192807 & N [v]{} & 74$\pm$19 & $\ge$14.87$\pm$0.10 & $ $\
Total(N [v]{}) & & & $\ge$15.33 & 16.55\
3.189788 & P [v]{} & 136 & $\ge$14.20$\pm$0.20 & $ $\
3.191109 & P [v]{} & 97 & $\ge$13.95$\pm$0.30 & $ $\
3.192757 & P [v]{} & 13 & $\ge$13.80$\pm$0.60 & $ $\
Total(P [v]{}) & & & $\ge$14.50 & 15.30\
\
\[hiion\]
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-------- --------------- ----------- -------------- ---------------
log U $n$$_{\rm H}$ $r$$_{0}$ $l$(H [i]{}) $l$(C [iv]{})
$ $ \[cm$^{-3}$\] \[pc\] \[pc\] \[pc\]
$-$1.1 710 579 2.3 3.0
$-$0.7 500 435 3.2 5.8
$-$0.3 355 326 4.6 14.6
+0.0 250 275 6.5 33.7
+0.3 180 229 9.1 80.0
-------- --------------- ----------- -------------- ---------------
: Hydrogen density ($n_{\rm H}$) in the cloud, its distance to the central AGN ($r$$_{0}$), size of the DLA ($l_{\rm
HI}$) and transverse size of the C [iv]{} phase for different values of the ionization parameter (U).
\[DLAsize\]
The ionizing spectrum incident on the cloud is taken as the combination of the standard AGN spectrum of Mathews & Ferland (1987), Haardt-Madau extragalactic spectrum (Haardt & Madau 2005, HM05) and the CMB radiation both at $z$ = 3.1910. Fig. \[Model\] summarizes the results of these calculations. Bottom panel gives resulting ionic ratios and top panel shows column densities. In both panels measurements and upper limits are indicated by dashed lines. For Al [ii]{}, we scale the Si [ii]{} column density assuming solar metallicity ratios.
The column density ratio log$N$(Al [ii]{})/$N$(Al [iii]{}) yields ionization parameter ranging from $-$1.1 to 0.3. Both log$N$(Si[ii]{})/$N$(Ar[i]{}) ratio and the limit on log$N$(Si[ii]{})/$N$(Si[iv]{}) are consistent with this range of ionization parameters. Our preferred value is log $U$ = $-0.3$ and we indicate the column densities for this model in Tables 1 and 2. Note that we detect a trough at the position of P[v]{}$\lambda$1117 with an absorption profile which is consistent with that of other high-ionization species (see Fig. 2). The second weaker line of the doublet, P[v]{}$\lambda$1128 is affected by noise. The fit to these two lines gives a column density which is consistent with the results of the preferred model. The same is true for N [v]{}. This strongly supports the fact that the cloud is highly ionized.
To determine the hydrogen density of this cloud (i.e. $n_{\rm H}$), a series of [cloudy]{} models with fixed ionization parameter (varying from log U = $-$1.1 to 0.3) and varying $n_{\rm H}$ were constructed. Note that in this series of models, the metallicity and incident radiation are the same as those considered above. By increasing the hydrogen density we are indeed trying to collisionally populate the excited states of the Si [ii]{} ground state to explain the observed Si [ii]{}$^*$ column density.
Knowing the ionization parameter and the density, we can derive the distance of the cloud to the quasar by estimating the number of ionizing photons emitted by the quasar. Since the quasar is reddened, we first estimate the flux at 20370Å (corresponding to H$\beta$ at the redshift of the quasar) by extrapolating with a power-law the continuum observed at 6125 and 8165 Å. Following Srianand & Petitjean (2000), we then assume that the de-reddened quasar continuum is a power law ($f_{\lambda}$ $\sim$ $\lambda^{\alpha_{\lambda}}$; with $\alpha_{\lambda}$ = $-$1.5) and we consider that the flux at $\sim$ 20370 $\textup{\AA}$ is not affected by the reddening. We thus estimate the flux at the Lyman limit in the rest frame of the absorber to be $f_{912{\tiny\textup{\AA}}}$ = 3.40$\times$10$^{-17}$ erg s$^{-1}$cm$^{-2}$$\textup{\AA}^{-1}$. This flux corresponds to a luminosity of $L_{912{\tiny\textup{\AA}}}$ = 3.25$\times$10$^{42}$ erg s$^{-1}$$\textup{\AA}$$^{-1}$ at the Lyman limit. We can now estimate the rate at which hydrogen ionizing photons are impinging upon the face of the cloud by integrating $L_{\nu}$/$h\nu$ over the energy range 1 to 20 Ryd. If we assume a flat spectrum for the quasar over this energy range, we get $Q$ = 6.78$\times$10$^{55}$ photons per second. From the definition of the ionization parameter U,
$$U = \frac{Q}{4 \pi r_{0}^{2} n_{\rm H} c}$$
one can see that for given values of $n_{\rm H}$, U, and $Q$, one can uniquely estimate the distance, $r_{0}$, from the quasar to the absorbing cloud ($c$ is the speed of light).
We can then derive the size of the cloud along the line of sight. For the neutral part, the size along the line of sight is $l$ $\sim$ $N_{\rm HI}$/$n_{\rm H}$. Results are summarized in Table \[DLAsize\] which gives the hydrogen density, distance to the quasar and size of the neutral phase of the cloud for different values of log $U$. The longitudinal size range from 2.3 to 9.1 pc for a distance of, respectively, 579 to 229 pc from the quasar and a hydrogen density of 710 to 180 cm$^{-3}$. Using the structure given by the model and assuming that the cloud is spherical we can derive the transverse size of the C [iv]{} phase. It is given in the last column of Table \[DLAsize\] and range from 3 to 80 pc. These estimates for the transverse size of the high-ionization zone are only rough estimates because our model is very simple (only one density) and we assume spherical geometry.
Residual flux in the bottom of the DLA trough
---------------------------------------------
We observe residual flux in the bottom of the DLA trough extending in velocity well beyond the narrow emission line. This can be seen in Fig. \[Lya\_residual\_2PA\] where we overplot zooms of the Lyman-$\alpha$ regions observed along the two PAs. It is apparent that the flux is never at zero in the trough. We checked in the Lyman-$\alpha$ forest that the bottom of saturated lines have on an average zero flux. The residual flux is consistent in these two spectra as is demonstrated in the bottom panel of the figure which gives the difference between the two spectra.
This means that the DLA cloud does not cover the background source completely. Since the source of the quasar continuum is much smaller than the broad line region (e.g. Hainline et al. 2013), this excess is due either to the cloud not covering the BLR completely or to a second continuum source.
We investigate here if this excess can be due to an additional continuum source. If this excess is due to the continuum of the host galaxy then we would expect some residual at the bottom of metal lines. Indeed C [ii]{}$\lambda$1334 is saturated and seems to show some weak residual (see Fig. 2). We thus have scaled the LBG mean continuum as given by Kornei et al. (2010) to the bottom of C [ii]{}$\lambda$1334. We find that this could explain 20 to 50% of the DLA residual. The corresponding magnitude of the galaxy would be 23.22 which would be very bright. In any case this possibility cannot explain all the residual.
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It is probable that part of the residual flux in the bottom of the DLA trough is due to the fact that part of the Lyman-$\alpha$ emission from the BLR is not covered by the DLA. The size of the broad line region (BLR) can be inferred from time delay measurements between variations in the continuum and in the broad lines. Recent investigations of low-redshift AGNs show a tight relation between this size and the luminosity of the AGN, $R=A\times (L/10^{43})^B$, where $R$ is the radius of the BLR, $A$ is a typical distance in light-days and $L$ is the luminosity either in an emission line (H$\beta$ or C [iv]{}) or in the continuum. The index is found to have a value close to B$\sim$0.6-0.7 when the typical distance A is in the range 20-80 light-days for local AGNs (Wu et al. 2004; Kaspi et al. 2005). More recently Bentz et al. (2009) find log $(R_{\rm
BLR})=K+\alpha {\rm log}(\lambda L_{\lambda}$(5100Å)) with $\alpha$ = 0.519$^{+0.063}_{-0.066}$ and K = $-$21.3$^{+2.9}_{-2.8}$. The slope suggests that brighter AGNs have to a first approximation the same structure as fainter AGNs with only larger dimensions. Therefore, extending the Kaspi et al. (2005) relation to higher luminosities yields a typical radius of the order of 1 pc for the BLR of bright high-z quasars. In the present case, luminosity is $L_{\rm CIV}~\sim~9~\times~10^{44}$ erg s$^{-1}$ which gives a size of the BLR of $\le$ 1.1 pc. We have estimated in the previous Section that the longitudinal size of the DLA is of order 2-9 pc. This is larger than the size of the BLR. It therefore would mean either that the cloud is much smaller in the transverse direction, corresponding possibly to a filamentary structure or that some holes are present in the cloud or that the cloud is not centered on the quasar.
QSO BLR Lyman-$\alpha$ emission and covering factor
---------------------------------------------------
To determine the H [i]{} column density accurately, one needs to reconstruct the shape of the quasar spectrum at the position of the DLA profile. We can apply principal component (PCA) reconstruction of the quasar flux using the red part of the spectrum to estimate the shape of the Lyman-$\alpha$$-$N [v]{} emission (Pâris et al. 2011, 2013). To do so, we first subtract the residual flux seen at the bottom of the DLA (i.e. residuals from BEL and NEL regions) so that we get zero flux in the DLA trough. We then de-redden this spectrum and add again the residuals subtracted above. We now have the complete de-reddened observed spectrum. Applying the Principle Component Analysis (PCA) method on this spectrum, we derive the PCA spectrum shown as the dotted red curve in the upper panel of Fig. \[simulateHI\]. Then, we subtract the continuum from the PCA spectrum (green dashed curve) and scale it so that it is consistent with the residual flux seen at the bottom of the DLA trough (solid blue curve). This residual flux is only $\sim$7 % of that of the Lyman-$\alpha$ broad emission, indicating that $\sim$93 % of the Lyman-$\alpha$ BLR is covered by the cloud.
To obtain the flux seen by the DLA, we subtract from the dashed green curve the residual flux seen at the bottom of the DLA (i.e. the solid blue curve) and re-add the continuum. We then fit the DLA to obtain the solid red curve in the lower panel. We also checked that we get the same result if, before fitting the DLA, we first re-redden the PCA continuum. As can be seen in Fig. \[simulateHI\], the fit is slightly high near the N [v]{} emission line although within errors. This is possibly because the N [v]{} emission line in this quasar may be weaker than what is predicted by the PCA method. It should be reminded that the PCA reconstruction is an estimate of the quasar spectrum which stays close to the median spectrum in the overall quasar population (see Pâris et al. 2011). The result (log $N$(H [i]{}) = 21.7$\pm$0.1) is consistent with what was derived previously.
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[c]{}
Figure \[simulateQSO\] shows the quasar spectrum as it would be observed if no DLA were present. In this figure, we have added the Lyman-$\alpha$ narrow emission component seen in the DLA trough to the PCA spectrum. The final spectrum is degraded to the SNR $\sim$ 20 by adding Gaussian noise to it. It is apparent from this spectrum that the emission from the narrow line region is quite strong in this quasar.
The quasar narrow line region
=============================
We call here the narrow line region (NLR), the region of the quasar host galaxy that is located within the PSF of the observations. This corresponds to about 1 arcsec or 7.1 kpc at the redshift of the quasar (or a distance of 3.55 kpc on both sides of the quasar). The emission seen beyond this will be called extended emission. We have shown that the DLA is of much smaller dimension so that most of the Lyman-$\alpha$ NLR is not covered and is detected as Lyman-$\alpha$ emission in the bottom of the DLA trough. We can extract this emission which is shown in the top panel of Fig. \[CIV\_NEL\_II\]. It can be seen that the emission is spread over more than 1200 km s$^{-1}$ with FWHM $\sim$ 900 km s$^{-1}$.
CIV partial coverage
--------------------
The large rest equivalent width of the C [iv]{} and Si [iv]{} absorption doublets and the flat-bottomed structure of their profiles suggest that these lines are saturated (see Fig. \[CIV\_NEL\_II\]). However, the flux at the bottom of the C [iv]{} doublet absorption lines apparently does not reach the zero flux level, indicating that the absorbing cloud is only partially covering the background emission-line region.
When partial coverage occurs, the residual intensity seen at the bottom of absorption lines can be written as at each wavelength: $$I(\lambda)~=~I_{0}(\lambda)(1~-~C_{\rm f})~+~C_{\rm
f}~I_{0}(\lambda)~exp~[-\tau(\lambda)]$$
where $I_{0}(\lambda)$ is the incident (unabsorbed) intensity, $\tau(\lambda)$ is the optical depth of the cloud at the considered wavelength, and $C_{\rm f}$ is the fraction of the background emitting region that is covered by the absorbing cloud (i.e. the covering factor). In the case of doublets, and assuming the covering factor is the same for each component of the doublet, we can write:
$$C_{\rm f}~=~\frac{1~+~R_{2}^{2}~-~2R_{2}}{1~+~R_{1}~-~2R_{2}}$$
where $R_{1}$ and $R_{2}$ are the normalized residual intensities in the two absorption lines of the doublet (Petitjean & Srianand 1999; Srianand & Shankaranarayanan 1999). In Fig. \[CIV\_Cf\], we show the covering factor of the C [iv]{}, Si [iv]{}, and N [v]{} doublets. Here, the red and blue histograms are the profiles of the weaker and stronger transitions of each doublet, respectively. The green solid line is the covering factor at each point of the profile calculated from Eq.(3). The green vertical dotted lines mark the regions used to calculate the mean values of $C_{\rm f}$, avoiding the wings of the profiles. The green horizontal dotted lines along with the green filled squares show these mean values. Error bars are calculated as the standard deviation of $C_{\rm f}$ in different velocity bins. The measured values of $C_{\rm f}$ for C [iv]{} and Si [iv]{} are $\sim$ 0.85 and 0.90 respectively suggesting that the size of the corresponding BEL region may be similar for the two high-ionization species. Note also that N [v]{} on the contrary seems fully covered.
The above numbers are however only indicative because the resolution of our spectrum is not high ($R\sim 4000$). In addition there is a possible FeII$\lambda$2586 line at $z_{\rm abs}~=~1.5140$ blended with the red wing of C [iv]{}$\lambda$1550 (cyan curve in the C [iv]{}$\lambda$1550 panel in Fig \[j08231D\]). In Fig. \[CIV\_NEL\_II\], we demonstrate the effect of partial coverage on the observed C [iv]{} doublet profiles taking into account the resolution of the spectrum and the presence of the FeII line. The contribution of the Fe [ii]{}$\lambda$2586 absorption to this fit is robustly determined by fitting together Fe [ii]{}$\lambda$$\lambda$2344,2600 in the same system. In panel (b) of Fig. \[CIV\_NEL\_II\] the red curve is the fit conducted with VPFIT[^4] on the original data. Here, we can see that there is no way to fit most of the profile without invoking partial coverage.
Reconstructing C [iv]{} narrow emission line (NEL) profile
----------------------------------------------------------
As mentioned earlier, the flux at the bottom of the C [iv]{} doublet absorption lines does not reach the zero level. This residual flux could be due to the partial coverage of either the C [iv]{} BLR or NLR. However, the radius of the C [iv]{} phase associated with the DLA is larger and sometimes much larger than the size of the BLR (see Section 3.2) which strongly suggests that the residual flux is due to the C [iv]{} NLR.
We thus tried to reproduce the residual by C [iv]{} emission, including [both]{} lines of the doublet. Doing this we would like to ask the question whether we can associate the emission with the Lyman-$\alpha$ emission. Indeed, this could bring important clues on the origin of the Lyman-$\alpha$ emission. In case the Lyman-$\alpha$ emission [*cannot*]{} be associated with C [iv]{} emission then this would strongly suggest that the Lyman-$\alpha$ emission corresponds to scattered light after radiative transfer.
We thus first try to simply scale the Lyman-$\alpha$ profile. This is bound to fail as it is apparent that the residuals do not follow the shape of the Lyman-$\alpha$ emission. We therefore decomposed the Lyman-$\alpha$ profile in three Gaussian functions (see top panel of Fig. \[CIV\_NEL\_II\]) as suggested by the profile itself.
This allows us to scale different parts of the Lyman-$\alpha$ profile differently. For each pixel we define the ratio $R$ = C [iv]{}/Lyman-$\alpha$ by combining the three Gaussian functions using the following equation:
$$R~=~\frac{R_{1}\times G_{1}~+~R_{2}\times G_{2}~+~R_{3}\times
G_{3}}{G_{1}~+~G_{2}~+~G_{3}}
\label{eq_CIV_NEL}$$
where $R_{1}$, $R_{2}$, and $R_{3}$ are the weights for each component directly related to the C [iv]{}/Lyman-$\alpha$ ratio in this component. By assigning different values to $R_{1}$, $R_{2}$, and $R_{3}$, we can scale each Gaussian individually. For instance, if $R_{1}$ = $R_{2}$ = $R_{3}$ = 1.0, this results in the red profile shown in panel (a) of Fig. \[CIV\_NEL\_II\]. which is simply the combination of the three Gaussian functions. One can now use the factor $R$ = C [iv]{}/Lyman-$\alpha$ defined for each wavelength to properly scale the Lyman-$\alpha$ NEL profile. We thus have to adjust the $R_{1}$, $R_{2}$, and $R_{3}$ parameters until the C [iv]{} emission is consistent with the residual seen at the bottom of the C [iv]{} doublet absorption lines.
We find that the residual can be reproduced (see panel (c) of Fig. \[CIV\_NEL\_II\]) with $R_{1}$ = 0.29, $R_{2}$ = 0.045, and $R_{3}$ = 0.18. The variation of $R$ through the profile is given as a dashed pink line in panels (c) and (e) of Fig. \[CIV\_NEL\_II\]. The corresponding fit of C [iv]{} after removing the residuals is given in panels (d) and (f).
Since the weight of the second emission component is much smaller than the two other ones, we ask the question whether it would be possible that the second component has no C [iv]{} associated. To test this, we impose $R_{2}$ = 0.001. The result of the fit is given in panels (e) and (f). It is apparent that we can find a solution with no C [iv]{} associated with the second component.
We thus conclude from all this that (i) the C [iv]{} emission is strongest around $v$ $\sim$ $-$200 km s$^{-1}$; this position could indicate the redshift of the quasar, implying that most of the Lyman-$\alpha$ emission and the DLA are redshifted; (ii) at least part of the NLR Lyman-$\alpha$ emission has no C [iv]{} emission associated (predominantly around $v=+100$ km s$^{-1}$) which means that the Lyman-$\alpha$ emission in this component is due to scattered light or that the emitted gas is located within a distance from the quasar smaller than the transverse size of the C [iv]{} phase associated with the cloud so that the corresponding C [iv]{} emission is absorbed by the high-ionization phase of the DLA cloud.
[c]{}
Extended Lyman-$\alpha$ emission
================================
In the middle and bottom panels of Fig. \[2D\_both\_PAs\] we show the 2D spectra of the Lyman-$\alpha$ emission detected in the DLA trough for the two PAs. It is apparent that the Lyman-$\alpha$ emission is extended and slightly displaced relative to the quasar trace. To quantify this we integrate the whole Lyman-$\alpha$ profile in the spectral direction and compare the result to the spatial PSF derived from the integration of the quasar spectrum over the rest of the order beyond 6520 Å. This is shown in the right panels in Fig. \[2D\_both\_PAs\]. It is apparent that the emission is extended well beyond the PSF for both PAs and mostly in one direction implying that the total emission is shifted towards this direction.
We investigate whether the extension of the emission varies with the velocity position. For this, we split the velocity range over which the emission is seen in several regions, following the profile, and integrate the spatial emission profile over these regions. We then fit the profile by a Gaussian function. The results show that along PA = NS the spatial extent of the emission is larger than 5 arcsec over about 2000 km s$^{-1}$ and that the shift is about 0.2 arcsec towards the North direction in the same region. For PA = EW, extension is about the same but the shift is consistent with zero meaning that the extended emission is more symmetric around the trace. The Lyman-$\alpha$ emission is detected up to more than 25 kpc from the quasar and there is a strong excess emission along PA = NS to the North.
To better visualize the extended emission, we will subtract the emission associated with the central PSF e.g. the emission located on the quasar trace. To do so, the 2D spectral order outside the Lyman-$\alpha$ region of each PA is split into several chunks, and in each chunk the counts are integrated along the spectral axis. A Gaussian is then fitted on the profiles to get the central pixel values of the spatial profiles. Finally, we fit a straight line on these values to determine the center of the trace for each spectral pixel.
Once we know the position of the trace exactly, we extract the spatial profile in each wavelength pixel and we fit a Gaussian with width equal to the continuum PSF width. We then subtract this Gaussian from the profile. Results are given in Fig \[Lya\_vs\_trace\_extension\]. Top panels show the resulting 2D spectrum of the extension. Bottom panels show the Lyman-$\alpha$ profiles integrated along the spectral direction: in red the emission which was fitted on top of the trace and in black is the extension.
[c]{}
Along PA = NS, most of the extended emission is seen to the North up to projected distances of 3.5 arcsec or $\sim$25 kpc from the center. The striking observation here is that the emission is extended over more than 1000 km s$^{-1}$ at all distances from the quasar. Along PA = EW, extension is more apparent in one direction as well, to the West. However the velocity extension gets smaller when the distance to the quasar increases. Fig \[Distance\] shows the variation of the integrated flux with distance. In this Figure, the red and blue curves indicate variations of flux with distance following the relation $F$ = $F_{0}$($r$/$r_{0}$)$^{-2}$ relation. The data roughly follow such a law. In case this emission is due to recombination and since the ionization flux is decreasing with distance as $r^{-2}$ this would mean that the number of ionized hydrogen atoms intersected by the slit is the same at all distances. This could be the case if the gas is clumpy so that changes in density has little effect.
The velocity profiles of the extended Lyman-$\alpha$ emission at different distances from the AGN are shown in Fig \[4\_Extession\_profiles\] and Table \[flux\_luminosity\] summarizes the integrated flux and luminosity of the trace and the extension.
Discussion and conclusions
==========================
We have performed slit spectroscopic observations of QSO J0823+0529 with the MagE spectrograph mounted on the Magellan telescope along two PAs, in the North-South and East-West directions. The quasar is unique because a DLA is located at a redshift very similar to that of the quasar ($z_{\rm DLA}$ = 3.1910 and $z_{\rm CIV}$ = 3.1875, i.e. $\Delta v$ $\sim$330 km s$^{-1}$) and acts as a coronagraph blocking most of the flux from the central regions of the AGN. In the present case, the DLA cloud is small enough so that it covers only approximately 90% of the Lyman-$\alpha$ BLR. This puts us in a unique position to be able to directly observe the quasar narrow line region and the extended emission line region. Indeed, Lyman-$\alpha$ emission is detected up to more than 25 kpc from the quasar along both PAs.
The quasar NLR
--------------
Finley et al. (2013) have gathered a sample of 57 DLAs acting as a coronagraph in front of quasars; the DLAs have redshifts within 1500 km s$^{-1}$ from the quasar redshift. Their statistical sample of 31 quasars shows an excess of such DLAs compared to what is expected from the distribution of intervening DLAs. This can be explained if most of these DLAs are part of galaxies clustering around the quasar. However 25% of such DLAs reveal Lyman-$\alpha$ emission probably from the quasar host galaxy implying that these DLAs have sizes smaller than the quasar emission region. The Lyman-$\alpha$ luminosities are consistent with those of Lyman-$\alpha$ emitters in 75% of the cases and 25% have much higher luminosities. The later systems probably reveal the central NLR of the quasar. QSO J0823+0529 is part of this later class of coronagraph DLAs.
We have shown that the size of the neutral phase is of order 2-9 pc which means that the gas does not cover the Lyman-$\alpha$ NLR. Three main components of Lyman-$\alpha$ emission from the NLR are clearly detected at $\Delta V$ = $-$300, +100 and +400 km s$^{-1}$ relative to the DLA redshift (see Fig. 9). The corresponding C [iv]{} emission seems to be absent in the second strongest component. Either the Lyman-$\alpha$ emission is scattered or the associated C [iv]{} emission is absorbed by the high-ionization phase of the DLA. To be covered, the corresponding gas must be located within $\sim$3-80 pc from the quasar which, we have seen, corresponds to the extension of the high-ionization phase of the cloud assuming spherical geometry. At least part of the two other components could therefore be located beyond this distance from the quasar, although part of the third component could also be hidden, as the C [iv]{}/H [i]{} ratio is larger by a factor of about two in the first component compared to the third component (see Fig. 9). Note that the relative velocities of components 1 and 3 can be interpreted as the result of a conical outflow with mean projected velocities of about $\pm$300 km s$^{-1}$.
[c c]{}
Lyman-$\alpha$ emission from the center to the outskirts of the host galaxy
---------------------------------------------------------------------------
Extended Lyman-$\alpha$ emission has been observed around high-redshift radio galaxies (Heckman et al. 1991; van Ojik et al. 1996), as well as around radio-quiet quasars (Bunker et al. 2003; Christensen et al. 2006). For radio-loud quasars or radio-galaxies, the Lyman-$\alpha$ flux of the nebula is an order of magnitude higher (Christensen et al. 2006), presumably because the emission of radio-loud quasar gaseous envelopes is enhanced by interactions with the radio jets. More recently, North et al. (2012) have used a careful treatment of the spectral PSF to extract quasar traces. This revealed four detections of extended emission out of six radio-quiet quasars at $z\sim 4.5$ with extensions of diameter $16<d<64$ kpc down to a luminosity of 2$\times$10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$. The emission has 900 $<$ FWHM $<$ 2200 km s$^{-1}$. Our observations are in line with these numbers. The extended emission we detect in QSO J0823+0529 has a diameter of $\sim$50 kpc and FWHM $\sim$ 900 km s$^{-1}$. However, QSO J0823+0529 does not seem to follow the $L$(Lyman-$\alpha$) vs $L$(BLR) relation indicated by these authors. Indeed the Lyman-$\alpha$ luminosity in QSO J0823+0529 is more than an order of magnitude larger than what would be expected from this relation even after correcting for dust attenuation. It is still possible that dust is present closer to the quasar and further attenuates the BLR Lyman-$\alpha$ emission. It is also possible that the decomposition between the broad and narrow line region emissions was ambiguous in previous studies so that the NLR emission could have been underestimated.
---------------------- ---------------------------- ------------------------- --
$ $ Integrated flux Luminosity
$ $ \[erg s$^{-1}$ cm$^{-2}$\] \[erg s$^{-1}$\]
Trace(PA=NS) 3.59 $\times$ 10$^{-16}$ 3.43 $\times$ 10$^{43}$
Extension(PA=NS) 3.97 $\times$ 10$^{-16}$ 3.79 $\times$ 10$^{43}$
Trace(PA=EW) 3.36 $\times$ 10$^{-16}$ 3.21 $\times$ 10$^{43}$
Extension(PA=EW) 2.68 $\times$ 10$^{-16}$ 2.56 $\times$ 10$^{43}$
Lyman-$\alpha$(BELR) 7.10 $\times$ 10$^{-15}$ 6.78 $\times$ 10$^{44}$
---------------------- ---------------------------- ------------------------- --
: Lyman-$\alpha$ integrated fluxes and luminosities in the PSF centered on the quasar trace and in the extension. The last row gives the integrated flux and luminosity of the broad Lyman-$\alpha$ emission calculated using the simulated spectrum shown in Fig \[simulateQSO\].
\[flux\_luminosity\]
The emission is more extended to the North-West of the object (see Figs. 11, 13 and 15). There are two notable features in the spatial and velocity structure of the nebula. First, in the North-West direction, the kinematics are strikingly similar along the trace and 10 kpc away from the center (see Fig. 13) with a velocity spread of more than 1000 km s$^{-1}$. The emission is quite strong in this region. Secondly, there is a clear gradient of about 1000 km s$^{-1}$ between 15 kpc to the East and 20 kpc to the West. It is tempting to interpret these features as the superposition of the emission of gas in the disk of the galaxy, where the density is higher and turbulent kinematics prevent gas clouds to be well organized with emission from gas flowing out of the disk with velocities of the order of 500 km s$^{-1}$. This gas is best seen up to 20 kpc to the West and 10 kpc to the East. Such winds can be reproduced by recent models taking into account the effects of radiation trapping (Ishibashi & Fabian 2015).
Nature of the DLA
-----------------
It is well demonstrated that bright quasars are surrounded by large amounts of gas both in extended ionized halos up to 10 Mpc from the quasar (e.g. Rollinde et al. 2005) and extended (300 kpc) reservoirs of neutral gas in the halo of the host galaxy (Prochaska et al. 2013, 2014; Johnson et al. 2015). However, along the line of sight to quasars, the incidence of neutral gas is less (e.g. Shen & Ménard 2012) indicating that the ionizing emission from quasars is highly anisotropic. The DLA we discuss here is part of the so-called proximate DLAs in the sense that the absorption redshift is similar to that of the quasar ($z_{\rm abs}$ $\sim$ z$_{\rm em}$, see e.g. Ellison et al. 2010). However, this is the first time it can be demonstrated that the gas is located very close to the quasar ($<$ 400 pc) and probably associated with the central part of the host galaxy.
We have modelled the physical state of the gas in the DLA using photo-ionization models. The ionization parameter is found to be in the range $-1.1$ $<$ log$U$ $<$ $+0.3$ which means that the cloud is mostly highly ionized. A density of $n_{\rm
H}$ $\sim$ 710-180 cm$^{-3}$ is needed to explain the absorption lines from O [i]{} and Si [ii]{} ground state excited levels implying that the neutral phase should have dimensions of the order of 2 to 9 pc and be embedded in an ionized cloud of size $\sim$3-80 pc. From this, we could derive that the cloud is located between 230 and 580 pc from the quasar. The metallicity of the cloud, $Z$ = 0.16 $Z_{\odot}$, is typical of the metallicity of standard intervening DLAs (Rafelski et al. 2012).
[c]{}
It is intriguing to note that the high-ionization phase of the cloud we see has characteristics similar to those of some of the warm absorbers seen in many AGNs. Tombesi et al. (2013) argued that warm absorbers (WA) and ultra-fast outflows (UFO) could represent parts of a single large-scale stratified outflow observed at different locations from the black hole. The UFOs are likely launched from the inner accretion disc and the WAs at larger distances, such as the outer disc and/or torus. There are still significant uncertainties on the exact location of this material, which ranges from a few pc up to kpc scales, (e.g. Krolik & Kriss 2001; Blustin et al. 2005). The absorption lines are systematically blue-shifted, indicating outflow velocities of the WAs in the range 100-1000 km s$^{-1}$. King & Pounds (2013) have argued that the dense gas which surrounds the AGN when it starts shining is swept out by the fast winds powered by the accretion luminosity. The wind is halted by collisions near the radius where radiation pressure drops. The shocked gas must rapidly cool and mix with the swept-up ISM. Distance from the AGN and properties of the gas are similar to what is observed for warm-absorbers, as in our case. Therefore, in QSO J0843+0529, the DLA could be located in the galactic disk at the terminal position of the wind, at the limit of the interstellar medium of the host galaxy. The presence of dust in the gas can be considered as supporting this view as it is expected in the dense environment of AGNs (see e.g. Leighly et al. 2015).
[c c]{}
The only caveat with this idea is that the metallicity of the DLA is typical of intervening DLAs, when we would expect the gas in the ISM of the quasar host galaxy to have larger metallicity. This, together with the low outflow velocity (the DLA is centered on the Lyman-$\alpha$ emission), argues for another explanation for the origin of this gas. Given the distance to the central AGN (230 $<$ $r_0$ $<$ 580 pc), one is tempted to conjecture that it could be the final fate of infalling gas. Note that in that case, the presence of dust is not a problem as the observed amount is not unusual compared to what is seen in typical high column density DLAs. Indeed, accretion is believed to happen through cold flows which are expected to be of lower metallicity compared to the environment of the quasar (see Bouché et al. 2013). Little is known about how quasars at high redshift interact with cold infalling streams of gas, and in particular whether these collimated structures can survive the energy released by the AGN. Therefore the DLA cloud could be in a transitory phase before it is completely destroyed by the AGN ionizing radiation field (Namekata et al. 2014).
The two alternatives are actually not incompatible, since some of the warm absorbers could be part of the galactic ISM swept away from the center by the AGN and placed at large distances from the AGN (e.g. King & Pounds 2013). These outflowing warm absorbers may then intersect with accreted cold streams.
Concluding remark
-----------------
There is little doubt that DLAs acting as coronagraphs are important targets to be observed and analyzed. Besides revealing interesting characteristics of the quasar host galaxy they are, like in QSO J0823+0529, potentially part of the machinery that power the AGN. Finley et al. (2013) found 57 such systems out of about a third of the BOSS targets. With the on-going eBOSS-SDSSIV survey and the forseen DESI survey (Schlegel et al. 2011), the number of such systems will increase by a large factor in the future. It will thus be important to gather a large sample of these objects to be able to study their characteristics statistically.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the anonymous referee for their constructive comments, which helped us to improve the paper. We also thank George Becker for advices on MagE data reduction and Hadi Rahmani for useful discussion. H.F. was supported by the Agence Nationale pour la Recherche under program ANR-10-BLAN-0510-01-02. SL has been supported by FONDECYT grant number 1140838 and partially by PFB-06 CATA.
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[^1]: Based on data obtained with MagE at the Clay telescope of the Las Campanas Observatory (CNTAC Prgm. ID CN2012B-51 and CN2013A-121)
[^2]: <ftp://ftp.ast.cam.ac.uk/pub/gdb/mage_reduce/>
[^3]: <https://www.eso.org/sci/observing/tools/standards.html>
[^4]: <http://www.ast.cam.ac.uk/~rfc/vpfit.html>
| ArXiv |
---
abstract: 'For some estimations and predictions, we solve minimization problems with asymmetric loss functions. Usually, we estimate the coefficient of regression for these problems. In this paper, we do not make such the estimation, but rather give a solution by correcting any predictions so that the prediction error follows a general normal distribution. In our method, we can not only minimize the expected value of the asymmetric loss, but also lower the variance of the loss.'
author:
- 'Naoya Yamaguchi, Yuka Yamaguchi, and Ryuei Nishii'
bibliography:
- 'reference.bib'
title: Minimizing the expected value of the asymmetric loss and an inequality of the variance of the loss
---
Introduction {#S1}
============
For some estimations and predictions, we solve minimization problems with loss functions, as follows: Let $\{ (x_{i}, y_{i}) \mid 1 \leq i \leq n \}$ be a data set, where $x_{i}$ are $1 \times p$ vectors and $y_{i} \in \mathbb{R}$. We assume that the data relate to a linear model, $$y = X \beta + \varepsilon,$$ where $y = {}^{t}(y_{1}, \ldots, y_{n})$, $\varepsilon = {}^{t}(\varepsilon_{1}, \ldots, \varepsilon_{n})$, and $X$ is the $n \times p$ matrix having $x_{i}$ as the $i$th row. Let $L$ be a loss function and let $r_{i}(\beta) := y_{i} - x_{i} \beta$. Then we estimate the value: $$\begin{aligned}
\hat{\beta} := \arg\min_{\beta} \left\{ \sum_{i = 1}^{n} L(r_{i}(\beta)) \right\}. \end{aligned}$$ The case of $L(r_{i}(\beta)) = r_{i}(\beta)^{2}$ is well-known (see, e.g., Refs. [@doi:10.1111/j.1751-5823.1998.tb00406.x], [@legendre1805nouvelles], and [@stigler1981]). In the case of an asymmetric loss function, we refer the reader to, e.g., Refs. [@10.2307/2336317], [@10.2307/24303995], [@10.2307/1913643], and [@10.2307/2289234]. These studies estimate the parameter $\hat{\beta}$. In this paper, however, we do not make such the estimation, but instead give a solution to the minimization problems by correcting any predictions so that the prediction error follows a general normal distribution. In our method, we can not only minimize the expected value of the asymmetric loss, but also lower the variance of the loss.
Let $y$ be an observation value, and let $\hat{y}$ be a predicted value of $y$. We derive the optimized predicted value $y^{*} = \hat{y} + C$ minimizing the expected value of the loss under the assumption:
1. The prediction error $z := \hat{y} - y$ is the realized value of a random variable $Z$, whose density function is a generalized Gaussian distribution function (see, e.g., Refs. [@Dytso2018], [@doi:10.1080/02664760500079464], and [@Sub23]) with mean zero $$\begin{aligned}
f_{Z}(z) := \frac{1}{2 a b \G(a)} \exp{\left( - \left\lvert \frac{z}{b} \right\rvert^{\frac{1}{a}} \right)}, \end{aligned}$$ where $\G(a)$ is the gamma function and $a$, $b \in \mathbb{R}_{> 0}$.
2. Let $k_{1}$, $k_{2} \in \mathbb{R}_{> 0}$. If there is a mismatch between $y$ and $\hat{y}$, then we suffer a loss, $$\begin{aligned}
\Pe(z) :=
\begin{cases}
k_{1} z, & z \geq 0, \\
- k_{2} z, & z < 0.
\end{cases}\end{aligned}$$
That is, the solution to the minimization problem is $$\begin{aligned}
C = \arg\min_{c} \left\{ \operatorname{{E}}\left[ \Pe(Z + c) \right] \right\}. \end{aligned}$$
The motivation of our research is as follows: (1) Predictions usually cause prediction errors. Therefore, it is necessary to use predictions in consideration of predictions errors. Actually, in some cases, it is best not to act as predicted because of prediction errors. For example, the paper [@Yamaguchi2018] formulates a method for minimizing the expected value of the procurement cost of electricity in two popular spot markets: [*day-ahead*]{} and [*intra-day*]{}, under the assumption that the expected value of the unit prices and the distributions of the prediction errors for the electricity demand traded in two markets are known. The paper showed that if the procurement is increased or decreased from the prediction, in some cases, the expected value of the procurement cost is reduced. (2) In recent years, prediction methods have been black boxed by the big data and machine learning (see, e.g., Ref. [@10.1145/3236009]). The day will soon come, when we must minimize the objective function by using predictions obtained by such black boxed methods. In our method, even if we do not know the prediction $\hat{y}$, we can determine the parameter $C$ if we know the prediction error distribution $f$ and asymmetric loss function $L$.
To obtain $y^{*}$, we derive $\operatorname{{E}}[\Pe(Z + c)]$ for any $c \in \mathbb{R}$. Let $\G(a, x)$ and $\g(a, x)$ be the upper and the lower incomplete gamma functions, respectively (see, e.g., Ref. [@doi:10.1142/0653]). The expected value and the variance of $\Pe(Z + c)$ are as follows:
\[lem:1.1\] For any $c \in \mathbb{R}$, we have $$\begin{aligned}
(1)\quad
\operatorname{{E}}[\Pe(Z + c)]
&= \frac{(k_{1} - k_{2}) c}{2}
+ \frac{(k_{1} + k_{2}) \lvert c \rvert}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
+ \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right), \\
(2)\quad
\operatorname{{V}}[\Pe(Z + c)]
&= \frac{(k_{1} + k_{2})^{2} c^{2}}{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} b \lvert c \rvert}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} c^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \nonumber \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \nonumber\end{aligned}$$
We write the value of $c$ satisfying $\frac{d}{dc} \operatorname{{E}}[\Pe(Z + c)] = 0$ as $C$. Then, we find that $\operatorname{{E}}[\Pe(Z + c)]$ has a minimum value at $c = C$. Also, it follows from $$\begin{aligned}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
= \operatorname{{sgn}}(C) \frac{k_{2} - k_{1}}{k_{1} + k_{2}} \G(a) \end{aligned}$$ that $\operatorname{{sgn}}(C) = \operatorname{{sgn}}(k_{2} - k_{1})$, where $\operatorname{{sgn}}(c) := 1 \: (c \geq 0); -1 \: (c < 0)$, and $C = 0$ only when $k_{1} = k_{2}$. This equation implies that the ratio of $\G(a)$ and $\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)$ is $1 : \frac{\lvert k_{2} - k_{1}\rvert}{k_{1} + k_{2}}$. That is, the vertical axis $t = \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}}$ divides the area between $t^{a - 1} e^{- t}$ and the $t$-axis into $\frac{\lvert k_{2} - k_{1}\rvert}{k_{1} + k_{2}} : 1- \frac{\lvert k_{2} - k_{1}\rvert}{k_{1} + k_{2}}$.
Substituting $c = C$ in the equation $(1)$ of Lemma $\ref{lem:1.1}$, from the equation $(3)$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + C)]
= \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ This is the minimum value of $\operatorname{{E}}[\Pe(Z + c)]$. From this and the $c = 0$ case of the equation $(1)$ of Lemma $\ref{lem:1.1}$, we have the following corollary:
\[cor:1.2\] We have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z)] - \operatorname{{E}}[\Pe(Z + C)]
&= \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\g\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right), \\
\frac{\operatorname{{E}}[\Pe(Z + C)]}{\operatorname{{E}}[\Pe(Z)]}
&= \frac{1}{\G(2a)} \G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$
This corollary asserts that the expected value of the loss is reduced by correcting a predicted value $y$ to the optimized predicted value $y^{*}$. Moreover, the following holds:
\[thm:1.3\] We have $$\begin{aligned}
\operatorname{{V}}[\Pe(Z + C)] \leq \operatorname{{V}}[\Pe(Z)], \end{aligned}$$ where equality sign holds only when $C = 0$; that is, when $k_{1} = k_{2}$.
This theorem asserts that the variance of the loss is reduced by correcting the predicted value $y$ to the optimized predicted value $y^{*}$. To prove this theorem, we use the following lemma:
\[lem:1.4\] For $a > 0$ and $x > 0$, we have $$\begin{aligned}
x^{a} \g(a, x)^{2} - x^{a} \G(a)^{2} + 2 \g(a, x) \G(2a, x) > 0. \end{aligned}$$
To prove Lemma $\ref{lem:1.4}$, we use the following lemmas:
\[lem:1.5\] For $a > 0$, we have $$\begin{aligned}
2 \G(2a) - a \G(a)^{2} > 0. \end{aligned}$$
\[lem:1.6\] For $a > 0$, we have $$\begin{aligned}
4^{a} \G\left(a + \frac{1}{2} \right) > \sqrt{\pi} \G(a + 1). \end{aligned}$$
The remainder of this paper is organized as follows. In Section $2$, we set up the problem. In Section $3$, we introduce the expected value and the variance of $\Pe(Z + c)$, and we determine the value of $c = C$ that gives the minimum value of $\operatorname{{E}}[\Pe(Z + c)]$. In addition, we give a geometrical interpretation of the parameter $C$, and give the minimized expected value $\operatorname{{E}}[\Pe(Z + C)]$. In Section $4$, we prove Theorem $\ref{thm:1.3}$. In Section $5$, we give some inequalities for the gamma and the incomplete gamma functions, which used to derive the inequality for the variance of the loss in Theorem $\ref{thm:1.3}$. In Section $6$, we write the calculation of the expected value and the variance of the loss $\Pe(Z + c)$ for $c \in \mathbb{R}$.
Problem statement
=================
In this section, we set a problem. Let $y$ be an observation value, let $\hat{y}$ be a predicted value of $y$, and let $\G(a)$ be the gamma function (see, e.g., Ref. [@doi:10.1142/0653 p. 93]) defined by $$\begin{aligned}
\G(a) := \int_{0}^{+\infty} t^{a - 1} e^{- t} dt, \quad \text{Re}(a) > 0. \end{aligned}$$ We assume the following:
1. The prediction error $z := \hat{y} - y$ is the realized value of a random variable $Z$, whose density function is a generalized Gaussian distribution function (see, e.g., Refs. [@Dytso2018], [@doi:10.1080/02664760500079464], and [@Sub23]) with mean zero $$\begin{aligned}
f_{Z}(z) := \frac{1}{2 a b \G(a)} \exp{\left( - \left\lvert \frac{z}{b} \right\rvert^{\frac{1}{a}} \right)}, \end{aligned}$$ where $a$, $b \in \mathbb{R}_{> 0}$.
2. Let $k_{1}$, $k_{2} \in \mathbb{R}_{> 0}$. If there is a mismatch between $y$ and $\hat{y}$, then we suffer a loss, $$\begin{aligned}
\Pe(z) :=
\begin{cases}
k_{1} z, & z \geq 0, \\
- k_{2} z, & z < 0.
\end{cases}\end{aligned}$$
We derive the optimized predicted value $y^{*} = \hat{y} + C$ minimizing $\operatorname{{E}}[\Pe(Z + c)]$. For this purpose, we derive $\operatorname{{E}}[\Pe(Z + c)]$ for any $c \in \mathbb{R}$ in the next section.
Expected value and variance of the loss
=======================================
Here, we introduce the expected value and the variance of $\Pe(Z + c)$, and determine the value of $c = C$ that gives the minimum value of $\operatorname{{E}}[\Pe(Z + c)]$. In addition, we give a geometrical interpretation of the parameter $C$ and give the minimized expected value $\operatorname{{E}}[\Pe(Z + C)]$.
Expected value and variance of the loss
---------------------------------------
Let $\G(a, x)$ and $\g(a, x)$ be the upper and the lower incomplete gamma functions, respectively, defined by $$\begin{aligned}
\G(a, x) := \int_{x}^{+\infty} t^{a - 1} e^{-t} dt, \qquad
\g(a, x) := \int_{0}^{x} t^{a - 1} e^{-t} dt, \end{aligned}$$ where $\text{Re}(a) > 0$ and $x \geq 0$. These functions have the following properties:
\[lem:3.0\] For ${\rm Re}(a) > 0$ and $x \geq 0$,
&(1)(a, x) + (a, x) = (a);\
&(2)\_[x ]{} (a, x) = (a);\
&(3)(a, 0) = (a);\
&(4) (a, x) = x\^[a - 1]{} e\^[-x]{};\
&(5) (a, x) = - x\^[a - 1]{} e\^[-x]{}. &
Also, for $c \in \mathbb{R}$, let $\operatorname{{sgn}}(c) := 1 \: (c \geq 0); -1 \: (c < 0)$. Then, the expected value and the variance of $\Pe(Z + c)$ are as follows:
\[lem:3.1\] For any $c \in \mathbb{R}$, we have $$\begin{aligned}
(1)\quad
\operatorname{{E}}[\Pe(Z + c)]
&= \frac{(k_{1} - k_{2}) c}{2}
+ \frac{(k_{1} + k_{2}) \lvert c \rvert}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
+ \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right), \\
(2)\quad
\operatorname{{V}}[\Pe(Z + c)]
&= \frac{(k_{1} + k_{2})^{2} c^{2}}{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} b \lvert c \rvert}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} c^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \nonumber \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \nonumber\end{aligned}$$
See the last two sections for the proof of Lemma $\ref{lem:3.1}$.
From Lemma $\ref{lem:3.1}$, we have the following: $$\begin{aligned}
\operatorname{{E}}[\Pe(Z)]
&= \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left( 2a \right), \label{E[L(Z)]} \\
\operatorname{{V}}[\Pe(Z)]
&= \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
- \frac{(k_{1} + k_{2})^{2} b^{2} \G(2a)^{2}}{4 \G(a)^{2}}. \label{V[L(Z)]}\end{aligned}$$
Let $\operatorname{{erf}}(x)$ be the error function defined by $$\begin{aligned}
\operatorname{{erf}}(x) := \frac{2}{\sqrt{\pi}} \int_{0}^{x} \exp{\left( - t^{2} \right)} dt \end{aligned}$$ for any $x \in \mathbb{R}$. We give two examples of $\operatorname{{E}}[\Pe(Z + c)]$ and $\operatorname{{V}}[\Pe(Z + c)]$.
\[rei:3.2\] In the case of ${\rm Laplace}(0, b)$, since $a = 1$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= \Pe(c) + \frac{(k_{1} + k_{2}) b}{2} \exp{\left(- \left\lvert \frac{c}{b} \right\rvert \right)}, \\
\operatorname{{V}}[\Pe(Z + c)]
&= \left\{ k_{1}^{2} + k_{2}^{2} + \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2} ) \right\} b^{2} \\
&\quad - \operatorname{{sgn}}(c) (k_{1} + k_{2}) \left\{ \Pe(c) + b (k_{1} - k_{2}) \right\} b \exp{\left(- \left\lvert \frac{c}{b} \right\rvert \right)} \\
&\quad - \frac{(k_{1} + k_{2})^{2} b^{2}}{4} \exp{\left(- 2 \left\lvert \frac{c}{b} \right\rvert \right)}. \end{aligned}$$ In the case of $\mathcal{N}(0, \frac{1}{2} b^{2})$, since $a = \frac{1}{2}$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= \frac{(k_{1} - k_{2}) c}{2}
+ \frac{(k_{1} + k_{2}) c}{2} \operatorname{{erf}}{\left(\frac{c}{b} \right)}
+ \frac{(k_{1} + k_{2}) b}{2 \sqrt{\pi}} \exp{\left(- \frac{c^{2}}{b^{2}} \right)}, \\
\operatorname{{V}}[\Pe(Z + c)]
&= \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} }{4}
+ \frac{(k_{1} + k_{2})^{2} c^{2} }{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) b^{2} }{4} \operatorname{{erf}}{\left( \frac{c}{b} \right)}
- \frac{(k_{1} + k_{2})^{2} c^{2} }{4} \operatorname{{erf}}^{2}{\left( \frac{c}{b} \right)} \\
&\quad - \frac{(k_{1} + k_{2})^{2} b c}{2 \sqrt{\pi}} \operatorname{{erf}}{\left(\frac{c}{b} \right)} \exp{\left( - \frac{c^{2}}{b^{2}} \right)}
- \frac{(k_{1} + k_{2})^{2} b^{2} }{4 \pi} \exp{\left( - \frac{2 c^{2}}{b^{2}} \right)}. \end{aligned}$$ With the conditions fixed as $k_{1} = 50$ and $b = 1$, we can plot $\operatorname{{E}}[\Pe(Z)]$ and $\operatorname{{V}}[\Pe(Z)]$ for the Laplace and the Gauss distributions as follows:
[cc]{}
{height="4cm"} \[fig:winter\]
{height="4cm"} \[fig:fall\]
[cc]{}
{height="4cm"} \[fig:winter\]
{height="4cm"} \[fig:fall\]
Parameter value minimizing the expected value
---------------------------------------------
Here, we determine the value of $c = C$ that gives the minimum value of $\operatorname{{E}}[\Pe(Z + c)]$. Since $$\begin{aligned}
\frac{d}{dc} \G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
&= - \frac{c}{a b} \exp{\left(- \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)}; \\
\frac{d}{dc} \g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
&= \operatorname{{sgn}}(c) \frac{1}{a b} \exp{\left(- \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)}, \end{aligned}$$ we have $$\begin{aligned}
\frac{d}{dc} \operatorname{{E}}[\Pe(Z + c)] = \frac{k_{1} - k_{2}}{2}
+ \operatorname{{sgn}}(c) \frac{k_{1} + k_{2}}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ We will denote the value of $c$ satisfying $\frac{d}{dc} \operatorname{{E}}[\Pe(Z + c)] = 0$ as $C$. Then, from the first derivative test, we find that $\operatorname{{E}}[\Pe(Z + c)]$ has a minimum value at $c = C$.
$c$ Less than $C$ $C$ More than $C$
----------------------------------------------- --------------------- ----- ---------------------
$\frac{d}{dc} \operatorname{{E}}[\Pe(Z + c)]$ Negative $0$ Positive
$\operatorname{{E}}[\Pe(Z + c)]$ Strongly decreasing Strongly increasing
Also, it follows from $$\begin{aligned}
\label{C}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
= \operatorname{{sgn}}(C) \frac{k_{2} - k_{1}}{k_{1} + k_{2}} \G(a) \end{aligned}$$ that $\operatorname{{sgn}}(C) = \operatorname{{sgn}}(k_{2} - k_{1})$ and $C = 0$ only when $k_{1} = k_{2}$.
Moreover, equation $(\ref{C})$ implies that the ratio of $\G(a)$ and $\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)$ is $1 : \frac{\lvert k_{2} - k_{1}\rvert}{k_{1} + k_{2}}$. That is, the vertical axis $t = \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}}$ divides the area between $t^{a - 1} e^{- t}$, and the $t$-axis into $\frac{\lvert k_{2} - k_{1}\rvert}{k_{1} + k_{2}} : 1- \frac{\lvert k_{2} - k_{1}\rvert}{k_{1} + k_{2}}$.
{width="80mm"}
Let $\operatorname{{erf}}^{-1}(x)$ be the inverse error function. We give two examples of $C$.
\[rei:\] In the case of ${\rm Laplace}(0, b)$, since $a = 1$, we have $$\begin{aligned}
C = - \operatorname{{sgn}}(k_{2} - k_{1}) b \log{\left( 1 - \left\lvert \frac{k_{2} - k_{1}}{k_{1} + k_{2}} \right\rvert \right)}. \end{aligned}$$ In the case of $\mathcal{N}(0, \frac{1}{2} b^{2})$, since $a = \frac{1}{2}$, we have $$\begin{aligned}
C = \operatorname{{sgn}}(k_{2} - k_{1}) b \operatorname{{erf}}^{-1}\left( \left\lvert \frac{k_{2} - k_{1}}{k_{1} + k_{2}} \right\rvert \right). \end{aligned}$$ Fixing the conditions as $k_{1} = 50$ and $b = 1$, we can plot $C$ for the Laplace and the Gauss distributions as follows:
[cc]{}
{height="4cm"} \[fig:winter\]
{height="4cm"} \[fig:fall\]
Minimized expected value of the loss
------------------------------------
We give the minimum value of $\operatorname{{E}}[\Pe(Z + c)]$. Substituting $c = C$ in equation $(1)$ of Lemma $\ref{lem:3.1}$, from equation $(\ref{C})$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + C)]
= \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ This is the minimum value of $\operatorname{{E}}[\Pe(Z + c)]$. From this and equation $(\ref{E[L(Z)]})$, we have the following corollary:
\[cor:3.3\] We have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z)] - \operatorname{{E}}[\Pe(Z + C)]
&= \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\g\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right), \\
\frac{\operatorname{{E}}[\Pe(Z + C)]}{\operatorname{{E}}[\Pe(Z)]}
&= \frac{1}{\G(2a)} \G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$
Fixing the conditions as $k_{1} = 50$ and $b = 1$, we can plot the plots of $\operatorname{{E}}[\Pe(Z)] - \operatorname{{E}}[\Pe(Z + C)]$ for the Laplace and the Gauss distributions as follows:
[cc]{}
![Plots of $\operatorname{{E}}[\Pe(Z)] - \operatorname{{E}}[\Pe(Z + C)]$ for the Laplace and the Gauss distributions](Laplace-DE.pdf "fig:"){height="4cm"} \[fig:winter\]
![Plots of $\operatorname{{E}}[\Pe(Z)] - \operatorname{{E}}[\Pe(Z + C)]$ for the Laplace and the Gauss distributions](Gauss-DE.pdf "fig:"){height="4cm"} \[fig:fall\]
An inequality for the variance of the loss
==========================================
In this section, we derive an inequality for the variance of $\Pe(Z + c)$. Let $C$ be the value of $c$ giving the minimum value of $\operatorname{{E}}[\Pe(Z + c)]$. Then, the following holds:
\[thm:3.4\] We have $$\begin{aligned}
\operatorname{{V}}[\Pe(Z + C)] \leq \operatorname{{V}}[\Pe(Z)], \end{aligned}$$ where equality holds only when $C = 0$; that is, when $k_{1} = k_{2}$.
Fixing the conditions as $k_{1} = 50$ and $b = 1$, we can plot $\operatorname{{V}}[\Pe(Z)] - \operatorname{{V}}[\Pe(Z + C)]$ for the Laplace and the Gauss distributions as follows:
[cc]{}
![Plots of $\operatorname{{V}}[\Pe(Z)] - \operatorname{{V}}[\Pe(Z + C)]$ for the Laplace and the Gauss distributions](Laplace-DV.pdf "fig:"){height="4cm"} \[fig:winter\]
![Plots of $\operatorname{{V}}[\Pe(Z)] - \operatorname{{V}}[\Pe(Z + C)]$ for the Laplace and the Gauss distributions](Gauss-DV.pdf "fig:"){height="4cm"} \[fig:fall\]
To prove Theorem $\ref{thm:3.4}$, we use the following lemma:
\[lem:3.5\] For $a > 0$ and $x > 0$, we have $$\begin{aligned}
x^{a} \g(a, x)^{2} - x^{a} \G(a)^{2} + 2 \g(a, x) \G(2a, x) > 0. \end{aligned}$$
The proof of Lemma $\ref{lem:3.5}$ is presented in Section $5.2$. Now we can prove Theorem $\ref{thm:3.4}$.
It follows from the equation $(\ref{C})$ that $$\begin{aligned}
\frac{(k_{1}^{2} - k_{2}^{2}) b C}{2 \G(a)}
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
&= - \operatorname{{sgn}}(C) \frac{k_{2} - k_{1}}{k_{1} + k_{2}}
\frac{(k_{1} + k_{2})^{2} b \lvert C \rvert}{2 \G(a)}
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right) \\
&= - \frac{(k_{1} + k_{2})^{2} b \lvert C \rvert}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right), \allowdisplaybreaks \\
\operatorname{{sgn}}(C) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
&= - \operatorname{{sgn}}(C) \frac{k_{2} - k_{1}}{k_{1} + k_{2}}
\frac{(k_{1} + k_{2})^{2} b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right) \\
&= - \frac{(k_{1} + k_{2})^{2} b^{2}}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
\g\left(3a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ Hence, substituting $c = C$ in equation $(2)$ of Lemma $\ref{lem:3.1}$, we have $$\begin{aligned}
\operatorname{{V}}[\Pe(Z + C)]
&= \frac{(k_{1} + k_{2})^{2} C^{2}}{4}
- \frac{(k_{1} + k_{2})^{2} b \lvert C \rvert}{\G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad - \frac{(k_{1} + k_{2})^{2} C^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
\g\left(3a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ From this and equation $(\ref{V[L(Z)]})$, we obtain $$\begin{aligned}
&\!\!\!\!\operatorname{{V}}[\Pe(Z)] - \operatorname{{V}}[\Pe(Z + C)] \\
&= - \frac{(k_{1} + k_{2})^{2} C^{2}}{4}
+ \frac{(k_{1} + k_{2})^{2} b \lvert C \rvert}{\G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad + \frac{(k_{1} + k_{2})^{2} C^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
+ \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \\
&\quad - \frac{(k_{1} + k_{2})^{2} b^{2} \G(2a)^{2}}{4 \G(a)^{2}}
+ \frac{(k_{1} + k_{2})^{2} b^{2}}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right)
\g\left(3a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right) \\
&= \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
f \left(a, \left\lvert \frac{C}{b} \right\rvert^{\frac{1}{a}} \right), \end{aligned}$$ where, for $a > 0$ and $x \geq 0$, $f(a, x)$ is defined as $$\begin{aligned}
f(a, x)
&:= x^{2a} \g(a, x)^{2} - x^{2a} \G(a)^{2} + 4 x^{a} \g(a, x) \G(2a, x) \\
&\quad + \G(2a, x)^{2} - \G(2a)^{2} + 2 \g(a, x) \g(3a, x). \end{aligned}$$ Here, since $$\begin{aligned}
\frac{d}{dx} f(a, x)
&= 2 a x^{a - 1} \left\{x^{a} \g(a, x)^{2} - x^{a} \G(a)^{2} + 2\g(a, x) \G(2a, x) \right\} \\
&\quad + 2 x^{a - 1} e^{-x} \g(3a, x) + 2 x^{2a - 1} e^{-x} \G(2a, x), \end{aligned}$$ from Lemma $\ref{lem:3.5}$, we have $\frac{d}{dx} f(a, x) > 0$ ($a > 0$, $x > 0$). Also, $f(a, 0) = 0$ holds for $a > 0$. Therefore, we obtain $$\begin{aligned}
\operatorname{{V}}[\Pe(Z)] - \operatorname{{V}}[\Pe(Z + C)] \geq 0, \end{aligned}$$ where equality holds only when $C = 0$. Moreover, from equation $(\ref{C})$, we find that $C = 0$ holds only when $k_{1} = k_{2}$.
Inequalities for the gamma and the incomplete gamma functions
=============================================================
In this section, we give some inequalities for the gamma and the incomplete gamma functions, which we used to derive the inequality for the variance of the loss in Theorem $\ref{thm:3.4}$.
Inequalities for the gamma function
-----------------------------------
To prove Lemma $\ref{lem:3.5}$, we use the following:
\[lem:4-1-3\] For $a > 0$, we have $$\begin{aligned}
2 \G(2a) - a \G(a)^{2} > 0. \end{aligned}$$
Next, to prove Lemma $\ref{lem:4-1-3}$, we use the following:
\[lem:4-1-1\] For $a > 0$, we have $$\begin{aligned}
4^{a} \G\left(a + \frac{1}{2} \right) > \sqrt{\pi} \G(a + 1). \end{aligned}$$
Furthermore, to prove Lemma $\ref{lem:4-1-1}$, we need another lemma:
\[lem:4-1-2\] We have $$\begin{aligned}
\sum_{n = 1}^{\infty} \frac{1}{n (2n - 1)} = 2 \log{2}. \end{aligned}$$
Let $S_{n} := \sum_{k = 1}^{n} \frac{1}{k (2k - 1)}$. Accordingly, we have $$\begin{aligned}
S_{n}
&= \sum_{k = 1}^{n} \left(\frac{2}{2k - 1} - \frac{1}{k} \right) \allowdisplaybreaks \\
&= 2 \sum_{k = 1}^{n} \frac{1}{2k - 1} - \sum_{k = 1}^{n} \frac{1}{k} \allowdisplaybreaks \\
&= 2 \sum_{k = 1}^{n} \frac{1}{2k - 1}
+ \left(2 \sum_{k = 1}^{n} \frac{1}{2 k} - 2 \sum_{k = 1}^{n} \frac{1}{2 k} \right)
- \sum_{k = 1}^{n} \frac{1}{k} \allowdisplaybreaks \\
&= 2 \left(\sum_{k = 1}^{n} \frac{1}{2 k - 1} + \sum_{k = 1}^{n} \frac{1}{2 k} \right)
- 2 \sum_{k = 1}^{n} \frac{1}{k} \allowdisplaybreaks \\
&= 2 \sum_{k = 1}^{2n} \frac{1}{k} - 2 \sum_{k = 1}^{n} \frac{1}{k} \allowdisplaybreaks \\
&= 2 \sum_{k = n + 1}^{2n} \frac{1}{k} \allowdisplaybreaks \\
&= 2 \sum_{k = 1}^{n} \frac{1}{k + n}. \end{aligned}$$ Therefore, we find $$\begin{aligned}
\lim_{n \rightarrow \infty} S_{n}
&= 2 \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{1}{k + n} \allowdisplaybreaks \\
&= 2 \lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} \frac{1}{1 + \frac{k}{n}}
\allowdisplaybreaks \\
&= \int_{0}^{1} \frac{1}{1 + x} dx \allowdisplaybreaks \\
&= 2\log{2}.\end{aligned}$$ The lemma is thus proved.
Now we can prove Lemma $\ref{lem:4-1-1}$.
Let $$\begin{aligned}
g(a) := \frac{4^{a} \G\left(a + \frac{1}{2} \right)}{\sqrt{\pi} \G(a + 1)}. \end{aligned}$$ To prove $g(a) > 1$ for $a > 0$, we use the following formula [@andrews_askey_roy_1999 p.13, Theorem 1.2.5]: $$\begin{aligned}
\frac{d}{dx} \log{\G(x)}
= \frac{\G^{'}(x)}{\G(x)}
= - \g_{0} + \sum_{n = 1}^{\infty} \left(\frac{1}{n} - \frac{1}{x + n - 1} \right), \end{aligned}$$ where $\g_{0}$ is Euler’s constant given by $$\begin{aligned}
\g_{0} := \lim_{n \to \infty} \left(\sum_{k = 1}^{n} \frac{1}{k} - \log{n} \right). \end{aligned}$$ Taking the logarithmic derivative of $g(a)$, from the above formula, we have $$\begin{aligned}
\frac{d}{da} \log{g(a)}
&= 2\log{2} + \frac{d}{da} \log{\G\left(a + \frac{1}{2} \right)}
- \frac{d}{da} \log{\G(a + 1)} \allowdisplaybreaks \\
&= 2\log{2} + \sum_{n = 1}^{\infty} \left(\frac{1}{n} - \frac{1}{a - \frac{1}{2} + n} \right)
- \sum_{n = 1}^{\infty} \left(\frac{1}{n} - \frac{1}{a + n} \right) \allowdisplaybreaks \\
&= 2\log{2} - \frac{1}{2} \sum_{n = 1}^{\infty} \frac{1}{(a + n) \left(a - \frac{1}{2} + n \right)}
\allowdisplaybreaks \\
&> 2\log{2} - \frac{1}{2} \sum_{n = 1}^{\infty} \frac{1}{n \left(n - \frac{1}{2} \right)}
\allowdisplaybreaks \\
&= 2\log{2} - \sum_{n = 1}^{\infty} \frac{1}{n (2n - 1)}\end{aligned}$$ for $a > 0$. Moreover, using Lemma $\ref{lem:4-1-2}$, we obtain $\frac{d}{da} \log{g(a)} > 0$ for $a > 0$. This leads to $\frac{d}{da} g(a) > 0$ for $a > 0$. The lemma follows from this and $g(0) = 1$.
Now, we can prove Lemma $\ref{lem:4-1-3}$.
We use the following formula [@andrews_askey_roy_1999 p.22, Theorem 6.5.1]: $$\begin{aligned}
\G(2a) = \frac{2^{2a - 1}}{\sqrt{\pi}} \G(a) \G\left(a + \frac{1}{2} \right). \end{aligned}$$ From this and Lemma $\ref{lem:4-1-1}$, we have $$\begin{aligned}
2 \G(2a) - a \G(a)^{2}
&= \frac{2^{2a}}{\sqrt{\pi}} \G(a) \G\left(a + \frac{1}{2} \right)
- \G(a) \G(a + 1) \allowdisplaybreaks \\
&= \frac{1}{\sqrt{\pi}} \G(a)
\left\{4^{a} \G\left(a + \frac{1}{2} \right) - \sqrt{\pi} \G(a + 1) \right\} \\
&> 0. \end{aligned}$$ The lemma is thus proved.
Inequalities for the incomplete gamma functions
-----------------------------------------------
We will prove the following lemma:
\[lem:4-2-1\] For $a > 0$ and $x > 0$, we have $$\begin{aligned}
x^{a} \g(a, x)^{2} - x^{a} \G(a)^{2} + 2 \g(a, x) \G(2a, x) > 0. \end{aligned}$$
To prove Lemma $\ref{lem:4-2-1}$, we need to prove two other lemmas:
\[lem:4-2-2\] For $a > 0$ and $x \geq 0$, we have $$\begin{aligned}
a \g(a, x) \geq x^{a} e^{-x}. \end{aligned}$$
For $a > 0$ and $x \geq 0$, we define $$\begin{aligned}
u(a, x) := a \g(a, x) - x^{a} e^{-x}. \end{aligned}$$ Then, we have $$\begin{aligned}
\frac{d}{dx} u(a, x) = x^{a} e^{-x} \geq 0. \end{aligned}$$ The lemma follows from this and $u(a, 0) = 0$.
\[lem:4-2-3\] For $a > 0$ and $b \in \mathbb{R}$, we have $$\begin{aligned}
\lim_{x \to +\infty} x^{b} \G(a, x) = 0. \end{aligned}$$
When $b \leq 0$, it is easily obtained from the definition of $\G(a, x)$. When $b > 0$, using the L’Hôpital’s rule, we obtain $$\begin{aligned}
\lim_{x \to +\infty} \frac{\G(a, x)}{x^{-b}}
&= \lim_{x \to +\infty} \frac{x^{a - 1} e^{-x}}{b x^{- b - 1}} \allowdisplaybreaks \\
&= \lim_{x \to +\infty} \frac{x^{a + b} e^{-x}}{b} \\
&= 0. \end{aligned}$$
Now, we can prove Lemma $\ref{lem:4-2-1}$.
For $a > 0$ and $x \geq 0$, we define $$\begin{aligned}
y_{1} (a, x) := x^{a} \g(a, x)^{2} - x^{a} \G(a)^{2} + 2 \g(a, x) \G(2a, x). \end{aligned}$$ Let us prove $y_{1} (a, x) > 0$ ($a > 0$, $x > 0$). For $a > 0$ and $x \geq 0$, we define $$\begin{aligned}
y_{2} (a, x) &:= a \g(a, x)^{2} - a \G(a)^{2} + 2 e^{-x} \G(2a, x); \\
y_{3} (a, x) &:= a x^{a - 1} \g(a, x) - \G(2a, x) - x^{2a - 1} e^{-x}; \\
y_{4} (a, x) &:= a (a - 1) \g(a, x) + x^{a} e^{-x} (2x + 1 - a). \end{aligned}$$ Then, we have $$\begin{aligned}
\frac{d}{dx} y_{1} (a, x) &= x^{a - 1} y_{2} (a, x); \\
\frac{d}{dx} y_{2} (a, x) &= 2 e^{-x} y_{3} (a, x); \\
\frac{d}{dx} y_{3} (a, x) &= x^{a - 2} y_{4} (a, x); \\
\frac{d}{dx} y_{4} (a, x) &= x^{a} e^{-x} (3a + 1 - 2x). \end{aligned}$$ From these relations, we find that the (positive or negative) signs of $\frac{d}{dx} y_{i}(a, x)$ and $y_{i + 1}(a, x)$ ($i = 1, 2, 3$) are equal to each other for $a > 0$ and $x > 0$. Let $p_{i} (a)$ ($i = 2, 3, 4$) be the value of $x$ satisfying $y_{i}(a, x) = 0$. It is easily verified that $\lim_{x \to 0+} \frac{d}{dx}y_{4}(a, x) = \lim_{x \to +\infty}\frac{d}{dx}y_{4}(a, x) = \lim_{x \to 0+} y_{4}(a, x) = 0$ and $\lim_{x \to +\infty}y_{4}(a, x) = a (a - 1) \G(a)$ for $a > 0$. Therefore, from the first derivative test, we obtain Tables $1$ and $2$. Moreover, using Lemmas $\ref{lem:4-2-2}$, $\ref{lem:4-2-3}$, and L’Hôpital’s rule, we obtain $$\begin{aligned}
&\lim_{x \to 0+} \frac{d}{dx} y_{3}(a, x)
=
\begin{cases}
\infty & (0 < a < 1), \\
0 & (a \geq 1),
\end{cases}
&
&\lim_{x \to +\infty} \frac{d}{dx} y_{3}(a, x)
=
\begin{cases}
0 & (0 < a < 2), \\
2 & (a = 2), \\
\infty & (a > 2),
\end{cases} \allowdisplaybreaks \\
&\lim_{x \to 0+} y_{3}(a, x)
= - \G(2a) \quad (a > 0),
&
&\lim_{x \to +\infty} y_{3}(a, x)
=
\begin{cases}
0 & (0 < a < 1), \\
1 & (a = 1), \\
\infty & (a > 1),
\end{cases} \allowdisplaybreaks \\
&\lim_{x \to 0+} \frac{d}{dx} y_{2}(a, x)
= - 2\G(2a) \quad (a > 0),
&
&\lim_{x \to +\infty} \frac{d}{dx} y_{2}(a, x)
= 0 \quad (a > 0), \allowdisplaybreaks \\
&\lim_{x \to 0+} y_{2}(a, x)
= 2 \G(2a) - a\G(a)^{2} \quad (a > 0),
&
&\lim_{x \to +\infty} y_{2}(a, x)
= 0 \quad (a > 0), \allowdisplaybreaks \\
&\lim_{x \to 0+} \frac{d}{dx} y_{1}(a, x)
=
\begin{cases}
\infty & (0 < a < 1), \\
1 & (a = 1), \\
0 & (a > 1),
\end{cases}
&
&\lim_{x \to +\infty} \frac{d}{dx} y_{1}(a, x)
= 0 \quad (a > 0), \allowdisplaybreaks \\
&\lim_{x \to 0+} y_{1}(a, x)
= 0 \quad (a > 0),
&
&\lim_{x \to +\infty} y_{1}(a, x)
= 0 \quad (a > 0). \end{aligned}$$ From these results, Lemma $\ref{lem:4-1-3}$, and the fact that the signs of $\frac{d}{dx} y_{i}(a, x)$ and $y_{i + 1}(a, x)$ ($i = 1, 2, 3$) are equal to each other for $a > 0$ and $x > 0$, we obtain Tables $3$ and $4$. From Tables $3$ and $4$, we can verify that $y_{1}(a, x) > 0$ holds for $a > 0$ and $x > 0$. This completes the proof of the lemma.
$x$ $\;0\;$ $\cdots$ $\;\frac{3a + 1}{2}\;$ $\cdots$ $\;p_{4}(a)\;$ $\cdots$ $+\infty$
---------------------------- --------- ---------- ------------------------ ---------- ---------------- ---------- -----------
$\frac{d}{dx} y_{4}(a, x)$ $0$ $+$ $0$ $-$ $-$ $-$ $0$
$y_{4}(a, x)$ $0$ $+$ $+$ $+$ $0$ $-$ $-$
: Case of $0 < a < 1$
$x$ $\;0\;$ $\cdots$ $\;\frac{3a + 1}{2}\;$ $\cdots$ $\;+\infty\;$
---------------------------- --------- ---------- ------------------------ ---------- -------------------------------------------------------
$\frac{d}{dx} y_{4}(a, x)$ $0$ $+$ $0$ $-$ $0$
$y_{4}(a, x)$ $0$ $+$ $+$ $+$ $\begin{matrix}0\;\;(a = 1)\\ +\;(a > 1)\end{matrix}$
: Case of $a \geq 1$
$x$ $\;0\;$ $\cdots$ $\;p_{2}(a)\;$ $\cdots$ $\;p_{3}(a)\;$ $\cdots$ $\;p_{4}(a)\;$ $\cdots$ $\;+\infty\;$
---------------------------- ----------- ---------- ---------------- ---------- ---------------- ---------- ---------------- ---------- ---------------
$\frac{d}{dx} y_{3}(a, x)$ $+\infty$ $+$ $+$ $+$ $+$ $+$ $0$ $-$ $0$
$y_{3}(a, x)$ $-$ $-$ $-$ $-$ $0$ $+$ $+$ $+$ $0$
$\frac{d}{dx} y_{2}(a, x)$ $-$ $-$ $-$ $-$ $0$ $+$ $+$ $+$ $0$
$y_{2}(a, x)$ $+$ $+$ $0$ $-$ $-$ $-$ $-$ $-$ $0$
$\frac{d}{dx} y_{1}(a, x)$ $+$ $+$ $0$ $-$ $-$ $-$ $-$ $-$ $0$
$y_{1}(a, x)$ $0$ $+$ $+$ $+$ $+$ $+$ $+$ $+$ $0$
: Case of $0 < a < 1$
$x$ $\;0\;$ $\cdots$ $\;p_{2}(a)\;$ $\cdots$ $\;p_{3}(a)\;$ $\cdots$ $\;+\infty\;$
---------------------------- --------- ---------- ---------------- ---------- ---------------- ---------- ------------------------------------------------------------------------
$\frac{d}{dx} y_{3}(a, x)$ $0$ $+$ $+$ $+$ $+$ $+$ $\begin{matrix}0\;(a < 2)\\ +\;(a = 2)\\ +\infty\;(a > 2)\end{matrix}$
$y_{3}(a, x)$ $-$ $-$ $-$ $-$ $0$ $+$ $0$
$\frac{d}{dx} y_{2}(a, x)$ $-$ $-$ $-$ $-$ $0$ $+$ $0$
$y_{2}(a, x)$ $+$ $+$ $0$ $-$ $-$ $-$ $0$
$\frac{d}{dx} y_{1}(a, x)$ $+$ $+$ $0$ $-$ $-$ $-$ $0$
$y_{1}(a, x)$ $0$ $+$ $+$ $+$ $+$ $+$ $0$
: Case of $a \geq 1$
Calculation of the expected value and the variance of the loss
==============================================================
Here, we calculate the expected value and the variance of the loss $\Pe(Z + c)$ for $c \in \mathbb{R}$.
**[Expected value of the loss]{}**
----------------------------------
Here, let us put $\beta := (2 a b \G(a))^{-1}$; then, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= \int_{- \infty}^{+\infty} \Pe(z + c) f_{Z}(z) dz \\
&= k_{2} \beta \int_{- \infty}^{- c} (- z - c)
\exp{\left( - \left\lvert \frac{z}{b} \right\rvert^{\frac{1}{a}} \right)} dz
+ k_{1} \beta \int_{- c}^{+\infty} (z + c)
\exp{\left( - \left\lvert \frac{z}{b} \right\rvert^{\frac{1}{a}} \right)} dz. \end{aligned}$$ Replace $z$ with $b z$ to get $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
= k_{2} b \beta \int_{- \infty}^{- c / b} (- b z - c)
\exp{\left( - \lvert z \rvert^{\frac{1}{a}} \right)} dz
+ k_{1} b \beta \int_{- c / b}^{+\infty} (b z + c)
\exp{\left( - \lvert z \rvert^{\frac{1}{a}} \right)} dz. \end{aligned}$$ When $c \geq 0$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= k_{2} b \beta
\int_{- \infty}^{- c / b} (- b z - c) \exp{\left( - (-z)^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1} b \beta
\int_{- c / b}^{0} (b z + c) \exp{\left( - (-z)^{\frac{1}{a}} \right)} dz
+ k_{1} b \beta \int_{0}^{+\infty} (b z + c) \exp{\left( - z^{\frac{1}{a}} \right)} dz
\allowdisplaybreaks \\
&= k_{2} b \beta
\int_{c / b}^{+\infty} (b z - c) \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1} b \beta
\int_{0}^{c / b} (- b z + c) \exp{\left( - z^{\frac{1}{a}} \right)} dz
+ k_{1} b \beta \int_{0}^{+\infty} (b z + c) \exp{\left( - z^{\frac{1}{a}} \right)} dz
\allowdisplaybreaks \\
&= (k_{1} + k_{2}) b^{2} \beta
\int_{c / b}^{+\infty} z \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + (k_{1} - k_{2}) b c \beta
\int_{0}^{+\infty} \exp{\left( - z^{\frac{1}{a}} \right)} dz
+ (k_{1} + k_{2}) b c \beta \int_{0}^{c / b} \exp{\left( - z^{\frac{1}{a}} \right)} dz. \end{aligned}$$ When $c < 0$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= k_{2} b \beta
\int_{- \infty}^{0} (- b z - c) \exp{\left( - (-z)^{\frac{1}{a}} \right)} dz
+ k_{2} b \beta \int_{0}^{- c / b} (- b z - c) \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1} b \beta
\int_{- c / b}^{+\infty} (b z + c) \exp{\left( - z^{\frac{1}{a}} \right)} dz
\allowdisplaybreaks \\
&= k_{2} b \beta \int_{0}^{+\infty} (b z - c) \exp{\left( - z^{\frac{1}{a}} \right)} dz
+ k_{2} b \beta \int_{0}^{- c / b} (- b z - c) \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1} b \beta
\int_{- c / b}^{+\infty} (b z + c) \exp{\left( - z^{\frac{1}{a}} \right)} dz
\allowdisplaybreaks \\
&= (k_{1} + k_{2}) b^{2} \beta
\int_{- c / b}^{+\infty} z \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + (k_{1} - k_{2}) b c \beta
\int_{0}^{+\infty} \exp{\left( - z^{\frac{1}{a}} \right)} dz
- (k_{1} + k_{2}) b c \beta \int_{0}^{- c / b} \exp{\left( - z^{\frac{1}{a}} \right)} dz. \end{aligned}$$ From the above, for any $c \in \mathbb{R}$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= (k_{1} + k_{2}) b^{2} \beta
\int_{\lvert c / b \rvert}^{+\infty} z \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + (k_{1} - k_{2}) b c \beta
\int_{0}^{+\infty} \exp{\left( - z^{\frac{1}{a}} \right)} dz
+ (k_{1} + k_{2}) b \lvert c \rvert \beta
\int_{0}^{\lvert c / b \rvert} \exp{\left( - z^{\frac{1}{a}} \right)} dz. \end{aligned}$$ Now set $t := z^{\frac{1}{a}}$ to get $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]
&= (k_{1} + k_{2}) a b^{2} \beta \int_{c'}^{+\infty} t^{2a - 1} e^{-t} dt \\
&\quad + (k_{1} - k_{2}) a b c \beta \int_{0}^{+\infty} t^{a - 1} e^{-t} dt
+ (k_{1} + k_{2}) a b \lvert c \rvert \beta \int_{0}^{c'} t^{a - 1} e^{-t} dt
\allowdisplaybreaks \\
&= (k_{1} + k_{2}) a b^{2} \beta \G(2a, c')
+ (k_{1} - k_{2}) a b c \beta \G(a)
+ (k_{1} + k_{2}) a b \lvert c \rvert \beta \g(a, c'), \end{aligned}$$ where $c' := \lvert c / b \rvert^{\frac{1}{a}}$. Therefore, for any $c \in \mathbb{R}$, we have $$\begin{aligned}
\label{E[Pe(Z+c)]-2}
\operatorname{{E}}[\Pe(Z + c)]
= \frac{(k_{1} - k_{2}) c}{2}
+ \frac{(k_{1} + k_{2}) \lvert c \rvert}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
+ \frac{(k_{1} + k_{2}) b}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$
**[Variance of the loss]{}**
----------------------------
Now let us calculate the variance of the loss $\Pe(Z + c)$ for $c \in \mathbb{R}$. Put $\beta := (2 a b \G(a))^{-1}$; then, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= \int_{- \infty}^{+\infty} \Pe(z + c)^{2} f_{Z}(z) dz
\allowdisplaybreaks \\
&= k_{2}^{2} \beta \int_{- \infty}^{- c} (z + c)^{2}
\exp{\left( - \left\lvert \frac{z}{b} \right\rvert^{\frac{1}{a}} \right)} dz
+ k_{1}^{2} \beta \int_{- c}^{+\infty} (z + c)^{2}
\exp{\left( - \left\lvert \frac{z}{b} \right\rvert^{\frac{1}{a}} \right)} dz. \end{aligned}$$ Replace $z$ with $b z$ to get $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
= k_{2}^{2} b \beta \int_{- \infty}^{- c / b} (b z + c)^{2}
\exp{\left( - \lvert z \rvert^{\frac{1}{a}} \right)} dz
+ k_{1}^{2} b \beta \int_{- c / b}^{+\infty} (b z + c)^{2}
\exp{\left( - \lvert z \rvert^{\frac{1}{a}} \right)} dz. \end{aligned}$$ When $c \geq 0$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= k_{2}^{2} b \beta \int_{- \infty}^{- c / b} (b z + c)^{2}
\exp{\left( - (- z)^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1}^{2} b \beta \int_{- c / b}^{0} (b z + c)^{2}
\exp{\left( - (- z)^{\frac{1}{a}} \right)} dz
+ k_{1}^{2} b \beta \int_{0}^{+\infty} (b z + c)^{2}
\exp{\left( - z^{\frac{1}{a}} \right)} dz \allowdisplaybreaks \\
&= k_{2}^{2} b \beta \int_{c / b}^{+\infty} (- b z + c)^{2}
\exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1}^{2} b \beta \int_{0}^{c / b} (- b z + c)^{2}
\exp{\left( - z^{\frac{1}{a}} \right)} dz
+ k_{1}^{2} b \beta \int_{0}^{+\infty} (b z + c)^{2}
\exp{\left( - z^{\frac{1}{a}} \right)} dz \allowdisplaybreaks \\
&= (k_{1}^{2} + k_{2}^{2}) b^{3} \beta
\int_{0}^{+\infty} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz
+ (k_{1}^{2} - k_{2}^{2}) b^{3} \beta
\int_{0}^{c / b} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + 2 (k_{1}^{2} - k_{2}^{2}) b^{2} c \beta
\int_{c / b}^{+\infty} z \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + (k_{1}^{2} + k_{2}^{2}) b c^{2} \beta
\int_{0}^{+\infty} \exp{\left( - z^{\frac{1}{a}} \right)} dz
+ (k_{1}^{2} - k_{2}^{2}) b c^{2} \beta
\int_{0}^{c / b} \exp{\left( - z^{\frac{1}{a}} \right)} dz. \end{aligned}$$ When $c < 0$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= k_{2}^{2} b \beta
\int_{- \infty}^{0} (b z + c)^{2} \exp{\left( - (- z)^{\frac{1}{a}} \right)} dz
+ k_{2}^{2} b \beta
\int_{0}^{- c / b} (b z + c)^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1}^{2} b \beta
\int_{- c / b}^{+\infty} (b z + c)^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz
\allowdisplaybreaks \\
&= k_{2}^{2} b \beta
\int_{0}^{+\infty} (- b z + c)^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz
+ k_{2}^{2} b \beta
\int_{0}^{- c / b} (b z + c)^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + k_{1}^{2} b \beta
\int_{- c / b}^{+\infty} (b z + c)^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz
\allowdisplaybreaks \\
&= (k_{1}^{2} + k_{2}^{2}) b^{3} \beta
\int_{0}^{+\infty} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz
- (k_{1}^{2} - k_{2}^{2}) b^{3} \beta
\int_{0}^{- c / b} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + 2 (k_{1}^{2} - k_{2}^{2}) b^{2} c \beta
\int_{- c / b}^{+\infty} z \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + (k_{1}^{2} + k_{2}^{2}) b c^{2} \beta
\int_{0}^{+\infty} \exp{\left( - z^{\frac{1}{a}} \right)} dz
- (k_{1}^{2} - k_{2}^{2}) b c^{2} \beta
\int_{0}^{- c / b} \exp{\left( - z^{\frac{1}{a}} \right)} dz. \end{aligned}$$ From the above, for any $c \in \mathbb{R}$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= (k_{1}^{2} + k_{2}^{2}) b^{3} \beta
\int_{0}^{+\infty} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) b^{3} \beta
\int_{0}^{\lvert c / b \rvert} z^{2} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + 2 (k_{1}^{2} - k_{2}^{2}) b^{2} c \beta
\int_{\lvert c / b \rvert}^{+\infty} z \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + (k_{1}^{2} + k_{2}^{2}) b c^{2} \beta
\int_{0}^{+\infty} \exp{\left( - z^{\frac{1}{a}} \right)} dz \\
&\quad + \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) b c^{2} \beta
\int_{0}^{\lvert c / b \rvert} \exp{\left( - z^{\frac{1}{a}} \right)} dz. \end{aligned}$$ Now set $t := z^{\frac{1}{a}}$ to get $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= (k_{1}^{2} + k_{2}^{2}) a b^{3} \beta \int_{0}^{+\infty} t^{3a - 1} e^{-t} dt
+ \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) a b^{3} \beta \int_{0}^{c'} t^{3a - 1} e^{-t} dt \\
&\quad + 2(k_{1}^{2} - k_{2}^{2}) a b^{2} c \beta \int_{c'}^{+\infty} t^{2a - 1} e^{-t} dt \\
&\quad + (k_{1}^{2} + k_{2}^{2}) a b c^{2} \beta \int_{0}^{+\infty} t^{a - 1} e^{-t} dt
+ \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) a b c^{2} \beta \int_{0}^{c'} t^{a - 1} e^{-t} dt
\allowdisplaybreaks \\
&= (k_{1}^{2} + k_{2}^{2}) a b^{3} \beta \G(3a)
+ \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) a b^{3} \beta \g(3a, c') \\
&\quad + 2(k_{1}^{2} - k_{2}^{2}) a b^{2} c \beta \G(2a, c') \\
&\quad + (k_{1}^{2} + k_{2}^{2}) a b c^{2} \beta \G(a)
+ \operatorname{{sgn}}(c) (k_{1}^{2} - k_{2}^{2}) a b c^{2} \beta \g(a, c'), \end{aligned}$$ where $c' := \lvert c / b \rvert^{\frac{1}{a}}$. Therefore, for any $c \in \mathbb{R}$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)^{2}]
&= \frac{(k_{1}^{2} + k_{2}^{2}) c^{2}}{2}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) c^{2}}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{\G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \end{aligned}$$ Also, from $(\ref{E[Pe(Z+c)]-2})$, we have $$\begin{aligned}
\operatorname{{E}}[\Pe(Z + c)]^{2}
&= \frac{(k_{1} - k_{2})^{2} c^{2}}{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) c \lvert c \rvert}{2 \G(a)}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad + \frac{(k_{1} + k_{2})^{2} b \lvert c \rvert}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \\
&\quad + \frac{(k_{1} + k_{2})^{2} c^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
+ \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}. \end{aligned}$$ Therefore, for any $c \in \mathbb{R}$, we have $$\begin{aligned}
\operatorname{{V}}[\Pe(Z + c)]
&= \operatorname{{E}}[\Pe(Z + c)^{2}] - \operatorname{{E}}[\Pe(Z + c)]^{2} \\
&= \frac{(k_{1} + k_{2})^{2} c^{2}}{4}
+ \frac{(k_{1}^{2} - k_{2}^{2}) b c}{2 \G(a)}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} b \lvert c \rvert}{2 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right) \nonumber \\
&\quad - \frac{(k_{1} + k_{2})^{2} c^{2}}{4 \G(a)^{2}}
\g\left(a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2}
- \frac{(k_{1} + k_{2})^{2} b^{2}}{4 \G(a)^{2}}
\G\left(2a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right)^{2} \nonumber \\
&\quad + \frac{(k_{1}^{2} + k_{2}^{2}) b^{2} \G(3a)}{2 \G(a)}
+ \operatorname{{sgn}}(c) \frac{(k_{1}^{2} - k_{2}^{2}) b^{2}}{2 \G(a)}
\g\left(3a, \left\lvert \frac{c}{b} \right\rvert^{\frac{1}{a}} \right). \nonumber\end{aligned}$$
Naoya Yamaguchi\
Office for Establishment of an Information-related School\
Nagasaki University\
1-14 Bunkyo, Nagasaki City 852-8521\
Japan\
[email protected]
Yuka Yamaguchi\
Office for Establishment of an Information-related School\
Nagasaki University\
1-14 Bunkyo, Nagasaki City 852-8521\
Japan\
[email protected]
Ryuei Nishi\
Office for Establishment of an Information-related School\
Nagasaki University\
1-14 Bunkyo, Nagasaki City 852-8521\
Japan\
[email protected]
| ArXiv |
---
abstract: 'Massive stars ejected from their parent cluster and supersonically sailing away through the interstellar medium (ISM) are classified as exiled. They generate circumstellar bow shock nebulae that can be observed. We present two-dimensional, axisymmetric hydrodynamical simulations of a representative sample of stellar wind bow shocks from Galactic OB stars in an ambient medium of densities ranging from $n_{\rm ISM }=0.01$ up to $10.0\, \rm cm^{-3}$. Independently of their location in the Galaxy, we confirm that the infrared is the most appropriated waveband to search for bow shocks from massive stars. Their spectral energy distribution is the convenient tool to analyze them since their emission does not depend on the temporary effects which could affect unstable, thin-shelled bow shocks. Our numerical models of Galactic bow shocks generated by high-mass ($\approx 40\, \rm M_{\odot}$) runaway stars yield H$\alpha$ fluxes which could be observed by facilities such as the [*SuperCOSMOS H-Alpha Survey*]{}. The brightest bow shock nebulae are produced in the denser regions of the ISM. We predict that bow shocks [*in the field*]{} observed at H$\alpha$ by means of Rayleigh-sensitive facilities are formed around stars of initial mass larger than about $20\, \rm M_{\odot}$.'
date: 'Received January 18 2015; accepted Month day, 2015'
title: On the observability of bow shocks of Galactic runaway OB stars
---
\[firstpage\]
methods: numerical – circumstellar matter – stars: massive.
Introduction {#sect:introduction}
============
The estimate of massive star feedback is a crucial question in the understanding of the Galaxy’s functioning [@langer_araa_50_2012]. Throughout their short lives, they release strong winds [@holzer_araa_8_1970] and ionising radiation [@diazmiller_apj_501_1998] which modify their ambient medium. This results in diaphanous [H[ii]{} ]{}regions [@dyson_ass_35_1975], parsec-scale bubbles of stellar wind [@weaver_apj_218_1977], inflated [@petrovic_aa_450_2006] or shed [@woosley_rvmp_74_2002; @garciasegura_1996_aa_305] stellar envelopes that impact their close surroundings and which can alter the propagation of their subsequent supernova shock wave [@vanveelen_phd; @meyer_mnras_450_2015]. Understanding the formation processes of these circumstellar structures allows us to constrain the impact of massive stars, e.g. on the energetics or the chemical evolution of the interstellar medium (ISM). Moreover, it links studies devoted to the dynamical evolution of supernova remnants expanding into the ISM [@rozyczka_mnras_261_1993] with works focusing on the physics of the star forming ISM [@peters_apj_711_2010].
These arc-like structures of swept-up stellar wind material and ISM gas the distortion of their stellar wind bubble by the bulk motion of their central star [@weaver_apj_218_1977]. Their size and their morphology are governed by their stellar wind mass loss, the bulk motion of the runaway star and their local ambient medium properties [@comeron_aa_338_1998]. These distorted wind bubbles have been first noticed in optical \[O[iii]{}\] $\lambda
\, 5007$ spectral emission line around the Earth’s closest runaway star, the OB star $\zeta$ Ophiuchi [@gull_apj_230_1979]. Other noticeable fast-moving massive stars producing a stellar wind bow shock are, e.g. the blue supergiant Vela-X1 [@kaper_apj_475_1997], the red supergiant Betelgeuse [@noriegacrespo_aj_114_1997] and the very massive star BD+43${\ensuremath{^\circ}}$365 running away from Cygnus OB2 [@comeron_aa_467_2007].
$M_{\star}\, (\rm M_{\odot})$ $t_{\mathrm{ start}}\, (\rm Myr)$ $\textcolor{black}{\log(L_{\star}/\rm L_{\odot})}$ $\log(\dot{M}/\rm M_{\odot}\, \rm yr^{-1})$ $v_{\rm w}\, (\mathrm{km}\, \mathrm{s}^{-1})$ $T_{\rm eff}\, (\mathrm{K})$ $S_{\star} (\mathrm{photon}\, \mathrm{s}^{-1})$ $t_{\rm MS} (\mathrm{Myr})$
------------------------------- ----------------------------------- ---------------------------------------------------- --------------------------------------------- ----------------------------------------------- ------------------------------ ------------------------------------------------- -----------------------------
$10$ $5.0$ $3.80$ $-9.52$ $1082$ $25200$ $10^{45}$ $22.5$
$20$ $3.0$ $4.74$ $-7.38$ $1167$ $33900$ $10^{48}$ $\,\,\,8.0$
$40$ $0.0$ $5.34$ $-6.29$ $1451$ $42500$ $10^{49}$ $\,\,\,4.0$
\[tab:lum\_stars\]
Analysis of data from the [*Infrared Astronomical Satellite*]{} facility [[*IRAS*]{}, @neugebauer_278_apj_1984] later extended to measures taken with the [*Wide-Field Infrared Satellite Explorer*]{} [[ *WISE*]{}, @wright_aj_140_2010] led to the compilation of bow shock records, see e.g. [@buren_apj_329_1988]. Soon arose the speculation that those isolated nebulae can serve physics of these stars, to constrain the still highly debated mass loss of massive stars [@gull_apj_230_1979] and/or their ambient medium density [@huthoff_aa_383_2002]. This also raised questions related to the ejection mechanisms of OB stars from young stellar clusters [@hoogerwerf_aa_365_2001]. More recently, multi-wavelengths data led to the publication of the E-BOSS catalog of stellar wind bow shocks [@peri_aa_538_2012; @2015arXiv150404264P].
Early simulations discussed the general morphology of the bow shocks around OB stars [@brighenti_mnras_277_1995, and references therein], their (in)stability [@blondin_na_57_1998] and the general uncompatibility of the shape of stellar wind bow shocks with analytical approximations such as the one of @wilkin_459_apj_1996, see in @comeron_aa_338_1998. However, observing massive star bow shocks remains difficult and they are mostly serendipitously noticed in infrared observations of the neighbourhood of stellar clusters [@gvaramadze_aa_490_2008]. Moreover, their optical emission may be screened by the [H[ii]{} ]{}region which surrounds the driving star and this may affect their H$\alpha$ observations [@brown_aa_439_2005]. We are particularly interested in the prediction of the easiest bow shocks to observe, their optical emission properties and their location in the Galaxy.
In the present study, we extend our numerical investigation of the circumstellar medium of runaway massive stars [@meyer hereafter Paper I]. explores the effects of the ambient medium density on the emission properties of the bow-like nebulae around the most common runaway stars, in the spirit of works on bow shocks generated by low-mass stars [@villaver_apj_748_2012 and references therein].
Our paper is organised as follows. In Section \[sect:method\] we present the numerical methods and the microphysics that is included in our models. The resulting numerical simulations are presented and discussed in Section \[sect:results\]. We then analyze and discuss the emission properties of our bow shock models in Section \[sect:emission\]. Finally, we formulate our conclusions in Section \[section:cc\].
Method {#sect:method}
======
Governing equations {#subsect:goveq}
-------------------
Hydrodynamical simulations {#subsect:hydrosim}
--------------------------
We run two-dimensional, axisymmetric, hydrodynamical numerical simulations using the [ pluto]{} code [@mignone_apj_170_2007; @migmone_apjs_198_2012] in axisymmetric, cylindrical coordinates on a uniform grid $[z_{\rm min},z_{\rm
max}]\times[O,R_{\rm max}]$ of spatial resolution $\Delta=2.25\times 10^{-4}\,
\rm{pc}\, \rm{cell}^{-1}$ minimum. The stellar wind is injected into the computational domain filling a circle of radius 20 cells centered onto the origin $O$ [see e.g., @comeron_aa_338_1998; @meyer_mnras_2013 and references therein]. The interaction with the ISM is calculated in the reference frame of the moving star [@vanmarle_aa_469_2007; @vanmarle_apj_734_2011; @vanmarle_aa_561_2014]. Inflowing ISM gas mimicing the stellar motion is set at the $z=z_{\rm max}$ boundary whereas semi-permeable boundary conditions are set at $z=z_{\rm min}$ and at $R=R_{\rm max}$. Wind material is distinguished from the ISM using a passive tracer $Q$ that is advected with the gas and initially set to $Q=1$ in the stellar wind and to $Q=0$ in the ISM. The ISM composition is assumed to be solar [@asplund_araa_47_2009].
${\rm {Model}}$ $M_{\star}\, (\rm M_{\odot})$ $v_{\star}\, (\mathrm{km}\, \mathrm{s}^{-1})$ $n_{\rm ISM}\, (\mathrm{cm}^{-3})$
----------------- ------------------------------- ----------------------------------------------- ------------------------------------ -- --
MS1020n0.01 $10$ $20$ $0.01$
MS1040n0.01 $10$ $40$ $0.01$
MS1070n0.01 $10$ $70$ $0.01$
MS2040n0.01 $20$ $40$ $0.01$
MS2070n0.01 $20$ $70$ $0.01$
MS1020n0.1 $10$ $20$ $0.10$
MS1040n0.1 $10$ $40$ $0.10$
MS1070n0.1 $10$ $70$ $0.10$
MS2020n0.1 $20$ $20$ $0.10$
MS2040n0.1 $20$ $40$ $0.10$
MS2070n0.1 $20$ $70$ $0.10$
MS4070n0.1 $40$ $70$ $0.10$
MS1020n10 $10$ $20$ $10.0$
MS1040n10 $10$ $40$ $10.0$
MS1070n10 $10$ $70$ $10.0$
MS2020n10 $20$ $20$ $10.0$
MS2040n10 $20$ $40$ $10.0$
MS2070n10 $20$ $70$ $10.0$
MS4020n10 $40$ $20$ $10.0$
MS4040n10 $40$ $40$ $10.0$
MS4070n10 $40$ $70$ $10.0$
: The hydrodynamical models. Parameters $M_{\star}$ (in $\rm M_{\odot}$), $v_{\star}$ (in $\mathrm{km}\, \mathrm{s}^{-1}$) and $n_{\rm ISM}$ (in $\mathrm{cm}^{-3}$) are the initial mass of the considered moving star, its space velocity and its local ISM density, respectively.
\[tab:models\]
Microphysics {#subsect:phys}
------------
In order to proceed on our previous bow shock studies [Paper I, @meyer_mnras_450_2015], we include the same microphysics in our simulations of the circumstellar medium of runaway, massive stars, i.e. we take into account losses and gain of internal energy by optically-thin cooling and heating together with electronic thermal conduction. Optically-thin radiative processes are included into the model using the cooling and heating laws established for a fully ionized medium in Paper I. It mainly consist of cooling contributions from hydrogen and helium for temperatures $T<10^{6}\, \rm K$ whereas it is principally due to metals for temperatures $T \ge 10^{6}\, \rm K$ [@wiersma_mnras_393_2009]. A term representing the cooling from collisionally excited forbidden lines [@henney_mnras_398_2009] incorporates the effects of, among other, the \[O[iii]{}\] $\lambda \, 5007$ line emission. The heating contribution includes the reionisation of recombining hydrogen atoms by the starlight [@osterbrock_1989; @hummer_mnras_268_1994]. All our models include electronic thermal conduction [@cowie_apj_211_1977].
![ Stellar wind bow shocks from the main sequence phase of the $20\, \rm M_{\odot}$ star moving with velocity $70\, \mathrm{km}\,
\mathrm{s}^{-1}$ as a function of the ISM density, with $n_{\rm ISM}=0.01$ (a), $0.1$ (b), $0.79$ (c) and $10.0\, \mathrm{cm}^{-3}$ (d). The gas number density (in $\rm cm^{-3}$) is shown in the logarithmic scale. The dashed black contour traces the boundary between wind and ISM material. The cross indicates the position of the runaway star. The $R$-axis represents the radial direction and the $z$-axis the direction of stellar motion (in $\mathrm{pc}$). Only part of the computational domain is shown. []{data-label="fig:grid_density"}](./MS2070n001_legend.eps){width="100.00000%"}
\
![ Stellar wind bow shocks from the main sequence phase of the $20\, \rm M_{\odot}$ star moving with velocity $70\, \mathrm{km}\,
\mathrm{s}^{-1}$ as a function of the ISM density, with $n_{\rm ISM}=0.01$ (a), $0.1$ (b), $0.79$ (c) and $10.0\, \mathrm{cm}^{-3}$ (d). The gas number density (in $\rm cm^{-3}$) is shown in the logarithmic scale. The dashed black contour traces the boundary between wind and ISM material. The cross indicates the position of the runaway star. The $R$-axis represents the radial direction and the $z$-axis the direction of stellar motion (in $\mathrm{pc}$). Only part of the computational domain is shown. []{data-label="fig:grid_density"}](./MS2070n01_legend.eps){width="100.00000%"}
\
![ Stellar wind bow shocks from the main sequence phase of the $20\, \rm M_{\odot}$ star moving with velocity $70\, \mathrm{km}\,
\mathrm{s}^{-1}$ as a function of the ISM density, with $n_{\rm ISM}=0.01$ (a), $0.1$ (b), $0.79$ (c) and $10.0\, \mathrm{cm}^{-3}$ (d). The gas number density (in $\rm cm^{-3}$) is shown in the logarithmic scale. The dashed black contour traces the boundary between wind and ISM material. The cross indicates the position of the runaway star. The $R$-axis represents the radial direction and the $z$-axis the direction of stellar motion (in $\mathrm{pc}$). Only part of the computational domain is shown. []{data-label="fig:grid_density"}](./MS2070n10_legend.eps){width="100.00000%"}
\
![ Stellar wind bow shocks from the main sequence phase of the $20\, \rm M_{\odot}$ star moving with velocity $70\, \mathrm{km}\,
\mathrm{s}^{-1}$ as a function of the ISM density, with $n_{\rm ISM}=0.01$ (a), $0.1$ (b), $0.79$ (c) and $10.0\, \mathrm{cm}^{-3}$ (d). The gas number density (in $\rm cm^{-3}$) is shown in the logarithmic scale. The dashed black contour traces the boundary between wind and ISM material. The cross indicates the position of the runaway star. The $R$-axis represents the radial direction and the $z$-axis the direction of stellar motion (in $\mathrm{pc}$). Only part of the computational domain is shown. []{data-label="fig:grid_density"}](./MS2070n100_legend.eps){width="100.00000%"}
Parameter range {#subsect:para}
---------------
This work consists of a parameter study extending our previous investigation of stellar wind bow shock (Paper I) to regions of the Galaxy where the ISM has either lower or higher densities. The boundary conditions are unchanged, i.e. we consider runaway stars of $10$, $20$ and $40\, \rm M_{\odot}$ star moving with velocity $v_{\star}=20$, $40$ and $70\, \rm km\, \rm s^{-1}$, respectively. Differences come from the chosen ISM number density that ranges from $n_{\rm ISM}=0.01$ to $10.0\, \rm
cm^{-3}$ whereas our preceeding work exclusively focused on bow shocks models with $n_{\rm ISM}=0.79\, \rm cm^{-3}$.
Bow shocks morphology {#sect:results}
=====================
Bow shocks structure {#subsect:structure}
--------------------
In Fig. \[fig:grid\_density\] we show the density fields in our hydrodynamical simulations of our $20\, \rm M_{\odot}$ star moving with velocity $v_{\star}=70\, \rm
km\, \rm s^{-1}$ in a medium of number density $n_{\rm ISM}=0.01$ (panel a, model MS2070n0.01), $0.1$ (panel b, model MS2070n0.1), $0.79$ (panel c, model MS2070) and $10.0\, \rm cm^{-3}$ (panel d, model MS2070n10), respectively.
Bow shocks size {#subsect:scaling}
---------------
The bow shocks have a stand-off distance $R(0)$, i.e. the distance separating them from the star along the direction of motion predicted by @wilkin_459_apj_1996. It decreases as a function of (i) $v_{\star}$, (ii) $\dot{M}$ (c.f. Paper I) and (iii) $n_{\rm ISM}$ since $R(0) \propto n_{\rm ISM}^{-1/2}$. A dense ambient medium produces a large ISM ram pressure $n_{\rm ISM}v_{\star}^{2}$ which results in a compression of the whole bow shock and consequently in a reduction of $R(0)$. As an example, our simulations involving a $20\, \rm M_{\odot}$ star with $v_{\star}=70\,
\mathrm{km}\, \mathrm{s}^{-1}$ has $R(0)\approx 3.80$, $1.14$, $0.38$ and $0.07\rm pc$ when the driving star moves in $n_{\rm ISM}=0.01$, $0.1$, $0.79$ and $10\, \rm cm^{-3}$, respectively (Fig. \[fig:grid\_density\]a-d), which is reasonably in accordance with @wilkin_459_apj_1996. All our measures of $R(0)$ are taken at the contact discontinuity, because it is appropriate measure to compare models with Wilkin’s analytical solution [@mohamed_aa_541_2012].
![ Axis ratio $R(0)/R(90)$ of our bow shock models. The figure shows the ratio $R(0)/R(90)$ measured in the density field of our models measured at their contact discontinuity, as a function of their stand-off distance $R(0)$ (in $\rm pc$). Symbols distinguish models as a function of (i) the ISM ambient medium with $n_{\rm ISM}=0.01$ (triangles), $0.1$ (diamonds), $0.79$ (circles) and $10.0\, \rm cm^{-3}$ (squares) and (ii) of the initial mass of the star with $10\, \rm M_{\odot}$ (blue dots), $20\, \rm M_{\odot}$ (blue plus signs) and $40\, \rm M_{\odot}$ (dark green crosses), respectively. The thin horizontal black line corresponds to the analytic solution $R(0)/R(90)= 1/\sqrt{3}\approx 0.58$ of @wilkin_459_apj_1996. []{data-label="fig:axis_ratio"}](./R0_vs_R90.eps){width="100.00000%"}
Non-linear instabilities and mixing of material {#subsect:stability}
-----------------------------------------------
In Fig. \[fig:grid\_velocity\] we show a time sequence evolution of the density field in hydrodynamical simulations of $40\, \rm M_{\odot}$ zero-age main-sequence star moving with velocity $v_{\star}=70\, \rm km\, \rm s^{-1}$ in a medium of number density $n=10.0\, \rm cm^{-3}$ (model MS4070n10). The figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d), respectively. After $0.02\, \rm Myr$ the whole shell is sparsed with small size clumps which are the seeds of non-linear instabilities (Fig. \[fig:grid\_velocity\]b). The fast stellar motion ($v_{\star}=70\, \rm km\, \rm s^{-1}$) provokes a distortion of the bubble into an ovoid shape [see fig. 7 of @weaver_apj_218_1977] and the high ambient medium density ($n=10.0\, \rm cm^{-3}$) induces rapidly a thin shell after only about $0.01\, \rm Myr$.
![ Same as Fig. \[fig:grid\_density\] for our $40\, \rm M_{\odot}$ star moving through an ISM of density $n_{\rm ISM}=10.0\, \mathrm{cm}^{-3}$ with velocity $70\, \mathrm{km}\, \mathrm{s}^{-1}$ . Figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d) after the beginning of the main sequence phase of the star, respectively. It illustrates the development of the non-linear thin-shell instability in the bow shock. []{data-label="fig:grid_velocity"}](./MS2070n100Time2_legend.eps){width="100.00000%"}
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![ Same as Fig. \[fig:grid\_density\] for our $40\, \rm M_{\odot}$ star moving through an ISM of density $n_{\rm ISM}=10.0\, \mathrm{cm}^{-3}$ with velocity $70\, \mathrm{km}\, \mathrm{s}^{-1}$ . Figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d) after the beginning of the main sequence phase of the star, respectively. It illustrates the development of the non-linear thin-shell instability in the bow shock. []{data-label="fig:grid_velocity"}](./MS2070n100Time3_legend.eps){width="100.00000%"}
\
![ Same as Fig. \[fig:grid\_density\] for our $40\, \rm M_{\odot}$ star moving through an ISM of density $n_{\rm ISM}=10.0\, \mathrm{cm}^{-3}$ with velocity $70\, \mathrm{km}\, \mathrm{s}^{-1}$ . Figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d) after the beginning of the main sequence phase of the star, respectively. It illustrates the development of the non-linear thin-shell instability in the bow shock. []{data-label="fig:grid_velocity"}](./MS2070n100Time4_legend.eps){width="100.00000%"}
\
![ Same as Fig. \[fig:grid\_density\] for our $40\, \rm M_{\odot}$ star moving through an ISM of density $n_{\rm ISM}=10.0\, \mathrm{cm}^{-3}$ with velocity $70\, \mathrm{km}\, \mathrm{s}^{-1}$ . Figures are shown at times $0.02$ (a), $0.05$ (b), $0.11$ (c) and $0.12\, \rm Myr$ (d) after the beginning of the main sequence phase of the star, respectively. It illustrates the development of the non-linear thin-shell instability in the bow shock. []{data-label="fig:grid_velocity"}](./MS2070n100Time5_legend.eps){width="100.00000%"}
![ Bow shock volume ($z\ge0$) in our model (see Fig. \[fig:grid\_velocity\]a-e). The figure shows the volume of perturbed material (in $\rm pc^{3}$) in the computational domain (thick solid blue line), together with the volume of shocked ISM gas (thin solid red line) and shocked stellar wind (thick dotted orange line), respectively, as function of time (in $\rm Myr$). The large dotted black line represents the volume of the thin shell of shocked ISM. []{data-label="fig:volume"}](./volume.eps){width="100.00000%"}
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The bow shock then experiences a series of cycles in which small scaled eddies grow in the shell (Fig. \[fig:grid\_velocity\]b) and further distort its apex into wing-like structures (Fig. \[fig:grid\_velocity\]c) which are pushed sidewards because of the transverse component of the stellar wind acceleration (Fig. \[fig:grid\_velocity\]d). Our model MS4070n10 has both characteristics from the models E “ High ambient density” and G “ Instantaneous cooling” of @comeron_aa_338_1998. Thin-shelled stellar wind bow shocks develop non-linear instabilities, in addition to the Kelvin-Helmholtz instabilities that typically affect interfaces between shearing flows of opposite directions, i.e. the outflowing stellar wind and the ISM gas penetrating the bow shocks [@vishniac_apj_428_1994; @garciasegura_1996_aa_305; @vanmarle_aa_469_2007]. A detailed discussion of the development of such non-linearities affecting bow shocks generated by OB runaway stars is in @comeron_aa_338_1998.
In Fig. \[fig:volume\] we plot the evolution of the volume of the bow shock in our model MS4070n10 (thick solid blue line), separating the volume of shocked ISM gas (thin dotted red line) from the volume of shocked stellar wind (thick dotted orange line) in the apex ($z\ge0$) of the bow shock. Such a discrimination of the volume of wind and ISM gas is possible because a passive scalar tracer is is numerically advected simultaneously with the flow. The figure further illustrates the preponderance of the volume of shocked ISM in the bow shock compared to the stellar wind material, regardless the growth of eddies. Interestingly, the volume of dense shocked ISM gas (large dotted black line) does not have large time variations (see Section \[sect:emission\]).
Bow shock energetics and emission signatures {#sect:emission}
============================================
![ Bow shocks luminosities. The panels correspond to models with an ISM density The simulations labels are indicated under the corresponding values. []{data-label="fig:lum1"}](./luminosity_grid_001.eps "fig:"){width="44.00000%"} ![ Bow shocks luminosities. The panels correspond to models with an ISM density The simulations labels are indicated under the corresponding values. []{data-label="fig:lum1"}](./luminosity_grid_01.eps "fig:"){width="44.00000%"} ![ Bow shocks luminosities. The panels correspond to models with an ISM density The simulations labels are indicated under the corresponding values. []{data-label="fig:lum1"}](./luminosity_grid_100.eps "fig:"){width="44.00000%"}
Methods {#subsect:methods}
-------
In Fig. \[fig:lum1\] the total bow shock luminosity $L_{\rm total}$ (pale green diamonds) is calculated integrating the losses by optically-thin radiation in the $z \ge 0$ region of the computational domain [@mohamed_aa_541_2012 Paper I]. Shocked wind emission $L_{\rm wind}$ (orange dots) is discriminated from $L_{\rm total}$ with the help of the passive scalar $Q$ that is advected with the gas, Additionaly, we compute $L_{\rm H\alpha}$ (blue crosses) and $L_{[\rm O{\sc III}]}$ (dark green triangles) which stand for the bow shock luminosities at H$\alpha$ and at \[O[iii]{}\] $\lambda \, 5007$ spectral line emission using the prescriptions for the emission coefficients in @dopita_aa_29_1973 and @osterbrock_1989, respectively. The overall X-ray luminosity $L_{\rm X}$ (black right crosses) is computed with emission coefficients generated with the [xspec]{} program [@arnaud_aspc_101_1996] with solar metalicity and chemical abundances from @asplund_araa_47_2009.
Results {#subsect:results}
-------
### Optical luminosities {#subsect:luminosities}
![ Luminosities of our bow shock simulation of a $40\, \rm M_{\odot}$ star moving with velocity $v_{\star}=70\, \rm km\, \rm s^{-1}$ through a medium with $n_{\rm ISM}=10\, \rm cm^{-3}$ (see corresponding time-sequence evolution of its density field in Fig. \[fig:grid\_velocity\]a-e). Plotted quantities and color-coding are similar to Fig. \[fig:lum1\] and are shown as function of time (in $\rm Myr$). []{data-label="fig:lum3"}](./luminosity.eps){width="100.00000%"}
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In Fig. \[fig:lum1\] we display the bow shocks luminosities as a function of the initial mass of the runaway star, its space velocity $v_{\star}$ and its ambient medium density $n_{\rm ISM}$. At a given density of the ISM, all of our models have luminosities from optically-thin gas radiation which with respect to the stellar mass loss are as described in Paper I for the simulations with $n_{\rm ISM}\approx 0.79\, \rm
cm^{-3}$.
The behaviour of the optically-thin emission originating from the shocked stellar wind $L_{\rm wind}$, the \[O[iii]{}\] $\lambda \, 5007$ spectral line emission and the H$\alpha$ emission at fixed $n_{\rm ISM}$ are similar as described in @meyer_mnras_2013. The contribution of $L_{\rm wind}$ is smaller than $L_{\rm total}$ by several orders of magnitude for all models, e.g. our model MS1020n0.1 has $L_{\rm wind}/L_{\rm total} \approx 10^{-5}$. All our models have $L_{\rm H\alpha} < L_{[\rm O{\sc III}]} < L_{\rm
total}$ and the H$\alpha$ emission, the $[\rm O{\sc III}]$ spectral line emission and $L_{\rm wind}$ have variations which are similar to $L_{\rm ISM}$ with respect to $M_{\star}$, $v_{\star}$ and $n_{\rm ISM}$.
Fig. \[fig:lum3\] shows the lightcurve of our model MS4070n10 computed over the whole simulation and plotted as a function of time with the color coding from Fig. \[fig:lum1\]. Very little variations of the emission are present at the beginning of the calculation up to a time of about $0.004\, \rm Myr$ and it remains almost constant at larger times. This is in accordance with the volume of the dense ISM gas trapped into the nebula (see large dotted black line in Fig. \[fig:volume\]). The independence of $L_{\rm IR}$ with respect to the strong volume fluctuations of thin-shelled nebulae (Fig. \[fig:lum3\]) indicates that their spectral energy distributions is likely to be the appropriate tool to analyze them since it constitutes an observable which is not reliable to temporary effects.
### Infrared and X-rays luminosities {#subsect:thermalisation}
reprocessed starlight on dust grains penetrating the bow shocks, $L_{\rm IR}$, is larger than $L_{\rm total}$ by about $1-2$ orders of magnitude. This is possible because the reemission of starlight by dust grains is not taken into account in our simulations. for the models MS2040n0.01 and MS2040n10, respectively. $L_{\rm IR}$ increases with $M_{\star}$ (Figs. \[fig:lum1\]a-d). Particularly, we find that $L_{\rm IR} \gg L_{\rm H\alpha}$ and $L_{\rm IR} \gg L_{[\rm O{\sc III}]}$, and therefore we conclude that the infrared waveband is the best way to detect and observe bow shocks from massive main-sequence runaway stars regardless of $n_{\rm ISM}$ (see section \[sect:observability\]).
Several current and/or planned facilities are designed to observe at these wavelengths and may be able to detect bow shocks from runaway stars:
1. First, the [*James Webb Space Telescope*]{} (JWST) which [*Mid-Infrared Instrument*]{} [MIRI, @swinyard_2004] observes in the infrared ($5$$-$$28\, \mu \rm m$) that roughly corresponds to our predicted waveband of dust continuum emission from stellar wind bow shocks of runaway OB stars.
2. Secondly, the [*Stratospheric Observatory for Infrared Astronomy*]{} (SOFIA) airborne facility which [*Faint Object infraRed CAmera for the SOFIA Telescope*]{} [FORCAST, @adams_2008] instrument detects photons in the $5.4$$-$$37\, \mu \rm m$ waveband.
3. Then, the proposed [*Space Infrared Telescope for Cosmology and Astrophysics*]{} [SPICA, @kaneda_2004] satellite would be the ideal tool the observe stellar wind bow shock, since it is planed to be mounted with (i) a far-infrared imaging spectrometer ($30$$-$$210\, \mu \rm m$), (ii) a mid-infrared coronograph ($3.5/5$$-$$27\, \mu \rm m$) and (iii) a mid-infrared camera/spectrometer ($5$$-$$38\, \mu \rm m$).
4. Finally, we should mention the proposed [*The Mid-infrared E-ELT Imager and Spectrograph*]{} (METIS) on the planned [*European Extremely Large Telescope*]{} [E-ELT, @brandl_2006], that will be able to scan the sky in the $3$$-$$19\, \mu \rm m$ waveband.
Exploitation of the associated archives of these instruments in regions surroundings young stellar clusters and/or at the locations of previously detected bow-like nebulae [@buren_apj_329_1988; @vanburen_aj_110_1995; @noriegacrespo_aj_113_1997; @peri_aa_538_2012; @2015arXiv150404264P] are research avenues to be explored.
Finally, we notice that the X-rays emission are much smaller than any other emission lines or bands, e.g. the model MS2070 has $L_{\rm X}/L_{\rm H\alpha} \approx 10^{-5}$, and it is consequently not a relevant waveband to observe our bow shocks.
### Feedback {#subsect:feedback}
The ratio $\dot{E}_{\rm motion}/L_{\rm total}$ is shown as a function of the bow shock volume in Fig. \[fig:obser\]b.
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Discussion {#sect:discussion}
----------
### The appropriated waveband to observe stellar wind bow shocks in the Galaxy {#sect:observability}
In Fig. \[fig:paving\_lum\] we show the H$\alpha$ surface brightness (in $\rm erg\,\rm
s^{-1}\,\rm cm^{-2}\,\rm arcsec^{-2}$, panels a-c) and the infrared luminosity (in $\rm erg\, \rm s^{-1}$, panels d-f) for models with $M_{\star}=10\, \rm M_{\odot}$ (left panels), $20\, \rm M_{\odot}$ (middle panels) and $40\, \rm M_{\odot}$ (right panels). The surface brightness $\Sigma^{\rm max}_{\rm H \alpha}$ scales with $n^{2}$, see Appendix A of Paper I, therefore the lower the ISM background density of the star, i.e. the higher its Galactic latitude, the fainter the projected emission of the bow shocks and the lower the probability to observe them. The brightest bow shocks are generated both in infrared and H$\alpha$ by our most massive stars running in the denser regions of the ISM ($\rm n_{\rm ISM} =
10.0\, \rm cm^{-3}$). The estimate of the infrared luminosity confirms our result relative to bow shock models with $n_{\rm ISM}=0.79\, \rm
cm^{-3}$ in the sense that the brightest bow shocks are produced by high-mass, stars (Paper I) moving in a relatively dense ambient medium, i.e. within the Galactic plane (Fig. \[fig:paving\_lum\]d-f). At H$\alpha$, these bow shocks are associated to fast-moving stars ($v_{\star}=70\, \rm km\, \rm s^{-1}$) producing the strongest shocks, whereas in infrared they are associated to slowly-moving stars ($v_{\star}=20\, \rm km\, \rm s^{-1}$) generating the largest nebulae.
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### Synthetic optical emission maps {#sect:maps}
In Fig. \[fig:maps1\] we plot synthetic H$\alpha$ and \[O[iii]{}\] $\lambda \, 5007$ emission maps of the bow shocks generated by our $20\, \rm M_{\odot}$ star moving with velocity $70\,
\mathrm{km}\, \mathrm{s}^{-1}$ moving through a medium with $n_{\rm ISM}=0.1$ (left column of panels), $0.79$ (middle column of panels) and $10.0\, \rm
cm^{-3}$ (right column of panels). The region of maximum H$\alpha$ emission of the gas is located close to the apex of the bow shock and extended to its trail ($z\,
\le 0$). This broadening of the emitting region is due to the high space velocity of the star, see Paper I. Neither the shocked stellar wind nor the hot shocked ISM of the bow shock contributes significantly to these emission since the $\rm H\alpha$ emission coefficient $j_{\rm H\alpha} \propto T^{-0.9}$ and the contact discontinuity is the brightest part of the whole structure (Fig. \[fig:maps1\]a). The \[O[iii]{}\] $\lambda \, 5007$ emission is maximum at the same location but, however, slightly different dependence on the temperature of the corresponding emission coefficient $j_{\rm [OIII]} \propto
\exp(-1/T)/T^{1/2}$ [@dopita_aa_29_1973] induces a weaker extension of the emission to the tail of the structure (Fig. \[fig:maps1\]a). The unstable simulations with $v_{\star}\, \ge 40\, \mathrm{km}\, \mathrm{s}^{-1}$ and $n_{\rm ISM} \simeq
10\, \rm cm^{-3}$ have ring-like artefacts which dominate the emission (see Fig. \[fig:maps1\]e-h and Fig. \[fig:maps1\]i-l). They are artificially generated by the over-dense regions of the shell that are rotated and mapped onto the Cartesian grid. A tri-dimensional unstable bow shock would have brighter clumps of matters sparsed around its layer of cold shocked ISM rather than regular rings [@mohamed_aa_541_2012]. Regardless of the properties of their driving star, our bow shocks are brighter in large ambient medium, e.g. the model MS2070n0.1 has $\Sigma^{\rm max}_{[\rm H\alpha]} \approx 10^{-18}
\, \mathrm{erg}\, \mathrm{s}^{-1}\, \mathrm{cm}^{-2}\, \mathrm{arcsec}^{-2}$ whereas the model MS2070n10 has $\Sigma^{\rm max}_{[\rm H\alpha]} \approx
3\times 10^{-15} \, \mathrm{erg}\, \mathrm{s}^{-1}\, \mathrm{cm}^{-2}\,
\mathrm{arcsec}^{-2}$. The projected \[O[iii]{}\] $\lambda \, 5007$ emission behaves similarly.
![ Cross-sections taken along the direction of motion of our $20\, \rm M_{\odot}$ star moving with velocity $70\, \rm km\, \rm s^{-1}$ in an ambient medium of number density $n_{\rm ISM}=0.1\, \rm cm^{-3}$. The data are plotted for inclination angles $\phi=30{\ensuremath{^\circ}}$ (thin solid red line), $\phi=45{\ensuremath{^\circ}}$ (thin dotted blue line), $\phi=60{\ensuremath{^\circ}}$ (thick solid orange line) and $\phi=90{\ensuremath{^\circ}}$ (thick dotted dark green line) through their H$\alpha$ surface brightness (see Fig. \[fig:maps1\]a-d). The position of the star is located at the origin. []{data-label="fig:profiles"}](./cut_profile_2070n01_Ha.eps){width="45.00000%"}
![ Bow shock H$\alpha$ surface brightness (a) and ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/
\Sigma^{\rm max}_{[\rm H\alpha]}$ (b) as a function of its volume $R(0)^{3}$ (in $\rm pc^{3}$). Upper panel shows the H$\alpha$ surface brightness as a function of the detection threshold of the SuperCOSMOS H$\alpha$ Survey (SHS) of $\Sigma_{\rm SHS}\approx 1.1-2.8 \times 10^{-17}\,
\rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm arcsec^{-2}$ [@parker_mnras_362_2005]. Lower panel plots the ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm max}_{[\rm H\alpha]}$ of the same models. []{data-label="fig:vol_vs_S_Ha"}](./vol_vs_S_Ha.eps){width="100.00000%"}
![ Bow shock H$\alpha$ surface brightness (a) and ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/
\Sigma^{\rm max}_{[\rm H\alpha]}$ (b) as a function of its volume $R(0)^{3}$ (in $\rm pc^{3}$). Upper panel shows the H$\alpha$ surface brightness as a function of the detection threshold of the SuperCOSMOS H$\alpha$ Survey (SHS) of $\Sigma_{\rm SHS}\approx 1.1-2.8 \times 10^{-17}\,
\rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm arcsec^{-2}$ [@parker_mnras_362_2005]. Lower panel plots the ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm max}_{[\rm H\alpha]}$ of the same models. []{data-label="fig:vol_vs_S_Ha"}](./vol_vs_OIII_over_Ha.eps){width="100.00000%"}
\
In Fig. \[fig:profiles\] we show cross-sections of the H$\alpha$ surface brightness of the model MS2070n0.1. The cuts are taken along the symmetry axis of the figures and plotted as a function of the inclination angle $\phi$ with respect to the plane of the sky. The emission rises slightly as $\phi$ increases from for $\phi=30{\ensuremath{^\circ}}$ (thin red solid line) to $\phi=60{\ensuremath{^\circ}}$ (thick solid orange line) since $\Sigma^{\rm max}_{[\rm H\alpha]}$ peaks at about $6\times 10^{-19}$ and about $10^{-18}\, \mathrm{erg}\, \mathrm{s}^{-1}\,
\mathrm{cm}^{-2}\, \mathrm{arcsec}^{-2}$, respectively. The case with $\phi=90{\ensuremath{^\circ}}$ is different since the emission decreases to about $\approx
2\times 10^{-19}\, \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm arcsec^{-2}$ (see thick dotted green line in Fig. \[fig:profiles\]). The same is true for the \[O[iii]{}\] emission since its dependence on the post-shock density is similar. Large angles of inclination make the opening of the bow shocks larger (Fig. \[fig:profiles\]a-c, e-g, i-k) and the stand-off distance appears smaller (Fig. \[fig:profiles\]a-c). Note that bow shocks observed with a viewing angle of $\phi=90{\ensuremath{^\circ}}$ do not resemble an arc-like shape but rather an overlapping of iso-emitting concentric circles (Fig. \[fig:profiles\]d,h,l).
### Bow shocks observability at H$\alpha$ and comparison with observations {#sect:comp}
In Fig. \[fig:vol\_vs\_S\_Ha\] we show our bow shocks’ H$\alpha$ surface brightness (a) and their $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm
max}_{[\rm H\alpha]}$ ratio (b), both as a function of the volume of emitting gas ($z\ge0$). The color coding of both panels takes over the definitions adopted in Fig. \[fig:axis\_ratio\]. The models with a $10\, \rm
M_{\odot}$ have a volume smaller than about a few $\rm pc^{3}$ and have emission smaller than about $10^{-15}\, \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm
arcsec^{-2}$. The models with $M_{\star}=20\, \rm M_{\odot}$ have larger volume at equal $n_{\rm ISM}$ and can reach surface brightness of about a few $10^{-14}\, \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm arcsec^{-2}$ if $n_{\rm
ISM}=10\, \rm cm^{-3}$. Note that all models with $n_{\rm ISM} \ge 10.0\, \rm
cm^{-3}$ produce emission larger than the diffuse emission sensitivity threshold of the [*SuperCOSMOS H-Alpha Survey*]{} (SHS) of $\Sigma_{\rm SHS} \approx
1.1$-$2.8 \times 10^{−17}\, \rm erg\, \rm s^{-1}\, \rm cm^{-2}\, \rm
arcsec^{-2}$ [@parker_mnras_362_2005] and such bow shocks should consequently be observed by this survey (see horizontal black line in Fig. \[fig:vol\_vs\_S\_Ha\]a).
As discussed above, a significant fraction of our sample of bow shocks models have a H$\alpha$ surface brightness larger than the sensitivity limit of the SHS survey [@parker_mnras_362_2005]. This remark can be extended to other (all-sky) H$\alpha$ observations campaigns, especially if their detection threshold is lower than the SHS. This is the case of, e.g. the [*Virginia Tech Spectral-Line Survey*]{} survey [VTSS, @dennison_aas_195_1999] and the [*Wisconsin H-Alpha Mapper*]{} [WHAM, @reynolds_pasa_15_1998] which provide us with images of diffuse sensitivity detection limit that allow the revelation of structures associated with sub-Rayleigh intensity. Consequently, one can expect to find optical traces of stellar wind bow shocks from OB stars in these data. According to our study, their driving stars are more likely to be of initial mass $M_{\star} \ge 20\, \rm M_{\odot}$ (Fig. \[fig:vol\_vs\_S\_Ha\]a). This also implies that bow shocks [*in the field*]{} that are observed with such facilities are necessary produced by runaway stars of initial mass larger than $M_{\star} \ge 20\, \rm M_{\odot}$. Moreover, we find that the models involving an $10\, \rm M_{\odot}$ star and with $v_{\star}\, \ge 40\, \mathrm{km}\,
\mathrm{s}^{-1}$ have $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm
max}_{[\rm H\alpha]}>10$, whereas almost all of the other simulations do not satisfy this criterion (Fig. \[fig:vol\_vs\_S\_Ha\]b).
, we find a similarity between some of the cross-sections taken along the symmetry axis of the H$\alpha$ surface brightness of our bow shock models (Fig. \[fig:profiles\]) and the measure of the radial brightness in emission measure of the bow shock generated by the runaway O star HD 57061 [see fig. 5 of @brown_aa_439_2005]. This observable and our model authorize a comparison since H$\alpha$ emission and emission measures have the same quadratic dependence on the gas number density. The emission measure profile of HD 57061 slightly increases from the star to the bow shock and steeply peaks in the region close to the contact discontinuity, before to decrease close to the forward shock of the bow shock and reach the ISM background emission. Our H$\alpha$ profile with $\phi=60{\ensuremath{^\circ}}$ is consistent with (i) the above described variations and (ii) with the estimate of the inclination of the symmetry axis of HD 57061 with respect to the plane of the sky of about $75{\ensuremath{^\circ}}$, see table 3 of @brown_aa_439_2005. Note that according to our simulations, the emission peaks in the region separating the hot from the cold shocked ISM gas.
### Implication for the evolution of supernova remnants generated by massive runaway stars {#sect:pre_sn}
Massive stars evolve and die supernovae, a sudden and strong release of matter, energy and momentum taking place inside the ISM pre-shaped by their past stellar evolution [@langer_araa_50_2012]. In the case of a runaway progenitor, the circumstellar medium at the pre-supernova phase can be a bow shock nebula with which the shock wave interacts into the unperturbed ISM [@brighenti_mnras_270_1994]. The subsequent growing supernova remnant develops asymmetries since it is braked by the mass at the apex of the bow shock but expands freely in the cavity driven by the star in the opposite direction [@borkowski_apj_400_1992]. If the progenitor is slightly supersonic, the bow shock is mainly shaped during the main-sequence phase of the star; whereas if the progenitor is a fast-moving star then the bow shock is essentially made of material from the last pre-supernova evolutionary phase. In the Galactic plane ($n_{\rm ISM}=0.79\, \rm cm^{-3}$) such asymmetries arise if the apex of the bow shock accumulates at least $1.5\, \rm M_{\odot}$ of shocked material [@meyer_mnras_450_2015].
In Fig. \[fig:mass\] we present the mass trapped into the $z\ge0$ region of our bow shock models as a function of their volume. As in Fig. \[fig:vol\_vs\_S\_Ha\] the figure distinguishes the initial mass and the ambient medium density of each models. Amongst our bow shock simulations, 9 models have $M_{\rm bow} \gtrsim
1.5\, \rm M_{\odot}$ and 4 of them are generated by the runaway stars which asymmetric supernova remnant studied in detail in @meyer_mnras_450_2015. The other models with $v_{\star} \le 40\, \rm km\, \rm s^{-1}$ may produce asymmetric remnants because they will explode inside their main-sequence wind bubble. The model MS4070n0.1 has $v_{\star}=70\, \rm km\, \rm s^{-1}$ which indicates that the main-sequence bow shock will be advected downstream by the rapid stellar motion and the surroundings of the progenitor at the pre-supernova phase is made of, e.g. red supergiant material. Consequently, its shock wave may be unaffected by the presence of the circumstellar medium. We leave the examination via hydrodynamical simulations of this conjecture for future works. Interestingly, we notice that most of the potential progenitors of asymmetric supernova remnants are moving in a low density medium $n_{\rm ISM}
\le 0.1\, \rm cm^{-3}$, i. e. in the rather high latitude regions of the Milky Way. This is consistent with the interpretation of the elongated shape of, e.g. Kepler’s supernova remnant as the consequence of the presence of a massive bow shock at the time of the explosion [@velazquez_apj_649_2006; @toledoray_mnras_442_2014].
### The influence of the interstellar magnetic field on the shape of supernovae remnants
![ Bow shocks mass as a function of the bow shock volume. The figure shows the mass $M_{\rm bow}$ (in $M_{\odot}$) trapped in the $z \ge 0$ region of the bow shock as a function of its volume $R(0)^{3}$ (in $\rm pc^{3}$). The dots distinguish between models (i) as a function of the ISM ambient medium with $n_{\rm ISM}=0.01$ (triangles), $0.1$ (diamonds), $0.79$ (circles) and $10\, \rm cm^{-3}$ (squares), and (ii) as a function of the initial mass of the star with $10$ (blue dots), $20$ (red plus signs) and $40\, \rm M_{\odot}$ (green crosses). The thin horizontal black line corresponds to $M_{\rm bow} = 1.5\, \rm M_{\odot}$, i.e. the condition to produce an asymmetric supernova remnant if $n_{\rm ISM}=0.79\, \rm cm^{-3}$ [@meyer_mnras_450_2015]. []{data-label="fig:mass"}](./vol_vs_mass.eps){width="100.00000%"}
\
Conclusion {#section:cc}
==========
Our bow shock simulations indicate that no structural difference arise when changing the density of the background ISM in which the stars move, i.e. their internal organisation is similar as described in @comeron_aa_338_1998 and Paper I. The same is true for their radiative properties, governed by line cooling such as the \[O[iii]{}\] $\lambda \, 5007$ line and showing faint H$\alpha$ emission, both principally originating from outer region of shocked ISM gas. We also find that their X-rays signature is fainter by several orders of magnitude than their H$\alpha$ emission, and, consequently, it is not a good waveband to search for such structures.
The best way to observe bow shocks remains their infrared emission of starlight reprocessed by shocked ISM dust [@meyer_mnras_2013]. We find that the brightest infrared bow shocks, i.e. the most easily observable ones, are produced by high-mass ($M_{\star} \approx 40\, \rm M_{\odot}$) stars moving with a slow velocity ($v_{\star}\approx 20\, \rm km\, \rm s^{-1}$) in the relatively dense regions ($n_{\rm ISM}\approx 10\, \rm cm^{-3}$) of the ISM, whereas the brightest H$\alpha$ structures are produced by these stars when moving rapidly ($v_{\star}\approx 70\, \rm km\, \rm s^{-1}$). Thin-shelled bow shocks have mid-infrared luminosities which does not report the time-variations of their unstable structures. This indicates that spectral energy distributions of stellar wind bow shocks are the appropriate tool to analyze them since they do not depend on the temporary effects that affect their density field. .
A detailed analysis of our grid of simulations indicates that the H$\alpha$ surface brightness of Galactic stellar wind bow shocks increases if their angle of inclination with respect to the plane of the sky increases up to $\phi = 60{\ensuremath{^\circ}}$, however, edge-on viewed bow shocks are particularly faint. We find that all bow shocks generated by a $40\, \rm M_{\odot}$ runaway star could be observed with Rayleigh-sensitive H$\alpha$ facilities and that bow shocks observed [*in the field*]{} by means of these facilities should have an initial mass larger than about $20\, \rm M_{\odot}$. Furthermore, all of our bow shocks generated by a $10\, \rm M_{\odot}$ star moving with $v_{\star}\ge 40\, \rm km\,
\rm s^{-1}$ have a line ratio $\Sigma^{\rm max}_{[\rm O{\sc III}]}/\Sigma^{\rm max}_{[\rm H\alpha]}>10$. Our study suggests that slowly-moving stars of ZAMS mass $M_{\star} \ge 20\, \rm
M_{\odot}$ moving in a medium of $n_{\rm ISM }\ge 0.1\, \rm cm^{-3}$ generate massive bow shocks, i.e. are susceptible to induce asymmetries in their subsequent supernova shock wave. This study will be enlarged, e.g. estimating observability of red supergiant stars.
Acknowledgements {#acknowledgements .unnumbered}
================
D. M.-A. Meyer thanks P. F. Velazquez, F. Brighenti and L. Kaper for their advices, and F. P. Wilkin for useful comments on stellar wind bow shocks which partly motivated this work. This study was conducted within the Emmy Noether research group on “Accretion Flows and Feedback in Realistic Models of Massive Star Formation” funded by the German Research Foundation under grant no. KU 2849/3-1. A.-J. van Marle acknowledges support from FWO, grant G.0227.08, KU Leuven GOA/2008, 04 and GOA/2009/09. The authors gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JUROPA at J" ulich Supercomputing Centre (JSC).
| ArXiv |
---
author:
- 'S. Sabari$^{1,2}$ and R. Kishor Kumar$^{3}$'
title: 'Effect of an oscillating Gaussian obstacle in a Dipolar Bose-Einstein condensate'
---
Introduction {#sec1}
============
The remarkable observation of Bose-Einstein condensates (BECs) in $^{52}$Cr [@Lahaye:2007; @Griesmaier:2006], $^{164}$Dy [@Lu:2011; @Youn:2010], and $^{168}$Er [@Aikawa:2012; @rev1] with both dipole-dipole interaction (DDI) and *s-wave* contact interaction has opened a wholly new exciting field that continues to thrive [@dbec1; @dbec2; @FetterRMP]. Contrary to the short-range contact interaction, the DDI is a long-range anisotropic interaction that can be either repulsive or attractive. The *s-wave* contact interaction, $a_s$, is experimentally controllable by Feshbach resonance [@FBR]. It is therefore appealing to study the properties of dipolar BECs in variable short-range contact interaction regimes. However, the DDI is also inherently controllable, either via the magnitude of the external electric field, or by modulating the external aligning field in time, which allows to tune the magnitude and sign of the DDI [@tuneDDI]. Due to the long-range nature and anisotropic character of the DDI, the dipolar BEC possesses many distinct features and new phenomena such as the new dispersion relations of elementary excitations [@Wilson:2010; @Ticknor:2011], unusual equilibrium shapes, the roton-maxon character of the excitation spectrum [@Santos:2000; @Yi:2003; @Ronen:2007; @Parker:2008], quantum phases including supersolid and checkerboard phases [@Tieleman:2011; @Zhou:2010], anisotropic solitons [@equ2Db; @solRK; @solPM], vortices [@rev4; @rev5], hidden vortices [@Sabari2017] and distinct vortex lattices including crater-like structure, square lattices [@vor1; @vor2; @vor3].
The modern optical techniques help to control the parameters of the condensate and visualize topological defects such as rarefaction pulses and quantized vortices in BECs [@FetterRMP; @equ2Db; @solRK; @solPM; @rev4; @rev5; @vor1; @vor2; @vor3]. Recently, vortex tangles caused by an oscillatory perturbation were observed experimentally. The vortex tangle configuration is a signature of the presence of a quantum turbulent regime in the BEC cloud [@Henn09]. Moreover, recent studies on quantum turbulence are still concentrating on understanding the dynamics of quantized vortices [@PLTP]. Vibrating structures such as spheres, grids, and wires are used in superfluid $^3$He and $^4$He to create quantum turbulence [@PLTP; @Hanninen07]. Despite the differences between these structures, the experiments show surprisingly similar behavior.
Introducing an oscillating potential in atomic dipolar BECs will be helpful to analyze the intrinsic nucleation of topological defects and synergy dynamics of vortices and rarefaction pulses. Also, this technique suggests a powerfull method for making quantum turbulence in trapped dipolar BECs, in addition to the other methods that have been used so far [@Henn09; @Berloff02; @Kobayashi07] in alkali BECs. Eventually, the dynamics of vortices and rarefaction pulses can be visualized in atomic dipolar BECs, which enabling experimental and theoretical challenges for further analysis. Up to now, vortex dipoles caused by oscillating potentials in alkali BECs have been observed in experiments and compared to theoretical models [@osc1; @osc2; @Jackson00; @Raman99; @Onofrio20; @Neely10; @rev2; @rev3]. Nonlinear dynamical behaviors, critical velocity for vortex dipoles, hydrodynamic flow, vortices, rarefaction pulses and other interesting perspectives have been carried out in alkali BECs using the oscillating Gaussian potential [@osc1; @osc2; @Jackson00; @Raman99; @Onofrio20; @Neely10].
Inspite of the many experiments that have been carried out on $^{164}$Dy and $^{168}$Er condensates there has still been no experimental observation of vortices in dipolar BECs. So, investigating the dynamics of vortex dipoles in a dipolar BEC by introducing an oscillating potential will be a fascinating experimental exploration. Thus, this model will be helpful to perform new experiments with the aim of observing vortices in dipolar BECs. In the present work, we are interested in studying the nucleation and dynamics of vortex dipoles and rarefaction pulses.
The next sections are organized as follows. In Sec. \[sec:frame\], we present the general three-dimensional mean-field equation for the dipolar BECs and the corresponding two-dimensional (2D) reduction. In Sec. \[sec:numerical\], we present our numerical results, where we include plots on the critical velocity for the nucleation and dynamics of vortex-dipoles. Further, in this section, we show the rarefaction pulses due to the annihilation of vortex-dipoles. Finally, in Sec. \[sec:con\], we present a summary of our conclusions and perspectives.
The mean-field formalism {#sec:frame}
========================
At ultralow temperatures, a dipolar BEC is described by the time-dependent GP equation with a nonlocal integral corresponding to the DDI [@dbec1; @dbec2; @lasPM; @rev5; @rev6]
$$\begin{aligned}
i\hbar\frac{\partial \phi({\mathbf r},t)}{\partial t}& =\Big(-\frac{\hbar^2}{2m}\nabla^2+V({\mathbf r},t) + g \left\vert \phi({\mathbf r},t)\right\vert^2 \Big)\phi({\mathbf r},t)\notag \\ &+N \int U_{\mathrm{dd}}({\mathbf r}-{\mathbf r}')\left\vert\phi({\mathbf r}',t)\right\vert^2 d{\mathbf r}'\phi({\mathbf r},t), \label{eqn:dgpe}\end{aligned}$$
with $({\bf r},t)=({\bf \rho},t)$ and the radial coordinate being $\rho=\sqrt{x^2+y^2}$. The trapping potential $V({\mathbf r},t) = V_{ext} + V_G$ contains a cylindrically symmetric harmonic trap in addition to a Blue detuned Gaussian obstacle. The cylindrically symmetric trap is $V_{ext}({\mathbf r})=\frac{1}{2} m (\omega_\rho^2 (x^2+y^2)+ \omega_z^2 z^2)$, with $\omega_x = \omega_y = \omega_\rho$ and $\omega_z$ being the radial and axial trap frequencies respectively. The trap aspect ratio of the harmonic trap is $\lambda=\omega_{z}/\omega_{\rho}$. The Gaussian obstacle is
$$V_{G}(\rho,t) = V_{0} \exp\left(-\frac{\left[x-x_0(t)\right]^2+y^2}{w_0^2}\right),$$
where $V_0$, $x_0(t)$ and $w_0^2$ are the height, position and width of the Gaussian obstacle. The position of the obstacle $x_0(t)=\epsilon \sin(\omega t)$ provides parametric resonance with respect to the oscillating frequency $\omega$. One can control the velocity ($v=\epsilon\omega$) of oscillation of the obstacle with respect to the amplitude $\epsilon$ and the frequency $\omega$. In the present study, $\epsilon = 10 \mu$m and $\omega = 60 /s$. However, the velocity of the obstacle also depends on $V_{0}$ and $w_0$, and we keep these fixed: $V_0=80\,\hbar\omega_\rho$ and $w_{0}=0.25\,\mu m$. The two-body contact interaction strength is $g=4\pi$ $\hbar^2a_s N/m$ where $a_s$, $m$, and $N$ are atomic scattering length, mass of the atom and number of atoms respectively. We consider that the magnetic dipoles are polarized along $z$ direction and the corresponding dipolar interaction term is $ U_{\mathrm{dd}}(\bf R)=(\mu_0 \mu^2/4\pi)(1-3\cos^2 \theta/ \vert {\bf R} \vert ^3)$, where the relative position of the dipoles is ${\bf R= r -r'}$, $\theta$ is the angle between ${\bf R}$ and the direction of polarization $z$, $\mu_0$ is the permeability of free space and $\mu$ is the dipole moment of the condensate atom. In the present study, we consider the $^{168}$Er and $^{164}$Dy atoms, their corresponding dipole moments being $\mu=7\mu_B$ and $10\mu_B$ respectively. The normalization of the mean-field wavefunction is $\int d{\bf r}\vert\phi({\mathbf r},t)\vert ^2=1.$
It is convenient to use the GP equation (\[eqn:dgpe\]) in a dimensionless form. For this purpose, we introduce the dimensionless variables ${\bar {\bf r}}= {\bf r}/l,{\bar {\bf R}}={\bf R}/l, \bar a_s=a_s/l, \bar a_{\mathrm{dd}}=a_{\mathrm{dd}}/l, \bar t=t\bar \omega_\rho$, $\bar x=x/l, \bar y=y/l, \bar z=z/l, \bar \phi=l^{3/2}\phi$, $l=\sqrt{\hbar/(m \omega_\rho)}$. Eq. (\[eqn:dgpe\]) can be rewritten (after removing the overhead bar from all the variables) as $$\begin{aligned}
i \frac{\partial \phi({\mathbf r},t)}{\partial t} & = \Big(-\frac{1}{2}\nabla^2+V({\mathbf r},t) +g \vert {\phi({\mathbf{r}},{t})} \vert^2 \Big) \phi({\mathbf r},t) \\ \nonumber &+ 3N a_{\mathrm{dd}}\int \frac{1-3\cos^2\theta}{\vert \bf{R}\vert^3} \vert \phi({\mathbf{r}}',t) \vert^2 d{\mathbf{r}}' \phi({\mathbf r},t),\label{gpe3d} \end{aligned}$$
To compare the dipolar and contact interactions, often it is useful to introduce the length scale $a_{\mathrm{dd}}\equiv \mu_0 \mu^2 m/(12\pi \hbar^2 l)$ and its values for $^{164}$Er and $^{168}$Dy are $66a_0$ and $131a_0$ respectively, with $a_0$ being the Bohr radius [@dbec1].
The stability of dipolar BEC depends on the external trap geometry, *e.g.* a dipolar BEC is stable or unstable depending on whether the trap is pancake- or cigar-shaped, respectively. The instability usually can be overcome by applying a strong pancake trap and applying repulsive two-body contact interaction. The external trap helps to stabilize the dipolar BEC by imprinting anisotropy to the density distribution. Hence, we carry out the present study with axially-symmetric pancake-shaped magnetic trap.
So, we assume that the dynamics of the dipolar BEC in the axial direction is strongly confined with the ground state $\phi(z)=\exp(-z^2 /2d_z^2)/(\pi d_z^2)^{1/4}, \quad d_z= \sqrt{1/\lambda},$ and the wave function $$\begin{aligned}
\phi({\bf r})\equiv \phi(z) \psi(\rho,t)=(1/(\pi d_z^2)^{\frac{1}{4}})\exp(-z^2/2d_z^2) \psi(\rho,t),\nonumber \end{aligned}$$ where axial harmonic oscillator length $d_z=\sqrt{1/(\lambda)}$ and ${\bf\rho} \equiv (x,y)$. Therefore, it is more suitable to consider the system in quasi two-dimensions. One can obtain the effective 2D equation for the pancake-shaped dipolar BEC by integrating the equation (\[gpe3d\]) over the $z$ direction using the above wave function $\phi({\bf r})$ [@equ2Da; @equ2Db; @lasPM] $$\begin{aligned}
i\frac{\partial \psi(\rho,t) }{\partial t}& =\Big(-\frac{1}{2}\nabla_\rho^2+ V_{2D} + g_{2} \, \vert \psi(\rho,t) \vert^2 \Big) \psi(\rho,t) \\ \notag &+g_{d}\int \frac{d{\bf k}_\rho}{(2\pi)^2} e^{-i{\bf k}_\rho.{\bf\rho}}\widetilde n({\bf k}_\rho,t)h_{2D}\left(\frac{k_\rho d_z}{\sqrt{2}}\right) \psi(\rho,t).\label{gpef}\end{aligned}$$ where, two-body contact interaction $g_{2}= 4\pi Na_s/(\sqrt{2\pi}d_z)$, dipolar interaction $g_{d}=4\pi Na_{\mathrm{dd}}/(\sqrt{2\pi}d_z)$, and $k_\rho=(k_x^2+k_y^2)^{1/2}$. The dipolar term has been written in the Fourier space [@equ1; @equ2].
Dynamics of vortices {#sec:numerical}
====================
The results in this section were obtained within a full numerical approach to solve 2D nonlinear differential Eq. (4). When we have dipolar interactions, we have to combine the split-step Crank-Nicholson method and Fast Fourier Transform (FFT) for evaluating dipolar integrals in momentum space as in Ref. [@equ1; @equ2; @CUDA].
{width="14.0cm"}
For looking at the dynamics of vortex dipoles, the 2D numerical simulations were carried out in real time with a grid size having $512$ points for each dimension, where we have $\Delta x = \Delta y = 0.05$ for the space-steps and $\Delta t = 0.00025$ for the time-step. Also, the results were verified by doubling the aforementioned grid sizes.
![(color online) Dynamics of the vortices for (upper panel) $^{168}$Er BEC ($a_{dd}=66a_0$) and (lower panel) $^{164}$Dy BEC ($a_{dd}=131a_0$) and $a_s=50a_0$ for all cases. From left to right $t=0\,$ms, $t=0.4\,$ms, $t=2.0\,$ms, $t=2.6\,$ms, $t=4.9\,$ms, $t=5.9\,$ms, $t=6.7\,$ms, and $t=7.5\,$ms, respectively. []{data-label="f2"}](cs.eps "fig:"){width="9cm" height="3.5cm"} ![(color online) Dynamics of the vortices for (upper panel) $^{168}$Er BEC ($a_{dd}=66a_0$) and (lower panel) $^{164}$Dy BEC ($a_{dd}=131a_0$) and $a_s=50a_0$ for all cases. From left to right $t=0\,$ms, $t=0.4\,$ms, $t=2.0\,$ms, $t=2.6\,$ms, $t=4.9\,$ms, $t=5.9\,$ms, $t=6.7\,$ms, and $t=7.5\,$ms, respectively. []{data-label="f2"}](dy.eps "fig:"){width="9cm" height="3.5cm"}
First, we calculate the critical velocity for the nucleation of vortex dipoles with respect to scattering length as shown in in Fig. (\[f1\]) (a) for the three different cases, non-dipolar $a_{dd}=0\,a_0$ (Green line with circles), $^{168}$Er dipolar $a_{dd}=66\,a_0$ (Blue line with triangles), and $^{164}$Dy dipolar $a_{dd}=131\,a_0$ (Magenta line with stars) condensates, respectively. If, the velocity of the Gaussian potential ($V_p=\epsilon\,\, \omega$) is larger than the critical velocity $V_c$, vortex dipoles are nucleated in the condensate. As shown in Fig. (\[f1\]) (a), the critical velocity ($V_c$) for the nucleation of vortex dipoles increases with respect to increasing two-body contact and dipolar interaction strengths. This happens due to increasing interaction strengths leading to increase in the chemical potential of the condensate. Usually the height of the obstacle is determined with respect to the chemical potential. One needs to increase the height of the obstacle when the chemical potential increases. In this case, the critical velocity for nucleating vortices also depends on the height of the Gaussian obstacle. Also, the chemical potentials for the three different cases, alkali BEC (non-dipolar $a_{dd}=0$, $^{168}$Er dipolar BEC, and $^{164}$Dy dipolar BEC are 36.77, 44.65 and 64.55, respectively. Figure (\[f1\]) (a) shows that if we fix the amplitude of the obstacle then the critical velocity increases with respect to the interaction strengths. Usually, in rotating magnetic trap, for the nucleation of single vortex, the critical rotation frequency decreases with respect to increasing contact and dipolar interaction strengths. In the rotating trap, the vortex with same charge circulation is created whereas while stirring with an obstacle, we observe vortices with opposite charges circulations. Stirring beyond the critical velocity nucleates more vortices. This will be helpful to study the dynamics of multiple vortex dipoles. Another feature to look in Fig. (\[f1\]) (a) is the stability of the $^{164}$Dy dipolar BECs. We show the critical velocity of the $^{164}$Dy BEC from $35\,a_0$, because below this scattering length the obstacle creates the local collapse. On the other hand, $^{168}$Er BEC is stable from $20\,a_0$. Figure (\[f1\]) (b) illustrates the plot of the $V_c/c_s$ Vs $a_s$ for the three different cases, nondipolar (Green line with circles) [@ref1], $^{168}$Er (Blue line with triangles), and $^{164}$Dy (Magenta line with stars) condensates, respectively. The speed of sound ($c_s$) depends on the scattering length and dipolar interaction strength ($c=\sqrt{2 n_0 \sqrt{2\pi} (a+a_{dd})/d_z}$).
![(Color online) Density profile for nucleation of a vortex pair by the oscillating potential at (a) $t$ = 0 ms, (b) $t$ = 0.2 ms, and (c) $t$ = 0.4 ms are shown. The symbols $-$ and $+$ denote a vortex with clockwise or counterclockwise circulation, respectively. The black arrows indicate the direction of motion of the potential. A ghost vortex pair nucleates inside the potential (a), exits it (b), and finally fully leaves the potential (c).[]{data-label="f3"}](nucleationF.eps){width="0.7\linewidth"}
The dynamics of vortices in which they experience a lengthy migration are shown in Figs. (\[f2\]) for two different cases. Following the destiny of vortices nucleated by the oscillating potential enables us to survey their dynamics. The initial state in the static Gaussian potential in Fig. (\[f2\]) is obtained by an imaginary time step of the GP equation. A vortex pair is nucleated behind the Gaussian potential in Fig. (\[f2\])(b) as the potential starts to move. Then the oscillating potential nucleates vortex pairs whose impulses alternately change direction. They reconnect with each other to make new vortex pairs, leaving the potential in two cases. This phenomenon is not observed for the case of uniform motion of the potential, but only for an oscillating potential. Reaching the surfaces, the vortex pairs interact with ghost vortices, which are vortices in the low-density region. Then the vortices head toward the bow of the condensate along the surfaces. A vortex coming up from the left side reaches the bow to meet one from the right side, thus making a new vortex pair. Finally, the pair comes back to the center of the condensates. Thus the vortices nucleated by the potential enjoy a lengthy migration in the “sea” of BEC; the vortices are nucleated from the potential, reconnect, move away from it, reach the surface, head toward the bow and come back to the center. In the following, we illustrate elementary processes related to the synergy dynamics.
![(Color online) Density profile for reconnection of vortex pairs near the oscillating potential at (a) $t$ = 0.8 ms, (b) $t$ = 1.2 ms, and (c) $t$ = 2.3 ms are shown. The density profile before the collision between the potential and the vortex pair are shown in (a). Thereafter, another ghost vortex pair nucleates in (b), exiting the potential through the collision, which causes reconnection of the vortices. As a result, two pairs appear in (c).[]{data-label="f4"}](reconnection2.eps){width="0.7\linewidth"}
An oscillating potential creates vortex pairs, causes reconnection of pairs characterized by the oscillation, and causes the new pairs to leave for the surface of the condensate. Consequently, the surface becomes filled with vortices having positive and negative circulation, which leads to nucleation of rarefaction pulses and the migration of vortices. We call this sequence synergy dynamics of vortices and rarefaction pulses, which often occurs in cases where the amplitude of the oscillation is larger than the size of the oscillating potential. Ghost vortices, namely quantized vortices in a low density region, are important for the nucleation of the usual vortices in the bulk density region since nucleation requires seeds of vortices. In rotating BECs, ghost vortices are nucleated outside the condensate, entering it through the excitation of the surface waves, leading to the creation of usual vortices [@Tsubota02; @Kasamatsu03]. Thus, the periphery of the condensate provides seeds of topological defects. In our system, the oscillating potential provides seeds within itself. The potential starts to move, inducing a velocity field like back-flow, emitting phonons, and a ghost vortex pair is nucleated inside the potential as shown in Fig. (\[f3\])(a). The ghost pair tends to move away from the potential in Fig. (\[f3\])(b), and a usual vortex pair appears in the condensate in Fig. (\[f3\])(c). Thus, the ghost vortices work as seeds of usual vortices.
Reconnection of vortex pairs occurs near the oscillating potential. The new vortex pair has an impulse in the same direction as that of the potential. Then, the potential changes the direction of the velocity. Thus, the potential will collide with the pair in Fig. (\[f4\])(a). Then, a new ghost vortex pair is nucleated inside the potential whose impulse is opposite to that of the usual vortex pair, reconnecting with it as shown in Fig. (\[f4\])(b). Thus, two new vortex pairs appear in the condensate in Fig. (\[f4\])(c). Thereafter, the pairs move away from the potential, leaving for the surface of the condensate. This reconnection is characteristic of the oscillating potential because the potential repeatedly emits vortices of positive and negative circulation in opposite directions, which is not seen for potentials of uniform motion. This leads to nucleation of rarefaction pulses, as shown in the following.
![(Color online) Density profile for nucleation of rarefaction pulses at (a) $t$ = 2.5 ms, (b) $t$ = 3.2 ms, (c) $t$ = 3.8 ms and (d) $t$ = 4.7 ms are shown. Some vortices sit near the surface in (a) and reconnection of the vortices occurs in (b). While the new pairs move toward the center of the condensate, the pairs annihilate, which leads to nucleation of rarefaction pulses in (c) and (d).[]{data-label="f5"}](soliton2.eps){width="9cm" height="4.5cm"}
The vortex pairs separate as they approach the surface of the condensate shown in Fig. (\[f5\]). This behavior is qualitatively understood by applying the idea of an image vortex, which is often used in dynamics. The vortices induce a circular velocity field in a uniform system, but the field is distorted in a nonuniform system. This effect is strongly evident near the surface of the condensate where the density profile rapidly varies. As the vortices arrive at the surface, the normal component of the velocity field is suppressed. This situation is approximately equal to the relation between a vortex and a solid wall, so that the dynamics of vortices near the surface in Fig. (\[f5\]) can be shown by the image vortex. Note that this idea only gives a qualitative understanding since the surface is not exactly a solid wall. The vortices near the surface of the condensate have two fates. One is that a vortex pair transforms into rarefaction pulses through the annihilation of the pairs. The other is that the vortices migrate in the condensate. We show these dynamics in the following. Many vortices have accumulated near the surface of the condensate in Fig. (\[f5\])(a) since the oscillating potential continues to make vortices with positive and negative circulation. Hence, the vortices near the surface can reconnect with each other as shown in Fig. (\[f5\])(b), where we enclose the new vortex pairs with square dotted lines. These pairs have impulse toward the center of the condensate. As a pair approaches the center, the size of the pair diminishes. Consequently, pair annihilation of vortices occurs, making the rarefaction pulses shown in Figs. (\[f5\])(c) and (d). The low density parts in Figs. (\[f5\])(c) and (d) have these properties and hence we can identify them as rarefaction pulses. This kind of nucleation of rarefaction pulses is characteristic of an oscillating potential since it is caused by the potential emitting vortices in opposite directions.
Conclusion {#sec:con}
==========
We have reported the dynamics of vortices in a dipolar Bose-Einstein condensate by solving the two-dimensional, nonlocal, Gross-Pitaevskii equation numerically. We have calculated the critical velocity for vortex nucleation and found that the critical velocity increases when we increase the strength of the dipolar interaction in the atomic BEC. We have showed the formation of the rarefaction pulses during the dynamics of the vortices with opposite rotations. We have reported the teaming dynamics of vortices and rarefaction pulses in a BEC with dipole-dipole interaction, which has not been reported upto now.
0.5 cm [**Acknowledgement**]{} The authors thank professors, P. Muruganandam, K. Porsezian, Bishwajyoti Dey and A. Gammal for their help. SS wishes to thank DST-SERB for offering a Post-Doc through National Post Doctoral Fellowship (NPDF) Scheme (Grant No. PDF/2016/004106). The work is partially supported by the UGC through Dr. D.S. Kothari Post Doctoral Fellowship Scheme (No.F.4-2/2006 (BSR)/PH/14-15/0046). RKK acknowledges the financial support from FAPESP of Brazil (Contract number 2014/01668-8). 0.5 cm
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| ArXiv |
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abstract: 'A search is performed for collimated muon pairs displaced from the primary vertex produced in the decay of long-lived neutral particles in proton-proton collisions at $\sqrt{s}$ = 7 centre-of-mass energy, with the ATLAS detector at the LHC. In a [1.9 fb$^{-1}$]{}event sample collected during 2011, the observed data are consistent with the Standard Model background expectations. Limits on the product of the production cross section and the branching ratio of a Higgs boson decaying to hidden-sector neutral long-lived particles are derived as a function of the particles’ mean lifetime.'
author:
- The ATLAS Collaboration
title: '[Search for displaced muonic lepton jets from light Higgs boson decay in proton-proton collisions at [$\sqrt{s}$]{} = 7 with the ATLAS detector]{}'
---
A search is presented for long-lived neutral particles decaying to final states containing collimated muon pairs in proton-proton collisions at $\sqrt{s}$ = 7 centre-of-mass energy. The event sample, collected during 2011 at the LHC with the ATLAS detector, corresponds to an integrated luminosity of [1.9 fb$^{-1}$]{}. The model considered in this analysis consists of a Higgs boson decaying to a new hidden sector of particles which finally produce two sets of collimated muon pairs, but the search described is equally valid for other, distinct models such as heavier Higgs boson doublets, singlet scalars or a $Z^\prime$ that decay to a hidden sector and eventually produce collimated muon pairs.\
Recently, evidence for the production of a boson with a mass of about 126 has been published by ATLAS [@higgsatl] and CMS [@higgscms]. The observation is compatible with the expected production and decay of the Standard Model (SM) Higgs boson [@HIGGS1; @HIGGS2; @HIGGS3] at this mass. Testing the SM Higgs hypothesis is currently of utmost importance. To this end two effects may be considered: (i) additional resonances which arise in an extended Higgs sector found in many extensions of the SM, or (ii) rare Higgs boson decays which may deviate from those predicted by the SM. In this Letter we search for a scalar that decays to a light hidden sector, focusing on the 100 GeV to 140 GeV mass range. In doing so, we cover both of the above aspects, deriving constraints on additional Higgs-like bosons, as well as placing bounds on the branching ratio of the discovered 126 GeV resonance into a hidden sector of the kind described below.\
The phenomenology of light hidden sectors has been studied extensively over the past few years [@b1; @b2; @b3; @b4; @b5]. Possible characteristic topological signatures of such extensions of the SM are “lepton jets". A lepton jet is a cluster of highly collimated particles: electrons, muons and possibly pions [@b2; @b6; @b7; @b8]. These arise if light unstable particles with masses in the to range (for example dark photons, [$\gamma_{d}$]{}) reside in the hidden sector and decay predominantly to SM particles. At the LHC, hidden-sector particles may be produced with large boosts, causing the visible decay products to form jet-like structures. Hidden-sector particles such as [$\gamma_{d}$]{}may be long-lived, resulting in decay lengths comparable to, or larger than, the detector dimensions. The production of lepton jets can occur through various channels. For instance, in supersymmetric models, the lightest visible superpartner may decay into the hidden sector. Alternatively, a scalar particle that couples to the visible sector may also couple to the hidden sector through Yukawa couplings or the scalar potential. This analysis is focused on the case where the Higgs boson decays to the hidden sector [@b9; @b10]. The SM Higgs boson has a narrow width into SM final states if $m_{H} < 2 m_W$. Consequently, any new (non-SM) coupling to additional states, which reside in a hidden sector, may contribute significantly to the Higgs boson decay branching ratios. Even with new couplings, the total Higgs boson width is typically small, well below the order of one GeV. If a SM-like Higgs boson is confirmed, it will remain important to constrain possible rare decays, e.g. into lepton jets.\
Neutral particles with large decay lengths and collimated final states represent, from an experimental point of view, a challenge both for the trigger and for the reconstruction capabilities of the detector. Collimated particles in the final state can be hard to disentangle due to the finite granularity of the detectors; moreover, in the absence of inner tracking detector information and a primary vertex constraint, it is difficult to reconstruct charged-particle tracks from decay vertices far from the interaction point (IP). The ATLAS detector [@ATLASTDR] is equipped with a muon spectrometer (MS) with high-granularity tracking detectors that allow charged-particle tracks to be reconstructed in a standalone configuration using only the muon detector information (MS-only). This is a crucial feature for detecting muons not originating from the primary interaction vertex.\
The search presented in this Letter focuses on neutral particles decaying to the simplest type of muon jets (MJs), containing only two muons; prompt MJ searches have been performed both at the Tevatron [@tevatron1; @tevatron2] and at the LHC [@CMS]. Other searches for displaced decays of a light Higgs boson to heavy fermion pairs have also been performed at the LHC [@Hiddenv].\
The benchmark model used for this analysis is a simplified scenario where the Higgs boson decays to a pair of neutral hidden fermions ($f_{d2}$) each of which decays to one long-lived [$\gamma_{d}$]{}and one stable neutral hidden fermion ($f_{d1}$) that escapes the detector unnoticed, resulting in two lepton jets from the [$\gamma_{d}$]{}decays in the final state (see Fig. \[fig:model\]). The mass of the [$\gamma_{d}$]{}(0.4 ) is chosen to provide a sizeable branching ratio to muons [@b9].
![Schematic picture of the Higgs boson decay chain, H$\rightarrow$2($f_{d2}\rightarrow f_{d1}$[$\gamma_{d}$]{}). The Higgs boson decays to two hidden fermions ($f_{d2}$). Each hidden fermion decays to a [$\gamma_{d}$]{}and to a stable hidden fermion ($f_{d1}$), resulting in two muon jets from the [$\gamma_{d}$]{}decays in the final state.[]{data-label="fig:model"}](fig_01a.pdf){width="55mm"}
ATLAS is a multi-purpose detector [@ATLASTDR] at the LHC, consisting of an inner tracking system (ID) embedded in a superconducting solenoid, which provides a 2 T magnetic field parallel to the beam direction, electromagnetic and hadronic calorimeters and a muon spectrometer using three air-core toroidal magnet systems[^1]. The trigger system has three levels [@L1TRIG] called Level-1 (L1), Level-2 (L2) and Event Filter (EF). L1 is a hardware-based system using information from the calorimeter and muon spectrometer, and defines one or more Regions of Interest (ROIs), geometrical regions of the detector, identified by ($\eta$, $\phi$) coordinates, containing interesting physics objects. L2 and the EF (globally called the High Level Trigger, HLT) are software-based systems and can access information from all sub-detectors. The ID, consisting of silicon pixel and micro-strip detectors and a straw-tube tracker, provides precision tracking of charged particles for . The electromagnetic and hadronic calorimeter system covers and, at , has a total depth of 9.7 interaction lengths (22 radiation lengths in the electromagnetic part). The MS provides trigger information () and momentum measurements () for charged particles entering the spectrometer. It consists of one barrel and two endcap parts, each with 16 sectors in $\phi$, equipped with precision tracking chambers and fast detectors for triggering. Monitored drift tubes are used for precision tracking in the region and cathode strip chambers are used for 2.0 $\leq$ . The MS detectors are arranged in three stations of increasing distance from the IP: inner, middle and outer. The air core toroidal magnetic field allows an accurate charged particle reconstruction independent of the ID information. The three planes of trigger chambers (resistive plate chambers in the barrel and the thin gap chambers in the endcaps) are located in middle and outer (only in the barrel) stations. The L1 muon trigger requires hits in the middle stations to create a low tranverse momentum () muon ROI or hits in both the middle and outer stations for a high ROI. The muon ROIs have a spatial extent of () in the barrel and of in the endcap. L1 ROI information seeds, at HLT level, the reconstruction of muon momenta using the precision chamber information. In this way sharp trigger thresholds up to 40 can be obtained.
The set of parameters used to generate the signal Monte Carlo samples is listed in Table \[tab:param\]. The Higgs boson is generated through the gluon-gluon fusion production mechanism which is the dominant process for a low mass Higgs boson. The gluon-gluon fusion Higgs boson production cross section in [*pp*]{} collisions at [$\sqrt{s}$]{}= 7 , estimated at the next-to-next-to-leading order (NNLO) [@HiggsCrossS], is $\sigma_{\textrm{\small SM}} = $ 24.0 pb for $m_{H}=$ 100 and $\sigma_{\textrm{\small SM}} = $ 12.1 pb for $m_{H}=$ 140 . The [[PYTHIA]{}]{} generator [@PYTHIA] is used, linked together with [[MadGraph]{}]{}4.4.2 [@b12] and [[BRIDGE]{}]{} [@BRIDGE], for gluon-gluon fusion production of the Higgs boson and the subsequent decay to hidden-sector particles.\
As discussed in the introduction, the signal is chosen to enable a study of rare, non-SM, Higgs boson decays in the (possibly extended) Higgs sector. To do so we choose two points which envelope a mass range covering the 126 GeV resonance. The lower mass point, $m_H=$ 100 GeV, is chosen to be compatible with the decay-mode-independent search by OPAL at LEP [@opal]. The higher mass point, $m_H=$ 140 GeV, is chosen well below the $WW$ threshold, where a sizeable branching ratio into a hidden sector may be naturally achieved. The masses of $f_{d2}$ and $f_{d1}$ are chosen to be light relative to the Higgs boson mass, and far from the kinematic threshold at $m_{f_{d1}} + m_{\gamma_{d}} = m_{f_{d2}}$. For the chosen dark photon mass (0.4 ), the [$\gamma_{d}$]{}decay branching ratios are expected to be [@b9]: 45$\%~ \ee$, 45$\%~\mu^+\mu^-$, 10$\%~\pi^+\pi^-$. Thus $20\%$ of the Higgs [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}decays are expected to have the required four-muon final state.\
The mean lifetime $\tau$ of the [$\gamma_{d}$]{}(expressed throughout this Letter as $\tau$ times the speed of light $c$) is a free parameter of the model. In the generated samples [c$ \tau$]{}is chosen so that a large fraction of the decays occur inside the sensitive ATLAS detector volume, i.e. up to 7 m in radius and 13 m along the $z$-axis, where the trigger chambers of the middle stations are located. The detection efficiency can then be estimated for a range of [$\gamma_{d}$]{}mean lifetime through re-weighting of the generated samples.\
------------ -------------- -------------- ---------------------- --------------
Higgs mass $m_{f_{d2}}$ $m_{f_{d1}}$ [$\gamma_{d}$]{}mass [c$ \tau$]{}
$[\gev]$ $[\gev]$ $[\gev]$ $[\gev]$ \[mm\]
100 5.0 2.0 0.4 47
140 5.0 2.0 0.4 36
------------ -------------- -------------- ---------------------- --------------
: Parameters used for the Monte Carlo simulation. The last column is the [$\gamma_{d}$]{}mean lifetime $\tau$ multiplied by the speed of light $c$, expressed in mm.[]{data-label="tab:param"}
Potential backgrounds include all the processes which lead to real prompt muons in the final state such as the SM processes [*W*]{}+jets, [*Z*]{}+jets, , [*WW*]{}, [*WZ*]{}, and [*ZZ*]{}. However, the main contribution to the background is expected from processes giving a high production rate of secondary muons which do not point to the primary vertex, such as decays in flight of $K/\pi$ and heavy flavour decays in multi-jet processes, or muons due to cosmic rays. The prompt lepton background samples are generated using [[PYTHIA]{}]{} ([*W*]{}+jets, and [*Z*]{}+jets) and [[MC@NLO]{}]{} [@mcatlno] (, [*WW*]{}, [*WZ*]{}, and [*ZZ*]{}). The generated Monte Carlo events are processed through the full ATLAS simulation chain based on [[GEANT4]{}]{} [@GEANT4; @ATLSIM]. Additional [*pp*]{} interactions in the same and nearby bunch crossings (pile-up) are included in the simulation. All Monte Carlo samples are re-weighted to reproduce the observed distribution of the number of interactions per bunch crossing in the data. For the multi-jet background evaluation a data-driven method is used. The cosmic-ray background is also evaluated from data.
The main kinematic characteristics of the signal sample are:
- [The [$\gamma_{d}$]{}pair are emitted approximately back-to-back in $\phi$, with an angular spread of the distribution due to the emission of the $f_{d1}$. ]{}
- [The average of the $\gamma_{d}$ in the laboratory frame is about 20 for [$m_{H}=~$100 GeV]{}and 30 for [$m_{H}=~$140 GeV]{}; due to the small mass of the [$\gamma_{d}$]{}, large boost factors in the decay length should be expected.]{}
- [ Fig. \[fig:drmu\] shows the distribution of $\Delta R = \sqrt{(\Delta\eta)^2 + (\Delta\phi)^2}$ between the two muons from the [$\gamma_{d}$]{}decay. The $\Delta R$ is computed at the decay vertex of the [$\gamma_{d}$]{}from the vector momenta of the two muons. Due to the small mass of the [$\gamma_{d}$]{}the $\Delta R$ is almost always below 0.1.]{}
Since the two $f_{d1}$ are, like the two [$\gamma_{d}$]{}, emitted back-to-back in $\phi$, the observed missing transverse momentum , computed at the event-generator level, is small and cannot be used as a discriminating variable against the background.
![$\Delta R$ distribution between the two muons from the $\gamma_{d}$ decay for the signal Monte Carlo samples with $m_{H}$ = 100 and $m_{H}$ = 140 .[]{data-label="fig:drmu"}](fig_02a.pdf){width="70mm"}
The dataset used for this analysis was collected at a centre-of-mass energy of 7 during the first part of 2011, where a low level of pile-up events in the same bunch-crossing was present (an average of $\approx 6$ interactions per crossing). Only periods in which all ATLAS subdetectors were operational are used. The total integrated luminosity used is 1.94 $\pm$ $0.07$ fb$^{-1}$ [@LUMI1; @LUMI2]. All events are required to have at least one reconstructed vertex along the beam line with at least three associated tracks, each with $\geq$ 0.4 . The primary interaction vertex is defined to be the vertex whose constituent tracks have the largest $\Sigma$[$p_{\mathrm{T}}^{\mathrm{2}}$]{}. This analysis deals with displaced [$\gamma_{d}$]{}decays with final states containing only muons. Signal events are therefore characterized by a four-muon final state with the four muons coming from two displaced decay vertices. Due to the relatively low of the muons and to the displaced decay vertex, a low- multi-muon trigger with muons reconstructed only in the MS is needed. In order to have an acceptably low trigger rate at a low threshold, a multiplicity of at least three muons is required. Candidate events are collected using an unprescaled HLT trigger with three reconstructed muons of $ \geq $ 6 , seeded by a L1-accept with three different muon ROIs. These muons are reconstructed only in the MS, since muons originating from a neutral particle decaying outside the pixel detector will not have a matching track in the ID tracking system. The trigger efficiency for the Monte Carlo signal samples, defined as the fraction of events passing the trigger requirement with respect to the events satisfying the analysis selection criteria (described in Section 6) is 0.32$\pm 0.01_{\textrm{\small stat}}$ for [$m_{H}=~$100 GeV]{}and 0.31$\pm 0.01_{\textrm{\small stat}}$ for [$m_{H}=~$140 GeV]{}.\
The main reason for the relatively low trigger efficiency is the small opening $\Delta R$ between the two muons of the [$\gamma_{d}$]{}decay ($\Delta R \leq 0.1$) shown in Fig. \[fig:drmu\]. These values of $\Delta R$ are often smaller than the L1 trigger granularity; in this case the L1 produces only one ROI. The trigger only fires if at least one of the [$\gamma_{d}$]{}produces two distinct L1 ROIs. The single [$\gamma_{d}$]{}ROI efficiency, $\varepsilon_{\textrm{\footnotesize 2ROI}}$ ($\varepsilon_{\textrm{\footnotesize 1ROI}}$), defined as the fraction of [$\gamma_{d}$]{}passing the offline selection that give two (one) trigger ROIs is 0.296 $\pm~0.004_{\textrm{\small stat}}$ (0.626 $\pm~0.004_{\textrm{\small stat}}$) for [$m_{H}=~$100 GeV]{}and 0.269 $\pm~0.003_{\textrm{\small stat}}$ (0.653 $\pm~0.003_{\textrm{\small stat}}$) for [$m_{H}=~$140 GeV]{}. Fig. \[fig:treggeffVSetadr2\] shows the $\varepsilon_{\textrm{\footnotesize 2ROI}}$ as a function of the dark photon $\eta$ and of the [$\Delta R$]{}of the two muons from the [$\gamma_{d}$]{}decay. The increased trigger granularity in the endcap and the efficiency decrease at small values of [$\Delta R$]{}are clearly visible.\
The systematic uncertainty on the trigger efficiency is estimated with a sample of from collision data and a corresponding sample of Monte Carlo events, using the tag-and-probe (TP) method. A cut on $\Delta R \leq 0.1$ between the two muons is used to reproduce the small track-to-track spatial separation in the MS of the signal. The tag is a (MS+ID) combined muon, defined as a MS-reconstructed muon that is associated with a trigger object and combined with a matching “good ID track". Good ID tracks must have at least one hit in the pixel detector, at least six hits in the silicon micro-strip detectors and at least six hits in the straw-tube tracker. The probe is a good ID track which, when combined with the tag track, gives an invariant mass inside a 100 window around the mass. A muon ROI that matches the probe in $ \eta$ and $\phi$, and is different from the ROI associated with the tag, is searched for. The number of probes with a matched ROI divided by the number of probes without a matched ROI gives the $\varepsilon$$_{\textrm{\footnotesize 2ROI}}^{\textrm{\footnotesize TP}}$/$\varepsilon$$_{\textrm{\footnotesize 1ROI}}^{\textrm{\footnotesize TP}}$ ratio. Values of $\varepsilon$$_{\textrm{\footnotesize 2ROI}}^{\textrm{\footnotesize TP}}$/$\varepsilon$$_{\textrm{\footnotesize 1ROI}}^{\textrm{\footnotesize TP}}$ = 0.42$\pm 0.05_{\textrm{\small stat}}$ for the data and $\varepsilon$$_{\textrm{\footnotesize 2ROI}}^{\textrm{\footnotesize TP}}$/$\varepsilon$$_{\textrm{\footnotesize 1ROI}}^{\textrm{\footnotesize TP}}$ = 0.39$\pm 0.05_{\textrm{\small stat}}$ for the corresponding Monte Carlo sample are obtained. The relative statistical uncertainty on the difference between these two estimates is 17$\%$ and this is taken conservatively to be the systematic uncertainty on the trigger efficiency.
![$\varepsilon_{\textrm{\tiny 2ROI}}$ as a function (a) of the $\eta$ of the [$\gamma_{d}$]{}and (b) of the [$\Delta R$]{}of the muon pair for the Monte Carlo samples with Higgs boson masses of 100 and 140 . The errors are statistical only.[]{data-label="fig:treggeffVSetadr2"}](fig_03a.pdf "fig:"){width="70mm"} ![$\varepsilon_{\textrm{\tiny 2ROI}}$ as a function (a) of the $\eta$ of the [$\gamma_{d}$]{}and (b) of the [$\Delta R$]{}of the muon pair for the Monte Carlo samples with Higgs boson masses of 100 and 140 . The errors are statistical only.[]{data-label="fig:treggeffVSetadr2"}](fig_03b.pdf "fig:"){width="70mm"}
MJs from displaced [$\gamma_{d}$]{}decays are characterized by a pair of muons in a narrow cone, produced away from the primary vertex of the event. Consequently tracks reconstructed in the MS with a good quality track fit [@reco] are used. MJs are identified using a simple clustering algorithm that associates all the muons in cones of $\Delta R = 0.2$, starting with the muon with highest . The size of the cone takes into account the multiple scattering of the muons in the calorimeters. All the muons found in the cone are associated with a MJ. After this procedure, if any muons are unassociated with a MJ the search is repeated for this remainder, starting again with the highest muon. This continues until all possible MJs are formed. The MJ direction and momentum are obtained from the vector sum over all muons in the MJ. Only MJs with two reconstructed muons are accepted and only events with two MJs are kept for the subsequent analysis.\
The possible contribution to the background of SM processes which lead to real prompt muon pairs in the final state is evaluated using simulated samples. After the trigger and the requirement of having two MJs in the event, their contributions have been found to be negligible. The only significant background sources are expected to be from processes giving a high production rate of secondary muons which do not point to the primary vertex, such as decays in flight of $K/\pi$ and heavy flavour decays in multi-jet production, or cosmic-ray muons not pointing to the primary vertex.\
In order to separate the signal from the background, a number of discriminating variables have been studied. The multi-jet background can be significantly reduced by using calorimeter isolation requirements around the MJ direction. The calorimetric isolation variable [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}is defined as the difference between the transverse calorimetric energy in a cone of $\Delta R = 0.4$ around the highest muon of the MJ and the in a cone of $\Delta R = 0.2$; a cut [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$ \leq$ 5$~ \gev~$ keeps almost all the signal. The isolation modelling is validated for real isolated muons with a sample of muons coming from $Z \rightarrow \mu\mu$ decays. To further improve the signal-to-background ratio, two additional discriminating variables are used: [$\Delta \phi$]{} between the two MJs and [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}for the MJ, defined as the scalar sum of the transverse momentum of the tracks, measured in the ID, inside a cone $\Delta R = 0.4$ around the direction of the MJ. The muon tracks of the MJ in the ID, if any, are not removed from the isolation sum, so that prompt muons, which give a reconstructed track in both the ID and MS, will contribute to the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}. As a consequence a cut on [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}of a few will remove prompt MJs or MJs with very short decay length.
{width="60mm"} {width="60mm"} {width="60mm"}
For the background coming from cosmic-ray muons (mainly pairs of almost parallel cosmic-ray muons crossing the detector) a cut on the impact parameters of the muon tracks with respect to the primary interaction vertex is used.\
The final set of selection criteria used is the following:
- Topology cut: events are required to have exactly two MJs, $N_{\textrm{\footnotesize MJ}}$ = 2.
- MJ isolation: require MJ isolation with [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$ \leq$ 5$~ \gev~$ for both MJs in the event.
- Require $|\Delta \phi|$ $\geq$ 2 between the two MJs.
- Require opposite charges for the two muons in a MJ (Q$_{\textrm{\footnotesize MJ}} = 0$).
- Require a cut on the transverse and longitudinal impact parameters of the muons with respect to the primary vertex: $|d_0| <$ 200 mm and $|z_0|< $ 270 mm.
- Require [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}$< 3 \GeV$ for both MJs.
The distributions of the relevant variables at the different steps of the cut flow are shown in Fig. \[fig:sumpt\_scan1\]. The results are summarized in Table \[tab:cutflowABCD\]. No events survive the selection in the data sample whereas the expected signals from Monte Carlo simulation, assuming 100$\%$ branching ratio for [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}and the parameters given in Table \[tab:param\], are 75 or 48 events for Higgs boson masses of 100 and 140 respectively. The method used to estimate the cosmic-ray and multi-jet background yields, quoted in Table \[tab:cutflowABCD\], is discussed in Section 7.
cut cosmic-rays multi-jet total background [$m_{H}=~$100 GeV]{} [$m_{H}=~$140 GeV]{} data
------------------------------------------------------ ------------------- ---------------------------------- ------------------------------------- ------------------------ ---------------------- ------
$N_{\textrm{\scriptsize MJ}}= 2 $ $3.0\pm 2.1 $ N/A N/A 135$\pm11_{-21}^{+29}$ 90$\pm9_{-13}^{+17}$ 871
[$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$ \leq$ 5$~\gev$ $3.0\pm 2.1 $ N/A N/A 132$\pm11_{-21}^{+28}$ 88$\pm9_{-13}^{+17}$ 219
$|\Delta \phi|\geq$ 2 $1.5\pm 1.5 $ 153 $\pm$ 18 $\pm$ 9 155 $\pm$ 18 $\pm$ 9 123$\pm11_{-19}^{+26}$ 81$\pm9_{-12}^{+15}$ 104
Q$_{\textrm{\scriptsize MJ}}$ = 0 $1.5 \pm 1.5$ 57 $\pm$15$\pm$22 59 $\pm$ 15 $\pm$ 22 121$\pm11_{-19}^{+26}$ 79$\pm8_{-12}^{+15}$ 80
$|d_0|$, $|z_0|$ $0_{-0} ^{+1.64}$ 111$\pm$39$\pm$63 111$\pm$39$\pm$63 105$\pm10_{-16}^{+22}$ 66$\pm8_{-10}^{+12}$ 70
[$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}$< 3~\GeV$ $0_{-0} ^{+1.64}$ 0.06$\pm 0.02 ^{+0.66} _{-0.06}$ $0.06^{+1.64 +0.66} _{-0.02 -0.06}$ 75$\pm9_{-12}^{+16}$ 48$\pm7_{-7}^{+9}$ 0
The resulting single [$\gamma_{d}$]{}reconstruction efficiency for the mean lifetimes given in Table \[tab:param\] is shown in Fig. \[fig:Efficiencies\] as a function of $\eta$, the [$\Delta R$]{}separation of the two muons from the [$\gamma_{d}$]{}decay and the decay length in the transverse plane, , of the [$\gamma_{d}$]{}. The efficiency is defined as the number of [$\gamma_{d}$]{}passing the offline selection divided by the number of [$\gamma_{d}$]{}in the spectrometer acceptance () with both muons having $\geq $ 6 . The low reconstruction efficiency at very short is a consequence of the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut.
![[$\gamma_{d}$]{}reconstruction efficiency $\varepsilon_{\textrm{\footnotesize rec}}$ as a function (a) of $\eta$, (b) of [$\Delta R$]{}and (c) of the transverse decay length of the [$\gamma_{d}$]{}for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}and for the mean lifetimes given in Table \[tab:param\]. The reconstruction efficiency is defined as the number of [$\gamma_{d}$]{}passing the offline selection divided by the number of [$\gamma_{d}$]{}in the spectrometer acceptance () with both muons having $\geq $ 6 . The uncertainties are statistical only.[]{data-label="fig:Efficiencies"}](fig_05a.pdf "fig:"){width="70mm"} ![[$\gamma_{d}$]{}reconstruction efficiency $\varepsilon_{\textrm{\footnotesize rec}}$ as a function (a) of $\eta$, (b) of [$\Delta R$]{}and (c) of the transverse decay length of the [$\gamma_{d}$]{}for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}and for the mean lifetimes given in Table \[tab:param\]. The reconstruction efficiency is defined as the number of [$\gamma_{d}$]{}passing the offline selection divided by the number of [$\gamma_{d}$]{}in the spectrometer acceptance () with both muons having $\geq $ 6 . The uncertainties are statistical only.[]{data-label="fig:Efficiencies"}](fig_05b.pdf "fig:"){width="70mm"} ![[$\gamma_{d}$]{}reconstruction efficiency $\varepsilon_{\textrm{\footnotesize rec}}$ as a function (a) of $\eta$, (b) of [$\Delta R$]{}and (c) of the transverse decay length of the [$\gamma_{d}$]{}for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}and for the mean lifetimes given in Table \[tab:param\]. The reconstruction efficiency is defined as the number of [$\gamma_{d}$]{}passing the offline selection divided by the number of [$\gamma_{d}$]{}in the spectrometer acceptance () with both muons having $\geq $ 6 . The uncertainties are statistical only.[]{data-label="fig:Efficiencies"}](fig_05c.pdf "fig:"){width="70mm"}
The systematic uncertainty on the reconstruction efficiency is evaluated using a tag-and-probe method by comparing the reconstruction efficiency $\varepsilon$$_{\textrm{\footnotesize rec}}^{\textrm{\footnotesize TP}}$ for samples from collision data and Monte Carlo simulation. The tag-and-probe definitions and the cut on $\Delta R \leq 0.1$ between the two muons are the same as in Section 5. To measure the reconstruction efficiency the ID probe track is associated with a MS-only muon track, different from the one associated with the tag. The result is shown in Fig. \[fig:effi-DR-Reco\].\
The relative difference between the result obtained from the data and the Monte Carlo sample in the same range of $\Delta R \leq 0.1$, as for the signal, is taken as the systematic uncertainty on the reconstruction efficiency and amounts to 13$\%$.
![Tag-and-probe reconstruction efficiency $\varepsilon$$_{\textrm{\scriptsize rec}}^{\textrm{\scriptsize TP}}$ as a function of the [$\Delta R$]{}between the two muons, evaluated on a sample of from collision data and a corresponding sample of Monte Carlo events. The $\varepsilon$$_{\textrm{\scriptsize rec}}^{\textrm{\scriptsize TP}}$ for the signal Monte Carlo, evaluated with a similar tag-and-probe method, is also shown. The uncertainties are statistical only.[]{data-label="fig:effi-DR-Reco"}](fig_06a.pdf){width="70mm"}
To estimate the multi-jet background contamination in the signal region we use a data-driven ABCD method slightly modified to cope with the problem of the very low number of events in the control regions. The ABCD method assumes that two variables can be identified, which are relatively uncorrelated, and which can each be used to separate signal and background. It is assumed that the multi-jet background distribution can be factorized in the MJ [$E_{\mathrm{T}}^{\mathrm{isol}}$]{} – $|\Delta \phi|$ plane. The region A is defined by [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$\leq$ 5 and $|\Delta \phi|<$ 2; the region B, defined by [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$\leq$ 5 and $|\Delta \phi|\geq$ 2, is the signal region. The regions C and D are the anti-isolated regions ([$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$>$ 5 ) and they are defined by $|\Delta \phi|<$ 2 and $|\Delta \phi|\geq$ 2, respectively. Neglecting the signal contamination in regions A, C and D ([$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$>$ 5 or $|\Delta \phi|< $ 2) the number of multi-jet background events in the signal region can be evaluated as $N_B = N_D \times N_A/ N_C$. Due to the very low number of events in the control regions the values of $N_A$, $N_C$ and $N_D$ as a function of the cut on the final discriminant variable [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}are extracted by modelling the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}distributions with bifurcated Gaussian templates, with parameters fitted from the data in the corresponding regions, and by integrating the fitted function in the range 0 $<$ [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}$<$ 3 . The low statistics in the four regions at each step of the cut flow give rise to large fluctuations in the multi-jet background estimate. The extracted yields are $N_A = (7.1 \pm 1.5_{\textrm{\small stat}})\cdot 10^{-3}$, $N_C = (1.81 \pm 1.0_{\textrm{\small stat}})\cdot 10^{-2}$ and $N_D = (1.51 \pm 0.07_{\textrm{\small stat}})\cdot 10^{-1}$ and the estimated number of multi-jet background events in the signal region is $N_B = 0.06 \pm 0.02_{\textrm{\small stat}}$. Possible sources of systematic uncertainty related to the background estimation method are also evaluated. The functional form is changed and the procedure to estimate the number of multi-jet background events in the signal region is repeated. The difference in $N_B$ is taken as the systematic uncertainty in the modelling of the multi-jet background shape and it amounts to $^{+0.66} _{-0.06}$. The effect of possible signal leakage in the background regions is also considered and is found to be negligible.\
The background induced by muons from cosmic-ray showers is evaluated using events collected by the trigger active when there are no collisions (empty bunch crossings). The number of triggered events is rescaled by the collision to empty bunch crossing ratio and for the active time (since the trigger in the empty bunch crossing was not active in all the runs). No events survived the requirements on the impact parameters with respect to the primary vertex ($|d_0| <$ 200 mm and $|z_0|<$ 270 mm), resulting in a cosmic-ray contamination estimate of $0^{+1.64} _{-0}$. The final yields for the different background sources are summarized in Table \[tab:cutflowABCD\].
The following effects are considered as possible sources of systematic uncertainty:
- [**Luminosity**]{}\
The overall normalisation uncertainty of the integrated luminosity is $3.7\%$ [@LUMI1; @LUMI2].
- [**Muon momentum resolution**]{}\
The systematic uncertainty on the muon momentum resolution for MS-only muons has been evaluated by smearing and shifting the momenta of the muons by scale factors derived from data-Monte Carlo comparison, and by observing the effect of this shift on the signal efficiency. The overall effect of the muon momentum resolution uncertainty is negligible.
- [**Trigger**]{}\
The systematic uncertainty on the single [$\gamma_{d}$]{}trigger efficiency, evaluated using a tag-and-probe method is $17\%$ (see Section 5).
- [**Reconstruction efficiency**]{}\
The systematic uncertainty on the reconstruction efficiency, evaluated using a tag-and-probe method for the single [$\gamma_{d}$]{}reconstruction efficiency, is $13\%$ (see Section 6).
- [**Effect of pile-up**]{}\
The systematic uncertainty on the signal efficiency related to the effect of pile-up is evaluated by comparing the number of signal events after imposing all the selection criteria on the signal Monte Carlo sample increasing the average number of interactions per crossing from $\approx 6$ to $\approx 16$. This systematic uncertainty is negligible.
- [**Effect of [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut**]{}\
Since the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut could affect the minimum [c$ \tau$]{}value that can be excluded, the effect of this cut on the signal Monte Carlo has been studied. A variation of $10\%$ on the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut results in a relative variation of $<$1$\%$ on the signal, which can therefore be neglected.
- [**Background evaluation**]{}\
The systematic uncertainties that can affect the background estimation are related to the data-driven method used. The functional model used to fit the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}distribution is varied to evaluate the systematic uncertainty in the modelling of its shape, which also includes the effect of the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut on the background estimation. This systematic uncertainty amounts to $^{+0.66} _{-0.06}$ events. The effect of signal leakage is also negligible.
The efficiency of the selection criteria described above is evaluated for the simulated signal samples (see Table \[tab:param\]) as a function of the mean lifetime of the [$\gamma_{d}$]{}. Using pseudo-experiments with [c$ \tau$]{}ranging from 0 to 700 mm the number of [$\gamma_{d}$]{}that decay in each region of the detector is weighted by the corresponding total efficiency for that region. In this way the number of expected signal events is predicted as a function of the [$\gamma_{d}$]{}mean lifetime. These numbers, together with the expected number of background events (multi-jet and cosmic rays) and taking into account the zero data events surviving the selection criteria in [1.9 fb$^{-1}$]{}, are used as input to obtain limits at the 95$\%$ confidence level ([*CL*]{}). The [*CLs*]{} method [@CLspaper] is used to set 95$\%$ [*CL*]{} upper limits on the cross section times branching ratio ([$\sigma\times$BR ]{}) for the process [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}. Here the branching ratio of [[$\gamma_{d}$]{}$\rightarrow \mu~\mu$]{}is set to $45\%$ with the [$\gamma_{d}$]{}mass set to 0.4 , as previously discussed. The [$\sigma\times$BR ]{}is given as a function of the [$\gamma_{d}$]{}mean lifetime, expressed as [c$ \tau$]{}for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}. These limits are shown on Fig. \[fig:CLboth\]. Table \[tab:limits\] shows the ranges in which the $\gamma_d$ [c$ \tau$]{}is excluded at the $95\%$ [*CL*]{} for [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}branching ratios of $100\%$ and $10\%$.
------------------ ----------------------------- -----------------------------
Higgs boson mass excluded c$\tau$ $[mm]$ excluded c$\tau$ $[mm]$
$[\gev]$ BR($100\%$) BR($10\%$)
100 1 $\leq$ c$\tau$ $\leq$ 670 5 $\leq$ c$\tau$ $\leq$ 159
140 1 $\leq$ c$\tau$ $\leq$ 430 7 $\leq$ c$\tau$ $\leq$ 82
------------------ ----------------------------- -----------------------------
: Ranges in which [$\gamma_{d}$]{}[c$ \tau$]{}is excluded at $95\%$ [*CL*]{} for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}, assuming 100$\%$ and 10$\%$ branching ratio of [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}.[]{data-label="tab:limits"}
![ The 95$\%$ upper limits on the [$\sigma\times$BR ]{}for the process [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}as a function of the dark photon [c$ \tau$]{}for the benchmark sample with (a) [$m_{H}=~$100 GeV]{}and with (b) [$m_{H}=~$140 GeV]{}. The expected limit is shown as the dashed curve and the solid curve shows the observed limit. The horizontal lines correspond to the Higgs boson SM cross sections at the two mass values.[]{data-label="fig:CLboth"}](fig_07a.pdf "fig:"){width="70mm"} ![ The 95$\%$ upper limits on the [$\sigma\times$BR ]{}for the process [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}as a function of the dark photon [c$ \tau$]{}for the benchmark sample with (a) [$m_{H}=~$100 GeV]{}and with (b) [$m_{H}=~$140 GeV]{}. The expected limit is shown as the dashed curve and the solid curve shows the observed limit. The horizontal lines correspond to the Higgs boson SM cross sections at the two mass values.[]{data-label="fig:CLboth"}](fig_07b.pdf "fig:"){width="70mm"}
The ATLAS detector at the LHC was used to search for a light Higgs boson decaying into a pair of hidden fermions ($f_{d2}$), each of which decays to a [$\gamma_{d}$]{}and to a stable hidden fermion ($f_{d1}$), resulting in two muon jets from the [$\gamma_{d}$]{}decay in the final state. In a [1.9 fb$^{-1}$]{}sample of $\sqrt{s} =7$ TeV proton-proton collisions no events consistent with this Higgs boson decay mode are observed. The observed data are consistent with the Standard Model background expectations.\
Limits are set on the [$\sigma\times$BR ]{}to [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}as a function of the long-lived particle mean lifetime for $m_{H}=$ 100 and 140 with the chosen [$\gamma_{d}$]{}mass that gives a decay branching ratio of 45% for [[$\gamma_{d}$]{}$\rightarrow \mu~\mu$]{}. Assuming the SM production rate for a 140 Higgs boson, its branching ratio to two hidden-sector photons is found to be below 10%, at $95\%$ [*CL*]{}, for hidden photon [c$ \tau$]{}in the range 7 mm $\leq$ c$\tau$ $\leq$ 82 mm. Bounds on the [$\sigma\times$BR ]{}of a 126 Higgs boson may be conservatively extracted using the corresponding 140 exclusion curve.
We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.
We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWF and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF, DNSRC and Lundbeck Foundation, Denmark; EPLANET and ERC, European Union; IN2P3-CNRS, CEA-DSM/IRFU, France; GNSF, Georgia; BMBF, DFG, HGF, MPG and AvH Foundation, Germany; GSRT, Greece; ISF, MINERVA, GIF, DIP and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; BRF and RCN, Norway; MNiSW, Poland; GRICES and FCT, Portugal; MERYS (MECTS), Romania; MES of Russia and ROSATOM, Russian Federation; JINR; MSTD, Serbia; MSSR, Slovakia; ARRS and MVZT, Slovenia; DST/NRF, South Africa; MICINN, Spain; SRC and Wallenberg Foundation, Sweden; SER, SNSF and Cantons of Bern and Geneva, Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal Society and Leverhulme Trust, United Kingdom; DOE and NSF, United States of America.
The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide.
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[^1]: ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the $z$-axis coinciding with the beam pipe axis. The $x$-axis points from the IP to the centre of the LHC ring, and the $y$-axis points upward. Cylindrical coordinates ($r$,$\phi$) are used in the transverse plane, $\phi$ being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle $\theta$ as .
| ArXiv |
---
abstract: 'A scheme for a spin-polarized current separator is proposed by studying the spin-dependent electron transport of a fork-shaped nanostructure with Rashba spin-orbit coupling (SOC), connected to three leads with the same width. It is found that two spin-polarized currents are of the same magnitude but opposite polarizations can be generated simultaneously in the two output leads when the spin-unpolarized electrons injected from the input lead. The underlying physics is revealed to originate from the different spin-dependent conductance caused by the effects of Rashba SOC and the geometrical structure of the system. Further study shows that the spin-polarized current with strong a robustness against disorder, demonstrates the feasibility of the proposed nanostructure for a real application.'
address: |
$^1$ School of Computer, Jiangxi University of Traditional Chinese Medicine, Nanchang 330004, China.\
$^2$ Institute for Advanced Study, Nanchang University, Nanchang 330031, China.\
$^3$ Department of Physics, Tongji University, Shanghai 200092, China.
author:
- 'Xianbo Xiao$^{1,2}$ and Yuguang Chen$^{3\dag}$'
title: 'Spin-polarized current separator based on a fork-shaped Rashba nanostructure'
---
INTRODUCTION
============
In the past decades, spin-dependent electron transport in semiconductor nanostructures has drawn unprecedented attention because of its potential applications to semiconductor spintronics,$^1$ in which the electron spin rather than its charge is utilized for information processing. One of the primary tasks in the development of semiconductor spintronics is to be capable of generating and manipulating excess spin in semiconductor nanostructures, particularly by all electrical means. The Rashba spin-orbit coupling (SOC),$^2$ existing in asymmetric heterostructures and can be controlled by an external gate voltage,$^{3,4}$ may be an efficient method to satisfy this goal.
Various spin filtering devices have been proposed based on the Rashba semiconductor nanostructures without need for a magnetic element or an external magnetic field, such as T-shape electron waveguide,$^{5-9}$ quantum wires,$^{10-14}$ wire network,$^{15}$ two-dimensional electron gas (2DEG),$^{16}$ and quantum rings.$^{17-18}$ Recently, an interesting Fano-Rashba effect has been found in a straight quantum wire with local Rashba SOC.$^{19}$ This effect is attributed to the interference between the bound states formed by the Rashba SOC and the electrons in the conduction channel, giving rise to pronounced dips in the linear charge conductance. Apart from the SOC-induced bound states, the effects of structure-induced bound states on the electron and spin transport have also been concerned intensively.$^{6,20,21}$ In our recent works, we have investigated the spin-polarized electron transport properties of several typical Rashba quantum wires and found that they are very sensitive to the systems’ longitudinal symmetry. Spin-polarized current can be generated in the longitudinally asymmetry systems when spin-unpolarized injections. Especially, the magnitudes of the spin polarization around the structure-induced Fano resonances are very large.$^{22,23}$ However, in the longitudinally symmetrical system no spin-polarized current can be achieved, despite the existence of the SOC- or/and structure-induced bound states.$^{24}$
Inspired by the three works above, in this paper, we study the spin-dependent electron transport for a fork-shaped Rashba nanostructure with longitudinal-inversion symmetry. It is shown that two spin-polarized currents, with the same magnitude but different polarized directions, can be achieved in the two output leads in spite of spin-unpolarized injections and they still survive even in the presence of strong disorder. Therefore, a spin-polarized current separator device can be devised by using this system. The rest of this paper is organized as follows. In Section II, the theoretical model and analysis are presented. In Section III, the numerical results are illustrated and discussed. A conclusion is given in Section IV.
MODEL AND ANALYSIS
==================
The investigated system in present work is schematically depicted in Fig. 1, where a 2DEG in the $(x,y)$ plane is restricted to a fork-shaped nanostructure by a confining potential $V(x,y)$. The 2DEG is confined in an asymmetric quantum well, where the Rashba SOC is assumed to play a dominantly role. The nanostructure consists of three narrow regions and a wide region. The wide region has a length $L_{2}$ and a uniform width $W_{2}$, while the narrow region has a length $L_{1}$ and a uniform width $W_{1}$, connected to a semi-infinite lead with the same width. The three connecting leads are normal-conductor electrodes without SOC, since we are only interested in spin-unpolarized injection. Such kind of Rashba system can be described by the spin-resolved discrete lattice model. The tight-binding Hamiltonian including the Rashba SOC on a square lattice is given as follow, $$\begin{aligned}
H=H_0+H_{so}+V,\end{aligned}$$ where $$\begin{aligned}
H_0=\sum\limits_{lm\sigma}\varepsilon_{lm\sigma}c_{lm\sigma}^{\dag}c_{lm\sigma}-t\sum\limits_{lm\sigma}\{c_{l+1m\sigma}^{\dag}c_{lm\sigma}\nonumber\\
+c_{lm+1\sigma}^{\dag}c_{lm\sigma}+H.c\},\end{aligned}$$ $$\begin{aligned}
H_{so}=t_{so}\sum\limits_{lm\sigma\sigma'}\{c_{l+1m\sigma'}^{\dag}(i\sigma_{y})_{\sigma\sigma'}c_{lm\sigma}\nonumber\\
-c_{lm+1\sigma'}^{\dag}(i\sigma_{x})_{\sigma\sigma'}c_{lm\sigma}+H.c\},\end{aligned}$$ and $$\begin{aligned}
V=\sum\limits_{lm\sigma}V_{lm}c_{lm\sigma}^{\dag}c_{lm\sigma},\end{aligned}$$ in which $c_{lm\sigma}^{\dag}(c_{lm\sigma})$ is the creation (annihilation) operator of electron at site $(lm)$ with spin $\sigma$, $\sigma_{x(y)}$ is Pauli matrix, and $\varepsilon_{lm\sigma}=4t$ is the on-site energy with the hopping energy $t=\hbar^{2}/2m^{\ast}a^{2}$, here $m^{\ast}$ and $a$ are the effective mass of electron and lattice constant, respectively. $V_{lm}$ is the additional confining potential. The SOC strength is $t_{so}=\alpha/2a$ with the Rashba constant $\alpha$. The Anderson disorder can be introduced by the fluctuation of the on-site energies, which distributes randomly within the range width $w$ \[$\varepsilon_{lm\sigma}= \varepsilon_{lm\sigma}+w_{lm}$ with $-w/2<w_{lm}<w/2$\].
In the ballistic transport, the spin-dependent conductance is obtained from the Landauer-B$\ddot{u}$ttiker formula$^{25}$ with the help of the nonequilibrium Green function formalism.$^{26}$ In order to calculate the Green function of the whole system conveniently, the tight-binding Hamiltonian (1) is divided into two parts in the column cell $$\begin{aligned}
H=\sum\limits_{l\sigma\sigma'}H_{l}^{\sigma\sigma'}+\sum\limits_{l\sigma\sigma'}(H_{l,l+1}^{\sigma\sigma'}+H_{l+1,l}^{\sigma'\sigma}),\end{aligned}$$ where $H_{l}^{\sigma\sigma'}$ is the Hamiltonian of the $l$th isolated column cell, $H_{l,l+1}^{\sigma\sigma'}$ and $H_{l+1,l}^{\sigma'\sigma}$ are intercell Hamiltonian between the $l$th column cell and the $(l+1)$th column cell with $H_{l,l+1}^{\sigma\sigma'}=(H_{l+1,l}^{\sigma'\sigma})^\dag$. The Green function of the whole system can be computed by a set of recursive formulas, $$\begin{aligned}
\langle l+1|G_{l+1}|l+1\rangle^{-1}=E-H_{l+1}-H_{l+1,l}\langle l|G_{l}|l\rangle H_{l,l+1},\nonumber\\
\langle l+1|G_{l+1}|0\rangle=\langle l+1|G_{l+1}|l+1\rangle
H_{l+1,l}\langle l| G_{l}|0\rangle,\end{aligned}$$ where $\langle l+1|G_{l+1}|l+1\rangle$ and $\langle
l+1|G_{l+1}|0\rangle$ are respectively the diagonal and off-diagonal Green function, and $$\begin{aligned}
H_{l+1}=\left(
\begin{array}{cc}
H_{l+1}^{\sigma\sigma} & H_{l+1}^{\sigma\sigma'} \\
H_{l+1}^{\sigma'\sigma} & H_{l+1}^{\sigma'\sigma'} \end{array}
\right),~~ H_{l+1,l}=(H_{l,l+1})^\dag=\left(
\begin{array}{cc}
H_{l+1,l}^{\sigma\sigma} & H_{l+1,l}^{\sigma\sigma'} \\
H_{l+1,l}^{\sigma'\sigma} & H_{l+1,l}^{\sigma'\sigma'} \end{array}
\right).\end{aligned}$$
Utilizing the Green function of the whole system obtained above, the spin-dependent conductance from arbitrary lead $p$ to lead $q$ is given by $$\begin{aligned}
G^{\sigma\sigma'}_{pq}=e^2/hTr[\Gamma_{p}^{\sigma}G^{r}\Gamma_{q}^{\sigma'}G^{a}],\end{aligned}$$ where $\Gamma_{p(q)}=i[\sum_{p(q)}^{r}-\sum_{p(q)}^{a}]$ with the self-energy from the lead $\sum_{p(q)}^{r}=(\sum_{p(q)}^{a})^{\ast}$, the trace is over the spatial and spin degrees of freedom. $G^{r}(G^{a})$ is the retarded (advanced) Green function of the whole system, which can be computed by the spin-resolved recursive Green function method,$^{23}$ and $G^{a}=(G^{r})^\dag$.
In the following calculation, the structural parameters of the system are fixed at $L_1=L_2=10~a$, $W_1=10~a$, and $W_2=40~a$. All the energy is normalized by the hoping energy $t(t=1)$. And the $z$ axis is chosen as the spin-quantized axis so that $|\uparrow>=(1,0)^{T}$ represents the spin-up state and $|\downarrow>=(0,1)^{T}$ denotes the spin-down state, where $T$ means transposition. For simplicity, the hard-wall confining potential approximation is adopted to determine the boundary of the nanostructure since different confining potentials only alter the positions of the subbands and the energy gaps between them. The charge conductance and the spin conductance of $z$-component are defined as $G^e_{pq}=G^{\uparrow\uparrow}_{pq}+G^{\uparrow\downarrow}_{pq}+G^{\downarrow\downarrow}_{pq}+G^{\downarrow\uparrow}_{pq}$ and $G^{Sz}_{pq}=\frac{e}{4\pi}\frac{G^{\uparrow\uparrow}_{pq}+G^{\downarrow\uparrow}_{pq}-G^{\downarrow\downarrow}_{pq}-G^{\uparrow\downarrow}_{pq}}{e^2/h}$, respectively. Here the charge conductance means the transfer probability of electrons, and the spin conductance represents the change in local spin density between the input lead and the output lead caused by the transport of spin-polarized electrons.$^{27}$
RESULTS AND DISCUSSION
======================
In our numerical example, we choose the same material as that in Ref. \[23\], where the requirements of the parameters have been discussed. Figure 2 shows the electron energy ($E$) dependence of the charge and spin conductance when the spin-unpolarized electron injected from lead 1. The Rashba SOC strength $t_{so}=0.19$. The step-like structures, oscillation caused by interference, and SOC-induced Fano resonance dips (see the red circles in Fig. 2(a)) can be found in the charge conductance. In addition, due to the system is longitudinally symmetrical, electrons have the same chance be transmitted to the different output leads. Therefore, as shown in Fig. 2(a) and (b), the charge conductance from lead 1 to 2 is the same as that from lead 1 to 3. However, the corresponding spin conductance from lead 1 to 2 is quite different from that from lead 1 to 3, as depicted in Fig. 2(c), the magnitudes of the spin conductance from the injecting lead to the two outgoing leads are always equal but their signs are contrary. In particular, a very large spin-polarized current can be generated at the structure-induced Fano resonances (such as $E=0.16,~0.44,$ etc.). It has demonstrated in our previous papers$^{22,23}$ that the magnitude of this spin-polarized current can be tuned by both the strength of Rashba SOC and the structural parameters of the system so that we do not presented these results here.
The remarkable difference in the spin conductance between the upper and lower output leads can be utilized to design a spin-polarized current separator, i.e., if a spin-polarized current generated in one output lead, there must be another one in possession of the same magnitude but adverse polarized directions achieved in the other output lead. The physical mechanism of this device owing to the effect of the Rashba SOC and the geometrical structure of the system. The spin-dependent conductance from the input lead 1 to the output leads 2 and 3 as function of the electron energy is illustrated in Fig. 3(a) and (b), respectively. The strength of Rashba SOC is the same as that in Fig. 2. The fork-shaped Rashba nanostructure can be equivalently viewed as two coupled zigzag wire, whose longitudinal and transversal symmetries are broken.$^{12}$ So the relations $G^{\uparrow\uparrow}_{12(3)}=G^{\downarrow\downarrow}_{12(3)}$ and $G^{\uparrow\downarrow}_{12(3)}=G^{\downarrow\uparrow}_{12(3)}$ cannot be guaranteed, as shown in Fig. 3, leading to the nonzero spin conductance (see Fig. 2(c)) in respective lead. Furthermore, because the two output leads 2 and 3 lie symmetrically in the opposite direction with respect to the input lead 1, the total current must still be spin-unpolarized.$^{12,24}$ Therefore, the transmission probability of the spin-up (-down) electron from lead 1 to 2 always equals that of the spin-down (-up) electron from lead 1 to 3, that is, $G^{\sigma\sigma}_{12}=G^{\sigma'\sigma'}_{13}$ and $G^{\sigma\sigma'}_{12}=G^{\sigma'\sigma}_{13}$. As a consequence, the signs of the spin conductance from the lead 1 to leads 2 and 3 are contrary all along.
The above proposed spin-polarized current separator is based upon a perfectly clean system, where no elastic or inelastic scattering happens. However, in a realistic system, there are many impurities in the sample. The impurities in any semiconductor heterostructure may induce a random Rashba field, which gives rise to many new effects such as the realization of the minimal possible strength of SOC$^{28}$ and the localization of the edge electrons for sufficiently strong electron-electron interactions.$^{29}$ Thus the effect of disorder should be considered in practical application. The spin conductance from the input lead to the different output leads as function of the electron energy for (weak and strong) different disorders $w$ are plotted in Fig. 4. The SOC strength is also set as $t_{so}=0.19$. The spin conductance is destroyed when the impurities exist in the system and its magnitude become smaller with the increase of disorder. However, as shown in Fig. 4(c), the magnitude of the spin conductance around the structure-induced Fano resonances is still large when the disorder strength $w=0.6$, which means that a comparatively large spin-polarized current can be obtained in the output leads even in the presence of strong disorder.
CONCLUSION
==========
In conclusion, a scheme of a spin-polarized current separator is proposed by investigating the spin-dependent electron transport of a fork-shaped nanostructure under the modulation of the Rashba SOC. Two spin-polarized currents with the same magnitude but different polarizations can be generated synchronously in the two output leads due to the distinct spin-dependent conductance results from the effects of SOC and the geometrical structure. The opposite spin-polarized currents can be generated and controlled by electrical means and they are robust against disorder. Thus the proposed nanostructure does not require the application of magnetic fields, external radiation or ferromagnetic leads, and has great potential for real applications.
ACKNOWLEDGMENT {#acknowledgment .unnumbered}
==============
This work was supported by the National Natural Science Foundation of China under Grant No. 10774112.
References {#references .unnumbered}
==========
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$^{18}$P. F$\ddot{o}$ldi, O. K$\acute{a}$lm$\acute{a}$n, M. G. Benedict, and F. M. Peeters, Nano Lett. **8** 2556 (2008).\
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$^{20}$Y. P. Chen, X. H. Yan, and Y. E Xie, Phys. Rev. B **71** 245335 (2005).\
$^{21}$Y. P. Chen, Y. E Xie, and X. H. Yan, Phys. Rev. B **74** 035310 (2006).\
$^{22}$X. B. Xiao, X. M. Li, and Y. G. Chen, Phys. Lett. A **373** 4489 (2009).\
$^{23}$X. B. Xiao and Y. G. Chen, Europhys. Lett. **90** 47004 (2010).\
$^{24}$X. B. Xiao, X. M. Li, and Y. G. Chen, ACTA PHYSICA SINICA **58** 7909 (2009).\
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$^{26}$L. W. Molenkamp, G. Schmidt, and G. E.W. Bauer, Phys. Rev. B **62** 4790 (2000); T. P. Pareek and P. Bruno, Phys. Rev. B **63** 165424-1 (2001).\
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| ArXiv |
---
abstract: 'These notes discuss, in a style intended for physicists, how to average data and fit it to some functional form. I try to make clear what is being calculated, what assumptions are being made, and to give a derivation of results rather than just quote them. The aim is put a lot useful pedagogical material together in a convenient place. This manuscript is a substantial enlargement of lecture notes I prepared for the Bad Honnef School on “Efficient Algorithms in Computational Physics”, September 10–14, 2012.'
author:
- Peter Young
bibliography:
- 'refs.bib'
title: '[Everything you wanted to know about Data Analysis and Fitting but were afraid to ask]{}\'
---
Introduction
============
These notes describe how to average and fit numerical data that you have obtained, presumably by some simulation.
Typically you will generate a set of values $x_i,\, y_i, \cdots,\, i = 1,
\cdots N$, where $N$ is the number of measurements. The first thing you will want to do is to estimate various average values, and determine *error bars* on those estimates. As we shall see, this is straightforward if one wants to compute a single average, e.g. $\langle x \rangle$, but not quite so easy for more complicated averages such as fluctuations in a quantity, $\langle x^2 \rangle
- \langle x \rangle^2$, or combinations of measured values such as $\langle y
\rangle / \langle x \rangle^2$. Averaging of data will be discussed in Sec. \[sec:averages\].
Having obtained several good data points with error bars, you might want to fit this data to some model. Techniques for fitting data will be described in the second part of these notes in Sec. \[sec:fit\]
I find that the books on these topics usually fall into one of two camps. At one extreme, the books for physicists don’t discuss all that is needed and rarely *prove* the results that they quote. At the other extreme, the books for mathematicians presumably prove everything but are written in a style of lemmas, proofs, $\epsilon$’s and $\delta$’s, and unfamiliar notation, which is intimidating to physicists. One exception, which finds a good middle ground, is Numerical Recipes [@press:92] and the discussion of fitting given here is certainly influenced by Chap. 15 of that book. In these notes I aim to be fairly complete and also to derive the results I use, while the style is that of a physicist writing for physicists. I also include scripts in python, perl, and gnuplot to perform certain tasks in data analysis and fitting. For these reasons, these notes are perhaps rather lengthy. Nonetheless, I hope, that they will provide a useful reference.
Averages and error bars {#sec:averages}
=======================
Basic Analysis {#sec:basic}
--------------
Suppose we have a set of data from a simulation, $x_i, \, (i = 1, \cdots, N)$, which we shall refer to as a *sample* of data. This data will have some random noise so the $x_i$ are not all equal. Rather they are governed by a distribution $P(x)$, *which we don’t know*.
The distribution is normalized, $$\int_{-\infty}^\infty P(x) \, d x = 1,$$ and is usefully characterized by its moments, where the $n$-th moment is defined by $$\langle x^n \rangle = \int_{-\infty}^\infty x^n\, P(x) \, d x\, .$$ We will denote the average *over the exact distribution* by angular brackets. Of particular interest are the first and second moments from which one forms the mean $ \mu$ and variance $\sigma^2$, by
$$\begin{aligned}
\mu &\equiv \langle x \rangle \label{xavexact} \\
\sigma^2 &\equiv \langle \, \left(x - \langle x\rangle\right)^2 \,
\rangle = \langle x^2 \rangle - \langle x \rangle^2
\, .
\label{sigma}\end{aligned}$$
The term “standard deviation” is used for $\sigma$, the square root of the variance.
In this section we will estimate the mean $\langle x \rangle$, and the uncertainty in our estimate, from the $N$ data points $x_i$. The determination of more complicated averages and resulting error bars will be discussed in Sec. \[sec:advanced\]
In order to obtain error bars we need to assume that the data are uncorrelated with each other. This is a crucial assumption, without which it is very difficult to proceed. However, it is not always clear if the data points are truly independent of each other; some correlations may be present but not immediately obvious. Here, we take the usual approach of assuming that even if there are some correlations, they are sufficiently weak so as not to significantly perturb the results of the analysis. In Monte Carlo simulations, measurements which differ by a sufficiently large number of Monte Carlo sweeps will be uncorrelated. More precisely the difference in sweep numbers should be greater than a “relaxation time”. This is exploited in the “binning” method in which the data used in the analysis is not the individual measurements, but rather an average over measurements during a range of Monte Carlo sweeps, called a “bin”. If the bin size is greater than the relaxation time, results from adjacent bins will be (almost) uncorrelated. A pedagogical treatment of binning has been given by Ambegaokar and Troyer [@ambegaokar:09]. Alternatively, one can do independent Monte Carlo runs, requilibrating each time, and use, as individual data in the analysis, the average from each run.
The information *from the data* is usefully encoded in two parameters, the sample mean $\overline{x}$ and the sample standard deviation $s$ which are defined by[^1]
$$\begin{aligned}
\overline{x} & = {1 \over N} \sum_{i=1}^N x_i \, ,
\label{meanfromdata}
\\
s^2 & = {1 \over N - 1} \sum_{i=1}^N \left( x_i -
\overline{x}\right)^2 \, .
\label{sigmafromdata}
$$
In statistics, notation is often confusing but crucial to understand. Here, an average indicated by an over-bar, $\overline{\cdots}$, is an average over the *sample of $N$ data points*. This is to be distinguished from an exact average over the distribution $\langle \cdots \rangle$, as in Eqs. (\[xavexact\]) and (\[sigma\]). The latter is, however, just a theoretical construct since we *don’t know* the distribution $P(x)$, only the set of $N$ data points $x_i$ which have been sampled from it. Next we derive two simple results which will be useful later:
1. The mean of the sum of $N$ independent variables *with the same distribution* is $N$ times the mean of a single variable, and
2. The variance of the sum of $N$ independent variables *with the same distribution* is $N$ times the variance of a single variable.
The result for the mean is obvious since, defining $X = \sum_{i=1}^N x_i$, $$\langle X \rangle = \sum_{i=1}^N \langle x_i \rangle = N \langle x_i \rangle
\ \boxed{ = N \mu\, .}
\label{X}$$ The result for the standard deviation needs a little more work:
$$\begin{aligned}
\sigma_X^2 & \equiv \langle X^2 \rangle - \langle X \rangle^2 \\
&= \sum_{i,j=1}^N \left( \langle x_i x_j\rangle - \langle
x_i \rangle \langle x_j \rangle \right) \label{1} \\
& = \sum_{i=1}^N \left( \langle x_i^2 \rangle - \langle x_i
\rangle^2 \right) \label{2} \\
& = N \left(\langle x^2 \rangle - \langle x \rangle^2 \right)
\\
& \boxed{ = N \sigma^2 \, .}
\label{dXsq}\end{aligned}$$
To get from Eq. (\[1\]) to Eq. (\[2\]) we note that, for $i \ne j$, $\langle x_i x_j\rangle = \langle x_i \rangle \langle
x_j\rangle$ since $x_i$ and $x_j$ are assumed to be statistically independent. (This is where the statistical independence of the data is needed.)
If the means and standard deviations are not all the same, then the above results generalize to
$$\begin{aligned}
\langle X \rangle &= \sum_{i=1}^N \mu_i \, , \\
\langle \sigma_X^2 \rangle &= \sum_{i=1}^N \sigma_i^2 \, .\end{aligned}$$
Now we describe an important thought experiment. Let’s *suppose* that we could repeat the set of $N$ measurements *very many* many times, each time obtaining a value of the sample average $\overline{x}$. From these results we could construct a distribution, $\widetilde{P}(\overline{x})$, for the sample average as shown in Fig. \[Fig:distofmean\].
If we do enough repetitions we are effectively averaging over the exact distribution. Hence the average of the sample mean, $\overline{x}$, over very many repetitions of the data, is given by $$\langle \overline{x} \rangle = {1 \over N} \sum_{i=1}^N \langle x_i \rangle =
\langle x \rangle \equiv \mu \, ,
\label{xav}$$ i.e. it is the exact average over the distribution of $x$, as one would intuitively expect, see Fig. \[Fig:distofmean\]. Eq. also follows from Eq. by noting that $\overline{x} =
X/N$.
![ The distribution of results for the sample mean $\overline{x}$ obtained by repeating the measurements of the $N$ data points $x_i$ many times. The average of this distribution is $\mu$, the exact average value of $x$. The mean, $\overline{x}$, obtained from one sample of data typically differs from $\mu$ by an amount of order $\sigma_{\overline{x}}$, the standard deviation of the distribution $\widetilde{P}(\overline{x})$. []{data-label="Fig:distofmean"}](distofmean.eps){width="9.5cm"}
In fact, though, we have only the *one* set of data, so we can not determine $\mu $ exactly. However, Eq. (\[xav\]) shows that $$\boxed{ \mbox{the best estimate of\ } \mu \mbox{ is }
\overline{x},}
\label{xbarest}$$ i.e. the sample mean, since averaging the sample mean over many repetitions of the $N$ data points gives the true mean of the distribution, $\mu$. An estimate like this, which gives the exact result if averaged over many repetitions of the experiment, is said to be
We would also like an estimate of the uncertainty, or “error bar”, in our estimate of $\overline{x}$ for the exact average $\mu$. We take $ \sigma_{\overline{x}}$, the standard deviation in $\overline{x}$ (obtained if one did many repetitions of the $N$ measurements), to be the uncertainty, or error bar, in $\overline{x}$. The reason is that $ \sigma_{\overline{x}}$ is the width of the distribution $\widetilde{P}(\overline{x})$, shown in Fig. \[Fig:distofmean\], so a *single* estimate $\overline{x}$ typically differs from the exact result $\mu$ by an amount of this order. The variance $\sigma_{\overline{x}}^2$ is given by $$\sigma_{\overline{x}}^2 \equiv \langle \overline{x}^2 \rangle
- \langle \overline{x} \rangle^2 = {\sigma^2 \over N}\, ,
\label{dxsq}$$ which follows from Eq. since $\overline{x} =X / N$.
The problem with Eq. (\[dxsq\]) is that **we don’t know $\sigma^2$** since it is a function of the exact distribution $P(x)$. We do, however, know the *sample* variance $s^2$, see Eq. (\[sigmafromdata\]), and the average of this over many repetitions of the $N$ data points, is equal to $\sigma^2$ since
$$\begin{aligned}
\langle s^2 \rangle & =
{1 \over N-1} \sum_{i=1}^N \langle x_i^2 \rangle -
{1 \over N(N-1)} \sum_{i=1}^N \sum_{j=1}^N \langle x_i x_j \rangle
\label{3} \\
& ={N \over N-1}\langle x^2 \rangle - {1 \over N(N-1)} \left[ N(N-1) \langle x \rangle^2 +
N \langle x^2\rangle \right] \label{4}\\
& = \left[\langle x^2 \rangle - \langle x \rangle^2 \right] \\
& = \sigma^2 \, .
\label{5}\end{aligned}$$
To get from Eq. (\[3\]) to Eq. (\[4\]), we have separated the terms with $i=j$ in the last term of Eq. (\[3\]) from those with $i \ne j$, and used the fact that each of the $x_i$ is chosen from the same distribution and is statistically independent of the others. It follows from Eq. (\[5\]) that $$\boxed{
\mbox{the best estimate of\ } \sigma^2 \mbox{ is }
s^2 \, ,}
\label{sigmasamp}$$ since averaging $s^2$ over many repetitions of $N$ data points gives $\sigma^2$. The estimate for $\sigma^2$ in Eq. (\[sigmasamp\]) is therefore unbiased.
Combining Eqs. (\[dxsq\]) and (\[sigmasamp\]) gives $$\boxed{
\mbox{the best estimate of\ } \sigma_{\overline{x}}^2 \mbox{ is }
{s^2 \over N}\; \, ,}
\label{errorbar}$$ We have now obtained, using only information from the data, that the mean is given by $$\boxed{
\mu = \overline{x}\; \pm \; \sigma_{\overline{x}} \, ,}$$ where $$\boxed{
\sigma_{\overline{x}} = {s \over \sqrt{N}}, , }
\label{finalans}$$ which we can write explicitly in terms of the data points as $$\boxed{
\sigma_{\overline{x}} = \left[
{1 \over N(N-1)} \, \sum_{i=1}^N (x_i - \overline{x})^2 \right]^{1/2} \, .}
\label{finalans2}$$ Remember that $\overline{x}$ and $s$ are the mean and standard deviation of the (one set) of data that is available to us, see Eqs. (\[meanfromdata\]) and (\[sigmafromdata\]).
As an example, suppose $N=5$ and the data points are $$x_i = 10, 11, 12, 13, 14,$$ (not very random looking data it must be admitted!). Then, from Eq. (\[meanfromdata\]) we have $\overline{x} = 12$, and from Eq. (\[sigmafromdata\]) $$s^2 = {1 \over 4} \, \left[(-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2\right] =
{5 \over 2} .$$ Hence, from Eq. (\[finalans\]), $$\sigma_{\overline{x}} = {1 \over \sqrt{5}}\, \sqrt{5 \over 2} = {1\over
\sqrt{2}}.$$ so $$\mu = \overline{x} \pm \sigma_{\overline{x}} = 12 \pm {1\over \sqrt{2}}.$$
How does the error bar decrease with the number of statistically independent data points $N$? Equation (\[5\]) states that the expectation value of $s^2$ is equal to $\sigma^2$ and hence, from Eq. (\[finalans\]), we see that
Hence, to reduce the error bar by a factor of 10 one needs 100 times as much data. This is discouraging, but is a fact of life when dealing with random noise.
For Eq. (\[finalans\]) to be really useful we need to know the probability that the true answer $\mu$ lies more than $\sigma_{\overline{x}}$ away from our estimate $\overline{x}$. Fortunately, for large $N$, the central limit theorem, derived in Appendix \[sec:clt\], tells us (for distributions where the first two moments are finite) that the distribution of $\overline{x}$ is a Gaussian. For this distribution we know that the probability of finding a result more than one standard deviation away from the mean is 32%, more than two standard deviations is $4.5\%$ and more than three standard deviations is $0.3\%$. Hence we expect that most of the time $\overline{x}$ will be within $\sigma_{\overline{x}}$ of the correct result $\mu$, and only occasionally will be more than two times $\sigma_{\overline{x}}$ from it. Even if $N$ is not very large, so there are some deviations from the Gaussian form, the above numbers are often a reasonable guide.
However, as emphasized in appendix \[sec:clt\], distributions which occur in nature typically have much more weight in the tails than a Gaussian. As a result, the weight in the tails of the distribution *of the sum* can also be much larger than for a Gaussian even for quite large values of $N$, see Fig. \[Fig:converge\_to\_clt\]. It follows that the probability of an “outlier” can be much higher than that predicted for a Gaussian distribution, as anyone who has invested in the stock market knows well!
Advanced Analysis {#sec:advanced}
-----------------
In Sec. \[sec:basic\] we learned how to estimate a simple average, such as $\mu_x \equiv \langle x \rangle$, plus the error bar in that quantity, from a set of data $x_i$. Trivially this method also applies to a *linear* combination of different averages, $\mu_x, \mu_y, \cdots$ etc. However, we often need more complicated, *non-linear* functions of averages. One example is the fluctuations in a quantity, i.e. $\langle x^2 \rangle -
\langle x \rangle^2$. Another example is a dimensionless combination of moments, which gives information about the *shape* of a distribution independent of its overall scale. Such quantities are very popular in finite-size scaling (FSS) analyses since the FSS form is simpler than for quantities with dimension. An popular example, first proposed by Binder, is $\langle x^4 \rangle / \langle x^2 \rangle^2$, which is known as the “kurtosis” (frequently a factor of 3 is subtracted to make it zero for a Gaussian).
Hence, in this section we consider how to determine *non-linear functions* of averages of one or more variables, $f(\mu_y, \mu_z, \cdots)$, where $$\mu_y \equiv \langle y \rangle \, ,$$ etc. For example, the two quantities mentioned in the previous paragraph correspond to $$f(\mu_y, \mu_z) = \mu_y - \mu_z^2 \, ,$$ with $y=x^2$ and $z = x$ and $$f(\mu_y, \mu_z) = {\mu_y \over \mu_z^2} \, ,$$ with $y = x^4$ and $z = x^2$.
The natural estimate of $f(\mu_y, \mu_z)$ from the sample data is clearly $f(\overline{y}, \overline{z} )$. However, it will take some more thought to estimate the error bar in this quantity. The traditional way of doing this is called “error propagation”, described in Sec. \[sec:traditional\] below. However, it is now more common to use either “jackknife” or “bootstrap” procedures, described in Secs. \[sec:jack\] and \[sec:boot\]. At the price of some additional computation, which is no difficulty when done on a modern computer (though it would have been tedious in the old days when statistics calculations were done by hand), these methods automate the calculation of the error bar.
Furthermore, the estimate of $f(\mu_y, \mu_z)$ turns out to have some *bias* if $f$ is a non-linear function. Usually this is small effect because it is order $1/N$, see for example Eq. below, whereas the statistical error is of order $1/\sqrt{N}$. Since $N$ is usually large, the bias is generally much less than the statistical error and so can generally be neglected. In any case, the jackknife and bootstrap methods also enable one to eliminate the leading ($\sim 1/N$) contribution to the bias in a automatic fashion.
### Traditional method {#sec:traditional}
First we will discuss the traditional method, known as error propagation, to compute the error bar and bias. We expand $f(\overline{y}, \overline{z})$ about $f(\mu_y, \mu_z)$ up to second order in the deviations: $$f(\overline{y}, \overline{z}) = f(\mu_y, \mu_z)
+ (\partial_{\mu_y}f)\, \delta_{\overline{y}}
+ (\partial_{\mu_z}f)\, \delta_{\overline{z}}
+ {1\over 2}\, (\partial^2_{\mu_y\mu_y}f)\, \delta_{\overline{y}}^2
+ (\partial^2_{\mu_y\mu_z}f)\, \delta_{\overline{y}} \delta_{\overline{z}}
+ {1\over 2}\, (\partial^2_{\mu_z\mu_z}f)\, \delta_{\overline{z}}^2
+ \cdots \, ,
\label{expand}$$ where $$\delta_{\overline{y}} = \overline{y} - \mu_y ,$$ etc.
The terms of first order in the $\delta's$ in Eq. give the leading contribution to the error, but would average to zero if the procedure were to be repeated many times. However, the terms of second order do not average to zero and so give the leading contribution to the bias. We now estimate that bias.
Averaging Eq. over many repetitions, and noting that $$\langle \delta_{\overline{y}}^2 \rangle = \langle \overline{y}^2 \rangle - \langle
\overline{y} \rangle^2 \equiv \sigma_{\overline{y}}^2 , \quad
\langle \delta_{\overline{z}}^2 \rangle = \langle \overline{z}^2 \rangle - \langle
\overline{z} \rangle^2 \equiv \sigma_{\overline{z}}^2 , \quad
\langle \delta_{\overline{y}} \delta_{\overline{z}} \rangle =
\langle \overline{y}\, \overline{z} \rangle - \langle \overline{y} \rangle
\langle \overline{z} \rangle \equiv \sigma_{\overline{y}\,\overline{z}}^2 ,$$ we get $$\langle f(\overline{y}, \overline{z})\rangle - f(\mu_y, \mu_z) =
{1\over 2}\, (\partial^2_{\mu_y\mu_y}f)\, \sigma_{\overline{y}}^2
+ (\partial^2_{\mu_y\mu_z}f)\, \sigma_{\overline{y}\,\overline{z}}^2
+ {1\over 2}\, (\partial^2_{\mu_z\mu_z}f)\, \sigma_{\overline{z}}^2 \, .
\label{df}$$ As shown in Eq. our estimate of $\sigma_{\overline{y}}^2$ is $N^{-1}$ times the sample variance (which we now call $s_{yy}^2$), and similarly for $\sigma_{\overline{z}}^2$. In the same way, our estimate of $\sigma_{\overline{y}\,\overline{z}}^2$ is $N^{-1}$ times the sample *covariance* of $y$ and $z$, defined by $$s_{y z}^2 = {1 \over N-1}\, \sum_{i=1}^N \left(y_i -
\overline{y}\right)\, \left(z_i - \overline{z}\right) \, .$$
Hence, from Eq. , we have $$f(\mu_y, \mu_z)
=
\langle f(\overline{y}, \overline{z})\rangle
- {1\over N}\,
\left[{1\over 2}\, (\partial^2_{\mu_y\mu_y}f)\, s_{y y}^2
+ (\partial^2_{\mu_y\mu_z}f)\, s_{y z}^2
+ {1\over 2}\, (\partial^2_{\mu_z\mu_z}f)\, s_{z z}^2 \right]\, ,
\label{bias2}$$ where the leading contribution to the bias is given by the $1/N$ term. Note that the bias term is “self-averaging”, i.e. the fluctuations in it are small relative to the average (by a factor of $1/\sqrt{N}$) when averaging over many repetitions of the data. It follows from Eq. that if one wants to eliminate the leading contribution to the bias one should $$\boxed{
\mbox{estimate } f(\mu_y,\mu_z)\ \mbox{ from }
f(\overline{y}, \overline{z}) - {1\over N}\,
\left[{1\over 2}\, (\partial^2_{\mu_y\mu_y}f)\, s_{y y}^2
+ (\partial^2_{\mu_y\mu_z}f)\, s_{y z}^2
+ {1\over 2}\, (\partial^2_{\mu_z\mu_z}f)\, s_{z z}^2 \right].}
\label{bias}$$ As claimed earlier, the bias correction is of order $1/N$. Note that it vanishes if $f$ is a linear function, as shown in Sec. \[sec:basic\]. The generalization to functions of more than two averages, $f(\mu_y, \mu_z, \mu_w, \cdots)$, is obvious.
Next we discuss the leading *error* in using $f(\overline{y}, \overline{z})$ as an estimate for $f(\mu_y, \mu_z)$. This comes from the terms linear in the $\delta$’s in Eq. . Just including these terms we have
$$\begin{aligned}
\langle f(\overline{y}, \overline{z}) \rangle &=
f(\mu_y, \mu_z) \, , \\
\langle\, f^2(\overline{y}, \overline{z})\, \rangle &=
f^2(\mu_y, \mu_z) +
(\partial_{\mu_y}f)^2 \, \langle \delta_{\overline{y}}^2 \rangle
+ 2(\partial_{\mu_y}f)\, (\partial_{\mu_z}f) \,
\langle \delta_{\overline{y}} \delta_{\overline{z}} \rangle
+ (\partial_{\mu_z}f)^2 \, \langle \delta_{\overline{z}}^2 \rangle \, .\end{aligned}$$
Hence $$\begin{aligned}
\sigma_f^2
&\equiv
\langle\, f^2(\overline{y}, \overline{z})\, \rangle -
\langle f(\overline{y}, \overline{z}) \rangle^2
\nonumber \\
&=
(\partial_{\mu_y}f)^2 \, \langle \delta_{\overline{y}}^2 \rangle
+ 2(\partial_{\mu_y}f)\, (\partial_{\mu_z}f) \,
\langle \delta_{\overline{y}} \delta_{\overline{z}} \rangle
+ (\partial_{\mu_z}f)^2 \, \langle \delta_{\overline{z}}^2 \rangle \, .\end{aligned}$$ As above, we use $s_{y y}^2 / N$ as an estimate of $\langle
\delta_{\overline{y}}^2 \rangle$ and similarly for the other terms. Hence $$\boxed{
\mbox{the best estimate of }
\sigma_f^2
\mbox{ is }
{1 \over N}\,
(\partial_{\mu_y}f)^2 \, s_{y y}^2
+ 2(\partial_{\mu_y}f)\, (\partial_{\mu_z}f) \,
s_{y z}^2
+ (\partial_{\mu_z}f)^2 \, s_{z z}^2 \, .}
\label{sigma_f}$$ This estimate is unbiased to leading order in $N$. Note that we need to keep track not only of fluctuations in $y$ and $z$, characterized by their variances $s_{y y}^2$ and $s_{z z}^2$, but also cross correlations between $y$ and $z$, characterized by their covariance $s_{y z}^2$.
Hence, still to leading order in $N$, we get $$\boxed{f(\mu_y, \mu_z) = f(\overline{y}, \overline{z}) \pm \sigma_f\, ,}$$ where we estimate the error bar $\sigma_f$ from Eq. which shows that it is of order $1/\sqrt{N}$. Again, the generalization to functions of more than two averages is obvious.
Note that in the simple case studied in Sec. \[sec:basic\] where there is only one set of variables $x_i$ and $f =\mu_x$, Eq. tells us that there is no bias, which is correct, and Eq. gives an expression for the error bar which agrees with Eq. .
In Eqs. and we need to keep track how errors in the individual quantities like $\overline{y}$ propagate to the estimate of the function $f$. This requires inputting by hand the various partial derivatives into the analysis program, and keeping track of all the variances and covariances. In the next two sections we see how *resampling* the data automatically takes account of error propagation without needing to input the partial derivatives and keep track of variances and covariances. These approaches, known as jackknife and bootstrap, provide a *fully automatic* method of determining error bars and bias.
### Jackknife {#sec:jack}
We define the $i$-th jackknife estimate, $y^J_i\, (i = 1,2, \cdots, N)$ to be the average over all data in the sample *except the point* $i$, i.e. $$y^J_i \equiv {1 \over N-1}\, \sum_{j \ne i} y_j \, .$$ We also define corresponding jackknife estimates of the function $f$ (again for concreteness we will assume that $f$ is a function of just 2 averages but the generalization will be obvious): $$f^J_i \equiv f(y^J_i, z^J_i) \, .
\label{fJi}$$ In other words, we use the jackknife values, $y^J_i, z^J_i$, rather than the sample means, $\overline{y}, \overline{z}$, as the arguments of $f$. For example a jackknife estimate of the Binder ratio $\langle x^4 \rangle / \langle x^2 \rangle^2$ is $$f^J_i = {(N-1)^{-1} \sum_{j, (j \ne i)} x_j^4 \over
\left[(N-1)^{-1} \sum_{j \ne i}
x_j^2\right]^2 }$$ The overall jackknife estimate of $f(\mu_ y, \mu_z)$ is then the average over the $N$ jackknife estimates $f_i^J$: $$\boxed{
\overline{f^J} \equiv {1 \over N} \sum_{i=1}^N f_i^J \, .}
\label{fJ}$$ It is straightforward to show that if $f$ is a linear function of $\mu_y$ and $\mu_z$ then $\overline{f^J} = f(\overline{y},\overline{z})$, i.e. the jackknife and standard averages are identical. However, when $f$ is not a linear function, so there is bias, there *is* a difference, and we will now show the resampling carried out in the jackknife method can be used to determine bias and error bars in an automated way.
We proceed as for the derivation of Eq. , which we now write as $$f(\mu_y, \mu_z) = \langle f(\overline{y},\overline{z}) \rangle
- {A \over N} - {B\over N^2} + \cdots,$$ where $A$ is the term in rectangular brackets in Eq. , and we have added the next order correction. The jackknife data sets have $N-1$ points with the same distribution as the $N$ points in the actual distribution, and so the bias in the jackknife average will be of the same form, with the same values of $A$ and $B$, but with $N$ replaced by $N-1$, i.e. $$f(\mu_y, \mu_z) = \langle \overline{f^J} \rangle -
{A \over N-1} - {B \over (N-1)^2} \cdots \, .$$ We can therefore eliminate the leading contribution to the bias by forming an appropriate linear combination of $f(\overline{y},\overline{z})$ and $\overline{f^J}$, namely $$f(\mu_y, \mu_z) = N \langle f(\overline{y},\overline{z}) \rangle
- (N-1) \langle \overline{f^J} \rangle + O\left({1\over N^2}\right) \, .
$$ It follows that, to eliminate the leading bias without computing partial derivatives, one should $$\boxed{
\mbox{estimate }
f(\mu_y, \mu_z) \mbox{ from }
N f(\overline{y},\overline{z})
- (N-1) \overline{f^J} \, .
}
\label{bias_elim}$$ The bias is then of order $1/N^2$. However, as mentioned earlier, bias is usually not a big problem because, even without eliminating the leading contribution, the bias is of order $1/N$ whereas the statistical error is of order $1/\sqrt{N}$ which is much bigger if $N$ is large. In most cases, therefore, $N$ is sufficiently large that one can use *either* the usual average $f(\overline{y}, \overline{z})$, or the jackknife average $\overline{f^J}$ in Eq. , to estimate $f(\mu_y, \mu_z)$, since the difference between them will be much smaller than the statistical error. In other words, elimination of the leading bias using Eq. is usually not necessary.
Next we show that the jackknife method gives error bars, which agree with Eq. but without the need to explicitly keep track of the partial derivatives and the variances and covariances.
We define the variance of the jackknife averages by $$\sigma^2_{f^J} \equiv \overline{\left(f^J\right)^2} - \left(
\overline{f^J} \right)^2 \, ,
\label{sigmafJ}$$ where $$\overline{\left(f^J\right)^2} = {1 \over N} \sum_{i=1}^N \left(f_i^J\right)^2
\, .$$ Using Eqs. and , we expand $\overline{f^J}$ away from the exact result $f(\mu_y, \mu_z)$. Just including the leading contribution gives $$\begin{aligned}
\overline{f^J} - f(\mu_y, \mu_z) &= {1 \over N} \sum_{i=1}^N \left[
(\partial_{\mu_y} f)\, (y_i^J - \mu_y) +
(\partial_{\mu_z} f)\, (z_i^J - \mu_z) \right] \nonumber \\
&= {1 \over N(N-1)} \sum_{i=1}^N \left[
(\partial_{\mu_y} f)\, \left\{N(\overline{y} - \mu_y) - (y_i - \mu_y) \right\}
+
(\partial_{\mu_z} f)\, \left\{N(\overline{z} - \mu_z) - (z_i - \mu_z) \right\}
\right] \nonumber \\
&= (\partial_{\mu_y} f)\, (\overline{y} - \mu_y) +
(\partial_{\mu_z} f)\, (\overline{z} - \mu_z) \, .
\label{fJ-f}\end{aligned}$$ Similarly we find $$\begin{aligned}
\overline{\left(f^J\right)^2 } &= {1 \over N} \sum_{i=1}^N \left[
f(\mu_y, \mu_z) + (\partial_{\mu_y} f)\, (y_i^J - \mu_y) +
(\partial_{\mu_z} f)\, (z_i^J - \mu_z) \right]^2 \nonumber \\
&= f^2(\mu_y, \mu_z) + 2 f(\mu_y, \mu_z) \, \left[ (\partial_{\mu_y} f)\,
(\overline{y} - \mu_y) + (\partial_{\mu_z} f)\, (\overline{z} - \mu_z) \right] \nonumber \\
&\quad +
(\partial_{\mu_y} f)^2\, \left[(\overline{y} - \mu_y)^2 + {s_{yy}^2 \over N(N-1)}\right] +
(\partial_{\mu_z} f)^2\, \left[(\overline{z} - \mu_z)^2 + {s_{zz}^2 \over N(N-1)}\right]
\nonumber \\
&\qquad + 2(\partial_{\mu_y} f)(\partial_{\mu_z} f)\,\left[(\overline{y} - \mu_y) (\overline{z} - \mu_z) +
{s_{yz}^2 \over N(N-1)}\right]
\, .\end{aligned}$$ Hence, from Eqs. and , the variance in the jackknife estimates is given by $$\sigma^2_{f^J} = {1 \over N(N-1)} \, \left[
(\partial_{\mu_y} f)^2\, s_{yy}^2 +
(\partial_{\mu_z} f)^2\, s_{zz}^2
+ 2(\partial_{\mu_y} f)(\partial_{\mu_z} f) s_{yz}\right] \, ,$$ which is just $1/(N-1)$ times $\sigma_f^2$, the estimate of the square of the error bar in $f(\overline{y}, \overline{z})$ given in Eq. . Hence $$\boxed{
\mbox{the jackknife estimate for } \sigma_f \mbox{ is } \sqrt{N-1} \,
\sigma_{f^J}\, .}
\label{error_jack}$$ Note that this is directly obtained from the jackknife estimates without having to put in the partial derivatives by hand. Note too that the $\sqrt{N-1}$ factor is in the *numerator* whereas the factor of $\sqrt{N}$ in Eq. is in the *denominator*. Intuitively the reason for this difference is that the jackknife estimates are very close since they would all be equal except that each one omits just one data point.
If $N$ is very large, roundoff errors could become significant from having to subtract large, almost equal, numbers to get the error bar from the jackknife method. It is then advisable to group the $N$ data points into $N_\text{group}$ groups (or “bins”) of data and take, as individual data points in the jackknife analysis, the average of the data in each group. The above results clearly go through with $N$ replaced by $N_\text{group}$.
To summarize this subsection, to estimate $f(\mu_y, \mu_z)$ one can use either $f(\overline{y}, \overline{z})$ or the jackknife average $\overline{f^J}$ in Eq. . The error bar in this estimate, $\sigma_f$, is related to the standard deviation in the jackknife estimates $\sigma_{f^J}$ by Eq. .
### Bootstrap {#sec:boot}
The bootstrap, like the jackknife, is a resampling of the $N$ data points Whereas jackknife considers $N$ new data sets, each of containing all the original data points minus one, bootstrap uses ${{N_{\rm boot}}}$ data sets each containing $N$ points obtained by random (Monte Carlo) sampling of the original set of $N$ points. During the Monte Carlo sampling, the probability that a data point is picked is $1/N$ irrespective of whether it has been picked before. (In the statistics literature this is called picking from a set “with replacement”.) Hence a given data point $x_i$ will, *on average*, appear once in each Monte Carlo-generated data set, but may appear not at all, or twice, and so on. The probability that $x_i$ appears $n_i$ times is close to a Poisson distribution with mean unity. However, it is not exactly Poissonian because of the constraint in Eq. (\[constraint\]) below. It turns out that we shall need to include the deviation from the Poisson distribution even for large $N$. We shall use the term “bootstrap” data sets to denote the Monte Carlo-generated data sets.
More precisely, let us suppose that the number of times $x_i$ appears in a bootstrap data set is $n_i$. Since each bootstrap dataset contains exactly $N$ data points, we have the constraint $$\sum_{i=1}^N n_i = N \, .
\label{constraint}$$ Consider one of the $N$ variables $x_i$. Each time we generate an element in a bootstrap dataset the probability that it is $x_i$ is $1/N$, which we will denote by $p$. From standard probability theory, the probability that $x_i$ occurs $n_i$ times is given by a binomial distribution $$P(n_i) = {N! \over n_i! \, (N - n_i)!} \, p^{n_i} (1-p)^{N -n_i} \, .$$ The mean and standard deviation of a binomial distribution are given by $$\begin{aligned}
[ n_i ]{_{_{\rm MC}}}& = N p = 1 \, ,
\label{nimc} \\
{[ n_i^2 ]{_{_{\rm MC}}}} - [n_i]{_{_{\rm MC}}}^2 & = N p (1 - p) = 1 - {1 \over N} \, ,
\label{epsi_epsi}\end{aligned}$$ where $[ \dots ]{_{_{\rm MC}}}$ denotes an exact average over bootstrap samples (for a fixed original data set $x_i$). For $N \to\infty$, the binomial distribution goes over to a Poisson distribution, for which the factor of $1/N$ in Eq. (\[epsi\_epsi\]) does not appear. We assume that ${{N_{\rm boot}}}$ is sufficiently large that the bootstrap average we carry out reproduces this result with sufficient accuracy. Later, we will discuss what values for ${{N_{\rm boot}}}$ are sufficient in practice. Because of the constraint in Eq. (\[constraint\]), $n_i $ and $n_j$ (with $i \ne j$) are not independent and we find, by squaring Eq. and using Eqs. and , that $$[ n_i n_j ]{_{_{\rm MC}}}- [ n_i ]{_{_{\rm MC}}}[ n_j ]{_{_{\rm MC}}}= - {1 \over N} \qquad (i \ne j)\, .
\label{epsi_epsj}$$
First of all we just consider the simple average $\mu_x \equiv \langle x
\rangle$, for which, of course, the standard methods in Sec. \[sec:basic\] suffice, so bootstrap is not necessary. However, this will show how to get averages and error bars in a simple case, which we will then generalize to non-linear functions of averages.
We denote the average of $x$ for a given bootstrap data set by ${x^B}_\alpha$, where $\alpha$ runs from 1 to ${{N_{\rm boot}}}$, [[*i.e.*]{}]{}$${x^B}_\alpha = {1 \over N} \sum_{i=1}^N n_i^\alpha x_i \, .$$ We then compute the bootstrap average of the mean of $x$ and the bootstrap variance in the mean, by averaging over all the bootstrap data sets. We assume that ${{N_{\rm boot}}}$ is large enough for the bootstrap average to be exact, so we can use Eqs. (\[epsi\_epsi\]) and (\[epsi\_epsj\]). The result is $$\begin{aligned}
\label{xb}
\overline{{x^B}} \equiv
{1 \over {{N_{\rm boot}}}} \sum_{\alpha=1}^{{N_{\rm boot}}}{x^B}_\alpha
& = & {1\over N} \sum_{i=1}^N [n_i]{_{_{\rm MC}}}x_i = {1\over N}
\sum_{i=1}^N x_i = \overline{x} \\
\sigma^2_{{x^B}}
\equiv
\overline{\left({x^B}\right)^2} - \left(\overline{{x^B}}\right)^2
& = & {1\over N^2}
\left(1 - {1\over N}\right) \sum_i
x_i^2 - {1 \over N^3} \sum_{i \ne j} x_i x_j \, ,
\label{sigmab}\end{aligned}$$ where $$\overline{\left({x^B}\right)^2} \equiv {1 \over {{N_{\rm boot}}}} \sum_{\alpha=1}^{{N_{\rm boot}}}\left[ \left({x^B}_\alpha\right)^2\right]{_{_{\rm MC}}}\, .$$
We now average Eqs. (\[xb\]) and (\[sigmab\]) over many repetitions of the original data set $x_i$. Averaging Eq. (\[xb\]) gives $$\langle \overline{{x^B}} \rangle = \langle \overline{x} \rangle = \langle x
\rangle \equiv \mu_x \, .$$ This shows that the bootstrap average $\,\overline{{x^B}}\, $ is an unbiased estimate of the exact average $\mu_x$. Averaging Eq. (\[sigmab\]) gives $$\left\langle \sigma^2_{{x^B}} \right\rangle
= {N-1 \over N^2} \sigma^2 =
{N-1 \over N} \sigma^2_{\overline{x}} \, ,$$ where we used Eq. (\[dxsq\]) to get the last expression. Since $\sigma_{\overline{x}}$ is the uncertainty in the sample mean, we see that $$\boxed{\mbox{the bootstrap estimate of }\sigma_{\overline{x}} \mbox{ is }
\sqrt{N \over N-1}\, \sigma_{{x^B}} \, .}
\label{sigmaxb}$$ Remember that $\sigma_{{x^B}}$ is the standard deviation of the bootstrap data sets. Usually $N$ is sufficiently large that the square root in Eq. (\[sigmaxb\]) can be replaced by unity.
As for the jackknife, these results can be generalized to finding the error bar in some possibly non-linear function, $f(\mu_y, \mu_z)$, rather than for $\mu_x$. One computes the bootstrap estimates for $f(\mu_y, \mu_z)$, which are $${f^B}_\alpha = f({y^B}_\alpha, {z^B}_\alpha) \, .$$ In other words, we use the bootstrap values, ${y^B}_\alpha, {z^B}_\alpha$, rather than the sample means, $\overline{y}, \overline{z}$, as the arguments of $f$. The final bootstrap estimate for $f(\mu_y, \mu_z)$ is the average of these, [[*i.e.*]{}]{}$$\boxed{ \overline{{f^B}} = {1 \over {{N_{\rm boot}}}} \sum_{\alpha=1}^{{N_{\rm boot}}}{f^B}_\alpha \, .}
\label{fb}$$ Following the same methods in the jackknife section, one obtains the error bar, $\sigma_f$, in $f(\mu_y, \mu_z)$. The result is $$\boxed{\mbox{the bootstrap estimate for } \sigma_f \mbox{ is }
\sqrt{N \over N-1} \,\, \sigma_{{f^B}}},
\label{sigmafb}$$ where $$\boxed{
\sigma^2_{{f^B}} =
\overline{\left({f^B}\right)^2} - \left(\overline{{f^B}}\right)^2 \, ,}
$$ is the variance of the bootstrap estimates. Here $$\overline{\left({f^B}\right)^2} \equiv {1 \over {{N_{\rm boot}}}} \sum_{\alpha=1}^{{N_{\rm boot}}}\left({f^B}_\alpha\right)^2 \, .$$ Usually $N$ is large enough that the factor of $\sqrt{N/(N-1)}$ is Eq. can be replaced by unity. Equation (\[sigmafb\]) corresponds to the result Eq. (\[sigmaxb\]) which we derived for the special case of $f = \mu_x$.
Again, following the same path as in the jackknife section, it is straightforward to show that the bias of the estimates in Eqs. (\[fb\]) and (\[sigmafb\]) is of order $1/N$ and so vanishes for $N\to\infty$. However, if $N$ is not too large it may be useful to eliminate the leading contribution to the bias in the mean, as we did for jackknife in Eq. (\[bias\_elim\]). The result is that one should $$\boxed{\mbox{estimate } f(\mu_y, \mu_z) \mbox{ from }
2 f(\overline{y}, \overline{z}) - \overline{{f^B}} \, .}
\label{improved_boot}$$ The bias in Eq. (\[improved\_boot\]) is of order $1/N^2$, whereas $f(\overline{y}, \overline{z})$ and $\overline{{f^B}}$ each have a bias of order $1/N$. However, it is not normally necessary to eliminate the bias since, if $N$ is large, the bias is much smaller than the statistical error.
I have not systematically studied the values of ${{N_{\rm boot}}}$ that are needed in practice to get accurate estimates for the error. It seems that ${{N_{\rm boot}}}$ in the range 100 to 500 is typically chosen, and this seems to be adequate irrespective of how large $N$ is. To summarize this subsection, to estimate $f(\mu_y, \mu_z)$ one can either use $f(\overline{y}, \overline{z})$, or the bootstrap average in Eq. , and the error bar in this estimate, $\sigma_f$, is related to the standard deviation in the bootstrap estimates by Eq. .
### Jackknife or Bootstrap? {#sec:jorb}
The jackknife approach involves less calculation than bootstrap, and is fine for estimating combinations of moments of the measured quantities. Furthermore, identical results are obtained each time jackknife is run on the same set of data, which is not the case for bootstrap. However, the range of the jackknife estimates is very much smaller, by a factor of $\sqrt{N}$ for large $N$, than the scatter of averages which would be obtained from individual data sets, see Eq. . By contrast, for bootstrap, $\sigma_{{f^B}}$, which measures the deviation of the bootstrap estimates ${f^B}_\alpha$ from the result for the single actual data set $f(\overline{y}, \overline{z})$, *is equal to* $\sigma_f$, the deviation of the average of a single data set from the exact result $f(\mu_y,\mu_z)$ (if we replace the factor of $N/(N-1)$ by unity, see Eq. ). This is the main strength of the bootstrap approach; it samples the full range of the distribution of the sample distribution. Hence, if you want to generate data which covers the full range then should use bootstrap. This is useful in fitting, see for example, Sec. \[sec:resample\]. However, if you just want to generate error bars on combinations of moments quickly and easily, then use jackknife.
Fitting data to a model {#sec:fit}
=======================
A good reference for the material in this section is Chapter 15 of Numerical Recipes [@press:92].
Frequently we are given a set of data points $(x_i, y_i), i = 1, 2,
\cdots, N$, with corresponding error bars, $\sigma_i$, through which we would like to fit to a smooth function $f(x)$. The function could be straight line (the simplest case), a higher order polynomial, or a more complicated function. The fitting function will depend on $M$ “fitting parameters”, $a_\alpha$ and we would like the “best” fit obtained by adjusting these parameters. We emphasize that a fitting procedure should not only
1. \[give\_params\] give the values of the fit parameters, but also
2. \[give\_errors\] provide error estimates on those parameters, and
3. \[gof\] provide a measure of how good the fit is.
If the result of part \[gof\] is that the fit is very poor, the results of parts \[give\_params\] and \[give\_errors\] are probably meaningless.
The definition of “best” is not unique. However, the most useful choice, and the one nearly always taken, is “least squares”, in which one minimizes the sum of the squares of the difference between the observed $y$-value, $y_i$, and the fitting function evaluated at $x_i$, weighted appropriately by the error bars since if some points have smaller error bars than others the fit should be closer to those points. The quantity to be minimized, called “chi-squared”,[^2] and written mathematically as $\chi^2$, is therefore $$\boxed{
\chi^2 = \sum_{i=1}^N \left( \, {y_i - f(x_i) \over \sigma_i } \, \right)^2. }
\label{chisq}$$
Often we assume that the distribution of the errors is Gaussian, since, according to the central limit theorem discussed in Appendix \[sec:clt\], the sum of $N$ independent random variables has a Gaussian distribution (under fairly general conditions) if $N$ is large. However, distributions which occur in nature usually have more weight in the “tails” than a Gaussian, and as a result, even for moderately large values of $N$, the probability of an “outlier” might be much bigger than expected from a Gaussian, see Fig. \[Fig:converge\_to\_clt\].
If the errors *are* distributed with a Gaussian distribution, and if $f(x)$ has the *exact* values of the fit parameters, then $\chi^2$ in Eq. is a sum of squares of $N$ random variables with a Gaussian distribution with mean zero and standard deviation unity. However, when we have minimized the value of $\chi^2$ with respect to the $M$ fitting parameters $a_\alpha$ the terms are not all independent. It turns out, see Appendix \[sec:NDF\], that, at least for a linear model (which we define below), the distribution of $\chi^2$ at the minimum is that of the sum of the squares of $N-M$ (not $N$) Gaussian random variable with zero mean and standard deviation unity[^3]. We call $N-M$ the “number of degrees of freedom” ($N_\text{DOF}$). The $\chi^2$ distribution is discussed in Appendix \[sec:Q\]. The formula for it is Eq. .
The simplest problems are where the fitting function is a *linear function of the parameters*. We shall call this a *linear model*. Examples are a straight line ($M=2$), $$y = a_0 + a_1 x \, ,
\label{sl}$$ and an $m$-th order polynomial ($M=m+1$), $$y = a_0 + a_1 x + a_2 x^2 + \cdots + a_m x^m
= \sum_{\alpha=0}^m a_\alpha x^m \, ,
\label{poly}$$ where the parameters to be adjusted are the $a_\alpha$. (Note that we are *not* stating here that $y$ has to be a linear function of $x$, only of the fit parameters $a_\alpha$.)
An example where the fitting function depends *non*-linearly on the parameters is $$y = a_0 x^{a_1} + a_2 \, .$$
Linear models are fairly simply because, as we shall see, the parameters are determined by *linear* equations, which, in general, have a unique solution that can be found by straightforward methods. However, for fitting functions which are non-linear functions of the parameters, the resulting equations are *non-linear* which may have many solutions or none at all, and so are much less straightforward to solve. We shall discuss fitting to both linear and non-linear models in these notes.
Sometimes a non-linear model can be transformed into a linear model by a change of variables. For example, if we want to fit to $$y = a_0 x^{a_1} \, ,$$ which has a non-linear dependence on $a_1$, taking logs gives $$\ln y = \ln a_0 + a_1 \ln x \, ,$$ which is a *linear* function of the parameters $a'_0 = \ln a_0$ and $a_1$. Fitting a straight line to a log-log plot is a very common procedure in science and engineering. However, it should be noted that transforming the data does not exactly take Gaussian errors into Gaussian errors, though the difference will be small if the errors are “sufficiently small”. For the above log transformation this means $\sigma_i / y_i \ll 1$, i.e. the *relative* error is much less than unity.
Fitting to a straight line
--------------------------
To see how least squares fitting works, consider the simplest case of a straight line fit, Eq. (\[sl\]), for which we have to minimize $$\chi^2(a_0, a_1) = \sum_{i=1}^N \left({\, y_i - a_0 - a_1 x_i\,
\over \sigma_i} \right)^2 \, ,
\label{chisq_sline}$$ with respect to $a_0$ and $a_1$. Differentiating $\chi^2$ with respect to these parameters and setting the results to zero gives
\[sline\] $$\begin{aligned}
a_0\, \sum_{i=1}^N {1 \over \sigma_i^2} + a_1\, \sum_{i=1}^N
{x_i\over\sigma_i^2} &= \sum_{i=1}^N
{y_i\over \sigma_i^2} ,
\label{da0}\\
a_0\, \sum_{i=1}^N {x_i\over \sigma_i^2} +a_1\, \sum_{i=1}^N
{x_i^2\over \sigma_i^2} &=
\sum_{i=1}^N {x_i y_i \over \sigma_i^2} .
\label{da1}\end{aligned}$$
We write this as
$$\begin{aligned}
U_{00} \, a_0 + U_{01} \, a_1 &= v_0 , \\
U_{10} \, a_0 + U_{11} \, a_1 &= v_1 ,\end{aligned}$$
\[lssl\]
where $$\begin{aligned}
&\boxed{U_{\alpha\beta} = \sum_{i=1}^N {x_i^{\alpha + \beta}\over
\sigma_i^2}, } \quad \mbox{and}
\label{Uab} \\
&\boxed{v_\alpha = \sum_{i=1}^N{ y_i\, x_i^\alpha \over
\sigma_i^2 \, }. } \label{v}\end{aligned}$$ The matrix notation, while an overkill here, will be convenient later when we do a general polynomial fit. Note that $U_{10} = U_{01}$. (More generally, later on, $U$ will be a symmetric matrix). Equations (\[lssl\]) are two linear equations in two unknowns. These can be solved by eliminating one variable, which immediately gives an equation for the second one. The solution can also be determined from $$\boxed{
a_\alpha = \sum_{\beta=0}^m \left(U^{-1}\right)_{\alpha\beta} \, v_\beta , }
\label{soln}$$ (where we have temporarily generalized to a polynomial of order $m$). For the straight-line fit, the inverse of the $2\times 2$ matrix $U$ is given, according to standard rules, by $$U^{-1} = {1 \over \Delta} \,
\begin{pmatrix}
U_{11} & -U_{01} \\
-U_{01} & U_{00}
\end{pmatrix}
\label{Uinv}$$ where $$\boxed{
\Delta = U_{00} U_{11} - U_{01}^2 ,}
\label{Delta}$$ and we have noted that $U$ is symmetric so $U_{01} = U_{10}$. The solution for $a_0$ and $a_1$ is therefore given by
$$\begin{aligned}
&\boxed{a_0 = {U_{11}\, v_0 - U_{01}\, v_1 \over \Delta}, } \\
&\boxed{a_1 = {-U_{01}\, v_0 + U_{00}\, v_1 \over \Delta}. } \end{aligned}$$
\[soln\_sl\]
We see that it is straightforward to determine the slope, $a_1$, and the intercept, $a_0$, of the fit from Eqs. (\[Uab\]), (\[v\]), (\[Delta\]) and (\[soln\_sl\]) using the $N$ data points $(x_i,y_i)$, and their error bars $\sigma_i$.
Fitting to a polynomial
-----------------------
Frequently we need to fit to a higher order polynomial than a straight line, in which case we minimize $$\chi^2(a_0,a_1,\cdots,a_m) =
\sum_{i=1}^N \left({y_i - \sum_{\alpha=0}^m a_\alpha x_i^\alpha \over
\sigma_i} \right)^2
\label{chisq_poly}$$ with respect to the $(m+1)$ parameters $a_\alpha$. Setting to zero the derivatives of $\chi^2$ with respect to the $a_\alpha$ gives $$\boxed{
\sum_{\beta=0}^m U_{\alpha\beta}\, a _\beta = v_\alpha ,}
\label{lspoly}$$ where $U_{\alpha\beta}$ and $v_\alpha$ have been defined in Eqs. (\[Uab\]) and (\[v\]). Eq. (\[lspoly\]) represents $M = m+1$ *linear* equations, one for each value of $\alpha$. Their solution is again given by Eq. (\[soln\]), i.e. it is expressed in terms of the inverse matrix $U^{-1}$.
Error Bars {#sec:error_bars}
----------
In addition to the best fit values of the parameters we also need to determine the error bars in those values. Interestingly, this information is *also* contained in the matrix $U^{-1}$.
First of all, we explain the significance of error bars in fit parameters. We assume that the data is described by a model with a particular set of parameters $\vec{a}^\text{true}$ which, unfortunately, we don’t know. If we were, somehow, to have many real data sets each one would give a different set of fit parameters $\vec{a}^{(i)}, i = 0, 1, 2, \cdots$, because of noise in the data, *clustered about the true set* $\vec{a}^\text{true}$. Projecting on to a single fit parameter, $a_1$ say, there will be a distribution of values $P(a_1)$ centered on $a_1^\text{true}$ with standard deviation $\sigma_1$, see the top part of Fig. \[Fig:distofa1\]. Typically the value of $a_1$ obtained from our *one actual data set*, $a_1^{(0)}$, will lie within about $\sigma_1$ of $a_1$. Hence we define the error bar to be $\sigma_1$.
![ The top figure shows the distribution of one of the fit parameters $a_1$ if one could obtain many real data sets. The distribution has standard deviation $\sigma_1$ about the true value $a_1^\text{true}$ and is Gaussian if the noise on the data is Gaussian. In fact, however, we have only one actual data set which has fit parameter $a_1^{(0)}$, and this typically lies within about $\sigma_1$ of $a_1^\text{true}$. Hence we However, we because we have only one the one value, $a_1^{(0)}$. However, we can generate many *simulated* data sets from the one actual set and hence of the distribution of the resulting fit parameter $a_1^S$, which is shown in the lower figure. This distribution is centered about the value from the actual data, $a_1^{(0)}$, and has standard deviation, $\sigma_1^S$. The important point is that if one assumes a linear model then one can show that $\boxed{\sigma_1^S = \sigma_1 ,}$ see text. Even if the model is non linear, one usually assumes that the difference in the standard deviations is sufficiently small that one can still equate the true error bar with the standard deviation from the simulated data sets. We emphasize that and this is assumed to equal $\sigma_1$. Furthermore, as shown in Appendices \[sec:proof\] and \[sec:proof2\], if the noise on the data is Gaussian (and the model is linear) both the distributions in this figure are also Gaussian. []{data-label="Fig:distofa1"}](distofa1.eps){width="7.5cm"}
Unfortunately, we can’t determine the error bar this way because we have only one actual data set, which we denote here by $y_i^{(0)}$ to distinguish it from other data sets that we will introduce. Our actual data set gives one set of fit parameters, which we call $\vec{a}^{(0)}$. Suppose, however, we were to generate many *simulated* data sets from of the one which is available to us, by generating random values (possibly with a Gaussian distribution though this won’t be necessary yet) centered at the $y_i$ with standard deviation $\sigma_i$. Fitting each simulated dataset would give different values for $\vec{a}$, *clustered now about* $\vec{a}^{(0)}$, see the bottom part of Fig. . We now come to an important, but rarely discussed, point:
> We assume that the standard deviation of the fit parameters of these simulated data sets about $\vec{a}^{(0)}$, which we will be able to calculate from the single set of data available to us, is equal to the standard deviation of the fit parameters of real data sets $\vec{a}$ about $\vec{a}^\text{true}$. The latter is what we *really* want to know (since it is our estimate of the error bar on $\vec{a}^\text{true}$) but can’t determine directly. See Fig. \[Fig:distofa1\] for an illustration. In fact we show in the text below that this assumption is correct for a linear model (and for a non-linear model if the range of parameter values is small enough that it can be represented by an effective linear model). Even if the model is non linear, one usually assumes that the two standard deviations are sufficiently close that the difference is not important. Furthermore, we show in Appendices \[sec:proof\] and \[sec:proof2\] that if the noise on the data is Gaussian (and the model is linear), the two distributions in Fig. are also both Gaussian.
Hence, as stated above, to derive the error bars in the fit parameters we take simulated values of the data points, $y_i^S$, which vary by some amount $\delta
y_i^S$ about $y_i^{(0)}$, i.e. $\delta y_i^S = y_i^S - y_i^{(0)}$, with a standard deviation given by the error bar $\sigma_i$. The fit parameters of this simulated data set, $\vec{a}^S$, then deviate from $\vec{a}^{(0)}$ by an amount $\delta \vec{a}^S$ where $$\delta a_\alpha^S = \sum_{i=1}^N {\partial a_\alpha \over \partial y_i}\, \delta
y_i^S\, .$$ Averaging over fluctuations in the $y_i^S$ we get the variance of $a_\alpha^S$ to be $$\left(\sigma_\alpha^S\right)^2
\equiv \langle \left(\delta a_\alpha^S\right)^2 \rangle = \sum_{i=1}^N
\sigma_i^2 \, \left( {\partial a_\alpha \over \partial y_i} \right)^2 \, ,
\label{sigma_alpha}$$ since $\langle \left(\delta y_i^S\right)^2 \rangle = \sigma_i^2$, and the data points $y_i$ are statistically independent. Writing Eq. explicitly in terms of the data values, $$a_\alpha = \sum_\beta \left(U^{-1}\right)_{\alpha\beta} \sum_{i=1}^N { y_i\,
x_i^\beta \over \sigma_i^2 \, } \, ,$$ and noting that $U$ is independent of the $y_i$, we get $${\partial a_\alpha \over \partial y_i} = \sum_\beta
\left(U^{-1}\right)_{\alpha\beta} {x_i^\beta \over \sigma_i^2} \, .$$ Substituting into Eq. gives $$\left(\sigma_\alpha^S \right)^2 = \sum_{\beta, \gamma} \left(U^{-1}\right)_{\alpha\beta}
\left(U^{-1}\right)_{\alpha\gamma} \left[
\sum_{i=1}^N {x_i^{\beta + \gamma} \over \sigma_i^2}
\right]
\, .$$ The term in rectangular brackets is just $U_{\beta\gamma}$, and so, noting that $U$ is given by Eq. and is symmetric, the last equation reduces to $$\left(\sigma_\alpha^S \right)^2 =
\left(U^{-1}\right)_{\alpha\alpha} \, .
\label{error_params_s}$$ Recall that $\sigma_\alpha^S$ is the standard deviation of the fitted parameter values about the $\vec{a}^{(0)}$ when constructing simulated data sets from the one set of data that is available to us. However, the error bar is defined to be the standard deviation the fitted parameter values would have relative to $a_\alpha^\text{true}$ if we could average over many actual data sets. To determine this quantity we simply repeat the above calculation with $\delta y_i = y_i - y_i^\text{true}$ in which $y_i$ is the value of the $i$-th data point in one of the actual data sets. The result is identical to Eq. , namely $$\boxed{ \sigma_\alpha^2 =
\left(U^{-1}\right)_{\alpha\alpha} \, ,}
\label{error_params}$$ in which $U$ is the *same* in Eq. as in Eq. because $U$ is a constant, for a linear model, independent of the $y_i$ or the fit parameters $a_\alpha$. Hence $\sigma_\alpha$ in Eq. is the error bar in $a_\alpha$. In addition to error bars, we also need a parameter to describe the quality of the fit. A useful quantity is the probability that, given the fit, the data could have occurred with a $\chi^2$ greater than or equal to the value found. This is generally denoted by $Q$ and is given by Eq. assuming the data have Gaussian noise. Note that the effects of *non-Gaussian* statistics is to increase the probability of outliers, so fits with a fairly small value of $Q$, say around $0.01$, may be considered acceptable. However, fits with a *very* small value of $Q$ should not be trusted and the values of the fit parameters are probably meaningless in these cases.
![An example of a straight-line fit to a set of data with error bars.[]{data-label="fig:slinefit"}](fitdata3.eps){width="10cm"}
For the case of a straight line fit, the inverse of $U$ is given explicitly in Eq. (\[Uinv\]). Using this information, and the values of $(x_i, y_i, \sigma_i)$ for the data in Fig. \[fig:slinefit\], the fit parameters (assuming a straight line fit) are $$\begin{aligned}
a_0 &= 0.84 \pm 0.32 , \\
a_1 &= 2.05 \pm 0.11 ,\end{aligned}$$ in which the error bars on the fit parameters on $a_0$ and $a_1$, which are denoted by $\sigma_0$ and $\sigma_1$, are determined from Eq. (\[error\_params\]). The data was generated by starting with $y = 1 + 2x$ and then adding some noise with zero mean. Hence the fit should be consistent with $y = 1 +2x$ within the error bars, and it is. The value of $\chi^2$ is 7.44 so $\chi^2/N_\text{DOF} = 7.44 / 9 = 0.866$ and the quality of fit parameter, given by Eq. , is $Q = 0.592$ which is good.
We call $U^{-1}$ the “*covariance matrix*”. Its off-diagonal elements are also useful since they contain information about correlations between the fitted parameters. More precisely, one can show, following the lines of the above derivation of $\sigma_\alpha^2$, that the correlation of fit parameters $\alpha$ and $\beta$, known mathematically as their “covariance”, is given by the appropriate off-diagonal element of the covariance matrix, $$\text{Cov}(\alpha, \beta) \equiv \langle \delta a_\alpha \, \delta a_\beta
\rangle = \left(U^{-1}\right)_{\alpha\beta} \, .
\label{Covab}$$ The correlation coefficient, $r_{\alpha\beta}$, which is a dimensionless measure of the correlation between $\delta
a_\alpha$ and $\delta a_\beta$ lying between $-1$ and 1, is given by $$r_{\alpha\beta} = {\text{Cov}(\alpha, \beta) \over \sigma_\alpha \sigma_\beta}
\, .
\label{rab}$$ A good fitting program should output the correlation coefficients as well as the fit parameters, their error bars, the value of $\chi^2/N_\text{DOF}$, and the goodness of fit parameter $Q$.
For a linear model, $\chi^2$ is a quadratic function of the fit parameters and so the elements of the “*curvature matrix*”[^4], $(1/2)\, \partial^2 \chi^2 /
\partial {a_\alpha}\partial {a_\beta}$ are constants, independent of the values of the fit parameters. In fact, we see from Eqs. and that $${1\over 2}\, { \partial^2 \chi^2 \over
\partial {a_\alpha} \partial {a_\beta}} = U_{\alpha \beta} \, ,
\label{curv}$$ so *the curvature matrix is equal to $U$*, given by Eq. for a polynomial fit.
If we fit to a *general* linear model, writing $$f(x) = \sum_{\alpha=1}^M a_\alpha \, X_\alpha(x) ,
\label{general_lin}$$ where $X_1(x), X_2(x), \cdots, X_M(x)$ a fixed functions of $x$ called basis functions, the curvature matrix is given by $$\boxed{
U_{\alpha\beta} = \sum_{i=1}^N {X_\alpha(x_i)\, X_\beta(x_i) \over \sigma_i^2}
\, .}
\label{Uab_general}$$ Similarly, the quantities $v_\alpha$ in Eq. become $$\boxed{
v_\alpha = \sum_{i=1}^N {y_i\, X_\alpha(x_i) \over \sigma_i^2}
\, ,}
\label{v_general}$$ for a general set of basis functions, and best fit parameters are given by the solution of the $M$ linear equations $$\boxed{ \sum_{\beta=1}^M U_{\alpha\beta}\, a_\beta = v_\alpha \, , }
\label{lin_eq}$$ for $\alpha= 1, 2, \cdots, M$. Note that for a linear model the curvature matrix $U$ is a constant, independent of the fit parameters. However, $U$ is not constant for a non-linear model.
Fitting to a non-linear model {#sec:nlmodel}
-----------------------------
As for linear models, one minimizes $\chi^2$ in Eq. . The difference is that the resulting equations are non-linear so there might be many solutions or non at all. Techniques for solving the coupled non-linear equations invariably require specifying an initial value for the variables $a_\alpha$. The most common method for fitting to non-linear models is the Levenberg-Marquardt (LM) method, see e.g. Numerical Recipes [@press:92]. Implementing the Numerical Recipes code for LM is a little complicated because it requires the user to provide a routine for the derivatives of $\chi^2$ with respect to the fit parameters as well as for $\chi^2$ itself, and to check for convergence. Alternatively, one can use the fitting routines in the `scipy` package of `python` or use `gnuplot`. But see the comments in Appendix \[sec:ase\] about getting the error bars in the parameters correct. This applies when fitting to linear as well as non-linear models. Gnuplot and scipy scripts for fitting to a non-linear model are given in Appendix \[sec:scripts\].
One difference from fitting to a linear model is that the curvature matrix, defined by the LHS of Eq. , is not constant but is a function of the fit parameters. Hence it is no longer true that the standard deviations of the two distributions in Fig. \[Fig:distofa1\] are equal. However, it still generally assumed that the difference is small enough to be unimportant and hence that the covariance matrix, which is now defined to be the inverse of the curvature matrix *at the minimum of $\chi^2$*, still gives information about error bars on the fit parameters. This is discussed more in the next two subsections, in which we point out, however, that a more detailed analysis is needed if the model is non-linear and the spread of fitted parameters is sufficiently large that it cannot be represented by an effective linear model, i.e. $\chi^2$ is not well fitted by a parabola over the needed range of parameter values. As a reminder:
- The *curvature matrix* is defined in general by the LHS of Eq. , which, for a linear model, is equivalent to Eq. (Eq. for a polynomial fit.)
- The *covariance matrix* is the inverse of the curvature matrix at the minimum of $\chi^2$ (the last remark being only needed for a non-linear model). Its diagonal elements give error bars on the fit parameters according to Eq. (but see the caveat in the previous paragraph for non-linear models) and its off-diagonal elements give correlations between fit parameters according to Eqs. and .
Confidence limits {#sec:conf_limits}
-----------------
In the last two subsections we showed that the diagonal elements of the covariance matrix give an error bar on the fit parameters. In this section we extend the notion of error bar to embrace the concept of a “confidence limit”.
There is a theorem [@press:92] which states that, for a linear model, if we take simulated data sets assuming Gaussian noise in the data about the actual data points, and compute the fit parameters $\vec{a}^{S(i)}, i = 1, 2, \cdots$ for each data set, then the probability distribution of the $\vec{a}^S$ is given by the multi-variable Gaussian distribution $$\boxed{
P(\vec{a}^S) \propto \exp\left(-{1 \over 2} \, \sum_{\alpha, \beta}
\delta a_\alpha^S\, U_{\alpha\beta}\, \delta a_\beta^S \right) \, ,}
\label{theorem}$$ where $\delta \vec{a}^S \equiv \vec{a}^{S(i)} - \vec{a}^{(0)}$ and $U$, given by Eq. , is the curvature matrix which can also be defined in terms of the second derivative of $\chi^2$ according to Eq. . A proof of this result is given in Appendix \[sec:proof\]. It applies for a linear model with Gaussian noise, and also for a non-linear model if the uncertainties in the parameters do not extend outside a region where an effective linear model could be used. (In the latter case one still needs a non-linear routine to *find* the best parameters). Note that for a non-linear model, $U$ is not a constant and is the curvature *at the minimum* of $\chi^2$.
From Eq. the change in $\chi^2$ as the parameters are varied away from the minimum is given by $$\Delta \chi^2 \equiv \chi^2(\vec{a}^{S(i)}) -
\chi^2(\vec{a}^{(0)}) = \sum_{\alpha, \beta}
\delta a_\alpha^S\, U_{\alpha\beta}\, \delta a_\beta^S \, ,
\label{Dchisq}$$ in which the $\chi^2$ are all evaluated from the single (actual) data set $y_i^{(0)}$. Equation can therefore be written as $$P(\vec{a}^S) \propto \exp\left(-{1 \over 2} \Delta \chi^2 \right) \, .
\label{P_dalpha}$$ We remind the reader that we have assumed the noise in the data is Gaussian and that either the model is linear or, if non-linear, the uncertainties in the parameters do not extend outside a region where an effective linear model could be used.
Hence the probability of a particular deviation, $\delta \vec{a}^S$, of the fit parameters in a simulated data set away from the parameters in the *actual* data set, depends on how much this change increases $\chi^2$ (evaluated from the actual data set) away from the minimum. In general a “confidence limit” is the range of fit parameter values such that $\Delta \chi^2$ is less than some specified value. The simplest case, and the only one we discuss here, is the variation of *one* variable at a time, though multi-variate confidence limits can also be defined, see Numerical Recipes [@press:92].
We therefore consider the change in $\chi^2$ when one variable, $a_1^S$ say, is held at a specified value, and all the others $(\beta = 2, 3,\cdots, M)$ are varied in order to minimize $\chi^2$. Minimizing $\Delta \chi^2$ in Eq. with respect to $a_\beta^S$ gives $$\sum_{\gamma=1}^M U_{\beta\gamma}\, \delta a_\gamma^S = 0 , \qquad (\beta = 2, 3, \cdots,M) \, .$$ The corresponding sum for $\beta = 1$, namely $\sum_{\gamma=1}^M U_{1\gamma}\,
\delta a_\gamma^S$, is not zero because $\delta a_1$ is fixed. It will be some number, $c$ say. Hence we can write $$\sum_{\gamma=1}^M U_{\alpha\gamma}\, \delta a_\gamma^S = c_\alpha, \qquad
(\alpha = 1, 2, \cdots,M) \, ,$$ where $c_1 = c$ and $c_\beta = 0\, (\beta \ne 1)$. The solution is $$\delta a_\alpha^S = \sum_{\beta=1}^M \left(U^{-1}\right)_{\alpha\beta} c_\beta \, .
\label{aalpha}$$ For $\alpha = 1$ this gives $$c = \delta a_1^S / \left(U^{-1}\right)_{11} \, .
\label{c}$$ Substituting Eq. into Eq. , and using Eq. we find that $\Delta \chi^2 $ is related to $\left(\delta a_1^S\right)^2$ by $$\Delta \chi^2 = {(\delta a_1^S)^2 \over \left(U^{-1}\right)_{11} } .
\label{Dchi2}$$ (Curiously, the coefficient of $(\delta a_1)^2$ is one over the $11$ element of the inverse of $U$, rather than $U_{11}$ which is how it appears in Eq. in which the $\beta \ne 1$ parameters are free rather than adjusted to minimize $\chi^2$.)
From Eq. we finally get $$P(a_1^S) \propto \exp\left(-{1 \over 2} \,
{(\delta a_1^S)^2 \over\sigma_1^2}\right) \, ,
\label{Pa1S}$$ where $$\sigma_1^2 = \left(U^{-1}\right)_{11} \, .$$
As shown in Appendices \[sec:proof\] and \[sec:proof2\], Eqs. , and also apply, under the same conditions (linear model and Gaussian noise on the data) to the probability for $\delta a_1 \equiv
a_1^\text{true} - a_1^{(0)} $, where we remind the reader that $a_1^{(0)}$ is the fit parameter obtained from the actual data, and $a_1^\text{true}$ is the exact value. In other words the probability of the true value is given by $$\boxed{
P(\vec{a}^\text{true}) \propto \exp\left(-{1 \over 2} \Delta \chi^2 \right) \, ,}
\label{P_dalphatrue}$$ where $$\Delta \chi^2 \equiv \chi^2(\vec{a}^\text{true}) -
\chi^2(\vec{a}^{(0)}) \, ,$$ in which we remind the reader that both values of $\chi^2$ are evaluated from the single set of data available to us, $y_i^{(0)}$. Projecting onto a single parameter, as above, gives $$\boxed{
P(a_1^\text{true}) \propto \exp\left(-{1\over 2}\,
{(\delta a_1)^2 \over \sigma_1^2}\right) \, , }
\label{Pa1}$$ so $\langle \left(\delta a_1\right)^2 \rangle = \sigma_1^2 =
\left(U^{-1}\right)_{11}$, in agreement with what we found earlier in Eq. . We emphasize that Eqs. and assumes Gaussian noise on the data points, and either the model is linear or, if non-linear, that the range of uncertainty in the parameters is small enough that a description in terms of an effective linear model is satisfactory.
However we have done more than recover our earlier result, Eq. , by more complicated means since we have gained *additional* information. From the properties of a Gaussian distribution we now know that, from Eq. , the probability that $a_\alpha$ lies within one standard deviation $\sigma_\alpha$ of the value which minimizes $\chi^2$ is 68%, the probability of its being within two standard deviations is 95.5%, and so on. Furthermore, from Eq. , we see that
> *if a single fit parameter is one standard deviation away from its value at the minimum of $\chi^2$ (the other fit parameters being varied to minimize $\chi^2$), then $\Delta \chi^2 = 1$.*
This last sentence, and the corresponding equations Eqs. and , are not valid for a non-linear model if the uncertainties of the parameters extends outside the range where an effective linear model can be used. In this situation, to get confidence limits, is is necessary to do a bootstrap resampling of the data, as discussed in the next subsection.
![[**Left:**]{} The change in $\chi^2$ as a fit parameter $a_1$ is varied away from the value that minimizes $\chi^2$ for a *linear* model. The shape is a parabola for which $\Delta
\chi^2=1$ when $\delta a = \pm \sigma_1$, where $\sigma_1$ is the error bar.\
[**Right:**]{} The solid curve is a sketch of the change in $\chi^2$ for a *non-linear* model. The curve is no longer a parabola and is even non-symmetric. The dashed curve is a parabola which fits the solid curve at the minimum. The fitting program only has information about the *local* behavior at the minimum and so gives an error range $\pm \sigma_1$ where the value of the parabola is $1$. However, the parameter $a_1$ is clearly more tightly constrained on the plus side than on the minus side, and a better way to determine the error range is to look *globally* and locate the values of $\delta a_1$ where $\Delta \chi^2 = 1$. This gives an error bar $\sigma_1^+$ on the plus side, and a different error bar, $\sigma_1^-$, on the minus side, both of which are different from $\sigma_1$.[]{data-label="fig:chi2"}](chi2_linear.eps "fig:"){width="8cm"} ![[**Left:**]{} The change in $\chi^2$ as a fit parameter $a_1$ is varied away from the value that minimizes $\chi^2$ for a *linear* model. The shape is a parabola for which $\Delta
\chi^2=1$ when $\delta a = \pm \sigma_1$, where $\sigma_1$ is the error bar.\
[**Right:**]{} The solid curve is a sketch of the change in $\chi^2$ for a *non-linear* model. The curve is no longer a parabola and is even non-symmetric. The dashed curve is a parabola which fits the solid curve at the minimum. The fitting program only has information about the *local* behavior at the minimum and so gives an error range $\pm \sigma_1$ where the value of the parabola is $1$. However, the parameter $a_1$ is clearly more tightly constrained on the plus side than on the minus side, and a better way to determine the error range is to look *globally* and locate the values of $\delta a_1$ where $\Delta \chi^2 = 1$. This gives an error bar $\sigma_1^+$ on the plus side, and a different error bar, $\sigma_1^-$, on the minus side, both of which are different from $\sigma_1$.[]{data-label="fig:chi2"}](chi2_nonlinear.eps "fig:"){width="8cm"}
However, if one is not able to resample the data we argue that it is better to take the range where $\Delta \chi^2 \le 1$ as an error bar for each parameter rather than the error bar determined from the curvature of $\chi^2$ at the minimum, see Fig. \[fig:chi2\]. The left hand plot is for a linear model, for which the curve of $\Delta \chi^2$ against $\delta a_1$ is exactly a parabola, and the right hand plot is a sketch for a non-linear model, for which it is not a parabola though it has a quadratic variation about the minimum shown by the dashed curve. For the linear case, the values of $\delta a_1$ where $\Delta \chi^2 = 1$ are the *same* as the values $\pm \sigma_1$, where $\sigma_1$ is the standard error bar obtained from the *local* curvature in the vicinity of the minimum. However, for the non-linear case, the values of $\delta a_1$ where $\Delta \chi^2 = 1$ are *different* from $\pm \sigma_1$, and indeed the values on the positive and negative sides, $\sigma_1^+$ and $\sigma_1^-$, are not equal. For the data Fig. \[fig:chi2\], it is clear that the value of $a_1$ is more tightly constrained on the positive side than the negative side, and so it is better to give the error bars as $+\sigma_1^+$ and $-\sigma_1^-$, obtained from the range where $\Delta \chi^2 \le 1$, rather the symmetric range $\pm
\sigma_1$. However, if possible, in these circumstances error bars and a confidence limit should actually be obtained from a bootstrap resampling of the data as discussed in the next section.
Confidence limits by resampling the data {#sec:resample}
----------------------------------------
More work is involved if one wants to get error bars and a confidence interval in the case where the model is non-linear and the range of parameter uncertainty extends outside the region where an effective linear model is adequate. Even for a linear model, we cannot convert $\Delta \chi^2$ into a confidence limit with a specific probability if the noise is non-Gaussian.
To proceed in these cases, one can bootstrap the individual data points as follows. Each data point $(x_i, y_i)$ has error bar $\sigma_i$, which comes from averaging over $N$ measurements, say. Generating bootstrap datasets by Monte Carlo sampling the $N$ measurements, as discussed in Sec. \[sec:boot\], the distribution of the mean of each bootstrap dataset has a standard deviation equal to the estimate of standard deviation on the mean of the actual data set, see Eq. (replacing the factor of $\sqrt{N/(N-1)}$ by unity which is valid since $N$ is large in practice). Hence, if we generate $N_\text{boot}$ bootstrap data sets, and fit each one, the scatter of the fitted parameter values will be a measure of the uncertainty in the values from the *actual* dataset. Forming a histogram of the values of a single parameter we can obtain a confidence interval within which 68%, say, of the bootstrap datasets lie (16% missing on either side) and interpret this range as a 68% confidence limit for the actual parameter value. The justification for this interpretation has been discussed in the statistics literature, see e.g. the references in Ref. [@press:92], but I’m not able to go into the details here. Note that this bootstrap approach could also be applied usefully for a *linear* model if the noise is not Gaussian.
Unfortunately, use of the bootstrap procedure to get error bars in fits to non-linear models does not yet seem to be a standard procedure in the statistical physics community.
Another possibility for a non-linear model, if one is confident that the noise is close to Gaussian, is to generate *simulated* data sets, assuming Gaussian noise on the $y_i$ values with standard deviation given by the error bars $\sigma_i$. Each simulated dataset is fitted and the distribution of fitted parameters is determined. This corresponds to the analytical approach in Appendix \[sec:proof\] but without the assumption that the model can be represented by an effective linear one over of the needed parameter range.
A tale of two probabilities. When can one rule out a fit? {#sec:lin_or_quad}
---------------------------------------------------------
If the noise on the data is Gaussian, which we will assume throughout this subsection, we have, so far, considered two different probabilities.
Firstly, as discussed in Appendix \[sec:Q\], the value of $\chi^2$ is typically in the range $N_\text{DOF} \pm \sqrt{2 N_\text{DOF}}$. The quality of fit parameter $Q$ is the probability that, *given the fit*, the data could have this value of $\chi^2$ or greater, and is given mathematically by Eq. . It varies from unity when $\chi^2 \ll N_\text{DOF} - \sqrt{2 N_\text{DOF}}$ to zero when $\chi^2 \gg
N_\text{DOF} + \sqrt{2 N_\text{DOF}}$. We emphasize that
Secondly, in the context of error bars and confidence limits, we have discussed, in Eqs. and , the probability that a fit parameter, $a_1$ say, takes a certain value relative to the optimal one. Equation becomes very small when $\Delta \chi^2$ varies by much more than unity. Note that Eqs. and refer to the
![[**Left:**]{} A straight-line fit to a data set. The value of $Q$ is reasonable. However, one notices that the data is systematically above the fit for small $x$ and for large $x$ while it is below the fit for intermediate $x$. This is unlikely to happen by random chance. This remark is made more precise in the right figure.\
[**Right:**]{} A parabolic fit to the same data set. The value of $Q$ is larger than for the straight-line fit, but the main result is that the coefficient of the quadratic term is 5 $\sigma$ away from zero, showing that the straight-line fit in the left figure is much less likely than the parabolic fit. []{data-label="fig:lin_or_quad"}](lin_or_quad.eps "fig:"){width="8cm"} ![[**Left:**]{} A straight-line fit to a data set. The value of $Q$ is reasonable. However, one notices that the data is systematically above the fit for small $x$ and for large $x$ while it is below the fit for intermediate $x$. This is unlikely to happen by random chance. This remark is made more precise in the right figure.\
[**Right:**]{} A parabolic fit to the same data set. The value of $Q$ is larger than for the straight-line fit, but the main result is that the coefficient of the quadratic term is 5 $\sigma$ away from zero, showing that the straight-line fit in the left figure is much less likely than the parabolic fit. []{data-label="fig:lin_or_quad"}](lin_or_quad2.eps "fig:"){width="8cm"}
At first, it seems curious that the probability $Q$ remains significantly greater than zero if $\chi^2$ changes by an amount of order $\sqrt{N_\text{DOF}}$, whereas if a fit parameter is changed by an amount such that $\chi^2$ changes by of order $\sqrt{N_\text{DOF}}$, the probability of this value becomes extremely small, of order $\exp(-\text{const.}\,\sqrt{N_\text{DOF}})$, in this limit, see Eqs. . While there is no mathemematical inconsistency, since the two probabilities refer to different situations (one is the probability of the data given the fit and the other is the relative probability of two fits given the data), it is useful to understand this difference intuitively.
We take, as an example, a problem where we want to know whether the data can be modeled by a straight line, or whether a quadratic term needs to be included as well. A set of data is shown in Fig. \[fig:lin\_or\_quad\].
Looking at the left figure one sees that the data more or less agrees with the straight-line fit. However, one also sees systematic trends: the data is too high for small $x$ and for high $x$, and too low for intermediate $x$. Chi-squared just sums up the contributions from each data point and is insenstive to any systematic trend in the deviation of the data from the fit. Hence the value of $\chi^2$, in itself, does not tell us that this data is unlikely to be represented by a straight line. It is only when we add another parameter in the fit which corresponds to those correlations, that we realize the straight-line model is relatively very unlikely. In this case, the extra parameter is the coefficient of $x^2$, and the resulting parabolic fit is shown in the right figure.
The qualitative comments in the last paragraph are made more precise by the parameters of the fits. The straight-line fit gives $a_0 = 0.59 \pm 0.26, a_1 = 2.003 \pm 0.022$ with $Q = 0.124$, whereas the parabolic fit gives $a_0 = 2.04 \pm 0.40, a_1 = 1.588 \pm 0.090, a_2 = 0.0203 \pm 0.0042 $ with $Q = 0.924$. The actual parameters used to generate the data are $a_0 = 2, a_1 = 1.6, a_2 = 0.02$, and there is Gaussian noise with standard deviation equal to $0.8$. Although the quality of fit factor for the straight-line fit is reasonable, the quadratic fit strongly excludes having the fit parameter $a_2$ equal to zero, since zero is five standard deviations away from the best value. For a Gaussian distribution, the probability of a five-sigma deviation or greater is $\text{erfc}(5/\sqrt{2}) \simeq 6 \times
10^{-7}$. The difference in $\chi^2$ for the quadratic fit, between the best fit and the fit forcing $a_2 = 0$, is $(0.0203 / 0.0042)^2 \simeq 23$ according to Eqs. and . We conclude that, in this case, the straight-line model is unlikely to be correct.
The moral of this tale is that a reasonable value of $Q$ does not, in itself, ensure that you have the right model. Another model might be very much more probable.
Central Limit Theorem {#sec:clt}
=====================
In this appendix we give a proof of the central limit theorem.
We assume a distribution that falls off sufficiently fast at $\pm \infty$ that the mean and variance are finite. This *excludes*, for example, the Lorentzian distribution: $$P_{\rm Lor} = {1 \over \pi}
{1 \over 1+x^2} \, .$$ A common distribution which *does* have a finite mean and variance is the Gaussian distribution, $$P_{\rm Gauss} = {1 \over \sqrt{2 \pi}\, \sigma} \exp\left[-{(x-\mu)^2 \over 2
\sigma^2}\right] \, .
\label{Gauss}$$ Using standard results for Gaussian integrals you should be able to show that the distribution is normalized and that the mean and standard deviation are $\mu$ and $\sigma$ respectively.
Consider a distribution, *not necessarily Gaussian*, with a finite mean and distribution. We pick $N$ random numbers $x_i$ from such a distribution and form the sum $$X = \sum_{i=1}^N x_i.$$ Note, we are assuming that all the random numbers have the *same* distribution.
The determination of the distribution of $X$, which we call $P_N(X)$, uses the Fourier transform of $P(x)$, called the “characteristic function” in the context of probability theory. This is defined by $$Q(k) = \int_{-\infty}^\infty P(x) e^{i k x} \, d x \, .
$$ Expanding out the exponential we can write $Q(k)$ in terms of the moments of $P(x)$ $$Q(k) = 1 + i k\langle x \rangle + {(i k)^2 \over 2!} \langle x^2 \rangle +
{(i k)^3 \over 3!} \langle x^3 \rangle + \cdots \, .$$ It will be convenient in what follows to write $Q(k)$ as an exponential, i.e. $$\begin{aligned}
Q(k) & = & \exp \left[ \ln \left(1 +
i k\langle x \rangle + {(i k)^2 \over 2!} \langle x^2 \rangle +
{(i k)^3 \over 3!} \langle x^3 \rangle + \cdots \right) \right] \nonumber \\
& = & \boxed{ \exp\left[ i k \mu -{k^2 \sigma^2 \over 2!} +
{c_3(i k)^3 \over 3!} +
{c_4 (i k)^4 \over 4!} + \cdots \right] \, ,}
\label{cumulant}\end{aligned}$$ where $c_3$ involves third and lower moments, $c_4$ involves fourth and lower moments, and so on. The $c_n$ are called *cumulant* averages.
For the important case of a Gaussian, the Fourier transform is obtained by “completing the square”. The result is that the Fourier transform of a Gaussian is also a Gaussian, namely, $$\boxed{
Q_{\rm Gauss}(k) = \exp\left[ i k \mu -{k^2 \sigma^2 \over 2} \right] \, ,}
\label{Qgauss}$$ showing that the higher order cumulants, $c_3, c_4$, etc. in Eq. (\[cumulant\]) *all vanish* for a Gaussian.
The distribution $P_N(x)$ can be expressed as $$P_N(x) = \int_{-\infty}^\infty P(x_1) d x_1 \, \int_{-\infty}^\infty P(x_2) d x_2 \,
\cdots
\int_{-\infty}^\infty P(x_N) d x_N
\, \delta (X - \sum_{i=1}^N x_i) \, .$$ We evaluate this by using the integral representation of the delta function $$\delta(x) = {1 \over 2 \pi} \int_{-\infty}^\infty e^{i k x} \, d k \, ,$$ so $$\begin{aligned}
P_N(X)
&= \int_{-\infty}^\infty {d k \over 2 \pi}
\int_{-\infty}^\infty P(x_1) d x_1 \, \int_{-\infty}^\infty P(x_2) d x_2 \,
\cdots
\int_{-\infty}^\infty P(x_N) d x_N \,
\exp[i k (x_1 + x_2 + \cdots x_N - X)] \\
&= \int_{-\infty}^\infty {d k \over 2 \pi} Q(k)^N e^{-i k X} \, ,
\label{inv_FT}\end{aligned}$$ showing that the Fourier transform of $P_N(x)$, which we call $Q_N(k)$, is given by $$\boxed{Q_N(k) = Q(k)^N \, . }
\label{fourier_N}$$ Consequently $$Q_N(k) =
\exp\left[ i k N\mu -{N k^2 \sigma^2 \over 2} + {N c_3(i k)^3 \over 4!} +
{N c_4 (i k)^4 \over 4!} + \cdots \right] \, .
\label{cumulant_N}$$
Comparing with Eq. (\[cumulant\]) we see that
> the mean of the distribution of the sum of $N$ independent and identically distributed random variables (the coefficient of $-i k$ in the exponential) is $N$ times the mean of the distribution of one variable, and the variance of the distribution of the sum (the coefficient of $-k^2/2!$) is $N$ times the variance of the distribution of one variable.
These are general statements applicable for *any* $N$ and have already been derived in Sec. \[sec:basic\].
However, if $N$ is *large* we can now go further. The distribution $P_N(X)$ is the inverse transform of $Q_N(k)$, see Eq. , so $$P_N(X) =
{1 \over 2\pi} \int_{-\infty}^\infty
\exp\left[ -i k X'-{N k^2 \sigma^2 \over 2!} + N{c_3(i k)^3 \over 3!} +
{N c_4 (i k)^4 \over 4!} + \cdots \right] \, d k \, ,
\label{invtrans}$$ where $$X' = X - N \mu \, .
\label{x'}$$ Looking at the $-N k^2 / 2$ term in the exponential in Eq. (\[invtrans\]), we see that the integrand is significant for $ k < k^\star$, where $N \sigma^2 (k^\star)^2 = 1$, and negligibly small for $k \gg k^\star$. However, for $0 < k < k^\star$ the higher order terms in Eq. (\[invtrans\]), (i.e. those of order $k^3, k^4$ etc.) are very small since $N (k^\star)^3 \sim N^{-1/2}, N (k^\star)^4 \sim N^{-1}$ and so on. Hence the terms of higher order than $k^2$ in Eq. (\[invtrans\]), do not contribute for large $N$ and so $$\lim_{N \to \infty} P_N(X)
= {1 \over 2\pi} \int_{-\infty}^\infty
\exp\left[ -i k X'-{N k^2 \sigma^2 \over 2} \right] \, d k \, .
\label{invtransG}$$ In other words, for large $N$ the distribution is the Fourier transform of a Gaussian, which, as we know, is also a Gaussian. Completing the square in Eq. (\[invtransG\]) gives $$\begin{aligned}
\lim_{N \to \infty} P_N(X) & = &
{1 \over 2\pi} \int_{-\infty}^\infty
\exp\left[-{N \sigma^2 \over 2 } \left(k - {i X' \over N \sigma^2}\right)^2
\right] \, d k \ \exp\left[ -{(X')^2 \over 2 N \sigma^2} \right] \nonumber \\
& = &
\boxed{
{1 \over \sqrt{2 \pi N} \, \sigma} \exp\left[ -{(X-N\mu)^2 \over 2 N \sigma^2}
\right] \, ,}
\label{clt}\end{aligned}$$ where, in the last line, we used Eq. (\[x’\]). This is a Gaussian with mean $N \mu$ and variance $N \sigma^2$. Equation (\[clt\]) is the in statistics. It tells us that,
> for $N
> \to\infty$, the distribution of the sum of $N$ independent and identically distributed random variables is a *Gaussian* whose mean is $N$ times the mean, $\mu$, of the distribution of one variable, and whose variance is $N$ times the variance of the distribution of one variable, $\sigma^2$, *independent of the form of the distribution of one variable*, $P(x)$, provided only that $\mu$ and $\sigma$ are finite.
The central limit theorem is of such generality that it is extremely important. It is the reason why the Gaussian distribution has such a preeminent place in the theory of statistics. Note that if the distribution of the individual $x_i$ is Gaussian, then the distribution of the sum of $N$ variables is *always* Gaussian, even for $N$ small. This follows from Eq. and the fact that the Fourier transform of a Gaussian is a Gaussian.
In practice, distributions that we meet in nature, have a much broader tail than that of the Gaussian distribution, which falls off very fast at large $|x-\mu|/\sigma$. As a result, even if the distribution of the sum approximates well a Gaussian in the central region for only modest values of $N$, it might take a much larger value of $N$ to beat down the weight in the tail to the value of the Gaussian. Hence, even for moderate values of $N$, the probability of a deviation greater than $\sigma$ can be significantly larger than that of the Gaussian distribution which is 32%. This caution will be important in Sec. \[sec:fit\] when we discuss the quality of fits.
![ Figure showing the approach to the central limit theorem for the distribution in Eq. , which has mean, $\mu$, equal to 0, and standard deviation, $\sigma$, equal to 1. The horizontal axis is the sum of $N$ random variables divided by $\sqrt{N}$ which, for all $N$, has zero mean and standard deviation unity. For large $N$ the distribution approaches a Gaussian. However, convergence is non-uniform, and is extremely slow in the tails. []{data-label="Fig:converge_to_clt"}](dist_long_all.eps){width="11cm"}
We will illustrate the slow convergence of the distribution of the sum to a Gaussian in Fig. , in which the distribution of the individual variables $x_i$ is $$P(x) = {3 \over 2}\, {1 \over (1 + |x|)^4} \, .
\label{dist_long}$$ This has mean 0 and standard deviation 1, but moments higher than the second do not exist because the integrals diverge. For large $N$ the distribution approaches a Gaussian, as expected, but convergence is very slow in the tails.
The number of degrees of freedom {#sec:NDF}
================================
Consider, for simplicity, a straight line fit, so we have to determine the values of $a_0$ and $a_1$ which minimize Eq. . The $N$ terms in Eq. are not statistically independent at the minimum because the values of $a_0$ and $a_1$, given by Eq. , depend on the data points $(x_i,
y_i, \sigma_i)$.
Consider the “residuals” defined by $$\epsilon_i = {y_i - a_0 - a_1 x_i \over \sigma} \, .$$ If the model were exact and we use the exact values of the parameters $a_0$ and $a_1$ the $\epsilon_i$ would be independent and each have a Gaussian distribution with zero mean and standard deviation unity. However, choosing the *best-fit* values of $a_0$ and $a_1$ *from the data* according to Eq. implies that
$$\begin{aligned}
\sum_{i=1}^N {1\over \sigma_i}\, \epsilon_i &= 0\, ,\\
\sum_{i=1}^N {x_i \over \sigma_i}\, \epsilon_i &= 0\, ,\end{aligned}$$
which are are two *linear constraints* on the $\epsilon_i$. This means that we only need to specify $N-2$ of them to know them all. In the $N$ dimensional space of the $\epsilon_i$ we have eliminated two directions, so there can be no Gaussian fluctuations along them. However the other $N-2$ dimensions are unchanged, and will have the same Gaussian fluctuations as before. Thus $\chi^2$ has the distribution of a sum of squares of $N-2$ Gaussian random variables. We can intuitively understand why there are $N-2$ degrees of freedom rather than $N$ by considering the case of $N=2$. The fit goes perfectly through the two points so one has $\chi^2=0$ exactly. This implies that there are zero degrees of freedom since, on average, each degree of freedom adds 1 to $\chi^2$.
Clearly this argument can be generalized to any fitting function which depends *linearly* on $M$ fitting parameters, with the result that $\chi^2$ has the distribution of a sum of squares of $N_\text{DOF} = N-M$ Gaussian random variables, in which the quantity $N_\text{DOF}$ is called the “number of degrees of freedom”.
Even if the fitting function depends non-linearly on the parameters, this last result is often taken as a reasonable approximation.
The chi-squared distribution and the goodness of fit parameter $\textbf{Q}$ {#sec:Q}
===========================================================================
The $\chi^2$ distribution for $m$ degrees of freedom is the distribution of the sum of $m$ independent random variables with a Gaussian distribution with zero mean and standard deviation unity. To determine this we write the distribution of the $m$ variables $x_i$ as $$P(x_1, x_2, \cdots, x_m)\, dx_1 dx_2 \cdots dx_m
=
{1 \over (2 \pi)^{m/2}} \, e^{-x_1^2/2} \, e^{-x_2^2/2} \cdots e^{-x_m^2/2} \,
dx_1 dx_2 \cdots dx_m \, .$$ Converting to polar coordinates, and integrating over directions, we find the distribution of the radial variable to be $$\widetilde{P}(r) \, dr =
{S_m \over (2 \pi)^{m/2}} \, r^{m-1}\, e^{-r^2/2} \, dr \, ,
\label{Pr}$$ where $S_m$ is the surface area of a unit $m$-dimensional sphere. To determine $S_m$ we integrate Eq. over $r$, noting that $\widetilde{P}(r) $ is normalized, which gives $$S_m = {2 \pi^{m/2} \over \Gamma(m/2)} \, ,
\label{Sm}$$ where $\Gamma(x)$ is the Euler gamma function defined by $$\Gamma(x) = \int_0^\infty t^{x-1} \, e^{-t}\, dt \, .
\label{gamma}$$ From Eqs. and we have $$\widetilde{P}(r) = {1 \over 2^{m/2-1} \Gamma(m/2)} \, r^{m-1} e^{-r^2/2} \, .$$ This is the distribution of $r$ but we want the distribution of $\chi^2 \equiv
\sum_i x_i^2 = r^2$. To avoid confusion of notation we write $X$ for $\chi^2$, and define the $\chi^2$ distribution for $m$ variables as $P^{(m)}(X)$. We have $P^{(m)}(X) \, dX = \widetilde{P}(r) \, dr$ so the $\chi^2$ distribution for $m$ degrees of freedom is $$\begin{aligned}
P^{(m)}(X) &= {\widetilde{P}(r) \over dX / dr} \nonumber \\
&\boxed{ = {1 \over 2^{m/2} \Gamma(m/2)} \, X^{(m/2)-1}\,
e^{-X/2} \qquad (X > 0) \, .}
\label{chisq-dist}\end{aligned}$$ The $\chi^2$ distribution is zero for $X
< 0$. Using Eq. and the property of the gamma function that $\Gamma(n+1) = n \Gamma(n)$ one can show that
$$\begin{aligned}
\int_0^\infty P^{(m)}(X)\, d X &= 1 \, , \\
\langle X \rangle \equiv \int_0^\infty X\, P^{(m)}(X)\, d X &= m \, , \label{mean} \\
\langle X^2 \rangle \equiv \int_0^\infty X^2\, P^{(m)}(X)\, d X &= m^2 + 2m \, ,
\quad \mbox{so }\\
\langle X^2 \rangle - \langle X \rangle^2 &= 2 m \, \label{var} .\end{aligned}$$
From Eqs. and we see that typically $\chi^2$ lies in the range $m - \sqrt{2 m}$ to $m + \sqrt{2m}$. For large $m$ the distribution approaches a Gaussian according to the central limit theory discussed in Appendix \[sec:clt\]. Typically one focuses on the value of $\chi^2$ per degree freedom since this should be around unity independent of $m$.
The goodness of fit parameter is the probability that the specified value of $\chi^2$, or greater, could occur by random chance. From Eq. it is given by $$\begin{aligned}
Q &= {1 \over 2^{m/2} \Gamma(m/2)} \, \int_{\chi^2}^\infty\, X^{(m/2)-1}\,
e^{-X/2} \, d X\, , \\
& \boxed{ = {1 \over \Gamma(m/2)} \, \int_{\chi^2/2}^\infty\, y^{(m/2)-1}\, e^{-y} \, dy\, , }
\label{Q_expression}\end{aligned}$$ which is known as an incomplete gamma function. Code to generate the incomplete gamma function is given in Numerical Recipes [@press:92]. There is also a built-in function to generate the goodness of fit parameter in the `scipy` package of `python` and in the graphics program `gnuplot`, see the scripts in Appendix \[sec:scripts\].
Note that $Q=1$ for $\chi^2 = 0$ and $Q\to 0$ for $\chi^2 \to\infty$. Remember that $m$ is the number of degrees of freedom, written as $N_\text{DOF}$ elsewhere in these notes.
Asymptotic standard error and how to get correct error bars from gnuplot {#sec:ase}
========================================================================
Sometimes one does not have error bars on the data. Nonetheless, one can still use $\chi^2$ fitting to get an *estimate* of those errors (assuming that they are all equal) and thereby also get an error bar on the fit parameters. The latter is called the “asymptotic standard error”. Assuming the same error bar $\sigma_\text{ass}$ for all points, we determine $\sigma_\text{ass}$ from the requirement that $\chi^2$ per degree of freedom is precisely one, i.e. its mean value according to Eq. . This gives $$1 = {\chi^2 \over N_\text{DOF}} = {1 \over N_\text{DOF}} \, \sum_{i=1}^N
\left( \, {y_i - f(x_i) \over \sigma_\text{ass} } \, \right)^2 \, ,$$ or, equivalently, $$\boxed{
\sigma_\text{ass}^2 = {1 \over N_\text{DOF}} \, \sum_{i=1}^N
\left(y_i - f(x_i)\right)^2 \, .}
\label{sigma_ass}$$ The error bars on the fit parameters are then obtained from Eq. , with the elements of $U$ given by Eq. in which $\sigma_i$ is replaced by $\sigma_\text{ass}$. Equivalently, one can set the $\sigma_i$ to unity in determining $U$ from Eq. , and estimate the error on the fit parameters from $$\qquad\qquad\qquad\boxed{
\sigma^2_\alpha = \left(U\right)^{-1}_{\alpha\alpha}
\, \sigma^2_\text{ass}\, ,} \quad\text{(asymptotic standard error)}.
\label{assterr}$$
A simple example of the use of the asymptotic standard error in a situation where we don’t know the error on the data points, is fitting to a constant, i.e. *determining the average of a set of data*, which we already discussed in detail in Sec. \[sec:averages\]. In this case we have $$U_{00} = N, \qquad v_0 = \sum_{i=1}^N y_i ,$$ so the only fit parameter is $$a_0 = {v_0 \over U_{00}} = {1\over N} \, \sum_{i=1}^N y_i = \overline{y} ,$$ which gives, naturally enough, the average of the data points, $\overline{y}$. The number of degrees of freedom is $N-1$, since there is one fit parameter, so $$\sigma_\text{ass}^2 = {1 \over N - 1} \, \sum_{i=1}^N
\left(y_i - \overline{y}\right)^2 \, ,$$ and hence the square of the error on $a_0$ is given, from Eq. , by $$\sigma^2_0 = {1 \over U_{00}}\, \sigma^2_\text{ass} =
{1 \over N ( N - 1)} \, \sum_{i=1}^N \left(y_i - \overline{y}\right)^2 \, ,$$ which is precisely the expression for the error in the mean of a set of data given in Eq. .
I now mention that a popular plotting program, `gnuplot`, which also does fits, presents error bars on the fit parameters incorrectly if there are error bars on the data. Whether or not there are error bars on the points, `gnuplot` presents the “asymptotic standard error” on the fit parameters. `Gnuplot` calculates the elements of $U$ correctly from Eq. including the error bars, but then apparently also determines an “assumed error” from an expression like Eq. but including the error bars, i.e.
$$\sigma_\text{ass}^2 = {1 \over N_\text{DOF}} \, \sum_{i=1}^N
\left(\ {y_i - f(x_i) \over \sigma_i}\ \right)^2 \ = \ {\chi^2 \over
N_\text{DOF}}, \qquad \text{(\texttt{gnuplot})}\, .$$
Hence `gnuplot`’s $\sigma^2_\text{ass}$ is just the chi-squared per degree of freedom. The error bar (squared) quoted by `gnuplot` is $
\left(U\right)^{-1}_{\alpha\alpha} \, \sigma^2_\text{ass}$, as in Eq. . However, this is wrong since the error bars on the data points have *already* been included in calculating the elements of $U$, so the error on the fit parameter $\alpha$ should be $\left(U\right)^{-1}_{\alpha\alpha}$. Hence,
> to get correct error bars on fit parameters from `gnuplot` when there are error bars on the points, you have to divide `gnuplot`’s asymptotic standard errors by the square root of the chi-squared per degree of freedom (which gnuplot calls `FIT_STDFIT` and, fortunately, computes correctly).
I have checked this statement by comparing with results for Numerical Recipes routines, and also, for straight-line fits, by my own implementation of the formulae. It is curious that I found no hits on this topic when googling the internet. Can no one else have come across this problem? Correction of `gnuplot` error bars is implemented in the `gnuplot` scripts in Appendix \[sec:scripts\]
The need to correct `gnuplot`’s error bars applies to linear as well as non-linear models.
I recently learned that error bars on fit parameters given by the routine `curve_fit` of `python` also have to be corrected in the same way. This is shown in two of the python scripts in appendix \[sec:scripts\]. Curiously, a different python fitting routine, `leastsq`, gives the error bars correctly.
The distribution of fitted parameters determined from simulated datasets {#sec:proof}
========================================================================
In this section we derive the equation for the distribution of fitted parameters determined from simulated datasets, Eq. , assuming an arbitrary linear model, see Eq. . Projecting on to a single fitting parameter, as above, this corresponds to the lower figure in Fig. \[Fig:distofa1\].
We have *one* set of $y$-values, $y_i^{(0)}$, for which the fit parameters are $\vec{a}^{(0)}$. We then generate an *ensemble* of simulated data sets, $y_i^S$, assuming the data has Gaussian noise with standard deviation $\sigma_i$ centered on the actual data values $y_i^{(0)}$. We ask for the probability that the fit to one of the simulated data sets has parameters $\vec{a}^S$.
This probability distribution is given by $$P(\vec{a}^S) = \prod_{i=1}^N \left\{ {1 \over \sqrt{2 \pi} \sigma_i} \,
\int_{-\infty}^\infty \, d y_i^S\,
\exp\left[-{\left(y_i^S - y_i^{(0)}\right)^2 \over
2 \sigma_i^2}\right]\, \right\} \,
\prod_{\alpha=1}^M \delta \left( \sum_\beta U_{\alpha\beta} a_\beta^S -
v_\alpha^S\right) \, \det U \, ,
\label{dist_of_a}$$ where the factor in curly brackets is (an integral over) the probability distribution of the data points $y_i^S$, and the delta functions project out those sets of data points which have a particular set of fitted parameters, see Eq. . The factor of $\det U$ is a Jacobian to normalize the distribution. Using the integral representation of the delta function, and writing explicitly the expression for $v_\alpha$ from Eq. , one has $$\begin{aligned}
P(\vec{a}^S) =& \prod_{i=1}^N \left\{ {1 \over \sqrt{2 \pi} \sigma_i} \,
\int_{-\infty}^\infty \, d y_i^S\, \exp\left[-{\left(y_i^S - y_i^{(0)}
\right)^2 \over 2 \sigma_i^2}\right]\, \right\}
\times \qquad\qquad\qquad \\
& \ \prod_{\alpha=1}^M \left( {1 \over 2 \pi} \, \int_{-\infty}^\infty d k_\alpha
\exp\left[i k_\alpha\left( \sum_\beta U_{\alpha\beta} a_\beta^S -
\sum_{i=1}^N {y_i^S\, X_\alpha(x_i) \over \sigma_i^2} \right)\right] \right) \,
\det U \, .\end{aligned}$$ We carry out the $y$ integrals by “completing the square”, $$\begin{aligned}
P(\vec{a}^S) = \prod_{\alpha=1}^M \left( {1 \over 2 \pi} \,
\int_{-\infty}^\infty d k_\alpha \right) \,
\prod_{i=1}^N \left\{ {1 \over \sqrt{2 \pi} \sigma_i} \,
\int_{-\infty}^\infty \, d y_i^S\, \exp\left[-{\left(y_i^S - y_i^{(0)}
+ i \vec{k}\cdot \vec{X}(x_i) \right)^2
\over 2 \sigma_i^2}\right] \right\} \times \\
\exp\left[ -{1 \over 2 \sigma_i^2} \, \left(\,
\left(\vec{k}\cdot\vec{X}(i)\right)^2
+2 i \left(\vec{k} \cdot \vec{X}(x_i)\right) \, y_i^{(0)}
\right) \right] \times
\exp\left[i \sum_{\alpha,\beta} k_\alpha\, U_{\alpha\beta}\, a_\beta^S\right] \,
\det U \, .\end{aligned}$$ Doing the $y^S$-integrals, the factors in curly brackets are equal to unity. Using Eqs. and and the fact that the $U_{\alpha\beta}$ are independent of the $y_i^S$, we then get $$P(\vec{a}^S) = \prod_{\alpha=1}^M \left( {1 \over 2 \pi} \,
\int_{-\infty}^\infty d k_\alpha \right) \,
\exp\left[ -{1 \over 2} \sum_{\alpha,\beta} k_\alpha\, U_{\alpha\beta}\, k_\beta
+ i \sum_{\alpha,\beta} k_\alpha\,
\delta v_\alpha^S \right]
\, \det U \, ,$$ where $$\delta v_\beta^S \equiv v_\beta^S - v^{(0)}_\beta \, ,$$ with $$v_\alpha^{(0)} =
\sum_{i=1}^N {y_i^{(0)} \, X_\alpha(x_i) \over \sigma_i^2} \, .$$ We do the $k$-integrals by working in the basis in which $U$ is diagonal. The result is $$P(\vec{a^S}) = {\left( \det U \right)^{1/2} \over (2\pi)^{m/2}} \,
\exp\left[-{1 \over 2}\, \sum_{\alpha,\beta} \delta v_\alpha^S
\left(U^{-1}\right)_{\alpha\beta}
\delta v_\beta^S \right]
\, .$$ Using Eq. and the fact that $U$ is symmetric we get our final result $$\boxed{
P(\vec{a^S}) = {\left( \det U \right)^{1/2} \over (2\pi)^{m/2}} \,
\exp\left[-{1 \over 2}\, \sum_{\alpha,\beta} \delta a_\alpha^S\,
U_{\alpha\beta}\, \delta a_\beta^S \right]}
\, ,
\label{P_of_a}$$ which is Eq. , including the normalization constant in front of the exponential.
The distribution of fitted parameters from repeated sets of measurements {#sec:proof2}
========================================================================
In this section we derive the equation for the distribution of fitted parameters determined in the hypothetical situation that one has many actual data sets. Projecting on to a single fitted parameter, this corresponds to the upper figure in Fig. \[Fig:distofa1\].
The exact value of the data is $y_i^\text{true} =
\vec{a}^\text{true} \cdot \vec{X}(x_i)$, and the distribution of the $y_i$ in an actual data set, which differs from $y_i^\text{true}$ because of noise, has a distribution, assumed Gaussian here, centered on $y_i^\text{true}$ with standard deviation $\sigma_i$. Fitting each of these real data sets, the probability distribution for the fitted parameters is given by $$P(\vec{a}) = \prod_{i=1}^N \left\{ {1 \over \sqrt{2 \pi} \sigma_i} \,
\int_{-\infty}^\infty \, d y_i\,
\exp\left[-{\left(y_i - \vec{a}^\text{true} \cdot \vec{X}(x_i)\right)^2 \over
2 \sigma_i^2}\right]\, \right\} \,
\prod_{\alpha=1}^M \delta \left( \sum_\beta U_{\alpha\beta} a_\beta -
v_\alpha\right) \, \det U \, ,
\label{dist_of_a2}$$ see Eq. for an explanation of the various factors. Proceeding as in Appendix \[sec:proof\] we have $$\begin{aligned}
P(\vec{a}) = \prod_{i=1}^N \left\{ {1 \over \sqrt{2 \pi} \sigma_i} \,
\int_{-\infty}^\infty \, d y_i\, \exp\left[-{\left(y_i - \vec{a}^\text{true}
\cdot \vec{X}(x_i)\right)^2 \over 2 \sigma_i^2}\right]\, \right\}
\times \qquad\qquad\qquad \\
\prod_{\alpha=1}^M \left( {1 \over 2 \pi} \, \int_{-\infty}^\infty d k_\alpha
\exp\left[i k_\alpha\left( \sum_\beta U_{\alpha\beta} a_\beta -
\sum_{i=1}^N {y_i\, X_\alpha(x_i) \over \sigma_i^2} \right)\right] \right) \,
\det U \, ,\end{aligned}$$ and doing the $y$- integrals by completing the square gives $$\begin{aligned}
P(\vec{a})&= \prod_{\alpha=1}^M \left( {1 \over 2 \pi} \,
\int_{-\infty}^\infty d k_\alpha \right) \times
\\
&\exp\left[ -{1 \over 2 \sigma_i^2} \, \left(\,
\left(\vec{k}\cdot\vec{X}(i)\right)^2
+2 i \left(\vec{k} \cdot \vec{X}(x_i)\right) \, \left(\vec{a}^\text{true} \cdot
\vec{X}(x_i)\right) \right) \right] \times
\exp\left[i \sum_{\alpha,\beta} k_\alpha\, U_{\alpha\beta}\, a_\beta\right] \,
\det U \, .\end{aligned}$$ Using Eq. we then get $$P(\vec{a}) = \prod_{\alpha=1}^M \left( {1 \over 2 \pi} \,
\int_{-\infty}^\infty d k_\alpha \right) \,
\exp\left[ -{1 \over 2} \sum_{\alpha,\beta} k_\alpha\, U_{\alpha\beta}\, k_\beta
+ i \sum_{\alpha,\beta} k_\alpha\, U_{\alpha\beta}\, \delta a_\beta\right]
\, \det U \, ,$$ where $$\delta a_\beta \equiv a_\beta - a^\text{true}_\beta \, .$$ The $k$-integrals are done by working in the basis in which $U$ is diagonal. The result is $$\boxed{
P(\vec{a}) = {\left( \det U \right)^{1/2} \over (2\pi)^{m/2}} \,
\exp\left[-{1 \over 2}\, \sum_{\alpha,\beta} \delta a_\alpha\,
U_{\alpha\beta}\, \delta a_\beta \right]}
\, .
\label{P_of_a2}$$ In other words, the distribution of the fitted parameters obtained from many sets of actual data, about the *true* value $\vec{a}^\text{true}$ is a Gaussian. Since we are assuming a linear model, the matrix of coefficients $U_{\alpha\beta}$ is a constant, and so the distribution in Eq. is the *same* as in Eq. . Hence
> For a linear model with Gaussian noise, the distribution of fitted parameters, obtained from simulated data sets, relative to *value from the one actual data set*, is the same as the distribution of parameters from many actual data sets relative to *the true value*, see Fig. \[Fig:distofa1\].
This result is also valid for a non-linear model if the range of parameter values needed is sufficiently small that the model can be represented by an effective one. It is usually assumed to be a reasonable approximation even if this condition is not fulfilled.
Scripts for some data analysis and fitting tasks {#sec:scripts}
================================================
In this appendix I give sample scripts using perl, python and gnuplot for some basic data analysis and fitting tasks. I include output from the scripts when acting on certain datasets which are available on the web.
Note “`this_file_name`” refers to the name of the script being displayed (whatever you choose to call it.)
Scripts for a jackknife analysis
--------------------------------
The script reads in values of $x$ on successive lines of the input file and computes $\langle x^4\rangle / \langle x^2\rangle^2$, including an error bar computed using the jackknife method.
### Perl
#!/usr/bin/perl
#
# Usage: "this_file_name data_file"
# (make the script executable; otherwise you have to preface the command with "perl")
#
$n = 0;
$x2_tot = 0; $x4_tot = 0;
#
# read in the data
#
while(<>) # Note this very convenient perl command which reads each line of
# of each input file in the command line
{
@line = split;
$x2[$n] = $line[0]**2;
$x4[$n] = $x2[$n]**2;
$x2_tot += $x2[$n];
$x4_tot+= $x4[$n];
$n++;
}
#
# Do the jackknife estimates
#
for ($i = 0; $i < $n; $i++)
{
$x2_jack[$i] = ($x2_tot - $x2[$i]) / ($n - 1);
$x4_jack[$i] = ($x4_tot - $x4[$i]) / ($n - 1);
}
$x2_av = $x2_tot / $n; # Do the overall averages
$x4_av = $x4_tot / $n;
$g_av = $x4_av / $x2_av**2;
$g_jack_av = 0; $g_jack_err = 0; # Do the final jackknife estimate
for ($i = 0; $i < $n; $i++)
{
$dg = $x4_jack[$i] / $x2_jack[$i]**2;
$g_jack_av += $dg;
$g_jack_err += $dg**2;
}
$g_jack_av /= $n;
$g_jack_err /= $n;
$g_jack_err = sqrt(($n - 1) * abs($g_jack_err - $g_jack_av**2));
printf " Overall average is %8.4f\n", $g_av;
printf " Jackknife average is %8.4f +/- %6.4f \n", $g_jack_av, $g_jack_err;
Executing this file on the data in `http://physics.ucsc.edu/~peter/bad-honnef/data.HW2` gives
Overall average is 1.8215
Jackknife average is 1.8215 +/- 0.0368
### Python
#
# Program written by Matt Wittmann
#
# Usage: "python this_file_name data_file"
#
import fileinput
from math import *
x2 = []; x2_tot = 0.
x4 = []; x4_tot = 0.
for line in fileinput.input(): # read in each line in each input file.
# similar to perl's while(<>)
line = line.split()
x2_i = float(line[0])**2
x4_i = x2_i**2
x2.append(x2_i) # put x2_i as the i-th element in an array x2
x4.append(x4_i)
x2_tot += x2_i
x4_tot += x4_i
n = len(x2) # the number of lines read in
#
# Do the jackknife estimates
#
x2_jack = []
x4_jack = []
for i in xrange(n):
x2_jack.append((x2_tot - x2[i]) / (n - 1))
x4_jack.append((x4_tot - x4[i]) / (n - 1))
x2_av = x2_tot / n # do the overall averages
x4_av = x4_tot / n
g_av = x4_av / x2_av**2
g_jack_av = 0.; g_jack_err = 0.
for i in xrange(n): # do the final jackknife averages
dg = x4_jack[i] / x2_jack[i]**2
g_jack_av += dg
g_jack_err += dg**2
g_jack_av /= n
g_jack_err /= n
g_jack_err = sqrt((n - 1) * abs(g_jack_err - g_jack_av**2))
print " Overall average is %8.4f" % g_av
print " Jackknife average is %8.4f +/- %6.4f" % (g_jack_av, g_jack_err)
The output is the same as for the perl script.
Scripts for a straight-line fit
-------------------------------
### Perl, writing out the formulae by hand
#!/usr/bin/perl
#
# Usage: "this_file_name data_file"
# (make the script executable; otherwise preface the command with "perl")
#
# Does a straight line fit to data in "data_file" each line of which contains
# data for one point, x_i, y_i, sigma_i
#
$n = 0;
while(<>) # read in the lines of data
{
@line = split; # split the line to get x_i, y_i, sigma_i
$x[$n] = $line[0];
$y[$n] = $line[1];
$err[$n] = $line[2];
$err2 = $err[$n]**2; # compute the necessary sums over the data
$s += 1 / $err2;
$sumx += $x[$n] / $err2 ;
$sumy += $y[$n] / $err2 ;
$sumxx += $x[$n]*$x[$n] / $err2 ;
$sumxy += $x[$n]*$y[$n] / $err2 ;
$n++;
}
$delta = $s * $sumxx - $sumx * $sumx ; # compute the slope and intercept
$c = ($sumy * $sumxx - $sumx * $sumxy) / $delta ;
$m = ($s * $sumxy - $sumx * $sumy) / $delta ;
$errm = sqrt($s / $delta) ;
$errc = sqrt($sumxx / $delta) ;
printf ("slope = %10.4f +/- %7.4f \n", $m, $errm); # print the results
printf ("intercept = %10.4f +/- %7.4f \n\n", $c, $errc);
$NDF = $n - 2; # the no. of degrees of freedom is n - no. of fit params
$chisq = 0; # compute the chi-squared
for ($i = 0; $i < $n; $i++)
{
$chisq += (($y[$i] - $m*$x[$i] - $c)/$err[$i])**2;
}
$chisq /= $NDF;
printf ("chi squared / NDF = %7.4lf \n", $chisq);
Acting with this script on the data in `http://physics.ucsc.edu/~peter/bad-honnef/data.HW3` gives
slope = 5.0022 +/- 0.0024
intercept = 0.9046 +/- 0.2839
chi squared / NDF = 1.0400
### Python, writing out the formulae by hand
#
# Program written by Matt Wittmann
#
# Usage: "python this_file_name data_file"
#
# Does a straight-line fit to data in "data_file", each line of which contains
# the data for one point, x_i, y_i, sigma_i
#
import fileinput
from math import *
x = []
y = []
err = []
s = sumx = sumy = sumxx = sumxy = 0.
for line in fileinput.input(): # read in the data, one line at a time
line = line.split() # split the line
x_i = float(line[0]); x.append(x_i)
y_i = float(line[1]); y.append(y_i)
err_i = float(line[2]); err.append(err_i)
err2 = err_i**2
s += 1 / err2 # do the necessary sums over data points
sumx += x_i / err2
sumy += y_i / err2
sumxx += x_i*x_i / err2
sumxy += x_i*y_i / err2
n = len(x) # n is the number of data points
delta = s * sumxx - sumx * sumx # compute the slope and intercept
c = (sumy * sumxx - sumx * sumxy) / delta
m = (s * sumxy - sumx * sumy) / delta
errm = sqrt(s / delta)
errc = sqrt(sumxx / delta)
print "slope = %10.4f +/- %7.4f " % (m, errm)
print "intercept = %10.4f +/- %7.4f \n" % (c, errc)
NDF = n - 2 # the number of degrees of freedom is n - 2
chisq = 0.
for i in xrange(n): # compute chi-squared
chisq += ((y[i] - m*x[i] - c)/err[i])**2;
chisq /= NDF
print "chi squared / NDF = %7.4lf " % chisq
The results are identical to those from the perl script.
### Python, using a built-in routine from scipy
#
# Python program written by Matt Wittmann
#
# Usage: "python this_file_name data_file"
#
# Does a straight-line fit to data in "data_file", each line of which contains
# the data for one point, x_i, y_i, sigma_i.
#
# Uses the built-in routine "curve_fit" in the scipy package. Note that this
# requires the error bars to be corrected, as with gnuplot
#
from pylab import *
from scipy.optimize import curve_fit
fname = sys.argv[1] if len(sys.argv) > 1 else 'data.txt'
x, y, err = np.loadtxt(fname, unpack=True) # read in the data
n = len(x)
p0 = [5., 0.1] # initial values of parameters
f = lambda x, c, m: c + m*x # define the function to be fitted
# note python's lambda notation
p, covm = curve_fit(f, x, y, p0, err) # do the fit
c, m = p
chisq = sum(((f(x, c, m) - y)/err)**2) # compute the chi-squared
chisq /= n - 2 # divide by no.of DOF
errc, errm = sqrt(diag(covm)/chisq) # correct the error bars
print "slope = %10.4f +/- %7.4f " % (m, errm)
print "intercept = %10.4f +/- %7.4f \n" % (c, errc)
print "chi squared / NDF = %7.4lf " % chisq
The results are identical to those from the above scripts.
### Gnuplot
#
# Gnuplot script to plot points, do a straight-line fit, and display the
# points, fit, fit parameters, error bars, chi-squared per degree of freedom,
# and goodness of fit parameter on the plot.
#
# Usage: "gnuplot this_file_name"
#
# The data is assumed to be a file "data.HW3", each line containing
# information for one point (x_i, y_i, sigma_i). The script produces a
# postscript file, called here "HW3b.eps".
#
set size 1.0, 0.6
set terminal postscript portrait enhanced font 'Helvetica,16'
set output "HW3b.eps"
set fit errorvariables # needed to be able to print error bars
f(x) = a + b * x # the fitting function
fit f(x) "data.HW3" using 1:2:3 via a, b # do the fit
set xlabel "x"
set ylabel "y"
ndf = FIT_NDF # Number of degrees of freedom
chisq = FIT_STDFIT**2 * ndf # chi-squared
Q = 1 - igamma(0.5 * ndf, 0.5 * chisq) # the quality of fit parameter Q
#
# Below note how the error bars are (a) corrected by dividing by
# FIT_STDFIT, and (b) are displayed on the plot, in addition to the fit
# parameters, neatly formatted using sprintf.
#
set label sprintf("a = %7.4f +/- %7.4f", a, a_err/FIT_STDFIT) at 100, 400
set label sprintf("b = %7.4f +/- %7.4f", b, b_err/FIT_STDFIT) at 100, 330
set label sprintf("{/Symbol c}^2 = %6.2f", chisq) at 100, 270
set label sprintf("{/Symbol c}^2/NDF = %6.4f", FIT_STDFIT**2) at 100, 200
set label sprintf("Q = %9.2e", Q) at 100, 130
plot \ # Plot the data and fit
"data.HW3" using 1:2:3 every 5 with errorbars notitle pt 6 lc rgb "red" lw 2, \
f(x) notitle lc rgb "blue" lw 4 lt 1
The plot below shows the result of acting with this gnuplot script on a the data in `http://physics.ucsc.edu/~peter/bad-honnef/data.HW3`. The results agree with those of the other scripts. {width="11cm"}
Scripts for a fit to a non-linear model
---------------------------------------
We read in lines of data each of which contains three entries $x_i, y_i$ and $\sigma_i$. These are fitted to the form $$y = T_c + A / x^\omega \, ,$$ to determine the best values of $T_c, A$ and $\omega$.
### Python
#
# Python program written by Matt Wittmann
#
# Usage: "python this_file_name data_file"
#
# Does a fit to the non-linear model
#
# y = Tc + A / x**w
#
# to the data in "data_file", each line of which contains the data for one point,
# x_i, y_i, sigma_i.
#
# Uses the built-in routine "curve_fit" in the scipy package. Note that this
# requires the error bars to be corrected, as with gnuplot
#
from pylab import *
from scipy.optimize import curve_fit
from scipy.stats import chi2
fname = sys.argv[1] if len(sys.argv) > 1 else 'data.txt'
x, y, err = np.loadtxt(fname, unpack=True) # read in the data
n = len(x) # the number of data points
p0 = [-0.25, 0.2, 2.8] # initial values of parameters
f = lambda x, Tc, w, A: Tc + A/x**w # define the function to be fitted
# note python's lambda notation
p, covm = curve_fit(f, x, y, p0, err) # do the fit
Tc, w, A = p
chisq = sum(((f(x, Tc, w, A) - y)/err)**2) # compute the chi-squared
ndf = n -len(p) # no. of degrees of freedom
Q = 1. - chi2.cdf(chisq, ndf) # compute the quality of fit parameter Q
chisq = chisq / ndf # compute chi-squared per DOF
Tcerr, werr, Aerr = sqrt(diag(covm)/chisq) # correct the error bars
print 'Tc = %10.4f +/- %7.4f' % (Tc, Tcerr)
print 'A = %10.4f +/- %7.4f' % (A, Aerr)
print 'w = %10.4f +/- %7.4f' % (w, werr)
print 'chi squared / NDF = %7.4lf' % chisq
print 'Q = %10.4f' % Q
When applied to the data in `http://physics.ucsc.edu/~peter/bad-honnef/data.HW4` the output is
Tc = -0.2570 +/- 1.4775
A = 2.7878 +/- 0.8250
w = 0.2060 +/- 0.3508
chi squared / NDF = 0.2541
Q = 0.9073
### Gnuplot
#
# Gnuplot script to plot points, do a fit to a non-linear model
#
# y = Tc + A / x**w
#
# with respect to Tc, A and w, and display the points, fit, fit parameters,
# error bars, chi-squared per degree of freedom, and goodness of fit parameter
# on the plot.
#
# Here the data is assumed to be a file "data.HW4", each line containing
# information for one point (x_i, y_i, sigma_i). The script produces a
# postscript file, called here "HW4a.eps".
#
set size 1.0, 0.6
set terminal postscript portrait enhanced
set output "HW4a.eps"
set fit errorvariables # needed to be able to print error bars
f(x) = Tc + A / x**w # the fitting function
set xlabel "1/x^{/Symbol w}"
set ylabel "y"
set label "y = T_c + A / x^{/Symbol w}" at 0.1, 0.7
Tc = 0.3 # need to specify initial values
A = 1
w = 0.2
fit f(x) "data.HW4" using 1:2:3 via Tc, A, w # do the fit
set xrange [0.07:0.38]
g(x) = Tc + A * x
h(x) = 0 + 0 * x
ndf = FIT_NDF # Number of degrees of freedom
chisq = FIT_STDFIT**2 * ndf # chi-squared
Q = 1 - igamma(0.5 * ndf, 0.5 * chisq) # the quality of fit parameter Q
#
# Below note how the error bars are (a) corrected by dividing by
# FIT_STDFIT, and (b) are displayed on the plot, in addition to the fit
# parameters, neatly formatted using sprintf.
#
set label sprintf("T_c = %5.3f +/- %5.3f",Tc, Tc_err/FIT_STDFIT) at 0.25, 0.33
set label sprintf("{/Symbol w} = %5.3f +/- %5.3f",w, w_err/FIT_STDFIT) at 0.25, 0.27
set label sprintf("A = %5.2f +/- %5.2f",A, A_err/FIT_STDFIT) at 0.25, 0.21
set label sprintf("{/Symbol c}^2 = %5.2f", chisq) at 0.25, 0.15
set label sprintf("{/Symbol c}^2/NDF = %5.2f", FIT_STDFIT**2) at 0.25, 0.09
set label sprintf("Q = %5.2f", Q) at 0.25, 0.03
#
# Plot the data and the fit
#
plot "data.HW4" using (1/$1**w):2:3 with errorbars notitle lc rgb "red" lw 3 pt 8 ps 1.5, \
g(x) notitle lc rgb "blue" lw 3 lt 2 , \
h(x) notitle lt 3 lw 4
The plot below shows the result of acting with this gnuplot script on the data at `http://physics.ucsc.edu/~peter/bad-honnef/data.HW4`. The results agree with those of the python script above. {width="11cm"}
I’m grateful to Alexander Hartmann for inviting me to give a lecture at the Bad Honnef School on “Efficient Algorithms in Computational Physics”, which provided the motivation to write up these notes, and also for a helpful comment on the need to resample the data to get error bars from fits to non-linear models. I would also like to thank Matt Wittmann for helpful discussions about fitting and data analysis using `python` and for permission to include his python codes. I am also grateful to Wittmann and Christoph Norrenbrock for helpful comments on an earlier version of the manuscript.
[^1]: The factor of $N-1$ rather than $N$ in the expression for the sample variance in Eq. (\[sigmafromdata\]) needs a couple of comments. Firstly, the final answer for the error bar on the mean, Eq. below, will be independent of how this intermediate quantity is defined. Secondly, the $N$ terms in Eq. (\[sigmafromdata\]) are not all independent since $\overline{x}$, which is itself given by the $x_i$, is subtracted. Rather, as will be discussed more in the section on fitting, Sec. \[sec:fit\], there are really only $N-1$ independent variables (called the “number of degrees of freedom” in the fitting context) and so dividing by $N-1$ rather than $N$ has a rational basis. However, this is not essential and many authors divide by $N$ in their definition of the sample variance.
[^2]: $\chi^2$ should be thought of as a single variable rather than the square of something called $\chi$. This notation is standard.
[^3]: Although this result is only valid if the fitting model is linear in the parameters, it is usually taken to be a reasonable approximation for non-linear models as well.
[^4]: It is conventional to include the factor of $1/2$.
| ArXiv |
---
abstract: |
We propose a method that is able to analyze chaotic time series, gained from experimental data. The method allows to identify scalar time-delay systems. If the dynamics of the system under investigation is governed by a scalar time-delay differential equation of the form $\frac{dy(t)}{dt} = h(y(t),y(t-\tau_0))$, the delay time $\tau_0$ and the function $h$ can be recovered. There are no restrictions to the dimensionality of the chaotic attractor. The method turns out to be insensitive to noise. We successfully apply the method to various time series taken from a computer experiment and two different electronic oscillators.
P.A.C.S.: 05.45.+b
author:
- |
M. J. B"unner, M. Popp, Th. Meyer, A. Kittel, J. Parisi [^1]\
[*Physical Institute, University of Bayreuth, D-95440 Bayreuth, Germany*]{}
date: 'April, 24th, 1996'
title: 'A Tool to Recover Scalar Time-Delay Systems from Experimental Time Series'
---
Time series analysis of chaotic systems has gained much interest in recent years. Especially, embedding of time series in a reconstructed phase space with the help of time-delayed coordinates was widely used to estimate fractal dimensions of chaotic attractors [@takens; @grassberger] and Lyapunov exponents [@wolf]. It is the advantage of embedding techniques that the time series of only one variable has to be analyzed, even if the investigated system is multi-dimensional. Furthermore, it can be applied, in principle, to any dynamical system. Unfortunately, the embedding techniques only yield information, if the dimensionality of the chaotic attractor under investigation is low. Another drawback is the fact that it does not give any information about the structure of the dynamical system, in the sense, that one is able to identify the underlying instabilities. In the following, we propose a method which is taylor-suited to identify scalar systems with a time-delay induced instability. We will show that the differential equation can be recovered from the time series, if the investigated dynamics obeys a scalar time-delay differential equation. There are no restrictions to the dimensionality of the chaotic attractor. Additionally, the method has the advantage to be insensitive to noise.
We consider the time evolution of scalar time-delay differential equations $$\label{tdde}
\dot{y}(t) = h(y(t),y(t-\tau_0)),$$ with the initial condition $$y(t)=y_0(t), \hspace{1.0cm} -\tau_0<t<0. \nonumber$$ The dynamics is supposed to be bounded in the counter-domain $\cal{D}$, $y(t) \in \cal{D}, \forall$ $t$. In equation (\[tdde\]), the time derivative of $y(t)$ does not only depend on the state of system at the time $t$, but there also exist nonlocal correlations in time, because the function $h$ additionally depends on the time-delayed value $y(t-\tau_0)$. These nonlocal correlations in time enable scalar time-delay systems to exhibit a complex time evolution. The number of positive Lyapunov exponents increases with the delay time $\tau_0$ [@farmer]. Scalar time-delay systems, therefore, constitute a major class of dynamical systems which exhibit hyperchaos [@roessler]. In general, though, the nonlocal correlations in time are not at all obvious from the time series. A state of the system (\[tdde\]) is uniquely defined by a function on an interval of length $\tau_0$. Therefore, the phase space of scalar time-delay systems must be considered as infinite dimensional. The trajectory in the infinite dimensional phase space $\vec{y}(t)=\{y(t'),
t-\tau_0<t'<t\}$ is easily obtained from the time series. The scalar time series $y(t)$, therefore, encompasses the complete information about the trajectory $\vec{y}(t)$ in the infinite dimensional phase space.
The main idea of our analysis method is the following. We project the trajectory $\vec{y}(t)$ from the infinite dimensional phase space to a three-dimensional space which is spanned by the coordinates $(y_{\tau_0}=y(t-\tau_0),y=y(t),\dot{y}=\dot{y}(t))$. In the $(y_{\tau_0},y,\dot{y})$-space the differential equation (\[tdde\]) determines a two-dimensional surface $h$. The projected trajectory $\vec{y}_{\tau_0}(t)=(y(t-\tau_0),y(t),\dot{y}(t))$, therefore, is confined to the surface $h$ and is not able to explore other directions of the $(y_{\tau_0},y,\dot{y})$-space. From this, we conjecture, that the fractal dimension of the projected attractor has to be between one and two. Furthermore, it follows that any intersection of the chaotic attractor with a surface $k(y_{\tau_0},y,\dot{y})=0$ yields a curve. More precisely spoken, if one transforms the projected trajectory $\vec{y}_{\tau_0}(t)$ to a series of points $\vec{y}_{\tau_0}^i=(y^i_{\tau_0},y^i,\dot{y}^i)$ that fulfill the condition $k(y^i_{\tau_0},y^i,\dot{y}^i)=0$, the series of points $(y^i_{\tau_0},y^i,\dot{y}^i)$ contract to a curve and its dimension has to be less than or equal to one. In general, it cannot be expected that one is able to project a chaotic attractor of arbitrary dimension to a three-dimensional space, in the way that its projection is embedded in a two-dimensional surface. We, nevertheless, demonstrate that this is always possible for chaotic attractors of scalar time-delay systems (\[tdde\]).
In the following, we will show that such finding can be used to reveal nonlocal correlations in time from the time series. If the dynamics is of the scalar time-delay type (\[tdde\]), the appropriate delay time $\tau_0$ and the function $h(y,y_{\tau_0})$ can be recovered . The trajectory in the infinite dimensional phase space $\vec{y}(t)$ is projected to several three-dimensional $(y_{\tau},y,\dot{y})$-spaces upon variation of $\tau$. The appropriate value $\tau=\tau_0$ is just the one for which the projected trajectory $\vec{y}_{\tau}$ lies on a surface, representing a fingerprint of the time-delay induced instability. Projecting the trajectory $\vec{y}$ to the $(y_{\tau_0},y,\dot{y})$-space, the projected trajectory yields the surface $h(y,y_{\tau_0})$ in the counter-domain $\cal{D} \times \cal{D}$. With a fit procedure the yet unknown function $h(y,y_{\tau_0})$ can be determined in $\cal{D} \times
\cal{D}$. Therefore, the complete scalar time-delay differential equation has been recovered from the time series. In some cases, it is more convenient to intersect the trajectory $\vec{y}_{\tau}$ in the $(y_{\tau},y,\dot{y})$-space with a surface $k(y_{\tau},y,\dot{y})=0$ which yields a series of points $\vec{y}_{\tau}^i=(y^i_{\tau},y^i,\dot{y}^i)$. For $\tau=\tau_0$, the points come to lie on a curve and the fractal dimension of the point set has to be less than or equal to one.
The analysis method is also applicable if noise is added to the time series. The only effect of additional noise is that the projected time series in the $(y_{\tau_0},y,\dot{y})$-space is not perfectly enclosed in a two-dimensional surface, but the surface is somewhat blurred up. If the analysis is done with an intersected trajectory, the alignment of the noisy data is not perfect. The arguments presented above do not require the dynamics to be settled on its chaotic attractor. Therefore, it is also possible to analyze transient chaotic dynamics. Recently, the coexistence of attractors of time-delay systems has been pointed out [@losson]. The only requirement for the analysis method is that the trajectory obeys the time-evolution equation (\[tdde\]) which holds for all coexisting attractors in a scalar time-delay system. Therefore, the method is applicable, no matter in which attractor the dynamics has decided to settle. The analysis requires only short time series, which makes it well-suited to be applied on experimental situations. We successfully apply the method to time series gained from a computer experiment and from two different electronic oscillators. We show the robustness of the method to additional noise by analyzing noisy time series.
We numerically calculated the time series of the scalar time-delay differential equation $$\begin{aligned}
\label{tdde2}
\dot{y}(t) & = & f(y_{\tau_0})- g(y),\\
f(y_{\tau_0}) & = & \frac{2.7y_{\tau_0}}{1+y_{\tau_0}^{10}} +c_0 \nonumber \\
g(y) & = & -0.567y + 18.17y^2 -38.35y^3+28.56y^4-6.8y^5 -c_0 \nonumber\end{aligned}$$ with the initial conditon $$y(t) = y_0(t),\hspace{1.0cm} -\tau_0<t<0,\nonumber$$ which is of the form (\[tdde\]) with $h(y_{\tau_0},y)=f(y_{\tau_0})-g(y)$. The function $g$ has been chosen to be non-invertible in the counter-domain $\cal{D}$. The definition of the functions $f$ and $g$ is ambiguous in the sense that adding a constant $c_0$ to $f$ can always be cancelled by subtracting $c_0$ from $g$ without changing $h$ and, therefore, leaving the dynamics of equation (\[tdde2\]) unchanged. The control parameter is the delay time $\tau_0$. Equation (\[tdde2\]) is somewhat similar to the Mackey-Glass equation [@mkg], except for the function $g$, which is linear in the Mackey-Glass system. Part of the time series is shown in Fig. 1. We used $500,000$ data points with a time step of $0.01$ for the analysis. The dimension of the chaotic attractor was estimated with the help of the Grassberger-Procaccia algorithm [@grassberger] to be clearly larger than $5$ . To recover the delay time $\tau_0$ and the functions $f$ and $g$ from the time series, we applied the analysis method outlined above. We projected the trajectory $\vec{y}(t)$ from the infinite dimensional phase space to several $(y_{\tau},y,\dot{y})$-spaces under variation of $\tau$ and intersected the projected trajectory $\vec{y}_{\tau}$ with the $(y=1.1)$-plane, which is repeatedly traversed by the trajectory, as can be seen in Fig. 1. The results are the times $t^i$ where the trajectory traverses the $(y=1.1)$-plane and the intersection points $\vec{y}_{\tau}^i=(y^i_{\tau},1.1,\dot{y}^ i)$. For $\tau$ being the appropriate value $\tau_0$, the point set $\vec{y}_{\tau_0}^i$ is correlated via equation (\[tdde2\]) $$\label{f}
\dot{y}^i = f(y^i_{\tau_0})- g(1.1)$$ and, therefore, must have a fractal dimension less than or equal to one. Then, we ordered the $(y^i_{\tau},\dot{y}^ i)$-points with respect to the values of $y^i_{\tau}$. A simple measure for the alignment of the points is the length $L$ of a polygon line connecting all ordered points $(y^i_{\tau},\dot{y}^ i)$. The length $L$ as a function of $\tau$ is shown in Fig. 2. For $\tau=0$, $L(\tau)$ is minimal, because the points $(y^i_\tau,\dot{y}^ i)$ are ordered along the diagonal in the $(y^i_\tau,\dot{y}^ i)$-plane. $L(\tau)$ increases with $\tau$ and eventually reaches a plateau, where the points $(y^i_\tau,\dot{y}^ i)$ are maximally uncorrelated. This is due to short-time correlations of the signal. Eventually, $L(\tau)$ decreases again and shows a dip for $\tau$ reaching the appropriate value $\tau_0$. A further decrease of $L(\tau)$ is observed for $\tau=2\tau_0$. In Fig. 3(a)-(c), we show the projections $\vec{y}_{\tau}(t)$ of the trajectory $\vec{y}(t)$ from the infinite dimensional phase space to different $(y_{\tau},y,\dot{y})$-spaces under variation of $\tau$. Clearly, for $\tau$ approaching the appropriate value $\tau_0$, the appearance of the projected trajectory changes. In Fig. 3(c), the projected trajectory is embedded in a surface which is determined by the function $h$. In Fig. 3(d)-(f) we show the point set $(y^i_\tau,\dot{y}^ i)$ resulting from the intersection of the projected trajectory $\vec{y}_{\tau}$ with the $(y=1.1)$-plane. The point set is projected to the $(y_\tau,\dot{y})$-plane. According to equation (\[f\]), the points are aligned along the function $f$ for $\tau=\tau_0$. With the appropriate value $\tau_0$, we are in the position to recover the functions $f$ and $g$ from the time series. The functions $f$ and $g$ are ambiguous with respect to the addition of a constant term $c_0$, as has been outlined above. Therefore, one is free to remove the ambiguity by invoking an additional condition which we choose to be $$\label{norm}
g(1.1)=0.$$ Then, equation (\[f\]) reads $$\label{f_r}
\dot{y}^i = f(y^i_{\tau_0}).$$ Therefore, function $f$ is recovered by analyzing the intersection points $\vec{y}_{\tau_0}^i$ in the $(\dot{y},y_{\tau_0})$-plane. To recover the function $g$, we intersected the time series with the $(y_{\tau_0}=1.1)$-plane. The resulting point set $\vec{y}_{\tau_0}^j=(\dot{y}^j,y^j)$ is correlated via $$\label{g_r}
\dot{y}^j = f(1.1)-g(y^j).$$ The value $f(1.1)$ has been taken from the time series using equation (\[f\_r\]). In Fig. 4(a)-(b), we compare the functions $f$ and $g$, as they have been defined in equation (\[tdde2\]) with the recovery of the functions $f$ and $g$ from the time series. We emphasize that no fit parameter is involved.
We checked the robustness of the method to additional noise by analyzing noisy time series, which had been produced by adding gaussian noise to the time series of equation (\[tdde2\]). We analyzed two noisy time series with a signal-to-noise ratio (SNR) of $10$ and $100$. In both cases, the additional noise was partially removed with a nearest-neighbor filter (for SNR = 100, average over six neighbors; for SNR = 10, average over twenty neighbors). After that, the noisy time series were analyzed in the same way as has been described above. The inset of Fig. 2 shows the result of the analysis. The length $L$ of the polygon line exhibits a local minimum for $\tau=\tau_0$. In the case of the time series with a SNR of $10$, the local minimum is again sharp, but somewhat less pronounced. We conjecture that the method is robust with respect to additional noise and, therefore, well suited for the analysis of experimental data.
Finally, we successfully applied the method to experimental time series gained from two different types of electronic oscillators. The first one is the Shinriki oscillator [@shinriki; @reisner]. The dynamics of the second oscillator [@pyragas] is time-delay induced and mimics the dynamics of the Mackey-Glass equation. In both cases, we intersected the trajectory with the $(\dot{y}=0)$-plane. The resulting point set was represented in a $(y_\tau,y)$-space with different values of $\tau$. Then, we ordered the points with respect to $y_\tau$ and the length $L$ of a polygon line connecting all ordered points $(y^i_\tau,y^i)$ was measured. The results are presented in Fig. 5 (a) and Fig. 5 (b). In both cases, $L(\tau)$ has a local minimum for small values of $\tau$ as a result of short-range correlations in time. $L(\tau)$ increases in time and reaches a plateau. For the Shinriki oscillator, no further decrease of $L(\tau)$ is observed for increasing $\tau$ (Fig. 5(a)). Such finding clearly shows that the dynamics of the Shinriki oscillator is not time-delay induced. Analyzing the Mackey-Glass oscillator (Fig. 5(b)), one finds sharp dips in $L(\tau)$ for $\tau=\tau_0$ and $\tau=2\tau_0$. This is a direct evidence for correlations in time, which are induced by the time delay (for details see [@physletta96]). Obviously, the method is able to identify nonlocal correlations in time from the time series. Eventually, the nonlinear characteristics of the electronic oscillator is compared to its recovery from the time series (Fig. 5 (c)).
In conclusion, we have presented a method capable to reveal nonlocal correlations in time of scalar systems by analyzing the time series. If the dynamics of the investigated system is governed by a scalar time-delay differential equation, we are able to recover the scalar time-delay differential equation. There are no constraints on the dimensionality of the attractor. Since scalar time-delay systems are able to exhibit high-dimensional chaos, our method might pave the road to inspect high-dimensional chaotic systems, where conventional time-series analysis techniques already fail. Furthermore, the motion is not required to be settled on its attractor. The method is insensitive against additional noise. We have successfully applied the method to time series gained from a computer experiment and to experimental data gained from two different types of electronic oscillators.
While, in general, the verification of dynamical models is a highly complicated task, we have shown that the identification of scalar time-delay systems can be accomplished easily and, thus, allows a detailed comparison of the model equation with experimental time series. In several disciplines, e.g., hydrodynamics [@villermaux95] , chemistry [@khrustova95], laser physics [@ikeda87], and physiology [@mkg; @longtin90], time-delay effects have been proposed to induce dynamical instabilities. With the help of our method, there is a good chance to verify these models by analyzing the experimental time series. If the dynamics is indeed governed by a time delay, the delay time and the time-evolution equation can be determined. Current and future research activities of the authors concentrate on extending the time-series analysis method to non-scalar time-delay systems as well as to time-delay systems with multiple delay times.
We thankfully acknowledge valuable discussions with J. Peinke and K. Pyragas and financial support of the Deutsche Forschungsgemeinschaft.
[XXX]{} F. Takens, Lect. Notes Math. [**898**]{} (1981) 366. P. Grassberger, I. Procaccia, Physica D [ **9**]{} (1983) 189. A. Wolf, J. B. Swift, H. L. Swinney, J. Vastano, Physica D [**16**]{} (1985) 285. J. D. Farmer, Physica D [**4**]{} (1982) 366. O. E. Rössler, Z. Naturforsch. [**38a**]{} (1983) 788. J. Losson, M. C. Mackey, A. Longtin, Chaos (AIP), [**3**]{} (1993), 167. M. C. Mackey, L. Glass, Science [**197**]{} (1977) 287. M. Shinriki, M. Yamamoto, S. Movi, Proc. IEEE [**69**]{} (1981) 394. B. Reisner, A. Kittel, S. Lück, J. Peinke, J. Parisi, Z. Naturforsch.[ **50a**]{} (1995) 105. A. Namajunas, K. Pyragas, A. Tamasevicius, Phys. Lett. A [**201**]{} (1995) 42. M. J. Bünner, M. Popp, Th. Meyer, A. Kittel, U. Rau, J. Parisi, Phys. Lett. A [**211**]{} (1996) 345. E. Villermaux, Phys. Rev. Lett. [**75**]{} (1995) 4618. N. Khrustova, G. Veser, A. Mikhailov, Phys. Rev. Lett. [**75**]{} (1995) 3564. K. Ikeda, K. Matsumoto, Physica D [**29**]{} (1987) 223. A. Longtin, J. G. Milton, J. E. Bos, M. C. Mackey, Phys. Rev. A [**41**]{} (1990) 6992.
Figure captions {#figure-captions .unnumbered}
===============
- Time series of the scalar time-delay system (\[tdde2\]) obtained from a computer experiment ($\tau_0=40.00$).
- Length $L$ of the polygon line connecting all ordered points of the projected point set $(y^i_\tau,\dot{y}^ i)$ versus $\tau$. $L$ has been normalized so that a maximally uncorrelated point set has the value $L=1.0$. The inset shows a close-up of the $\tau$-axis around the local minimum at $\tau=\tau_0=40.00$. Additionally, $L(\tau)$-curves gained from the analysis of noisy time series are shown (no additional noise – straight line, signal-to-noise ratio of $100$ – open circles, and signal-to-noise ratio of $10$ – squares).
- (a)-(c): Trajectory $\vec{y}_{\tau}(t)$ which has been projected from the infinite dimensional phase space to the $(y_{\tau},y,\dot{y})$-space under variation of $\tau$. (a) $\tau=20.00$. (b) $\tau=39.60$. (c) $\tau=\tau_0=40.00$. (d)-(f): Projected point set $\vec{y}_{\tau}^i=(y^i_\tau,\dot{y}^ i)$ resulting from the intersection of the projected trajectory $\vec{y}_{\tau}(t)$ with the $(y=1.1)$- plane under variation of $\tau$. (d) $\tau=20.00$. (e) $\tau=39.60$. (f) $\tau=\tau_0=40.00$.
- \(a) Comparison of the function $f$ (line) of equation (\[tdde2\]) with its recovery from the time series (points). (b) Comparison of the function $g$ (line) of equation (\[tdde2\]) with its recovery from the time series (points).
- Length $L$ of the polygon line connecting all ordered points of the projected point set $\vec{y}_{\tau}^i=(y^i_\tau,y^ i)$ versus $\tau$ for (a) the Shinriki and (b) the Mackey-Glass oscillator. $L(\tau)$ has been normalized so that it has the value $L=1$ for an uncorrelated point set. (c) Comparison of the nonlinear characteristics of the Mackey-Glass oscillator, which is the function $f(y_{\tau_0})$ of an ansatz of the form $h(y,y_{\tau_0}) = f(y_{\tau_0}) + g(y)$, measured directly on the oscillator (line) with its recovery from the time series (dots).
[^1]: published in Phys. Rev. E [**54**]{} (1996) R3082.
| ArXiv |
---
author:
- |
(BES Collaboration)\
\
M. Ablikim
- 'J. Z. Bai'
- 'Y. Ban'
- 'X. Cai'
- 'H. F. Chen'
- 'H. S. Chen'
- 'H. X. Chen'
- 'J. C. Chen'
- Jin Chen
- 'Y. B. Chen'
- 'Y. P. Chu'
- 'Y. S. Dai'
- 'L. Y. Diao'
- 'Z. Y. Deng'
- 'Q. F. Dong'
- 'S. X. Du'
- 'J. Fang'
- 'S. S. Fang[^1]'
- 'C. D. Fu'
- 'C. S. Gao'
- 'Y. N. Gao'
- 'S. D. Gu'
- 'Y. T. Gu'
- 'Y. N. Guo'
- 'Z. J. Guo[^2]'
- 'F. A. Harris'
- 'K. L. He'
- 'M. He'
- 'Y. K. Heng'
- 'J. Hou'
- 'H. M. Hu'
- 'J. H. Hu'
- 'T. Hu'
- 'G. S. Huang[^3]'
- 'X. T. Huang'
- 'X. B. Ji'
- 'X. S. Jiang'
- 'X. Y. Jiang'
- 'J. B. Jiao'
- 'D. P. Jin'
- 'S. Jin'
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- 'G. Li, [^4]'
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- 'W. G. Li'
- 'X. L. Li'
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- 'X. Q. Li'
- 'Y. F. Liang'
- 'H. B. Liao'
- 'B. J. Liu'
- 'C. X. Liu'
- 'F. Liu'
- Fang Liu
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- 'J. Liu[^5]'
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- 'J. P. Liu'
- Jian Liu
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- 'Y. C. Lou'
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- Yiyun Zhang
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date: 'Received: date / Revised version: date'
title: '**Study of ${J/\psi}$ decaying into $\omega{p\bar{p}}$**'
---
Introduction
============
Decays of the $J/\psi$ meson are regarded as being well suited for searches for new types of hadrons and for systematic studies of light hadron spectroscopy. Recently, a number of new structures have been observed in $J/\psi$ decays. These include strong near-threshold mass enhancements in the $p{\bar{p}}$ invariant mass spectrum from ${J/\psi}\rightarrow\gamma p{\bar{p}}$ decays [@bes1860], the $p \bar \Lambda$ and $K^-\bar \Lambda$ threshold enhancements in the $p \bar \Lambda$ and $K^-\bar \Lambda$ mass spectra in $J/\psi \rightarrow p K^- \bar \Lambda$ decays [@pkl], the $\omega\phi$ resonance in the $\omega\phi$ mass spectrum in the double-OZI suppressed decay $J/\psi\to\gamma \omega\phi$ [@goph], and a new resonance, the $X(1835)$, in $J/\psi\to\gamma \pi^+\pi^-\eta'$ decays [@x1835].
The enhancement $X(1860)$ in ${J/\psi}\rightarrow\gamma p{\bar{p}}$ can be fitted with an $S$- or $P$-wave Breit-Wigner (BW) resonance function. In the case of the $S$-wave fit, the mass is $1859^{+3}_{-10}$$^{+5}_{-25}$ MeV/$c^2$ and the width is smaller than 30 MeV/$c^2$ at the 90$\%$ confidence level (C.L.). It is of interest to note that a corresponding mass threshold enhancement is not observed in either $p \bar p$ cross section measurements or in $B$-meson decays [@B-ppbar].
This surprising experimental observation has stimulated a number of theoretical interpretations. Some have suggested that it is a $p \bar p$ bound state ([*baryonium*]{}) [@ppbar; @theory; @gao; @yan; @baryonium]. Others suggest that the enhancement is primarily due to final state interactions (FSI) between the proton and antiproton [@fsi1; @fsi2].
The CLEO Collaboration published results on the radiative decay of the $\Upsilon (1S)$ to the $p\bar p$ system [@cleoc], where no $p \bar p$ threshold enhancement is observed and the upper limit of the branching fraction is set at $B(\Upsilon(1S)\rightarrow\gamma
X(1860))B(X(1860)\rightarrow p\overline{p})<5\times10^{-7}$ at 90$\%$ C.L.. This enhancement is not observed in BES2 $\psi(2S) \to
\gamma p \bar p$ data either [@bes2psip] and the upper limit is set at $B(\psi(2S)\rightarrow\gamma X(1860))B(X(1860)\rightarrow
p\overline{p})<5.4\times10^{-6}$ at 90$\%$ C.L..
The investigation of the near-threshold ${p\bar{p}}$ invariant mass spectrum in other $J/\psi$ decay modes will be helpful in understanding the nature of the observed new structures and in clarifying the role of ${p\bar{p}}$ FSI effects. If the enhancement seen in ${J/\psi}{\rightarrow}{\gamma}{p\bar{p}}$ is from FSI, it should also be observed in other decays, such as ${J/\psi}{\rightarrow}\omega{p\bar{p}}$, which motivated our study of this channel. In this paper, we present results from an analysis of ${J/\psi}{\rightarrow}{\pi^+}{\pi^-}{\pi^0}{p\bar{p}}$ using a sample of $5.8 \times 10^7 J/\psi$ decays recorded by the BESII detector at the Beijing Electron-Positron Collider (BEPC). BES is a conventional solenoidal magnetic detector that is described in detail in Ref. [@bes]. BESII is the upgraded version of the BES detector [@besii]. A twelve-layer Vertex Chamber (VC) surrounds a beryllium beam pipe and provides track and trigger information. A forty-layer main drift chamber (MDC) located just outside the VC provides measurements of charged particle trajectories over $85\%$ of the total solid angle; it also provides ionization energy loss ($dE/dx$) measurements that are used for particle identification (PID). A momentum resolution of $\sigma
_p/p =1.78\%\sqrt{1+p^2}$ ($p$ in GeV/$c$) and a $dE/dx$ resolution of $\sim$8% are obtained. An array of 48 scintillation counters surrounding the MDC measures the time of flight (TOF) of charged particles with a resolution of about 200 ps for hadrons. Outside of the TOF counters is a 12 radiation length, lead-gas barrel shower counter (BSC), that operates in self quenching streamer mode and measures the energies and positions of electrons and photons over $80\%$ of the total solid angle with resolutions of $\sigma_{E}/E=0.21/\sqrt{E}$ ($E$ in GeV/$c^{2}$), $\sigma_{\phi}=7.9$ mrad, and $\sigma_{z}=2.3$ cm. External to a solenoidal coil, which provides a 0.4 T magnetic field over the tracking volume, is an iron flux return that is instrumented with three double-layer muon counters that identify muons with momentum greater than 500 MeV$/c$.
Monte-Carlo simulation is used to determine the mass resolution and detection efficiency, as well as to estimate the contributions from background processes. In this analysis, a GEANT3-based Monte-Carlo program (SIMBES), with a detailed simulation of the detector performance, is used. As described in detail in Ref. [@SIMBES], the consistency between data and Monte-Carlo has been validated using many physics channels from both $J/\psi$ and $\psi(2S)$ decays.
Analysis of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$, $\omega \to {\pi^+}{\pi^-}{\pi^0}$ {#analysis}
=======================================================================================
For candidate ${J/\psi}{\rightarrow}{\pi^+}{\pi^-}{\pi^0}{p\bar{p}}$ events, we require four well reconstructed charged tracks with net charge zero in the MDC and at least two isolated photons in the BSC. Each charged track is required to be well fitted to a helix, be within the polar angle region $|\cos\theta| <
0.8$, have a transverse momentum larger than 70 MeV/$c$, and have a point of closest approach of the track to the beam axis that is within 2 cm of the beam axis and within 20 cm from the center of the interaction region along the beam line. For each track, the TOF and $dE/dx$ information is combined to form a particle identification confidence level for the $\pi, K$ and $p$ hypotheses; the particle type with the highest confidence level is assigned to each track. The four charged tracks are required to consist of an unambiguously identified $p$, $\bar{p}$, ${\pi^+}$ and ${\pi^-}$ combination. An isolated neutral cluster is considered as a photon candidate when the angle between the nearest charged track and the cluster is greater than 5$^{\circ}$, the angle between the $\bar{p}$ track and the cluster is greater than 25$^{\circ}$ [@angpb], the first hit is in the beginning of six radiation lengths of the BSC, the difference between the angle of the cluster development direction in the BSC and the photon emission direction is less than 30$^{\circ}$, and the energy deposited in the shower counter is greater than 50 MeV. A four-constraint kinematic fit is performed to the hypothesis $J/\psi \to
{p\bar{p}}{\pi^+}{\pi^-}{\gamma}{\gamma}$, and, in the cases where the number of photon candidates exceeds two, the combination with the smallest $\chi^{2}_{{p\bar{p}}{\pi^+}{\pi^-}{\gamma}{\gamma}}$ value is selected. We further require that $\chi^{2}_{{p\bar{p}}{\pi^+}{\pi^-}{\gamma}{\gamma}} < 20$.
Figure \[ompi0\] shows the $\gamma \gamma$ invariant mass of the events which survive the above-listed criteria, where a distinct ${\pi^0}\to {\gamma}{\gamma}$ signal is evident. Candidate ${\pi^0}$ mesons are selected by requiring $|M_{{\gamma}{\gamma}}-m_{{\pi^0}}|<0.04$ GeV/$c^2$. After this selection, a total of 15260 events is retained. The ${{\pi^+}{\pi^-}{\pi^0}}$ invariant mass spectrum for these events is shown as data points with error bars in Fig. \[momegafit\], where prominent $\omega$ and $\eta$ signals are observed.
The backgrounds in the selected event sample are studied with Monte-Carlo simulations. We generated ${J/\psi}\rightarrow$ ${p\bar{p}}{\pi^+}{\pi^-}{\pi^0}$ decays as well as a variety of processes that are potential sources of background: ${J/\psi}\to {p\bar{p}}{\eta^{\prime}}({\eta^{\prime}}\to{\pi^+}{\pi^-}\eta)$; ${p\bar{p}}{\eta^{\prime}}({\eta^{\prime}}\to\rho^{0}{\gamma})$; ${p\bar{p}}{\pi^+}{\pi^-}$; ${\Lambda\bar{\Lambda}}{\pi^0}$; $\Sigma^0\bar{\Sigma}^0$; $\Sigma(1385)^-\bar{\Sigma}^+$; $\eta_c{\gamma}$; $\Delta^{++}\Delta^{--}$; ${\gamma}{p\bar{p}}{\pi^+}{\pi^-}$; $\Delta^{++}\bar{p}{\pi^-}$; $\Lambda\bar{\Sigma}^-\pi^+$ (+ [*c.c.*]{}); $\Sigma^0\pi^0\bar{\Lambda}$; $\Sigma(1385)^0\bar{\Sigma}^0$; $\Delta^{++}\Delta^{--}{\pi^0}$; and $\Xi^0\bar{\Xi}^0$, in proportion to the branching fractions listed in the Particle Data Group (PDG) Tables [@pdg2004]. The main background sources are found to be the decays $J/\psi \to \Lambda\bar{\Sigma}^-\pi^+$ (+ [*c.c.*]{}) and $\Delta^{++}\Delta^{--}\pi^0$. The ${\pi^+}{\pi^-}{\pi^0}$ invariant mass spectrum for background events that survive the selection criteria is shown as a solid histogram in Fig. \[momegafit\]; here no signal for $\omega{p\bar{p}}$ is evident.
The branching fraction for $J/\psi \to \omega p \bar p$ is computed using the relation $$B({J/\psi}{\rightarrow}\omega{p\bar{p}}) = \frac{N_{obs}}{N_{{J/\psi}}\cdot {\varepsilon}\cdot
B(\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0})\cdot B({\pi^0}{\rightarrow}{\gamma}{\gamma})}.$$
Here, $N_{obs}$ is the number of observed events; $N_{{J/\psi}}$ is the number of ${J/\psi}$ events, $(57.7\pm 2.6)\times 10^6$ [@jpsinum]; ${\varepsilon}$ is the Monte-Carlo determined detection efficiency; and $B(\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0})$ and $B({\pi^0}{\rightarrow}{\gamma}{\gamma})$ are the $\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0}$ and ${\pi^0}{\rightarrow}{\gamma}{\gamma}$ branching fractions.
The ${{\pi^+}{\pi^-}{\pi^0}}$ invariant mass spectrum shown in Fig. \[momegafit\] is fitted using an unbinned maximum likelihood fit with resolution broadened BW functions to represent the $\omega$ and $\eta$ signal peaks. The mass resolutions are obtained from Monte-Carlo simulation to be 12 MeV$/c^{2}$ for the $\omega$ and and 14 MeV$/c^{2}$ for the $\eta$. The masses and widths of the $\omega$ and $\eta$ are fixed at their PDG values [@pdg2004]. A 4th-order Chebychev polynomial is used to describe the background. The fit gives an $\omega$ signal yield of 2449$\pm$69 events. The detection efficiency from a uniform-phase-space Monte-Carlo simulation of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ ($\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0}$, ${\pi^0}{\rightarrow}{\gamma}{\gamma}$) is $4.9 \pm 0.1)$%. The branching fraction is determined to be: $$B({J/\psi}{\rightarrow}\omega{p\bar{p}})=(9.8\pm0.3)\times 10^{-4},$$ where the error is statistical only.
We use this sample with $|M_{{\pi^+}{\pi^-}{\pi^0}}-0.783|<0.03$ GeV/$c^2$ to study the near-threshold region of the ${p\bar{p}}$ invariant mass spectrum. Figure \[dalitz\] shows a Dalitz plot for the selected ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ candidates, where no obvious structure is observed although it is not a uniform distribution. Figure \[mppbfit\] shows the threshold behavior of the $p \bar p$ invariant mass distribution. The dotted curve in the figure indicates how the acceptance varies with invariant mass.
The backgrounds in the $p \bar p$ threshold region mainly come from the decays of $J/\psi \to \Lambda\bar{\Sigma}^-\pi^+$ (+ [*c.c.*]{}) and $\Delta^{++}\Delta^{--}\pi^0$. The $M({p\bar{p}})$ dependence of this background can be modeled by appropriately scaled data from the $\omega$ sidebands (0.663 GeV/c$^2$ $<M_{{{\pi^+}{\pi^-}{\pi^0}}}<$0.723 GeV/c$^2$ and 0.843 GeV/c$^2$ $<M_{{{\pi^+}{\pi^-}{\pi^0}}}<$0.903 GeV/c$^2$). The contributions of sideband and non-resonant $\omega{p\bar{p}}$ events can be well described by a function of the form $$f(\delta) =
N(\delta^{\frac{1}{2}}+a_1\delta^{\frac{3}{2}}+a_2\delta^{\frac{5}{2}})$$ with $\delta\equiv M_{{p\bar{p}}} - 2m_p $.
In Fig. \[mppbfit\], no significant excess over the background plus non-resonant terms is evident. A Bayesian approach [@pdg2004] is employed to extract the upper limit on the branching fraction of ${J/\psi}{\rightarrow}\omega X(1860)$. An acceptance-weighted $S$-wave BW function $$BW(M) \propto \frac{q^{(2l+1)}k^3}{(M^2-M_0^2)^2-M_0^2\Gamma^2}\cdot\varepsilon(M)$$ is used to represent the low-mass enhancement. Here, $\Gamma$ is a constant width, $q$ is the momentum of proton in the ${p\bar{p}}$ rest frame, $l$ is the relative orbital angular momentum of $p$ and $\bar
p$, $k$ is the momentum of $\omega$, and $\varepsilon(M)$ is the detection efficiency obtained from Monte-Carlo simulation. The mass and width of the BW signal function are fixed to 1860 MeV/c$^2$ and 30 MeV/c$^2$, respectively. The contributions of background and non-resonant $\omega{p\bar{p}}$ events are presented by the function form $f(\delta)$, where the parameters $a_1$ and $a_2$ are allowed to float. As shown in Fig. \[mppbfit\], the solid curve is the fit of the $M_{{p\bar{p}}}$ - 2$m_p$ with the BW signal function and $f(\delta)$ function described above. Using the Bayesian method, the 95% C.L. upper limit on the number of observed signal events is 29. Since the $J^{PC}$ of X(1860) is unknown, we use simulated events distributed uniformly in phase space to determine a detection efficiency of ${J/\psi}{\rightarrow}\omega
X(1860)$ ($X(1860){\rightarrow}{p\bar{p}}$, $\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0}$, ${\pi^0}{\rightarrow}{\gamma}{\gamma}$) of $(4.7 \pm 0.1)$%. The upper limit of the branching fraction, without considering the systematic errors, is then: $$B({J/\psi}{\rightarrow}\omega X(1860))\cdot B( X(1860){\rightarrow}{p\bar{p}}))$$ $$< \frac{N_{obs}^{UL}}{N_{{J/\psi}}\cdot {\varepsilon}\cdot B(\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0})\cdot B({\pi^0}{\rightarrow}{\gamma}{\gamma})} = 1.2\times 10^{-5}.$$
Systematic errors {#syserrs}
=================
The systematic errors on the branching fractions are mainly due to uncertainties in the MDC tracking, kinematic fitting, particle identification (PID), photon detection, background estimation, the model used to describe hadronic interactions in the material of the detector, and the uncertainty of the total number of $J/\psi$ decays in the data sample.
The systematic error associated with the tracking efficiency has been carefully studied [@SIMBES]. The difference of the tracking efficiencies between data and Monte-Carlo is 2% per charged track; an 8% contribution to the systematic error associated with the efficiency for detecting the four-track final state is assigned. In Ref. [@SIMBES; @pnpi], the efficiencies for charged particle identification and photon detection are analyzed in detail. The systematic errors from PID and photon detection are 2% per proton (antiproton), 1% per pion and 2% per photon. In this analysis, with four charged tracks and two isolated photons; 6% is taken as the systematic error due to PID and 4% due to photon detection. The uncertainty due to kinematic fitting is studied using a number of exclusive $J/\psi$ and $\psi(2S)$ decay channels that are cleanly isolated without a kinematic fit [@rhopi; @etanpi]. It is found that the Monte-Carlo simulates the kinematic fit efficiency at the 5% or less level of uncertainty for almost all channels tested. Therefore, we take 5% as the systematic error due to the kinematic fit.
The background uncertainties come from the uncertainty of the background shape. For the branching fraction measurement of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$, changing the order of the polynomial background causes an uncertainty in the number of background events. For the upper limit determination of ${J/\psi}{\rightarrow}\omega X(1860)$, the uncertainty of background shape can be determined by the fitting results with the background shape fixed to the function form $f(\delta)$, derived from fitting the scaled $\omega$ sideband data plus phase-space generated $\omega{p\bar{p}}$ MC events. Respectively, 5% and 10% are taken as the systematic errors due to the background uncertainties in the branching fraction measurement of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ and the upper limit determination of ${J/\psi}{\rightarrow}\omega X(1860)$.
Different simulation models for the hadronic interactions in the material of the detector (GCALOR/FLUKA) [@gcalor; @fluka] give different efficiencies. Respectively, 4.8% and 11.4% are taken as the systemic errors due to the different hadronic models in the branching fraction measurement of ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ and the upper limit determination of ${J/\psi}{\rightarrow}\omega X(1860)$. In addition, if the $J^{P}$ of X(1860) is $0^{-}$ , the angular distribution of the $\omega$ would be $1+cos^{ 2}\theta$. A Monte-Carlo sample generated with the $\omega$ produced with a $1+cos^{2}\theta$ distribution and a uniform distribution for the X(1860) decay into ${p\bar{p}}$ results in an 8.5% reduction in detection efficiency. This difference is taken as the systematic error associated with the production model.
The branching fractions of $\omega{\rightarrow}{\pi^+}{\pi^-}{\pi^0}$ and ${\pi^0}{\rightarrow}{\gamma}{\gamma}$ are taken from the PDG tables. The errors of the intermediate decay branching fractions, as well as the uncertainty of the number of $J/\psi$ events [@jpsinum] also result in the systematic errors in the measurements.
The systematic errors from the different sources are listed in Table \[syserr\]. The total systematic errors for the branching fractions are obtained by adding up all the systematic sources in quadrature.
B(${J/\psi}{\rightarrow}\omega{p\bar{p}}$) Upper Limit
------------------------- -------------------------------------------- ------------- --
Tracking efficiency 8 8
Photon efficiency 4 4
Particle ID 6 6
Kinematic fit 5 5
Background uncertainty 5 10
Hadronic model 4.8 11.4
Production model - 8.5
Intermediate decays 0.8 0.8
Total ${J/\psi}$ events 4.7 4.7
Total systematic error 14.6 21.6
: Systematic error sources and contributions (%).
\[syserr\]
Summary
=======
With a $5.8 \times 10^7 {J/\psi}$ event sample in the BESII detector, the branching fraction ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ is measured as: $$B({J/\psi}{\rightarrow}\omega{p\bar{p}})=(9.8\pm 0.3\pm 1.4)\times 10^{-4}.$$
No obvious near-threshold ${p\bar{p}}$ mass enhancement in ${J/\psi}{\rightarrow}\omega{p\bar{p}}$ is observed, and the FSI interpretation of the $p \bar p$ enhancement in $J/\psi \to \gamma p \bar p$ is disfavored. A conservative estimate of the upper limit is determined by lowering the efficiency by one standard deviation. In this way, a 95% confidence level upper limit on the branching fraction $$B({J/\psi}{\rightarrow}\omega X(1860))\cdot B( X(1860){\rightarrow}{p\bar{p}}))< 1.5 \times 10^{-5}$$ is determined. The absence of the enhancement $X(1860)$ in $J/\psi \to \omega p \bar p$, $\Upsilon(1S) \to \gamma p \bar p$ and $\psi(2S) \to \gamma p \bar p$ also indicates its similar production property to that of $\eta'$ [@ichep06; @klempt], [*i.e.*]{}, $X(1860)$ is only largely produced in $J/\psi$ radiative decays.
Acknowledgments
===============
The BES collaboration thanks the staff of BEPC and computing center for their hard efforts. This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 10491300, 10225524, 10225525, 10425523, 10625524, 10521003, the Chinese Academy of Sciences under contract No. KJ 95T-03, the 100 Talents Program of CAS under Contract Nos. U-11, U-24, U-25, and the Knowledge Innovation Project of CAS under Contract Nos. U-602, U-34 (IHEP), the National Natural Science Foundation of China under Contract No. 10225522 (Tsinghua University), and the Department of Energy under Contract No. DE-FG02-04ER41291 (U. Hawaii).
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[^1]: Current address: DESY, D-22607, Hamburg, Germany
[^2]: Current address: Johns Hopkins University, Baltimore, MD 21218, USA
[^3]: Current address: University of Oklahoma, Norman, OK 73019, USA
[^4]: Current address: Universite Paris XI, LAL-Bat. 208–BP34, 91898 ORSAY Cedex, France
[^5]: Current address: Max-Plank-Institut fuer Physik, Foehringer Ring 6, 80805 Munich, Germany
[^6]: Current address: University of Toronto, Toronto M5S 1A7, Canada
[^7]: Current address: CERN, CH-1211 Geneva 23, Switzerland
[^8]: Current address: Laboratoire de l’Acc[é]{}l[é]{}rateur Lin[é]{}aire, Orsay, F-91898, France
[^9]: Current address: University of Michigan, Ann Arbor, MI 48109, USA
| ArXiv |
8.5in -30pt ł c
[FERMILAB–Pub–94/XXX-A\
March 1994]{}
[**Quantum Cosmology and Higher-Order Lagrangian Theories**]{}\
Henk van Elst$^{1a}$, James E. Lidsey$^{2b}$ & Reza Tavakol$^{1c}$\
$^1$[*School of Mathematical Sciences\
Queen Mary & Westfield College\
Mile End Road\
London E1 4NS, UK\
*]{} $^2$[*NASA/Fermilab Astrophysics Center\
Fermi National Accelerator Laboratory\
Batavia IL 60510, USA*]{}
> In this paper the quantum cosmological consequences of introducing a term cubic in the Ricci curvature scalar $R$ into the Einstein–Hilbert action are investigated. It is argued that this term represents a more generic perturbation to the action than the quadratic correction usually considered. A qualitative argument suggests that there exists a region of parameter space in which neither the tunneling nor the no-boundary boundary conditions predict an epoch of inflation that can solve the horizon and flatness problems of the big bang model. This is in contrast to the $R^2$–theory.
>
> e-mail: $^[email protected]; $^[email protected]; $^[email protected]
>
> PACS number(s): 98.80.Hw; 04.50.+h; 04.60.Kz; 98.80.Cq
Introduction
============
An important motivation for the development of the quantum cosmology programme has been to explain the initial conditions for the emergence of the Universe as a classical outcome. In principle one must find the form of the wave function $\Psi$ satisfying the Wheeler–DeWitt equation [@wdw]. This equation describes the annihilation of the wave function by the Hamiltonian operator and since it admits an infinite number of solutions, one must also choose the boundary conditions in order to specify the wave function uniquely. Such boundary conditions must be viewed as an additional physical law since, by definition, there is nothing external to the Universe. In practice one assumes, at least implicitly, that a finite subset of all possible boundary conditions is favoured by cosmological observations, in the sense that the wave functions corresponding to such boundary conditions predict outcomes which are compatible with observations. For example, if one believes in the inflationary scenario, the requirement that sufficient inflation occurred, in order to solve the assorted problems of the standard big bang model can, in principle, restrict the number of plausible boundary conditions.
Among the set of all possible choices the Vilenkin, or [*tunneling from nothing*]{}, boundary condition [@v1; @v2] and the Hartle–Hawking, or [*no-boundary*]{}, boundary condition [@HH83] have been the subject of intense discussion. Given the non-uniqueness of such conditions, the question arises as to the consequences of choosing different boundary conditions for the resulting wave function of the Universe and its corresponding probability measures. An important study in this regard is due to Vilenkin [@v2], who considered the effects of the above boundary conditions within the context of Einstein gravity minimally coupled to a self-interacting scalar field. He restricted his analysis to the minisuperspace corresponding to the spatially closed, isotropic and homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) Universe and showed that the tunneling wave function predicts initial states that are likely to lead to sufficient inflation, whereas the Hartle–Hawking wave function does not.
It is sometimes argued that this result indicates that observations favour the tunneling as opposed to the no-boundary boundary condition. However, the precise relation between the boundary conditions and the observations is determined by the specific models employed and since such models always involve idealisations in the form of a set of simplifying assumptions, it follows that the above conclusion can not be made [*a priori*]{}. Indeed it only makes sense in general if the correspondence between the observations and the boundary conditions is robust under physically motivated perturbations to the underlying quantum cosmological model.
Consequently, it is important to consider the ‘stability’ of the above conclusions. In particular, are the conclusions robust under higher-order perturbations to the Einstein–Hilbert action? Quadratic and higher-order terms in the Riemann curvature tensor and its traces appear in the low-energy limit of superstrings [@canetal85] and they also arise when the usual perturbation expansion is applied to General Relativity [@barchr83; @anttom86]. Such terms diverge as the initial singularity is approached, but can in principle be eliminated if higher-order corrections are included in the action. In four-dimensional space-times the Hirzebrucht signature and Euler number imply that the most general, four-dimensional gravitational action to quadratic order is S = d\^4x , where $R$ is the Ricci curvature scalar of the space-time with metric tensor $g_{\mu\nu}$, $g={\rm det}\,g_{\mu\nu}$, $C_{\alpha\beta\gamma\delta}$ is the Weyl conformal curvature tensor, $\kappa^{2}$ is the gravitational coupling constant and $\epsilon_1$ and $\gamma$ are coupling constants of dimension $(\mbox{length})^{2}$. The action simplifies further for spatially homogeneous and isotropic four-geometries, since the conformal flatness of these space-times implies that the Weyl tensor vanishes. The effects of including quadratic terms have been investigated in Refs. [@bg93; @hawlut84; @mijetal89]. In particular Mijić et al [@mijetal89] studied the effects of such perturbations on Vilenkin’s result [@v2] and found that those results remain robust in the sense that the inflationary scenario still favours the tunneling boundary condition in the presence of quadratic terms in the action. On the other hand Biswas and Guha have recently arrived at the opposite conclusion [@bg93].
The renormalisation of higher loop contributions introduces terms into the effective action that are higher than quadratic order. Consequently it is important to also study the effects of these additional terms. In this paper we shall investigate what happens to the wave function if an $R^3$-contribution is present. By employing the conformal equivalence of higher-order gravity theories with Einstein gravity coupled to matter fields, we argue that this term represents a more general perturbation to the Einstein–Hilbert action than the $R^2$-correction, at least within the context of four-dimensional FLRW space-times. We then consider the conditional probability that an inflationary epoch of sufficient duration can occur. We estimate how the qualitative behaviour of this quantity changes when higher-order perturbations to the action are included. Our main result is that for the $R^3$–theory there exists a finite region of parameter space in which neither of the boundary conditions discussed above predict an epoch of inflationary expansion that leads to the observed Universe. We use (dimensionless) Planckian units defined by $\hbar
= c = G = 1$ throughout and define $\kappa^2 = 8\pi$.
Higher-Order Lagrangians as Einstein Gravity plus Matter {#Lagrange}
========================================================
The wave function of the Universe in higher-order Lagrangian theories can be determined in one of two ways. It is well known that theories with a Lagrangian given by a differentiable function of the Ricci curvature scalar are conformally equivalent to Einstein gravity with a matter sector containing a minimally coupled, self-interacting scalar field [@whitt84; @m]. The precise form of the self-interaction is uniquely determined by the higher-derivative metric terms in the field equations. It follows that one can start either from the original action or the conformal action and derive the corresponding Wheeler–DeWitt equation [@hall91]. One takes the related Lagrangian as the defining feature of the theory and then applies the canonical quantisation rules. The advantage of the conformal transformation is that it allows the known results from Einstein gravity to be carried over to the higher-order examples and we shall follow such an approach in this paper.
Consider the general, $D$-dimensional, vacuum theory S = d\^Dx , where the Lagrangian $f(R)$ is some arbitrary differentiable function of the Ricci curvature scalar satisfying $\{ f(R),
df(R)/dR \} > 0$ and $g_D$ is the determinant of the $D$-dimensional space-time metric $g_{D\,\mu\nu}$. If we perform the conformal transformation [@m] ł[conf]{} \_[D]{} = \^2g\_[D]{} \^2 = ( 2\^2)\^[2/(D-2)]{} , and define a new scalar field ł[phi]{} | ( )\^[1/2]{} , the conformally transformed action takes the Einstein–Hilbert form ł[conformal]{} S = d\^D x , where the self-interaction potential is given by ł[\*]{} U(|) ( 2\^2 )\^[-D/(D-2)]{}( R(|) -f\[R(|)\] ) . Definition (\[phi\]) yields a correspondence between the values of the Ricci curvature scalar $R$ and the values of the scalar field $\bar{\phi}$. We shall consider the quadratic and cubic Lagrangians \[lagr\] f\_2 (R) & = & (R + \_1R\^2)\
f\_3 (R) & = & (R + \_1R\^2 +\_2R\^[3]{}) , in four dimensions, where the parameters $\epsilon_{1}$ and $\epsilon_{2}$ have dimensions $(\mbox{length})^{2}$ and $(\mbox{length})^{4}$ respectively before the introduction of Planckian units. The corresponding potentials for positive $\epsilon_1$ and $\epsilon_2$ are given by [@mijetal89; @b]: \[pot2\] U\_[f\_[2]{}]{}(|) & = & \^2\
\[pot3\] U\_[f\_[3]{}]{}(|) & = & (-2|) , and are semi-positive definite for all values of $\bar{\phi}$.
**Figures 1a & 1b**
In the classical $R^{3}$–theory the requirement that the inflationary epoch lasts sufficiently long implies that the coupling constants must satisfy $|\epsilon_2| \ll \epsilon_1\!^2$ [@b]. Moreover, the observed isotropy of the cosmic microwave background radiation requires that $\epsilon_{1}\approx 10^{11}$ [@mijetal89]. In view of these constraints we specify $\epsilon_1=10^{11}$ in the subsequent numerical calculations. Figures 1a and 1b illustrate the behaviour of the potentials (\[pot2\]) and (\[pot3\]) for $\epsilon_1 \approx
10^{11}$ and $\epsilon_2 \approx 10^{20}$. The effect of decreasing the value of the parameter $\epsilon_1$ is to increase the height of the plateau and the relative maximum of the potentials in the quadratic and cubic cases respectively. This reflects the fact that decreasing this parameter is equivalent to increasing the energy scales involved. In this sense there exists no continuous transformation from an $R^{2}$–theory to the ordinary Einstein–Hilbert action as this parameter approaches zero. In the neighbourhood of the origin of $\bar{\phi}$ corresponding to smaller values of $R$ the quadratic term in the action dominates and the potentials in this region are equivalent. This can be seen by expanding the last of the three terms in the square brackets of Eq. (\[pot3\]). The first-order contribution cancels the remaining terms in $U_{f_3}$ and the second-order term reduces the form of $U_{f_3}$ to that of $U_{f_2}$. Hence the two potentials are effectively identical if the third- and higher-order terms in the expansion can be neglected. It is straightforward to show that this is a consistent approximation if ł[consistent]{} | |\_[limit]{} ( ) . For polynomial Lagrangians with $f(R)=\left(\,\sum_{k=1}^{n}\,\epsilon_{k-1}\,
R^k\,\right)\,/\,2\kappa^{2}$, the detailed form of the corresponding potential $U(\bar{\phi} )$ is extremely complicated and generally not expressible in an analytically closed form. Nevertheless, one can determine the qualitative behaviour of the potential at small and large $\bar{\phi}$. Close to the origin the quadratic term in the action again dominates and the potential in this region is therefore similar to Eq. (\[pot2\]). The asymptotic behaviour at infinity, however, depends critically upon the combination of the highest degree $n$ of the polynomial and the dimensionality $D$ of the space-time [@m]. More precisely, for $D>2n$ the potential is unbounded from above, for $D=2n$ it flatens into a plateau and for $D<2n$ the potential has an exponentially decaying tail [@b]. In particular, if $D<2n$ the effective scalar field potential $U(\bar{\phi})$ is qualitatively equivalent to the cubic potential (\[pot3\]). As a result, when $D=4$ the qualitative behaviour of $U(\bar{\phi})$ does not change relative to the cubic case as terms with $n>3$ are considered, although the relative position of the maximum of $U(\bar{\phi})$ will be $n$-dependent. This implies that the $n=2$ contribution is rather special in four dimensions, whereas the $R^3$-term is in fact a more generic perturbation. Thus, it is instructive to consider this case further.
Behaviour of the Wave Function {#Psi}
==============================
Within the context of the spatially closed FLRW minisuperspace, the Wheeler–DeWitt equation derived from theory (\[conformal\]) has been solved for an arbitrary potential, subject to the condition that the momentum operator for the scalar field can be neglected [@v1; @v2]. This is self-consistent if $|dV/d\phi | \ll {\rm
max} \{ |V|, a^{-2} \}$, where $a$ represents the cosmological scale factor and ł[rescale]{} V U | .
The WKB approximations of the wave functions satisfying the quantum tunneling boundary condition ($\Psi_{V}$) and the Hartle–Hawking no-boundary proposal ($\Psi_{HH}$) then take the forms [@v2] \_[V]{} &=& (1-a\^2V)\^[-1/4]{} \
\_[HH]{} &=& (1-a\^2V)\^[-1/4]{} in the classically forbidden (Euclidian signature) region defined by $a^2\,V<1$, and \_[V]{} &=& e\^[i/4]{}(a\^2V-1)\^[-1/4]{} \
\_[HH]{} &=& 2(a\^2V-1)\^[-1/4]{} in the classically allowed (Lorentzian signature) region $a^2\,V>1$. Substituting for $V(\phi)$ from the potentials of the quadratic and cubic Lagrangians of Section \[Lagrange\], it can readily be seen that the wave functions corresponding to the quadratic and cubic theories have very different types of behaviour, at least for large $\phi$. In the quadratic case both $\Psi_{V}$ and $\Psi_{HH}$ remain bounded. However, for the cubic case $\Psi_{HH}$ becomes divergent in the classically allowed region whilst $\Psi_{V}$ remains regular. In this sense then the qualitative behaviour of the wave function satisfying the no-boundary proposal is fragile with respect to cubic perturbations to the action. This is significant because often the quadratic corrections to the action are taken as representative of higher-order perturbations.
To proceed it is important to ensure that for the regimes under consideration the conformal transformation (\[conf\]) remains non-singular. This is the case if the condition $df(R)/dR\neq 0$ is valid for all values of $R$. The conformal transformation is singular at the point R = - , in the $R^2$–theory and at the point R = -for the $R^{3}$–theory. Since $\epsilon_{1}$ and $\epsilon_{2}$ are taken to be positive, these conditions imply that in both cases the problematic values of $R$ lie in the region $R<0$. However, for a classical, spatially closed FLRW model, the Ricci curvature scalar is given by R = 6(1-q)()\^[2]{} + , where $q \equiv -\ddot{a}\,a / \dot{a}^{2}$ defines the deceleration parameter and a dot denotes differentiation with respect to cosmic proper time. Now if, as is generally assumed, the Universe tunnels into the Lorentzian region in an inflationary phase ($q<0$), it follows that $R$ will be positive-definite. Thus, the conformal transformation is self-consistent in these theories.
Interpretation of the Wave Function
===================================
In the previous section we saw that the wave functions corresponding to the tunneling and the Hartle–Hawking boundary conditions have qualitatively different modes of behaviour for the quadratic and cubic theories. To see what predictive effects such changes might have, we employ the notion of a probability density $\rho$ as is usually done. For the cases of the tunneling and the Hartle–Hawking boundary conditions respectively, $\rho$ takes the form [@v2] \_[V]{}(a,) &=& C\_[V]{}\[rhot\]\
\_[HH]{}(a,) &=& C\_[HH]{}\[rhohh\] on surfaces of constant scale factor in the classically allowed region of minisuperspace, where the normalisation constants $C_{V}$ and $C_{HH}$ are given by C\_[V]{}\^[-1]{} &=& \_[V()>0]{} d \
C\_[HH]{}\^[-1]{} &=& \_[V()>0]{} d . Since $\rho (\phi)$ is usually not normalisable, the common practice is to employ the notion of a conditional probability [@hall91]. One argues that the initial values of the scalar field must lie in the range $\phi_{min}<\phi_{i}<\phi_{P}$. The lower limit $\phi_{min}$ follows from the requirement that the Universe expands at least until the formation of large-scale structure and the upper bound follows from the condition that $V(\phi_P )\approx 1$, since the minisuperspace approximation is unlikely to be valid when the potential energy of the matter sector exceeds the Planck density. However, in a chaotic inflationary scenario there is a critical value of the scalar field, $\phi_{suf}$, and sufficient inflation occurs if $\phi_i >
\phi_{suf}$ but not for $\phi_i<\phi_{suf}$. We must therefore calculate the conditional probability that sufficient inflation occurs given that $\phi_i$ is bounded by $\phi_{min}$ and $\phi_P$. This quantity takes the form [@hall91] \[condprob\] P(\_[i]{}>\_[suf]{}|\_[min]{}<\_[i]{}<\_[P]{}) = , and allows us to determine which of the two boundary conditions considered here “naturally” predicts a phase of sufficiently long inflationary expansion. Sufficient inflation is a prediction of a theory if $P\approx 1$, whereas it is not if $P \ll 1$.
For standard reheating the minimum amount of inflation that solves the horizon problem is determined by the condition $N\equiv \ln
(a_f / a_i ) \approx 65$, where subscripts $i$ and $f$ denote the values of the scale factor at the onset and end of inflation respectively [@guth]. It is then straightforward to deduce from the classical field equations that \[efold\] N 65 6 \_[\_[f]{}]{}\^[\_[suf]{}]{} V() ( )\^[-1]{}d , where the value of the scalar field at the end of inflation, $\phi_f$, is computed from the relation ł[end]{} \_[= \_f]{}\^[2]{} = 1 . This condition corresponds to the breakdown of the slow-roll approximation [@st1984].[^1] Once $\phi_f$ is known, the value of $\phi_{suf}$ can be determined numerically by evaluating the integral in Eq. (\[efold\]).
To understand how the probability densities (\[rhot\]) and (\[rhohh\]) change in the quadratic and cubic cases, we shall consider them in turn. Since (\[rhot\]) and (\[rhohh\]) are usually [*not*]{} normalisable (unless the range of values that $\phi$ can take is bounded), we set the “normalisation constants” equal to one as is the common practice.
The Quadratic case {#quadratic}
------------------
To begin with, we note that the shape of $V(\phi)$ does not qualitatively change with changes in the coupling constant $\epsilon_{1}$. This parameter only fixes the height of the plateau and as a result leaves the shapes of the two probability densities unchanged. Consequently the qualitative behaviours of the probability densities are robust with respect to changes in $\epsilon_{1}$. Figure 2a gives a plot of $\rho_{V}$ showing that it starts at zero when $\phi=0$ and asymptotically approaches a constant value. On the other hand, as can be seen from Figure 2b, $\rho_{HH}$ decreases from infinity and asymptotically approaches a constant value. We should emphasise here that since the probability distribution functions () and () typically take values of the order $\exp(\pm 10^{14})$, we, for the sake of graphical representation, applied non-linear scalings of the kinds $\tilde{\rho}_{V}={\rho_V}^{1/C}$ and $\tilde{\rho}_{HH}
=\ln\left({\rho_{HH}}^{1/C}\right)$ respectively (where $C$ is a constant) to the two probability distribution functions. Note, however, that the values of the argument $\phi$ remain uneffected by this scaling.
Contrary to the claim of Biswas and Guha [@bg93], the two probability distribution functions reveal [*no*]{} qualitative changes as compared to the case of “chaotic” type potentials (e.g. $V(\phi)=m^{2}\,\phi^{2}/2$) as discussed by Vilenkin [@v2] and Halliwell [@hall91]. This means that the tunneling wave function has its maximum nucleation probability for the Universe coming into existence somewhere on the plateau of the potential $V(\phi)$, whereas the Hartle–Hawking wave function peaks near the true minimum of the potential at $\phi=0$. Translated into initial values of the Ricci curvature scalar, this means that the tunneling wave function prefers values of $ R_{i}$ near the Planck scale, whereas the no-boundary wave function favours a Universe of large initial size, i.e. small $R_{i}$ [@mijetal89].
**Figures 2a & 2b**
We now consider the conditional probability (\[condprob\]). The range of values of $\phi_{i}$ is specified by the range of initial values $R_i$ . In Planckian units, where $R_{P}=1$, we deduce that $\phi_{P}=13.0$. The value of $\phi_f$ is calculated from (\[end\]) to be $\phi_f =0.38$ and condition (\[efold\]) is therefore satisfied for $\phi_{suf} = 2.27$. Since the conditional probability measure (\[condprob\]) essentially amounts to a comparison of areas between the $\rho(\phi)$ curve and the positive $\phi$-axis in Figures 2a and 2b, it seems obvious that the tunneling wave function leads to sufficient inflation whereas the no-boundary wave function does not. This is in line with the conclusions of Vilenkin [@v2] and Mijić et al [@mijetal89] and in contrast to what is claimed by Biswas and Guha [@bg93].
The Cubic Case {#cubic}
--------------
We now consider the effects of adding a cubic term to the action. In general $\rho_{V}$ is peaked around the maximum of $V(\phi)$ at $\phi_{max}$ and falls off to zero on both sides. In contrast $\rho_{HH}$ decreases from infinity near $\phi=0$ to a minimum at $\phi_{max}$ and diverges again as $\phi\rightarrow\infty$. In this sense the presence of the cubic term drastically alters the shapes of the two probability distributions. This qualitative behaviour is illustrated in Figures 3a and 3b for $\epsilon_1 =10^{11}$ and $\epsilon_2 =10^{20}$.
**Figures 3a & 3b**
Now, regarding the location of the maximum nucleation probability, the tunneling case is unambiguous since there is only a single peak in the probability distribution function. Note, however, that in the cubic case this wave function favours [*smaller*]{} values of the initial curvature $R_i$ (viz. $\phi_i$) as compared to those in the quadratic case, where they are of Planckian order. On the other hand, the case of the Hartle–Hawking boundary condition is ambiguous because of the presence of two peaks in the probability distribution function, corresponding respectively to low and high values of $R_{i}$.
From a practical point of view, the question arises as to whether the Vilenkin wave function still predicts a phase of sufficiently long inflationary expansion immediately after tunneling into the Lorentzian signature region. To investigate this, we confined ourselves to the region on the left of the maximum in the potential (\[pot3\]), i.e. $\phi \le \phi_{max}$. Although inflation occurs on both sides of the turning point, there is no end to the superluminal expansion if the field rolls down the right-hand side and consequently there is no reasonable mechanism of reheating [@b]. On the basis of these physical considerations it is therefore more appropriate to [*identify*]{} the upper limit $\phi_P$ of the integrals in Eq. (\[condprob\]) with $\phi_{max}$ rather than with the Planck limit.
The specific value of the conditional probability depends on the magnitude of $\epsilon_2$ and it is therefore necessary to determine the relevant range of values for this parameter. We noted in Section that $\epsilon_2$ is bounded from above by the condition $\epsilon_2 \ll {\epsilon_1}^2$. As $\epsilon_2$ is decreased relative to a [*fixed*]{} $\epsilon_1$, the location of the maximum is shifted to larger values of $\phi$ and eventually beyond the Planck limit $\phi_{P}$. This follows since the model reduces to the $R^2$–theory for which the potential exhibits a plateau, i.e. the maximum is effectively located at infinity in this case. However, according to condition (\[consistent\]) the region over which the cubic and quadratic potentials are equivalent also increases as $\epsilon_2$ decreases. The question then is whether $\phi_{limit}$ grows faster or slower than $\phi_{max}$. By explicitly calculating the values of $\phi_{max}$ and $\phi_{limit}$ it is found that $\phi_{limit}$ exceeds $\phi_{max}$ for all parameter values $\epsilon_2 \le 10^{20}$. This implies that the $R^2$– and $R^3$–theories are equivalent for $\phi <
\phi_{max} $ in this range. Hence the results in Section for $R^2$–theory may be carried over directly to the cubic case in this region of the variable $\phi$, although there is the important difference that the upper bound on $\phi_i$ is now identified with $\phi_{max}$ and not $\phi_P$.
For any given $\epsilon_2$ the end of inflation occurs at $\phi_f
=0.38$ as in the $R^2$-case, since the $R^3$-contribution is negligible at very small $\phi$. Unfortunately a direct numerical integration of Eq. (\[condprob\]) can not be performed, because the integrands are typically of the orders of of $\exp(\pm 10^{14})$. However, since the probability density $\rho$ is a single valued, positive-definite function of $\phi$, it follows that a handle on the qualitative behaviour of the conditional probability can be obtained by investigating how the area under the $\rho (\phi)$ curve changes as $\epsilon_2$ changes. The problem then reduces to determining how the limits of the integrals in the numerator and denominator vary as the parameters of the theory are altered.
The dependences of the parameters of interest on $\epsilon_2$ are summarised in Table . We find that $\phi_{suf}$ for the potential (\[pot3\]) settles at the same value as in the quadratic case when $\epsilon_{2}$ is of order $10^{18}$ or smaller. We also find that $\phi_{max}$ rapidly approaches $\phi_{suf}$ in the region $10^{18}\le\epsilon_{2}\le10^{20}$. This implies that the integral in the numerator of the conditional probability (\[condprob\]) becomes [*much smaller*]{} than the term in the denominator for $\epsilon_2 \ge 10^{18}$. Consequently the Vilenkin scheme does not predict a phase of sufficiently long inflation in this region, contrary to the results for the $R^{2}$–model. We further note that for the same range of initial values of $\phi$, the Hartle–Hawking wave function shows no qualitative change from the quadratic case. Consequently, it appears that neither boundary condition predicts inflation for this choice of the parameters $\epsilon_{1}$ and $\epsilon_{2}$. This behaviour occurs because the presence of the cubic perturbation severely restricts the range of initial field values $\phi_i$ for which a phase of sufficiently long inflationary expansion is likely.
Including the full range of values of $\phi_{i}$ up to the Planck limit $\phi_{P}$ would not significantly improve this result in the Vilenkin scheme. In the Hartle–Hawking case, however, the integral in the numerator of (\[condprob\]) would have a large contribution from the second peak in $\rho_{HH}$. However, this range of $\phi_{i}$ was excluded, as discussed above, in order to avoid the problem of exiting the inflationary expansion.
---------------- ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- --
$\epsilon_{2}$ $10^{20}$ $10^{18}$ $10^{16}$ $10^{14}$ $10^{12}$ $10^{10}$ $10^{8}$ $10^{6}$
$\phi_{P}$ $23.6$ $21.3$ $19.0$ $16.7$ $14.4$ $13.1$ $13.0$ $13.0$
$\phi_{limit}$ $2.65$ $4.95$ $7.25$ $9.56$ $11.9$ $14.2$ $16.5$ $18.8$
$\phi_{max}$ $1.59$ $2.68$ $3.78$ $4.94$ $7.06$ $9.34$ $11.7$ $13.0$
$\phi_{suf}$ $1.59$ $2.24$ $2.27$ $2.27$ $2.27$ $2.27$ $2.27$ $2.27$
---------------- ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- --
: Summarising, for different values of $\epsilon_2$, the values of the scalar field corresponding to $R_P=1$ $(\phi_P)$, the limit of $\phi$ below which the $R^2$– and $R^3$–potentials are equivalent $(\phi_{limit})$, the location of the maximum in the potential $(\phi_{max})$ and the values of the field that just lead to sufficient inflation $(\phi_{suf})$. We specify $\epsilon_1
=10^{11}$ throughout due to microwave background considerations. As $\epsilon_2$ increases to order of $10^{18}$, the magnitudes of the quantities $\phi_{max}$ and $\phi_{suf}$ become comparable to one another and this implies that the numerator in the conditional probability approaches zero. This suggests that the conditional probability will become significantly smaller than unity for values of $\epsilon_2 \ge 10^{18}$.
ł[tab1]{}
Even though the conditional probability $P$ of Eq. () cannot be estimated numerically in this case, nevertheless, we present a set of values of “scaled conditional probabilities” in Appendix A which are obtained by applying a non-linear scaling to the probability distribution functions as discussed in Section . These values, which may be treated as qualitative indicators of $P$, also support the conclusions given in this section.
Discussion and Conclusions
==========================
In this paper we have investigated how the probability of realising sufficient inflation from quantum cosmology is altered when higher-order corrections to the Einstein–Hilbert action are introduced. Our results confirm that the addition of quadratic terms to the action does not reverse the conclusions of Vilenkin [@v1; @v2] regarding the effects of boundary conditions on the likelihood of sufficient inflation, in contrast to some recent claims [@bg93]. On the other hand, cubic perturbations can produce qualitative changes to the nature of the probability distribution function $\rho(\phi)$. From a physical point of view one is confined to consider initial values of the scalar field that allow an exit from the inflationary expansion. As a result the important physical (as opposed to purely mathematical) consequences of cubic perturbations are that they restrict the measure of allowed initial field values $\phi_{i}$ that lead to sufficient inflation. This is in agreement with the classical arguments [@b]. By considering the conditional probability (\[condprob\]) (see also Appendix A) we have argued that if the coupling constant $\epsilon_{2}$, which determines the strength of the $R^{3}$-contribution to the Lagrangian, exceeds a critical value, neither the tunneling nor the no-boundary boundary conditions predict an epoch of sufficient inflation, in the sense that the conditional probability is significantly less than unity in both cases.
Our results appear to exhibit some generality in four-dimensions. As discussed in Section , the qualitative shape of the self-interaction potential $V(\phi)$ remains unaltered if general polynomial perturbations with a highest order term $\epsilon_{n-1}\,R^n$ are considered. In general this result is true when $D<2n$. This immediately implies that neither of the two probability distributions $\rho_V$ and $\rho_{HH}$ for the $n=3$ case will be qualitatively affected under $n>3$ perturbations. The qualitative conclusions drawn for the case of cubic perturbations in Section therefore remain robust under higher-order perturbations to the action, although of course the details of what happens will depend on how the precise location of the maximum in the potential $V(\phi)$ is related to the highest-order term.
However, the consequences of the quadratic and the cubic perturbations (as well as those of general polynomial types) depend crucially on the values of the free parameters of the system, namely $\epsilon_k~(k=1, \ldots , n-1)~,~D,~n$, as well as on the initial field values $\phi_i$. In particular, the dimensionality $D$ of the space-time is crucial in deciding the maximum degree $n$ of perturbations allowed ($D<2n$ say) above which the perturbations would be qualitatively inconsequential, i.e. the system would be robust.
Finally we remark that inflation is possible, at least at the classical level, if the field is initially placed to the right of the maximum in Eq. (\[pot3\]) and given sufficient kinetic energy to travel over the hill towards $\phi=0$. Unfortunately, our analysis can not consider this possibility since the scalar field momentum operator in the Wheeler–DeWitt equation then becomes important and the solutions (\[rhot\]) and (\[rhohh\]) are no longer valid. Furthermore, if one is prepared to include the effects of the $R^3$-contribution in the action, the cubic term $R
{\Box} R$ should also be considered. In this case the effective theory resembles Einstein gravity minimally coupled to two scalar fields after a suitable conformal transformation on the metric [@b] and in principle a similar analysis to the one presented here can be followed for this more general case. We shall return to some of these questions in future.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Jonathan J Halliwell for helpful remarks. HvE is supported by a Grant from the Drapers’ Society at QMW. JEL is supported by the Science and Engineering Research Council (SERC), UK, and is supported at Fermilab by the DOE and NASA under Grant No. NAGW-2381. RT is supported by the SERC, UK, under Grant No. H09454.
Appendix A {#appendix-a .unnumbered}
==========
As was pointed out in Section , the integrands involved in the definition of conditional probability typically have magnitudes of order $\exp(\pm 10^{14})$, which makes the numerical calculation of the integrals not possible in practice. Now due to the nature of these numbers no linear scaling of the probability function $\rho$ can bypass this difficulty. The question then arises as to whether appropriate non-linear scalings exist which keep the conditional probability $P$ invariant. To see that there do [*not*]{}, recall that the only scalings that leave the Wheeler–DeWitt equation, $H\,\Psi=0$, of the $D$-dimensional minisuperspace models of Quantum Cosmology invariant are given by $\tilde{H}=\Omega^{-2}\,H,\
\tilde{\Psi}=\Omega^{\gamma}\,\Psi
\hspace{0.5mm} \rightarrow \hspace{0.5mm}\tilde{H}\,\tilde{\Psi}=
\Omega^{\gamma-2}\,H\,\Psi=0$, ($\Omega(q)$ is an arbitrary function of the minisuperspace co-ordinates $q$) provided $\gamma$ and $\xi$ (a free parameter in the Wheeler–DeWitt equation) are given by $\gamma=(2-D)/2$ and $\xi=-(D-2)/8(D-1)$ respectively [@hall88]. Effectively this amounts to a redefinition of the potential $U(q)$ and the DeWitt metric of minisuperspace $f^{\alpha\beta}(q)$, which occur in the Hamilton operator $H$. More importantly, under such scale transformations the conserved probability current density $j^{\alpha}$ defined from $\Psi$ remains unchanged. This freedom, however, is not of much use in bypassing the numerical difficulty mentioned above in order to obtain quantitative values for $P$. Nevertheless, if we confine ourselves to qualitative information, we may choose non-linear (but monotonic) scalings of $\rho$, which, while violating the invariance properties of the model, would nevertheless supply us with a qualitative indicator of $P$. This is not dissimilar to the way non-linear scalings of functions are employed for the purpose of graphical representation.
To calculate a qualitative indicator of $P$ we define the [*non-linearly scaled conditional probability*]{} $\tilde{P}$ as \[scalconpr\] P(\_[i]{}>\_[suf]{}|\_[min]{}<\_[i]{}<\_[P]{}) , where $C$ is the index of non-linear scaling. Clearly such a scaling will not change the qualitative behaviour of $\rho_{V}$ and the values of its argument $\phi$, and therefore the values of the boundaries of the integrals occurring in Eq. () (as listed in Table ) remain the same. Furthermore, such scalings leave $P$ invariant in the limiting cases where $P=0$ and $P=1$.
Here we chose $C=10^{14}$. Table gives the values of $\tilde{P}$ as a function of $\epsilon_{2}$ for the Vilenkin wave function in the case of the $R^3$–model, calculated for $\epsilon_{1}=10^{11}$ and the boundary values of $\phi$ given in Table . We approximated $\phi_{min}$ by $\phi_{f}=0.38$. For the corresponding value of $\tilde P$ for the Vilenkin model in the $R^{2}$-case of Section we found $\tilde{P}=0.85$.
---------------- ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- --
$\epsilon_{2}$ $10^{20}$ $10^{18}$ $10^{16}$ $10^{14}$ $10^{12}$ $10^{10}$ $10^{8}$ $10^{6}$
$\tilde P$ $0.00$ $0.20$ $0.45$ $0.59$ $0.72$ $0.79$ $0.84$ $0.85$
---------------- ----------- ----------- ----------- ----------- ----------- ----------- ---------- ---------- --
: Behaviour of the non-linearly scaled conditional probability distribution $\tilde{P}$ for the Vilenkin wave function $\Psi_V$ in the $R^3$–model of Section . We specify $\epsilon_1=10^{11}$ throughout.
ł[tab2]{}
As can be seen from Table , the behaviour of $\tilde P$ supports the conclusions drawn in Section on the basis of qualitative analysis.
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Figure Captions {#figure-captions .unnumbered}
===============
[*Figure 1:*]{} (a) The effective self-interaction potential (\[pot2\]) corresponding to the $R^2$–theory with $\epsilon_{1}=10^{11}$. The scalar field and magnitude of the potential have been rescaled via Eq. (\[rescale\]) to enable easy comparison with the results of Section 4; (b) The rescaled effective interaction potential (\[pot3\]) corresponding to the $R^3$–theory with $\epsilon_{1}=10^{11}$ and $\epsilon_{2}=10^{20}$.
[*Figure 2:*]{} (a) The Vilenkin probability distribution $\rho_V(\phi)$ for the $R^{2}$–theory with a rescaling $\rho_V(\phi) = \left[\ \exp (- 2/3V )\ \right]^{10^{-14}}$; (b) The Hartle–Hawking probability distribution $\rho_{HH} (\phi)$ for the $R^{2}$–theory with a rescaling $\rho_{HH}(\phi) =
\ln \left[\ \exp ( 2/3V)\ \right]^{10^{-14}}$. We choose these particular rescaled values of $\rho(\phi)$ in order to obtain easily interpretable plots from our numerical programme.
[*Figure 3:*]{} (a) The Vilenkin probability distribution $\rho_V(\phi)$ for the $R^{3}$–theory with the same rescaling as for Figure 2a; (b) The Hartle–Hawking probability distribution $\rho_{HH} (\phi)$ for the $R^{3}$–theory with the same rescaling as for Figure 2b.
[^1]: Strictly speaking, conditions (\[efold\]) and (\[end\]) are only valid in spatially flat FLRW models, but we are considering spatially closed cases in this work. However, during inflation the curvature term in the Friedmann equation is redshifted to zero within one Hubble expansion time and the Universe effectively becomes spatially flat at an exponentially fast rate. For our purposes, therefore, these expressions remain valid.
| ArXiv |
---
abstract: 'In the scheme of a quantum nondemolition (QND) measurement, an observable is measured without perturbing its evolution. In the context of studies of decoherence in quantum computing, we examine the ‘open’ quantum system of a two-level atom, or equivalently, a spin-1/2 system, in interaction with quantum reservoirs of either oscillators or spins, under the QND condition of the Hamiltonian of the system commuting with the system-reservoir interaction. For completeness, we also examine the well-known non-QND spin-Bose problem. For all these many-body systems, we use the methods of functional integration to work out the propagators. The propagators for the QND Hamiltonians are shown to be analogous to the squeezing and rotation operators, respectively, for the two kinds of baths considered. Squeezing and rotation being both phase space area-preserving canonical transformations, this brings out an interesting connection between the energy-preserving QND Hamiltonians and the homogeneous linear canonical transformations.'
address:
- 'Raman Research Institute, Bangalore - 560 080, India'
- 'School of Physical Sciences, Jawaharlal Nehru University, New Delhi - 110 067, India'
author:
- Subhashish Banerjee
- R Ghosh
title: Functional integral treatment of some quantum nondemolition systems
---
Introduction
============
In the scheme of a quantum nondemolition (QND) measurement, an observable is measured without perturbing its free motion. Such a scheme was originally introduced in the context of the detection of gravitational waves [@caves80]. It was to counter the quantum mechanical unpredictability that in general would disturb the system being measured. The dynamical evolution of a system immediately following a measurement limits the class of observables that may be measured repeatedly with arbitrary precision, with the influence of the measurement apparatus on the system being confined strictly to the conjugate observables. Observables having this feature are called QND or back-action evasion observables [@bo96; @vo98; @zu84]. In addition to its relevance in ultrasensitive measurements, a QND scheme provides a way to prepare quantum mechanical states which may otherwise be difficult to create, such as Fock states with a specific number of particles. One of the original proposals for a quantum optical QND scheme was that involving the Kerr medium [@walls], which changes its refractive index as a function of the number of photons in the ‘signal’ pump laser. The advent of experimental methods for producing Bose-Einstein condensation (BEC) enables us to make progress in the matter-wave analogue of the optical QND experiments. In the context of research into BEC, QND schemes with atoms are particularly valuable, for instance, in engineering entangled states or Schrödinger’s cat states. A state preparation with BEC has recently been performed in the form of squeezed state creation in an optical lattice [@science01]. In a different context, it has been shown that the accuracy of atomic interferometry can be improved by using QND measurements of the atomic populations at the inputs to the interferometer [@kbm98].
No system of interest, except the entire universe, can be thought of as an isolated system – all subsets of the universe are in fact ‘open’ systems, each surrounded by a larger system constituting its environment. The theory of open quantum systems provides a natural route for reconciliation of dissipation and decoherence with the process of quantization. In this picture, friction or damping comes about by the transfer of energy from the ‘small’ system (the system of interest) to the ‘large’ environment. The energy, once transferred, disappears into the environment and is not given back within any time of physical relevance. Ford, Kac and Mazur [@fkm65] suggested the first microscopic model describing dissipative effects in which the system was assumed to be coupled to a reservoir of an infinite number of harmonic oscillators. Interest in quantum dissipation, using the system-environment approach, was intensified by the works of Caldeira and Leggett [@cl83], and Zurek [@wz91] among others. The path-integral approach, developed by Feynman and Vernon [@fv63], was used by Caldeira and Leggett [@cl83], and the reduced dynamics of the system of interest was followed taking into account the influence of its environment, quantified by the influence functional. In the model of the fluctuating or “Brownian" motion of a quantum particle studied by Caldeira and Leggett [@cl83], the coordinate of the particle was coupled linearly to the harmonic oscillator reservoir, and it was also assumed that the system and the environment were initially factorized. The treatment of the quantum Brownian motion has since been generalized to the physically reasonable initial condition of a mixed state of the system and its environment by Hakim and Ambegaokar [@ha85], Smith and Caldeira [@sc87], Grabert, Schramm and Ingold [@gsi88], and by us for the case of a system in a Stern-Gerlach potential [@sb00], and also for the quantum Brownian motion with nonlinear system-environment couplings [@sb03-2].
An open system Hamiltonian is of the QND type if the Hamiltonian $H_S$ of the system commutes with the Hamiltonian $H_{SR}$ describing the system-reservoir interaction, i.e., $H_{SR}$ is a constant of motion generated by $H_S$. Interestingly, such a system may still undergo decoherence or dephasing without any dissipation of energy [@gkd01; @sgc96].
In this paper, we study such QND ‘open system’ Hamiltonians of particular interest in the context of decoherence in quantum computing, and obtain the propagators of the composite systems explicitly using path integral methods, for two different models of the environment. The aim is to shed some light on the problem of QND measurement schemes. Can one draw upon any familiar symmetries to connect with the time-evolution operation of these QND systems of immense physical importance?
We take our system to be a two-level atom, or equivalently, a spin-1/2 system. We consider two types of environment, describable as baths of either oscillators or spins. One cannot in general map a spin-bath to an oscillator-bath (or vice versa); they constitute distinct ‘universality classes’ of quantum environment [@rpp00]. The first case of oscillator-bath models (originated by Feynman and Vernon [@fv63]) describes delocalized environmental modes. For the spin-bath, on the other hand, the finite Hilbert space of each spin makes it appropriate for describing the low-energy dynamics of a set of localized environmental modes. A difficulty associated with handling path integrals for spins comes from the discrete matrix nature of the spin-Hamiltonians. This difficulty is overcome by bosonizing the Hamiltonian by representing the spin angular momentum operators in terms of boson operators following Schwinger’s theory of angular momentum [@schwin].
We then use the Bargmann representation [@vb47] for all the boson operators. The Schrödinger representation of quantum states diagonalizes the position operator, expressing pure states as wave functions, whereas the Bargmann representation diagonalizes the creation operator $b^{\dagger}$, and expresses each state vector $|\psi \rangle$ in the Hilbert state ${\cal
H}$ as an entire analytic function $f(\alpha)$ of a complex variable $\alpha$. The association $|\psi \rangle
\longrightarrow f(\alpha)$ can be written conveniently in terms of the normalized coherent states $|\alpha \rangle$ which are the right eigenstates of the annihilation operator $b$: $$\begin{aligned}
b|\alpha \rangle & = & \alpha |\alpha \rangle , \nonumber \\
\langle \alpha '|\alpha \rangle & = & \exp \left( -\frac{1}{2}|
\alpha '|^2 - \frac{1}{2}|\alpha |^2 + \alpha '^* \alpha
\right) , \nonumber \end{aligned}$$ giving $$f(\alpha ) = e^{-|\alpha |^2/2} ~\langle \alpha ^* |\psi
\rangle .$$ We obtain the explicit propagators for these many-body systems from those of the expanded bosonized forms by appropriate projection.
The propagators for the QND Hamiltonians with an oscillator bath and a spin bath are shown to be analogous to the squeezing and rotation operators, respectively, which are both phase space area-preserving canonical transformations. This suggests an interesting connection between the energy-preserving QND Hamiltonians and the homogeneous linear canonical transformations, which would need further systematic probing.
The plan of the paper is as follows. In section 2 we take up the case of a QND-type of open system Hamiltonian where the bath is a bosonic one of harmonic oscillators. In section 2.1 we consider a case, which is a variant of the previous one, wherein we include an external mode in resonance with the atomic transition and obtain its propagator. In section 2.2 we discuss the non-QND variant of the Hamiltonian which usually occurs in the literature in discussions of the spin-Bose problem [@papa86; @lc87]. In section 3 we treat the case of a QND-type of open system Hamiltonian where the bath is composed of two-level systems or spins. The structure of the propagators in the two cases of the oscillator and spin baths is discussed in section 4, and in section 5 we present our conclusions.
Bath of harmonic oscillators
============================
We first take the case where the system is a two-level atom interacting with a bosonic bath of harmonic oscillators with a QND type of coupling. Such a model has been studied [@unruh95; @ps96; @dd95] in the context of the influence of decoherence in quantum computation. The total system evolves under the Hamiltonian, $$\begin{aligned}
H_1 & = & H_S + H_R + H_{SR} \nonumber\\ & = & {\hbar \omega
\over 2} \sigma_z + \sum\limits^M_{k=1} \hbar \omega_k
b^{\dagger}_k b_k + \left( {\hbar \omega \over 2} \right)
\sum\limits^M_{k=1} g_k (b_k + b^{\dagger}_k) \sigma_z.
\label{h1} \end{aligned}$$ Here $H_S, H_R$ and $H_{SR}$ stand for the Hamiltonians of the system, reservoir, and system-reservoir interaction, respectively. We have made use of the equivalence of a two-level atom and a spin-1/2 system, $\sigma_x, \sigma_z$ denote the standard Pauli spin matrices and are related to the spin-flipping (or atomic raising and lowering) operators $S_+$ and $S_-$: $\sigma_x = S_+ + S_-$, $\sigma_z = 2 S_+ S_- - 1$. In (\[h1\]) $b^{\dagger}_k, b_k$ denote the Bose creation and annihilation operators for the $M$ oscillators of frequency $\omega_k$ representing the reservoir, $g_k$ stands for the coupling constant (assumed real) for the interaction of the field with the spin. Since $[H_S, H_{SR}]=0$, the Hamiltonian (\[h1\]) is of QND type.
The explicit propagator $\exp (-{i H t \over \hbar})$ for the Hamiltonian (\[h1\]) is obtained by using functional integration and bosonization [@papa86; @sb03-1]. In order to express the spin angular momentum operators in terms of boson operators, we employ Schwinger’s theory of angular momentum [@schwin] by which any angular momentum can be represented in terms of a pair of boson operators with the usual commutation rules. The spin operators $\sigma_z$ and $\sigma_x$ can be written in terms of the boson operators $a_{\beta}$, $a_{\beta}^{\dagger}$ and $a_{\gamma}$, $a_{\gamma}^{\dagger}$ as $$\begin{aligned}
\sigma_z & = & a_{\gamma}^{\dagger} a_{\gamma} -
a_{\beta}^{\dagger} a_{\beta}, \nonumber \\ \sigma_x & = &
a_{\gamma}^{\dagger} a_{\beta} + a_{\beta}^{\dagger} a_{\gamma}
. \nonumber \end{aligned}$$ In the Bargmann representation [@vb47] the actions of $b$ and $b^{\dagger}$ are $$\begin{aligned}
b^{\dagger} f(\alpha) & = & \alpha^* f(\alpha), \nonumber\\ b
f(\alpha) & = & {df(\alpha) \over d\alpha^*}, \end{aligned}$$ where $|\alpha \rangle$ is the normalized coherent state. The spin operator becomes $$\sigma_z \longrightarrow \left( \gamma^* {\partial \over
\partial \gamma^*} - \beta^* {\partial \over \partial \beta^*}
\right).$$ Here the variable $\beta^*$ is associated with the spin-down state and the variable $\gamma^*$ with the spin-up state.
The bosonized form of the Hamiltonian (1) is $$\begin{aligned}
H_{B_{1}} & = & {\hbar \omega \over 2} \left( \gamma^*
{\partial \over \partial \gamma^*} - \beta^* {\partial \over
\partial \beta^*} \right) + \sum\limits^M_{k=1} \hbar \omega_k
\alpha^*_k {\partial \over \partial \alpha^*_k} \nonumber\\ & &
+ {\hbar \omega \over 2} \sum\limits^M_{k=1} g_k \left(
\alpha^*_k + {\partial \over \partial \alpha^*_k} \right)
\left( \gamma^* {\partial \over \partial \gamma^*} - \beta^*
{\partial \over \partial \beta^*} \right). \end{aligned}$$ Here $\alpha^*_k$, ${\partial \over \partial \alpha^*_k}$ are the Bargmann representations for $b^{\dagger}_k$ and $b_k$, respectively. A particular solution of the Schr$\ddot{o}$dinger equation for the bosonized Hamiltonian (4) is $$U_1 = U_{00} \beta^* \beta' + U_{01} \beta^* \gamma' + U_{10}
\gamma^* \beta' + U_{11} \gamma^* \gamma',$$ where the amplitude $U_{ij}$ are functions of time as well as the coherent state variables associated with the boson oscillators, with the initial condition $$U_{ij} (t=0) = \exp \left\{ \sum\limits^M_{k=1} \alpha^*_k
\alpha'_k \right\} \delta_{ij} ~~~~~(i,j=0,1).$$ The initial state for the expanded propagator associated with the bosonized Hamiltonian (5) is $$U(t=0) = \exp \left\{ \sum\limits^M_{k=1} \alpha^*_k \alpha'_k
\right\} \exp \left\{ \beta^* \beta' + \gamma^* \gamma'
\right\}.$$ If the Hamiltonian is in the normal form given by $H(\alpha^*,
{\partial \over \partial \alpha^*}, t)$, the associated propagator is given as a path integral over coherent state variables as [@klauder] $$%%\fl
U(\alpha^*,t;\alpha',0) = \int {\bf D} \{ \alpha \} \exp
\left\{ \sum\limits_{0\leq \tau < t} \alpha^* (\tau +) \alpha
(\tau) - {i \over \hbar} \int\limits^t_0 d\tau H \left(
\alpha^* (\tau +), \alpha (\tau), \tau \right) \right\}.$$ Here $\sum\limits_{0\leq \tau <t} \alpha^* (\tau +) \alpha
(\tau)$ stands for $\sum\limits^{N-1}_{j=0} \alpha^*
(\tau_{j+1}) \alpha (\tau_j)$ in the subdivision of the internal $[0,t]$, i.e., where $\tau$ stands for $\tau_j$, $\tau
+$ stands for the next point $\tau_{j+1}$ in the subdivision. Also, in the subdivision scheme, $$\int\limits^t_0 d\tau
H\left( \alpha^* (\tau +), \alpha (\tau), \tau \right) =
\sum\limits^{N-1}_{j=0} H\left( \alpha^* (\tau_{j+1}), \alpha
(\tau_j), \tau_j \right) \Delta \tau_j.$$ Here the path differential in (8) is $${\bf D}^2 \{ \alpha \} = \prod_{0<\tau <t} D^2 \alpha (\tau),$$ where the weighted differential is $$D^2 \alpha (\tau) = {1 \over \pi} \exp \left( -|\alpha
(\tau)|^2 \right) d^2 \alpha (\tau).$$ Using (8), the propagator for the bosonized Hamiltonian (4) is $$\begin{aligned}
%%\fl
u_1 (\mbox{\boldmath $\alpha^*$}, \beta^*, \gamma^*, t;
\mbox{\boldmath $\alpha'$}, \beta', \gamma', 0) & = & \int {\bf
D}^2 \{ \mbox{\boldmath $\alpha$} \} {\bf D}^2 \{\beta\} {\bf
D}^2 \{\gamma\} \nonumber \\ & & \times \exp \Bigg\{
\sum\limits_{0\leq \tau <t} \Bigg[ \sum\limits^M_{k=1}
\alpha^*_k (\tau+) \alpha_k (\tau) \nonumber \\ & & + \beta^*
(\tau+) \beta (\tau) + \gamma^* (\tau+) \gamma (\tau) \Bigg]
\nonumber \\ & & - i\sum\limits^M_{k=1} \int\limits^t_0 d\tau
\omega_k \alpha^*_k (\tau+) \alpha_k (\tau) \nonumber \\ & & -
i {\omega \over 2} \int\limits^t_0 d\tau \Bigg[ \gamma^*(\tau+)
\gamma (\tau) - \beta^* (\tau+) \beta(\tau) \Bigg] \nonumber\\
& & - i {\omega \over 2} \sum\limits^M_{k=1} \int\limits^t_0
d\tau g_k \Bigg[ \alpha^*_k (\tau+) + \alpha_k (\tau) \Bigg]
\Bigg[ \gamma^* (\tau+) \gamma(\tau) \nonumber \\ & & -
\beta^*(\tau+) \beta(\tau) \Bigg] \Bigg\} . \end{aligned}$$ In Eq. (11) [$\alpha$]{} is a vector with components $\{ \alpha_k\}$, and ${\bf D}^2\{ \mbox{\boldmath $\alpha$}\} =
\prod^M_{k=1} {\bf D}^2 \{ \alpha_k \}$.
Now we introduce a complex auxiliary field $f(\tau)$ to decouple the interaction term in (11) as $$\begin{aligned}
& & \exp \left( -{i\omega \over 2} \sum\limits^M_{k=1}
\int\limits^t_0 d\tau g_k \left[ \alpha^*_k (\tau +) + \alpha_k
(\tau) \right] \left[ \gamma^* (\tau +) \gamma (\tau) - \beta^*
(\tau +) \beta (\tau) \right] \right) \nonumber\\ & & = \int
{\bf D}^2 \{ f\} \exp \left[ -i \sum\limits^M_{k=1}
\int\limits^t_0 d\tau f^* (\tau) g_k \left( \alpha^*_k (\tau +)
+ \alpha_k (\tau) \right) \right] \nonumber \\ & & \times \exp
\left[ \int\limits^t_0 d\tau f(\tau) {\omega \over 2} \left(
\gamma^* (\tau +) \gamma (\tau) - \beta^* (\tau +) \beta (\tau)
\right) \right]. \end{aligned}$$ Here we have used the $\delta$-functional identify, [@papa86] $$\int {\bf D}^2 \{ x \} P[x^*(t)] \exp \left\{ \int\limits^t_0
d\tau y(\tau)x(\tau) \right\} = P[y(t)],$$ where ${\bf D}^2\{ x\}$ is the functional differential $${\bf D}^2 \{ x\} = \exp \left( -\int\limits^t_0 d\tau
|x(\tau)|^2 \right) \prod_{0\leq \tau <t} \left( {d\tau \over
\pi} \right) d^2 x(\tau),$$ and ${\bf P}[x^*(t)]$ is an explicit functional of $x^*$ only. Using (12), the bosonized propagator (11) can be written as $$\begin{aligned}
u_1 (\mbox {\boldmath $ \alpha^* $}, \beta^*, \gamma^*, t;
\mbox{\boldmath $\alpha'$}, \beta', \gamma', 0) & = & \int {\bf
D}^2 \{f\} G_1 (\mbox{\boldmath $\alpha^*$}, t; \mbox{\boldmath
$\alpha'$}, 0; [f^*]) \nonumber \\ & & \times N_1 \left(
\beta^*, \gamma^*, t; \beta', \gamma', 0; [f] \right). \end{aligned}$$ Here $G_1$ stands for the propagator for $$H_{G_{1}} = \hbar \sum\limits^M_{k=1} \left[ \omega_k
\alpha^*_k {\partial \over \partial \alpha^*_k} + f^*(t) g_k
\alpha^*_k + f^*(t) g_k \alpha_k \right],$$ $N_1$ is the propagator for $$H_{N_{1}} = {\hbar \omega \over 2}\left( \gamma^* {\partial
\over \partial \gamma^*} - \beta^* {\partial \over \partial
\beta^*} \right) + {i\hbar \omega \over 2} f(t) \left( \gamma^*
{\partial \over \partial \gamma^*} - \beta^* {\partial \over
\partial \beta^*} \right).$$ These obey the Schrödinger equations $i\hbar {\partial
\over \partial t} G_1=H_{G_{1}} G_1, i\hbar {\partial \over
\partial t} N_1=H_{N1} N_1$ with the initial conditions $$\begin{aligned}
G_1 (t=0) & = & \exp \left\{ \sum\limits^M_{k=1} \alpha^*_k
\alpha'_k \right\}, \nonumber\\ N_1 (t=0) & = & \exp \left\{
\beta^* \beta' + \gamma^* \gamma' \right\}. \end{aligned}$$ The propagator $G_1$ is given by $$\begin{aligned}
G_1 & = & \exp \Bigg\{ \sum\limits^M_{k=1} \alpha^*_k \alpha'_k
e^{-i\omega_kt} - \sum\limits^M_{k=1} \Bigg[ i \alpha^*_k g_k
\int\limits^t_0 d\tau f^* (\tau) e^{-i\omega_k(t-\tau)}
\nonumber\\ & & + i\alpha'_k g_k \int\limits^t_0 d\tau e^{-
i\omega_k\tau} f^* (\tau) \nonumber \\ & & + g^2_k
\int\limits^t_0 d\tau \int\limits^{\tau}_0 d\tau' e^{-i\omega_k
(\tau-\tau')} f^* (\tau) f^* (\tau') \Bigg] \Bigg\}. \end{aligned}$$ The propagator $N_1$ is given by $$\begin{aligned}
N_1 & = & \exp \left\{ Q_{00} \beta^* \beta' + Q_{01} \beta^*
\gamma' + Q_{10} \gamma^* \beta' + Q_{11} \gamma^* \gamma'
\right\} \nonumber \\ & = & \sum\limits^{\infty}_{l=0} {1 \over
l!} \left[ (\beta^*,\gamma^*) Q \pmatrix{\beta' \cr \gamma'}
\right]^l. \end{aligned}$$ Here $Q(t)$ is given by $$Q(t) = \exp \left( {i\omega \over 2} \sigma_z t - {\omega \over
2} \sigma_z \int\limits^t_0 d\tau f(\tau) \right) ,$$ with $Q_{ij}(0) = \delta_{ij}, Q(0)=I$.
Thus the propagator for the bosonized Hamiltonian (4) as given by (15) becomes $$u_1 = \sum\limits^{\infty}_{l=0} \int {\bf D}^2 \{ f\} G_1 {1
\over l!} \left[ (\beta^*, \gamma^*) Q \pmatrix{\beta' \cr
\gamma'} \right]^l.$$ The propagator for the Hamiltonian (1) is obtained from (22) by taking the $l=1$ term in the above equation. By making use of the $\delta$-functional identity (13) the amplitudes of the propagator for the Hamiltonian (1) are obtained in matrix form as $$\begin{aligned}
u_1 = \pmatrix{U_{00} & U_{01} \cr U_{10} & U_{11}} & = & \exp
\left\{ \sum\limits^M_{k=1} \alpha^*_k \alpha'_k e^{-
i\omega_kt} \right\} \nonumber\\ & & \times e^A \pmatrix{e^B &
0 \cr 0 & e^{-B}}, \end{aligned}$$ where $$A = i \left( {\omega \over 2} \right)^2 \sum\limits^M_{k=1}
{g^2_k \over \omega_k} t - \left( {\omega \over 2} \right)^2
\sum\limits^M_{k=1} {g^2_k \over \omega^2_k} \left( 1-e^{-
i\omega_kt} \right),$$ $$B = \sum\limits^M_{k=1} \phi_k \left( \alpha^*_k + \alpha'_k
\right) + i {\omega \over 2} t,$$ $$\phi_k = {\omega \over 2} {g_k \over \omega_k} \left( 1-e^{-
i\omega_kt} \right).$$
Here we associate the values $\alpha^*$ with time $t$ and $\alpha'$ with time $t=0$ as is also evident from (8). The simple form of the last term on the right-hand side of (23) reveals the QND nature of the system-reservoir coupling. Since we are considering the unitary dynamics of the complete Hamiltonian (\[h1\]) there is no decoherence, and the propagator (23) does not have any off-diagonal terms. In a treatment of the system alone, i.e., an open system analysis of Eq. (\[h1\]) after the tracing over the reservoir degrees of freedom, it has been shown [@ps96] that the population, i.e., the diagonal elements of the reduced density matrix of the system remain constant in time while the off-diagonal elements that are a signature of the quantum coherences decay due to decoherence, as expected.
Note that though the commonly used coordinate-coupling model describing a free particle in a bosonic bath, explicitly solved by Hakim and Ambegaokar [@ha85], with $$H = {P^2 \over 2} + {1 \over 2} \sum\limits^M_{j=1} \left(
p^2_j + \omega^2_j (q_j - Q)^2 \right) , \label{h14}$$ is seemingly not of the QND type, it can be shown to be unitarily equivalent to a Hamiltonian of the QND type as follows: $$\begin{aligned}
U_2 U_1 H U^{\dagger}_1 U^{\dagger}_2 & = & {P^2 \over 2} + P
\sum\limits^M_{j=1} \omega_j q_i + {1 \over 2}
\sum\limits^M_{j=1} \left( p^2_j + \omega^2_j q^2_j \right)
\nonumber\\ & & + {1 \over 2} \left( \sum\limits^M_{j=1}
\omega_j q_j \right)^2 , \label{ha1} \end{aligned}$$ where $U_1$ and $U_2$ are the unitary operators $$U_1 = \exp \left[ {i\pi \over 2\hbar} \sum\limits^M_{j=1}
\left( {p^2_j \over 2\omega_j} + {1 \over 2} \omega_j q^2_j
\right) \right],$$ $$U_2 = \exp \left[ {-i \over \hbar} Q \sum\limits^M_{j=1}
\omega_j q_j \right].$$ The above Hamiltonian (\[ha1\]) is of the QND type with $[H_S, H_{SR}]=[P^2/2, P \sum\limits^M_{j=1} \omega_j q_j]=0$. It is commonly known as the velocity-coupling model [@flc88].
An external mode in resonance with the atomic transition
--------------------------------------------------------
In this subsection we consider a Hamiltonian which is a variant of the one in (1): $$\begin{aligned}
H_2 & = & {\hbar \omega \over 2} \sigma_z + \hbar \Omega
a^{\dagger}a - {\hbar \Omega \over 2} \sigma_z \nonumber \\ & &
+ \sum\limits^M_{k=1} \hbar \omega_k b^{\dagger}_k b_k + {\hbar
\omega \over 2} \sum\limits^M_{k=1} g_k (b_k + b^{\dagger}_k)
\sigma_z. \end{aligned}$$ Here $$\Omega = 2 \vec{\epsilon}.\vec{d}^*,$$ where $\vec{d}$ is the dipole transition matrix element and $\vec{\epsilon}$ comes from the field strength of the external driving mode $\vec{E}_L(t)$ such that $$\vec{E}_L(t) = \vec{\epsilon} e^{-i\omega t} + \vec{\epsilon}^*
e^{i\omega t}.$$ Here we have used the form $-{\Omega \over 2} \sigma_z$, associated with the external mode, instead of the usual form $-
{\Omega \over 2} \sigma_x$ and (31) is of a QND type. Proceeding as in section 2 and introducing the symbol $\nu^*$ for the Bargmann representation of the external mode $a^{\dagger}$ we have the amplitudes of the propagator for (31) in matrix form as $$\begin{aligned}
u_2 = \pmatrix{U_{00} & U_{01} \cr U_{10} & U_{11}} & = & \exp
\left\{ \sum\limits^M_{k=1} \alpha^*_k \alpha'_k e^{-i\omega_k
t} \right\} \nonumber\\ & & \times \exp \left\{ \nu^* \nu' e^{-
i\Omega t} \right\} e^A \pmatrix{e^{B_{2}} & 0 \cr 0 & e^{-
B_{2}}}, \end{aligned}$$ where $A$ is as in Eq. (24), $$B_2 = \sum\limits^M_{k=1} \phi_k \left( \alpha^*_k + \alpha'_k
\right) + i \left( {\omega - \Omega \over 2} \right) t,$$ and $\phi_k$ is as in Eq. (26).
Non-QND spin-Bose problem
-------------------------
In this subsection we consider a Hamiltonian that is a variant of the spin-Bose problem [@papa86; @sb03-1; @lc87]. This addresses a number of problems of importance such as the interaction of the electromagnetic field modes with a two-level atom [@cm78; @rn82]. Another variant of the spin-Bose problem has been used for treating problems of phase transitions [@be70; @cl84] and also to the tunnelling through a barrier in a potential well [@uw93]. Our Hamiltonian is $$\begin{aligned}
H_3 & = & {\hbar \omega \over 2} \sigma_z + \sum\limits^M_{k=1}
\hbar \omega_k b^{\dagger}_k b_k \nonumber\\ & & + {\hbar
\omega \over 2} \sum\limits^M_{k=1} g_k (b_k + b^{\dagger}_k)
\sigma_x. \label{p1} \end{aligned}$$ This could describe, for example, the interaction of $M$ modes of the electromagnetic field with a two-level atom via a dipole interaction. This has a form similar to Eq. (1) except that here the system-environment coupling is via $\sigma_x$ rather than $\sigma_z$. This makes the Hamiltonian (\[p1\]) a non-QND variant of the Hamiltonian (1). We proceed as in Section II with $H_{N_{1}}$ (17) now given by $$H_{N_{1}} = {\hbar \omega \over 2} \left( \gamma^* {\partial
\over \partial \gamma^*} - \beta^* {\partial \over \partial
\beta^*} \right) + i {\hbar \omega \over 2} f(t) \left(
\gamma^* {\partial \over \partial \beta^*} + \beta^* {\partial
\over \partial \gamma^*} \right). \label{p2}$$ The propagator for $H_{N_{1}}$ (\[p2\]) has the same form as $N_1$ (20) but with $Q$ now satisfying the equation $${\partial \over \partial t} Q = {i\omega \over 2} \sigma_z Q +
{\omega \over 2} f(t) \sigma_x Q,$$ with $Q_{ij}(0)=\delta_{ij}, Q(0)=I$. This is solved recursively to yield the series solution $$\begin{aligned}
Q(t) & = & \sum\limits^{\infty}_{n=0} Q^{(n)} (t), \nonumber \\
Q^{(n)} (t) & = & \left( {i\omega \over 2} \sigma_z \right)^n
\int\limits^t_0 d\tau_n \int\limits^{\tau_n}_0 d\tau_{n-1}...
\int\limits^{\tau_{2}}_0 d\tau_1 \nonumber\\ & & \times \exp
\left[ {\omega \over 2} \sigma_x \left(
\int\limits^{\tau_{1}}_0 - \int\limits^{\tau_{2}}_{\tau_{1}} +
... + (-1)^n \int\limits^t_{\tau_{n}} \right) d\tau f(\tau)
\right]. \label{p3} \end{aligned}$$ Using Eq. (\[p3\]) and proceeding as before, we obtain the amplitudes of the propagator for the Hamiltonian (\[p1\]) in matrix form as $$\begin{aligned}
u_3 & = & \pmatrix{U_{00} & U_{01} \cr U_{10} & U_{11}}
\nonumber\\ & = & \exp \left\{ \sum\limits^M_{k=1} \alpha^*_k
\alpha'_k e^{-i\omega_kt} \right\} \nonumber\\ & & \times
\sum\limits^{\infty}_{n=0} \left( {i\omega \over 2} \right)^n
\int\limits^t_0 d\tau_n \int\limits^{\tau_{n}}_0 d\tau_{n-1}
... \int\limits^{\tau_{2}}_0 d\tau_1 \exp \left\{ \kappa^{(n)}
\right\} \nonumber\\ & & \times \pmatrix{\cosh \left(
\chi^{(n)} \right) & \sinh \left( \chi^{(n)} \right) \cr (-1)^n
\sinh \left( \chi^{(n)} \right) & (-1)^n \cosh \left(
\chi^{(n)} \right)}, \end{aligned}$$ where $$\begin{aligned}
\kappa^{(n)} & = & - \left( {\omega \over 2} \right)^2
\sum\limits^M_{k=1} {g^2_k \over \omega^2_k} \Bigg[ (2n+1) -
i\omega_kt + (-1)^{n+1} e^{-i\omega_kt} \nonumber \\ & & - 2
\sum\limits^n_{l=1} (-1)^{l+1} e^{-i\omega_k \tau_l} + 2(-1)^n
\sum\limits^n_{l=1} (-1)^{l+1} e^{-i\omega_k(t-\tau_l)}
\nonumber \\ & & + 4 \sum\limits^n_{p=2} \sum\limits^{p-1}_{q=1}
(-1)^{p+q} e^{-i\omega_k (\tau_p-\tau_q)} \Bigg], \end{aligned}$$ and $$\begin{aligned}
\chi^{(n)} & = & - {\omega \over 2} \sum\limits^M_{k=1} {g_k
\over \omega_k} \Bigg[ \left( \alpha'_k + (-1)^n \alpha^*_k
\right) \left( 1+(-1)^{n+1} e^{-i\omega_kt} \right) \nonumber
\\ & & + 2\alpha^*_k \sum\limits^n_{l=1} (-1)^{l+1} e^{-
i\omega_k(t-\tau_l)} - 2\alpha'_k \sum\limits^n_{l=1} (-
1)^{l+1} e^{-i\omega_k\tau_l} \Bigg]. \end{aligned}$$ This agrees with the results obtained in [@papa86; @sb03-1]. The matrix on the right-hand side of Eq. (40) contains diagonal as well as off-diagonal terms in contrast to the matrix on the right-hand side of Eq. (23) in which only diagonal elements are present. This is due to the non-QND nature of the system-bath interaction of the Hamiltonian described by Eq. (\[p1\]) whose propagator is given by Eq. (40), whereas Eq. (23) is the propagator of the Hamiltonian given by Eq. (1) where the system-bath interaction is of the QND type. The simpler form of the structure of the propagator (23) compared to the non-QND propagator (40) reflects on the simplification in the dynamics due to the QND nature of the coupling.
Bath of spins
=============
Now we consider the case where the reservoir is composed of spin-half or two-level systems, as has been dealt with by Shao and collaborators in the context of QND systems [@sgc96] and also quantum computation [@sh97], and for a nanomagnet coupled to nuclear and paramagnetic spins [@rpp00]. The total Hamiltonian is taken as $$\begin{aligned}
H_4 & = & H_S + H_R + H_{SR} \nonumber\\ & = & {\hbar \omega
\over 2} S_z + \sum\limits^M_{k=1} \hbar \omega_k \sigma_{zk} +
{\hbar \omega \over 2} \sum\limits^M_{k=1} c_k \sigma_{xk}S_z .
\label{h10} \end{aligned}$$ Here we use $S_z$ for the system and $\sigma_{zk}, \sigma_{xk}$ for the bath. Since $[H_S, H_{SR}]=0$, we have a QND Hamiltonian. In the Bargmann representation, we associate the variable $\beta^*$ with the spin-down state and the variable $\gamma^*$ with the spin-up state for the bath variables, and we have $$\begin{aligned}
\sigma_z & \longrightarrow & \gamma^* {\partial \over \partial
\gamma^*} - \beta^* {\partial \over \partial \beta^*},
\nonumber \\ \sigma_x & \longrightarrow & \gamma^* {\partial
\over \partial \beta^*} + \beta^* {\partial \over \partial
\gamma^*}. \end{aligned}$$ Similarly, the bosonization of the system variable gives $$S_z \longrightarrow \xi^* {\partial \over \partial \xi^*} -
\theta^* {\partial \over \partial \theta^*} ,$$ where the variable $\theta^*$ is associated with the spin-down state and the variable $\xi^*$ with the spin-up state. The bosonized form of the Hamiltonian (43) is given by $$\begin{aligned}
H_{B_{4}} & = & {\hbar \omega \over 2} \left( \xi^* {\partial
\over \partial \xi^*} - \theta^* {\partial \over \partial
\theta^*} \right) + \sum\limits^M_{k=1} \hbar \omega_k \left(
\gamma^*_k {\partial \over \partial \gamma^*_k} - \beta^*_k
{\partial \over \partial \beta^*_k} \right) \nonumber \\ & & +
{\hbar \omega \over 2} \sum\limits^M_{k=1} c_k \left(
\gamma^*_k {\partial \over \partial \beta^*_k} + \beta^*_k
{\partial \over \partial \gamma^*_k} \right) \left( \xi^*
{\partial \over \partial \xi^*} - \theta^* {\partial \over
\partial \theta^*} \right). \end{aligned}$$ A particular solution of the Schrödinger equation for the bosonized Hamiltonian (46) is obtained by attaching amplitudes to the polynomial parts in the products $$U_4 = (\theta^* + \xi^*) (\theta' + \xi') \prod^M_{k=1} \left(
\beta^*_k + \gamma^*_k \right) \left( \beta'_k + \gamma'_k
\right).$$ The initial state for the expanded propagator associated with the bosonized Hamiltonian (46) is $$U (t=0) = \exp \left\{ \theta^* \theta' + \xi^* \xi' \right\}
\prod^M_{k=1} \exp \left\{ \beta^*_k \beta'_k + \gamma^*_k
\gamma'_k \right\}.$$ Using (8), the propagator for the bosonized Hamiltonian (46) is $$\begin{aligned}
\fl u_4 (\theta^*, \xi^*, \mbox{\boldmath $\beta^*$},
\mbox{\boldmath $\gamma^*$}, t; \theta', \xi', \mbox{\boldmath
$\beta'$}, \mbox{\boldmath $\gamma'$}, 0) & = & \prod^M_{k=1}
\int {\bf D}^2 \{\theta\} {\bf D}^2 \{\xi \} {\bf D}^2
\{\beta_k \} {\bf D}^2 \{\gamma_k \} \nonumber \\ & & \times
\exp \Bigg\{ \sum\limits_{0\leq \tau <t} \Bigg[ \theta^*
(\tau+) \theta (\tau) + \xi^* (\tau +) \xi (\tau) \nonumber \\
& & + \beta^*_k (\tau+) \beta_k (\tau) + \gamma^*_k (\tau+)
\gamma_k (\tau) \Bigg] \nonumber \\ & & - i {\omega \over 2}
\int\limits^t_0 d\tau \Bigg[ \xi^*(\tau+) \xi (\tau) - \theta^*
(\tau+) \theta(\tau) \Bigg] \nonumber \\ & & - i
\int\limits^t_0 d\tau \omega_k \Bigg[ \gamma^*_k(\tau+)
\gamma_k (\tau) - \beta^*_k (\tau+) \beta_k(\tau) \Bigg]
\nonumber \\ & & - i {\omega \over 2} \int\limits^t_0 d\tau c_k
\Bigg[ \gamma^*_k(\tau+) \beta_k (\tau) + \beta^*_k(\tau+)
\gamma_k (\tau) \Bigg] \nonumber \\ & & \times \Bigg[ \xi^*(\tau+)
\xi (\tau) - \theta^* (\tau+) \theta(\tau) \Bigg] \Bigg\}. \end{aligned}$$ On the left-hand side of Eq. (49), [$\beta^*$]{}, [$\gamma^*$]{} are vectors with components $\{
\beta_k\}$ and $\{ \gamma_k\}$, respectively. Now we introduce a complex auxiliary field $f(\tau)$ to decouple the interaction term in (49) as $$\begin{aligned}
& & \exp \left( -i {\omega \over 2} \int\limits^t_0 d\tau c_k
\left[ \gamma^*_k (\tau +) \beta_k (\tau) + \beta^*_k (\tau +)
\gamma_k (\tau) \right] \left[ \xi^* (\tau +) \xi (\tau) -
\theta^* (\tau +) \theta (\tau) \right] \right) \nonumber \\ &
& = \int D^2 \{ f\} \exp \left[ -i \int\limits^t_0 d\tau f^*
(\tau) c_k \left( \gamma^*_k (\tau +) \beta_k (\tau) +
\beta^*_k (\tau +) \gamma_k (\tau) \right) \right] \nonumber \\
& & \times \exp \left[ \int\limits^t_0 d\tau f(\tau) {\omega
\over 2} \left( \xi^* (\tau +) \xi (\tau) - \theta^* (\tau +)
\theta (\tau) \right) \right]. \end{aligned}$$ Using (50) in (49) the propagator for the bosonized Hamiltonian (46) becomes $$\begin{aligned}
\fl u_4 \left(
\theta^*,\xi^*, \mbox{\boldmath $\beta^*$},
\mbox{\boldmath $\gamma^*$},t;\theta',\xi', \mbox{\boldmath
$\beta'$}, \mbox{\boldmath $\gamma'$},0
\right) & = & \prod^M_{k=1} \int D^2 \{ f\} M_1 \left(
\theta^*,\xi^*,t;\theta',\xi',0;[f] \right) \nonumber \\ & &
\times N_{1_{k}} \left( \beta^*_k, \gamma^*_k, t; \beta'_k,
\gamma'_k, 0; [f^*] \right), \end{aligned}$$ where $M_1$ is the propagator for $$H_{M_{1}} = {\hbar \omega \over 2} \left( \xi^* {\partial \over
\partial \xi^*} - \theta^* {\partial \over \partial \theta^*}
\right) + {i\hbar \omega \over 2} f(t) \left( \xi^* {\partial
\over \partial \xi^*} - \theta^* {\partial \over \partial
\theta^*} \right),$$ and $N_{1_{k}}$ is the propagator for $$H_{N_{1k}} = \hbar \omega_k \left( \gamma^*_k {\partial \over
\partial \gamma^*_k} - \beta^*_k {\partial \over \partial
\beta^*_k} \right) + \hbar f^*(t) c_k \left( \gamma^*_k
{\partial \over \partial \beta^*_k} + \beta^*_k {\partial \over
\partial \gamma^*_k} \right).$$ Here the propagator $M_1$ is $$M_1 = \sum\limits^{\infty}_{p=1} {1 \over p!} \left[ \left(
\theta^*, \xi^* \right) \widetilde{Q} \pmatrix{\theta' \cr
\xi'} \right]^p,$$ where $\widetilde{Q}$ is given by $$\widetilde{Q} (t) = \exp \left( {i\omega \over 2} S_z t -
{\omega \over 2} S_z \int\limits^t_0 d\tau f(\tau) \right) ,$$ with $\widetilde{Q}_{ij}(0) = \delta_{ij}, \widetilde{Q}(0)=I$.
The propagator $N_{1_{k}}$ is $$N_{1_{k}} = \sum\limits^{\infty}_{l=0} {1 \over l!} \left[
\left( \beta^*_k, \gamma^*_k \right) Q^k \pmatrix{\beta'_k \cr
\gamma'_k} \right]^l,$$ where $Q^k$ satisfies the equation $${\partial \over \partial t} Q^k = i \left( \omega_k
\sigma_{z_{k}} - f^* (t) c_k \sigma_{x_{k}} \right) Q^{(k)}.$$ This equation can be solved recursively to give $$Q^k (t) = \sum\limits^{\infty}_{n=0} Q^{k(n)} (t) ,$$ with $$Q^{k(0)} (0) = I,~~~ Q^{k(n)} (0) (n \neq 0) = 0,$$ $$\begin{aligned}
Q^{k(n)} (t) & = & \left( i\omega_k \sigma_{z_{k}} \right)^n
\int\limits^t_0 d\tau_n \int\limits^{\tau_{n}}_0 d\tau_{n-1}
... \int\limits^{\tau_{2}}_0 d\tau_1 \nonumber \\ & & \times
\exp \left( -i\sigma_{x_{k}} c_k \left(
\int\limits^{\tau_{1}}_0 - \int\limits^{\tau_{2}}_{\tau_{1}} +
... + (-1)^n \int\limits^t_{\tau_{n}} \right) d\tau f^* (\tau)
\right). \end{aligned}$$ Using Eqs. (54), (56) with $p=1$, $l=1$, respectively, in Eq. (51) and making use of Eqs. (55), (58), (59), (60) along with the $\delta$-functional identity (13), the amplitudes of the propagator for the Hamiltonian (43) are obtained in matrix form (in the Hilbert space of $H_R$) as $$\begin{aligned}
\fl u_4 = \pmatrix{U_{00} & U_{01} \cr U_{10} & U_{11}} & = &
\prod^M_{k=1} \sum\limits^{\infty}_{n=0} (i\omega_k)^n
\int\limits^t_0 d\tau_n \int\limits^{\tau_{n}}_0 d\tau_{n-1}...
\int\limits^{\tau_{2}}_0 d\tau_1 \nonumber\\ & & \times
e^{i{\omega \over 2} S_z t} \pmatrix{\cos (\Theta^{k(n)}) & i
\sin (\Theta^{k(n)}) \cr (-1)^ni \sin (\Theta^{k(n)}) & (-1)^n
\cos (\Theta^{k(n)})}, \end{aligned}$$ where $$\Theta^{k(n)} = {\omega \over 2} S_z c_k A_n,$$ $$A_n = \sum\limits^n_{j=1} (-1)^{j+1} 2\tau_j + (-1)^nt.$$
Now if we expand the terms containing $S_z$, i.e., make an expansion in the system space, in Eq. (61) we get terms such as $$e^{i{\omega \over 2}S_zt} \cos(\Theta^{k(n)}) = \cos({\omega
\over 2} c_k A_n) \pmatrix{ e^{i{\omega \over 2}t} & o \cr 0 &
e^{-i{\omega \over 2}t} }.$$ Here we have used the fact that $$e^{S_z A} = \pmatrix{ e^{A} & o \cr 0 & e^{-A} }.$$ Similarly, $$e^{i{\omega \over 2}S_zt} i \sin(\Theta^{k(n)}) = i
\sin({\omega \over 2} c_k A_n) \pmatrix{ e^{i{\omega \over 2}t}
& o \cr 0 & -e^{-i{\omega \over 2}t} }.$$ The above equations have only diagonal elements. We can see from the above equations that there are 16 amplitudes of the propagator for each mode $k$ of the reservoir out of which only the energy-conserving terms are present due to the QND nature of the system-reservoir coupling.
Discussions
===========
We look closely at the forms of the propagators (23) and (61) of the QND type Hamiltonians (\[h1\]) and (43), respectively. In the first case with an oscillator bath, Eq. (23) involves the matrix $$\pmatrix{e^B & 0 \cr 0 & e^{-B}},$$ where $B$ is given by Eq. (25). This can be used to generate the following transformation in phase space: $$\pmatrix{X \cr P} = \pmatrix{e^B & 0 \cr 0 & e^{-B}} \pmatrix{x
\cr p}. \label{s1}$$ It can be easily seen from Eq. (\[s1\]) that the Jacobian of the transformation is unity and it is a phase space area-preserving transformation. The first matrix on the right-hand side of (\[s1\]) has the form of a ‘squeezing’ operation [@km91], which is an area-preserving (in phase space) canonical transformation coming out as an artifact of homogeneous linear canonical transformations [@bk05].
In the second case of a spin bath, Eq. (61) involves the matrix $$R \equiv \pmatrix{\cos \Theta^{k(n)} & i \sin \Theta^{k(n)} \cr
(-1)^ni \sin \Theta^{k(n)} & (-1)^n \cos \Theta^{k(n)}},
\label{h16}$$ where $\Theta^{k(n)}$ is given by Eq. (62). For particular $n$ and $k$, we write $\Theta^{k(n)}$ as $\Theta$. For $n$ even, the above matrix (\[h16\]) becomes $$\pmatrix{\cos \Theta & i\sin \Theta \cr i\sin \Theta & \cos
\Theta} = e^{i\Theta \sigma_x}. \label{h17}$$ Using the Campbell-Baker-Hausdorff identity [@qsrl] this matrix can be shown to transform the spin vector $\sigma =
(\sigma_x, \sigma_y, \sigma_z)$ as $$e^{i\Theta \sigma_x} \pmatrix{\sigma_x \cr \sigma_y \cr
\sigma_z} e^{-i\Theta \sigma_x} = \pmatrix{1 & 0 & 0 \cr 0 &
\cos 2\Theta & -\sin 2\Theta \cr 0 & \sin 2\Theta & \cos
2\Theta} \pmatrix{\sigma_x \cr \sigma_y \cr \sigma_z},
\label{h18}$$ i.e., the abstract spin vector is ‘rotated’ about the $x$-axis by an angle $2\Theta$. For $n$ odd, (\[h16\]) becomes (again writing $\Theta^{k(n)}$ for particular $n$ and $k$ as $\Theta$) $$\pmatrix{\cos \Theta & i\sin \Theta \cr -i\sin \Theta & -\cos
\Theta} = \sigma_z \pmatrix{\cos \Theta & i\sin \Theta \cr
i\sin \Theta & \cos \Theta} = \sigma_z e^{i\Theta \sigma_x}.
\label{h19}$$ Thus the $n$-odd matrix is related to the $n$-even matrix by the spin-flipping energy. The above matrix transforms the spin vector $\sigma$ as $$\sigma_z e^{i\Theta \sigma_x} \pmatrix{\sigma_x \cr \sigma_y
\cr \sigma_y} e^{-i\Theta \sigma_x} \sigma_z = e^{i\pi}
\pmatrix{1 & 0 & 0 \cr 0 & \cos 2\Theta & \sin 2\Theta \cr 0 &
\sin 2\Theta & -\cos 2\Theta} \pmatrix{\sigma_x \cr \sigma_y
\cr \sigma_z}. \label{r1}$$ It can be easily seen from the right-hand side of the Eq. (\[r1\]) that the determinant of the transformation of the spin vectors brought about by the $n$-odd matrix (\[h19\]) has the value unity. It is well known that the determinant of a rotation matrix is unity [@gms]. Thus we see that the above transformation has the form of a rotation. Specifically, it can be seen that $$\sigma_z \pmatrix{\cos 2\Theta & \sin 2\Theta \cr \sin 2\Theta
& -\cos 2\Theta} = \pmatrix{ \cos 2\Theta & \sin 2\Theta \cr -
\sin 2\Theta & \cos 2\Theta},$$ and $$\pmatrix{\cos 2\Theta & \sin 2\Theta \cr -\sin 2\Theta & \cos
2\Theta}^T = \pmatrix{ \cos 2\Theta & -\sin 2\Theta \cr \sin
2\Theta & \cos 2\Theta}. \label{h20}$$ Here $T$ stands for the transpose operation. From the above it is seen that the matrix (\[h16\]) has the form of the operation of ‘rotation’, which is also a phase space area-preserving canonical transformation [@km91] and comes out as an artifact of homogeneous linear canonical transformations [@bk05]. Any element of the group of homogeneous linear canonical transformations can be written as a product of a unitary and a positive transformation [@barg; @bk052], which in turn can be shown to have unitary representations (in the Fock space) of rotation and squeezing operations, respectively [@bk05]. It is interesting that the propagators for the Hamiltonians given by Eqs. (\[h1\]) and (43), one involving a two-level system coupled to a bath of harmonic oscillators and the other with a bath of two-level systems, are analogous to the squeezing and rotation operations, respectively.
Conclusions
===========
In this paper we have investigated the forms of the propagators of some QND Hamiltonians commonly used in the literature, for example, for the study of decoherence in quantum computers. We have evaluated the propagators using the functional integral treatment relying on coherent state path integration. We have treated the cases of a two-level system interacting with a bosonic bath of harmonic oscillators (section 2), and a spin bath of two-level systems (section 3). In each case the system-bath interaction is taken to be of the QND type, i.e., the Hamiltonian of the system commutes with the Hamiltonian describing the system-bath interaction. We have shown the commonly occuring free-particle coordinate coupling model to be unitarily equivalent to the free-particle velocity coupling model which is of the QND type. For the variants of the model in section 2, we have examined (a) the case where the two-level system in addition to interacting with the bosonic bath of harmonic oscillators is also acted upon by an external mode in resonance with the atomic transition (section 2.1), and (b) the non-QND spin-Bose problem (section 2.2), which could be used to describe the spin-Bose problem of the interaction of a two-level atom with the electromagnetic field modes in a cavity via a dipole interaction.
The evaluation of the exact propagators of these many body systems could, apart from their technical relevance, also shed some light onto the structure of QND systems. We have found an interesting analogue of the propagators of these many-body Hamiltonians to squeezing and to rotation, for the bosonic and spin baths, respectively. Every homogeneous linear canonical transformation can be factored into the rotation and squeezing operations and these cannot in general be mapped from one to the other – just as one cannot in general map a spin bath to an oscillator bath (or vice versa) – but together they span the class of homogeneous linear canonical transformations and are ‘universal’. Squeezing and rotation, being artifacts of homogeneous linear canonical transformations, are both phase-space area-preserving transformations, and thus this implies a curious analogy between the energy-preserving QND Hamiltonians and the homogeneous linear canonical transformations. This insight into the structure of the QND systems would hopefully lead to future studies into this domain.
It is a pleasure to acknowledge useful discussions with Joachim Kupsch. The School of Physical Sciences, Jawaharlal Nehru University, is supported by the University Grants Commission, India, under a Departmental Research Support scheme.
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| ArXiv |
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abstract: |
The present works deals with gravitational collapse of cylindrical viscous heat conducting anisotropic fluid following the work of Misner and Sharp. Using Darmois matching conditions, the dynamical equations are derived and the effect of charge and dissipative quantities over the cylindrical collapse are analyzed. Finally, using the Miller-Israel-Steward causal thermodynamic theory, the transport equation for heat flux are derived and its influence on collapsing system has been studied.\
Keywords : Cylindrical collapse, Dissipation, heat flux, Junction conditions, Dynamical equations .
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0.2cm 0.2cm
\
Introduction
============
A challenging but curious issue in gravitational physics as well as in relativistic astrophysics is to know the final fate of a continual gravitational collapse. The stable configuration of a massive star persists as long as the inward pull of gravity is neutralized by the outward pressure of the nuclear fuel at the core of the star. Subsequently, when the star has exhausted its nuclear fuel there is no longer any thermonuclear burning and there will be endless gravitational collapse. However, depending on the mass of the collapsing star, the compact objects such as white dwarfs, neutron stars and black holes are formed. In white dwarf and neutron star gravity is counter balanced by electron and neutron degeneracy pressure respectively while black hole is an example of the end state of collapse.\
The study of gravitational collapse was initiated long back in 1939 by Oppenheimer and Snyder \[1\]. They have studied the collapse of a homogeneous spherical dust cloud in the frame work of general relativity. Then after a quarter century, a more realistic investigation was done by Misner and Sharp\[2\] with perfect fluid in the interior of a collapsing star. In both the studies, the exterior of the collapsing star was chosen as vacuum. Vaidya\[3\] formulated the non-vacuum exterior of a star having radiating fluid in the interior. An inhomogeneous spherically symmetric dust cloud was analytically studied by Joshi and Singh \[4\] and they have shown that the final fate of the collapsing star depend crucially on the initial density profile and the radius of the star. Debnath etal\[5\] investigated collapse dynamics of the non-adiabatic fluid, considering quasi-spherical Szekeres space-time in the interior and plane symmetric Vaidya solution in the exterior region.\
Although most of the works on collapse dynamics are related to spherical objects, still there are interesting information about self-gravitating fluids for collapsing object with different symmetries. The natural choice for non-spherical symmetry is axis symmetric objects. The vacuum solution for Einstein field equations in cylindrically symmetric space-time was obtained first by Levi-Civita \[6\] but still it is a challenging issue of interpreting two independent parameters in the solution. Herrera etal \[7\] studied cylindrical collapse of non-dissipative fluid with exterior Einstein -Rosen space-time and showed wrongly a non-vanishing radial pressure on the boundary surface and subsequently in collaboration with M.A.H. Maccallum \[8\] they corrected the result. Then Herrera and collaborators investigated cylindrical collapse of matter with\[9\] or without shear \[10\].\
Further, the junction conditions due to Darmois \[11\] has a very active role in dealing collapsing problems. Sharif etal \[12-14\] showed the effect of positive cosmological constant on the collapsing process by using junction conditions between static exterior and non-static interior with a cosmological constant. Also Herrera etal \[15\], using junction conditions were able to prove that any conformally flat cylindrically symmetric static source cannot be matched to the Levi-Civita space-time. Then Kurita and Nakao \[16\] formulated naked singularity along the axis of symmetry, considering cylindrical collapse with null dust.\
Moreover from realistic point of view it is desirable to consider dissipative matter in the context of collapse dynamics \[17-19\]. Considering collapse of a radiating star with dissipation in the form of radial heat flow and shear viscosity, Chan\[20\] has showed that shear viscosity plays a significant role in the collapsing process. Collapse dynamics with dissipation of energy as heat flow and radiation has been studied by Herrera and Santos\[18\]. Subsequently, by Considering of causal transport equations related to different dissipative components (heat flow, radiation, shear and bulk viscosity) Herrera etal \[15,21,22\] investigated the collapse dynamics. The same collapsing process with plane symmetric geometry or others has been examined by Sharif etal \[23,24\] .\
On the other-hand, in the context of gravitational waves, the sources must have non spherical symmetry. Further, cylindrical collapse of non-dissipative fluid with exterior containing gravitational waves shows non-vanishing pressure on the boundary surface by using Darmois matching conditions. Recently, it has been verified \[25\] in studying cylindrical collapse of anisotropic dissipative fluid with formation of gravitational waves outside the collapsing matter.\
In the present work, following Misner and Sharp collapse dynamics of viscous, heat conducting charged anisotropic fluid in cylindrically symmetric background will be studied. The paper is organized as follows. Section 2 deals with basic equations related to interior and exterior space-time. The junction conditions are evaluated and discussed in Section 3. The dynamical equations are derived and studied in Section 4. Finally, the process of mass, heat and momentum transfer through transport equation is discussed in section 5.\
Interior and exterior space-time: Basic equations.
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Mathematically, the whole four dimensional space-time manifold having a cylindrical collapsing process can be written as $M=M^+U \Sigma U M^-$ with $M^{-}\bigcap M^{+}=\phi$. Here, $\Sigma$, the collapsing cylindrical surface is a time-like three surface and is the boundary of the two four dimensional sub-manifolds $M^-$ (interior) and $M^+$ (exterior).\
In $M^-$ choosing co-moving coordinates the line element can be written as \[25\] $$d{s_-^2}=-{A^2}d{t^2} +{B^2 }d{r^2} +{C^2}d{\phi^2} +{D^2}d{z^2}$$\
where the metric coefficients are functions of t and r i.e. A=A(t,r) and so on. Also due to cylindrical symmetry, the coordinates are restricted as:\
$-\infty\leq t\leq +\infty,~~~r\geq 0,~~~-\infty<z<+\infty,~~~0\leq \phi \leq 2\pi$\
For compact notation we write $\lbrace x^{-\mu} \rbrace \equiv[t,r,\phi,z]~~~,~~~ (\mu=0,1,2,3) $.\
The anisotropic fluid having dissipation in the form of shear viscosity and heat flow has the energy- momentum tensor of the form \[7,9\] $$T_{\mu\nu}=(\rho+p_t){v_\mu}{v_\nu}+{p_t}g_{\mu\nu}+({p_r}-{p_t}){\chi_\mu}{\chi_\nu}-2\eta\sigma_{\mu\nu}+2q_{(\mu}v_{\nu)}$$\
Here $\rho,~p_r,~p_t~ ~~\eta~~~ and~~q_{\mu}$ stands for energy density ,the radial pressure, the tangential pressure, coefficient of shear viscosity and radial heat flux vector respectively. Also $v_\mu$ and $\chi_\mu$ are unit time-like and space-like vectors satisfying the following relations $$v_\mu v^\mu =-\chi_\mu\chi^\mu=-1~~~,~~~~\chi^\mu v_\mu=0~~~,~~~q_{\mu}v^{\mu}=0~~~~$$\
Moreover, the shear tensor $\sigma_{\mu\nu}$ has the expression $$\sigma_{\mu\nu}= v_{(\mu;\nu)}+a_{(\mu}v_{\nu)}-\frac{1}{3}\Theta(g_{\mu\nu}+v_\mu v_\nu)$$\
where $a_\mu=v_{\mu;\nu}v^\nu $ is the acceleration vector and $\Theta=v^\mu;_\mu$ is the expansion scalar.\
For the above metric one may choose the unit time-like vector, space-like vector and heat flux vector in a simple form as $$v^\mu=A^{-1}\delta_0^\mu~~~,~~~~~~~~~~~~~~\chi^\mu=B^{-1}\delta_1^\mu~~~,~~~~~q^{\mu}=q\delta^{\mu}_{1}~~~~~$$\
The shear tensor has only non-zero diagonal components as $$\sigma_{11}=\frac{B^2}{3A}[\Sigma_1-\Sigma_3]~~,~~~~~~\sigma_{22}=\frac{C^2}{3A}[\Sigma_2-\Sigma_1]~~~and~~~\sigma_{33}=\frac{D^2}{3A}[\Sigma_3-\Sigma_2]~~~with~~~~~\sigma^2=\frac{1}{6A^2}[\Sigma_1^2+\Sigma_2^2+\Sigma_3^2]$$\
where $ \Sigma_1=\frac{\dot{B}}{B}-\frac{\dot{C}}{C}~~,~~~\Sigma_2=\frac{\dot{C}}{C}-\frac{\dot{D}}{D}~~,~~~\Sigma_3=\frac{\dot{D}}{D}-\frac{\dot{B}}{B}
$\
Also the acceleration vector and the expansion scalar have the explicit expressions $$a_1=\frac{A^\prime}{A}~~,\Theta=\frac{1}{A}(\frac{\dot{B}}{B}+\frac{\dot{C}}{C}+\frac{\dot{D}}{D})$$\
In the above, by notation we have used $\cdot$ $\equiv\frac{\partial}{\partial t}$ and $^\prime$ $\equiv\frac{\partial}{\partial r}.$\
If in addition we assume the above fluid distribution to be charged then the energy-momentum tensor for the electromagnetic field has the form $$E_{\alpha\beta}=\frac{1}{4\pi}(F^{\alpha}_{\mu}F_{\nu\alpha}-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}g_{\mu\nu})~~~~~$$
where the Maxwell field tensor $F_{\alpha\beta}$ is related to the four potential $\phi_{\alpha}$ as $$F_{\alpha\beta}=\phi_{\beta , \alpha}-\phi_{\alpha , \beta}$$ and the evolution of the field tensor corresponds to Maxwell equations $$F^{\alpha\beta}_{;\beta}=4\pi J^{\alpha}$$ where $J^{\alpha}$ the four current vector.\
As the charge per unit length of the cylinder is at rest with respect to comoving co-ordinates so the magnetic field will be zero in this local coordinate system \[26,27\]. Hence the four potential and the four current takes the simple form $$\phi_{\alpha}=\phi\delta^{0}_{\alpha} ,~~~~J^{\alpha}=\epsilon v^{\alpha}$$ where $\phi=\phi(t,r)$ is the scalar potential and $\epsilon=\epsilon(t,r)$ is the charge density.\
From the law of conservation of charge : $J^{\alpha}_{;\alpha}=0$, one obtains the total charge distribution interior to radius $r$ and per unit length of the cylinder as $$s(r)=2\pi\int^{r}_{0}\epsilon BCD dr$$ Now the explicit form of the Maxwell’s equations (10) for the interior space-time $M^{-}$ are given by $$\phi^{''}-(\frac{A^{'}}{A}+\frac{B^{'}}{B}-\frac{C^{'}}{C}-\frac{D^{'}}{D})\phi^{'}=4\pi\epsilon AB^{2}$$ and $$\dot{\phi}^{'}-(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}-\frac{\dot{C}}{C}-\frac{\dot{D}}{D})\phi^{'}=0$$ A first integral of equation (13) gives $$\phi^{'}=\frac{2sAB}{CD}$$ which satisfies identically the other Maxwell’s equation(14). Hence one obtains the electric field intensity as $E(t,r)=\frac{s(r)}{2\pi C}$
Further, in the interior space-time $M^{-}$ the Einstein field equations $G_{\alpha\beta}=8\pi(T_{\alpha\beta}+E_{\alpha\beta})$ have the explicit form
$$\frac{A^{2}}{B^{2}}(-\frac{C^{''}}{C}-\frac{D^{''}}{D}+\frac{B^{'}}{B}(\frac{C^{'}}{C}+\frac{D^{'}}{D})-\frac{C^{'}D^{'}}{CD})+(\frac{\dot{B}\dot{C}}{BC}+\frac{\dot{B}\dot{D}}{BD}+\frac{\dot{C}\dot{D} }{CD})=8\pi(\rho A^{2}-2\eta\sigma_{00})+4\frac{s^{2}A^{2}}{C^{2}D^{2}}$$
\
$$-\frac{B^{2}}{A^{2}}(\frac{\ddot{C}}{C}+\frac{\ddot{D}}{D}+\frac{\dot{C}\dot{D}}{CD}-\frac{\dot{A}\dot{C}}{AC}-\frac{\dot{A}\dot{D}}{AD})+(\frac{C^{'}D^{'}}{CD}+\frac{A^{'}C^{'}}{AC}+\frac{A^{'}D^{'}}{AD})=8\pi(p_{r}B^{2}-2\eta\sigma_{11})-4\frac{s^{2}B^{2}}{C^{2}D^{2}}$$
$$-\frac{C^2}{A^2}[\frac{\ddot{B}}{B}+\frac{\ddot{D}}{D}-\frac{\dot{A}}{A}(\frac{\dot{B}}{B}+\frac{\dot{D}}{D})+\frac{\dot{B}\dot{D}}{BD}]+\frac{C^2}{B^{2}}[\frac{A^{''}}{A}+\frac{D^{''}}{D}-\frac{A^{'}}{A}(\frac{B^{'}}{B}-\frac{D^{'}}{D})-\frac{D^{'}}{D}\frac{B^{'}}{B}]=8\pi(p_{t}C^2-2\eta\sigma_{22})+4\frac{s^{2}}{D^{2}}$$
$$-\frac{D^{2}}{A^{2}}[\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}-\frac{\dot{A}}{A}(\frac{\dot{B}}{B}+\frac{\dot{C}}{C})+\frac{\dot{B}\dot{C}}{BC}]+\frac{D^{2}}{B^{2}}[\frac{A^{''}}{A}+\frac{C^{''}}{C}-\frac{A^{'}}{A}(\frac{B^{'}}{B}-\frac{C^{'}}{C})-\frac{C^{'}}{C}\frac{B^{'}}{B}]=8\pi(p_{t}D^{2}-2\eta\sigma_{33})+4\frac{s^{2}}{C^{2}}$$
and $$\frac{1}{AB}(\frac{\dot{C^{'}}}{C}+\frac{\dot{D^{'}}}{D}-\frac{C^{'}}{C}\frac{\dot{B}}{B}-\frac{\dot{B}}{B}\frac{D^{'}}{D}-\frac{A^{'}}{A}\frac{\dot{C}}{C}-\frac{A^{'}}{A}\frac{\dot{D}}{D})=8\pi q$$
The gravitational energy per specific length in cylindrically symmetric space-time is defined as \[28-30\] $$E=\frac{(1-l^{-2}\nabla^{a}r \nabla_{a}r)}{8}$$ In principle, $E$ is the charge associated with a general current which combines the energy-momentum of the matter and gravitational waves. It is usually referred in the literature as $C$-energy for the cylindrical symmetric space-time. For cylindrically symmetric model with killing vectors the circumference radius $\rho$ and specific length $l$ are defined as \[28-30\]\
$\rho^{2}=\xi_{(1)a}\xi_{(1)}^{a}$ , $l^{2}=\xi_{(2)a}\xi_{(2)}^{a}$, so that $r=\rho l$ is termed as areal radius.\
For the present model with the contribution of electromagnetic field in the interior region the $C-$energy takes the form
$$E'=\frac{l}{8}+\frac{1}{8D}[\frac{1}{A^{2}}(C\dot{D}+\dot{C}D)^{2}-\frac{1}{B^{2}}(CD^{'}+C^{'}D)^{2}]+\frac{s^{2}}{2C}$$
\
It should be noted that the above energy is also very similar to Tabu’s mass function in the plane symmetric space-time\[31\]\
The exterior space-time manifold $(M^{+})$ of the cylindrical surface $\Sigma$ is described by the metric in the retarded time co-ordinate as \[32,33\]
$$ds_{+}^{2}=-(\frac{-2M(v)}{R}+\frac{Q^{2}(v)}{R^{2}})dv^{2}-2dRdv+R^{2}(d\phi^{2}+\lambda^{2}dz^{2})$$
where $v$ is the usual retarded time, $M(v)$ is the total mass inside $\Sigma$, $Q(v)$ is the total charge bounded by $\Sigma$ and $\lambda$ is an arbitrary constant. Further, from the point of view of the interior manifold $(M^{-})$ the bounding three surface $\Sigma$ (comoving surface) is described as $$f_{-}(t,r)=r-r_{\Sigma}=0$$ and hence the interior metric on $\Sigma$ takes the form $$ds_-^2\stackrel{\Sigma}{=}-d\tau^2 +C^2dz^2+D^2d\phi^2$$ where $$d\tau\stackrel{\Sigma}{=}Adt,$$\
defines the time co-ordinate on $\Sigma$ and $\stackrel{\Sigma}{=}$ by notation implies the equality of both sides on the surface $\Sigma$.\
Similarly, from the perspective of the exterior manifold the boundary three surface $\Sigma$ is characterized by
$$f_{+}(v,R)\equiv R-R_{\Sigma}(v)=0$$
so that the exterior metric on $\Sigma$ takes the form
$$ds_{+}^{2}\stackrel{\Sigma}{=}-(\frac{-2M(v)}{R}+\frac{Q^{2}(v)}{R^{2}}+\frac{2dR_{\Sigma}(v)}{dv})dv^{2}+R^{2}(d\phi^{2}+\lambda^{2}dz^{2})$$
Here by notation we write $[x^{+\mu}]=[v,R,\phi, z]$
Junction conditions
===================
In order to have a smooth matching of the interior and exterior manifolds over the bounding three surface (not a surface layer), the following conditions due to Darmois \[11\] are to be satisfied:\
(i)The continuity of the first fundamental form i.e.
$$(ds^2)_{\Sigma}=(ds^2_{-})_{\Sigma}=(ds^{2}_{+})_{\Sigma}$$
(ii)The continuity of the second fundamental form i.e $K_{ij}d\xi^{i}d\xi^{j}$. This implies the continuity of the extrinsic curvature $K_{ij}$ over the hypersurface \[11\] i.e. $$[K_{ij}]\equiv K_{ij}^+ -K_{ij}^-=0$$
where $K_{ij}^\pm$ is given by
$$K_{ij}^\pm=-n_\sigma^\pm[\frac{\partial^2x_\pm ^\sigma}{\partial\xi^i \partial\xi^j}+\Gamma_{\mu\nu}^\sigma \frac{\partial x_\pm^\mu}{\partial\xi^i}\frac{\partial x_\pm^\nu}{\partial\xi^j}],~~~(\sigma,~\mu,~\nu~=0,1,2,3)$$
In the above expression for extrinsic curvature, $n_{\sigma}^{\pm}$ are the components of the outward unit normal to the hyper-surface with respect to the manifolds $M^{\pm}$ (i.e. in the co-ordinates $x^{\pm \mu}$) and have explicit expressions\
$n_\sigma^- \stackrel{\Sigma}{=}(0,B,0,0)~~and~~~~n_\sigma^+ \stackrel{\Sigma}{=}\mu(\frac{-dR}{dv},1,0,0)$ with $\mu=[\frac{-2M(v)}{R}+\frac{Q^{2}(v)}{R^{2}}+2\frac{dR}{dv}]^{\frac{-1}{2}}$\
Also in the above the christoffel symbols are evaluated for the metric in $M^{-}$ or $M^{+}$ accordingly and we choose $\xi^{0}=\tau$, $\xi^{2}=z$, $\xi^{3}=\phi$ as the intrinsic co-ordinates on $\Sigma$ for convenience.\
The continuity of the 1st fundamental form gives
$$C(t,r_{\Sigma})\stackrel{\Sigma}{=}R_{\Sigma}(v)~~~,~~~D(t,r_{\Sigma})\stackrel{\Sigma}{=}\lambda R_{\Sigma}(v)$$
$$\frac{dt}{d\tau}=1/A~~~~\frac{dv}{d\tau}=\mu$$
Now the non vanishing components of extrinsic curvature $K^{\pm}_{ij}$ are\
$$K_{00}^-{=}-(\frac{A^\prime}{AB})_{\Sigma}$$ $$K_{00}^+{=}[(\frac{d^{2}v}{d\tau^{2}})(\frac{dv}{d\tau})^{-1}-(\frac{dv}{d\tau})(\frac{M}{R^{2}}-\frac{Q^{2}}{R^{3}})]_{\Sigma}$$\
$$K_{22}^-{=}(\frac{C{C}^\prime}{B})_{\Sigma}$$ $$K_{33}^-{=}(\frac{D{D}^\prime}{B})_{\Sigma}$$
$$K_{22}^+{=}[R(\frac{dR}{d\tau})-\frac{(dv)}{d\tau}(2M-\frac{Q^{2}}{R})]_{\Sigma}=\lambda ^{-2}K^{+}_{33}$$
\
and\
Hence continuity of the extrinsic curvature together with equations (32) and (33) gives the following relations over $\Sigma$ \[33,34\] $$M(v)\stackrel{\Sigma}{=}\frac{R}{2}[(\frac{\dot{R}}{A})^{2}-(\frac{R^{'}}{B})^{2}]+\frac{Q^{2}}{2R}$$\
$$E\stackrel{\Sigma}{=}\frac{l}{8}+\lambda M$$\
and $$q\stackrel{\Sigma}{=}p_{r}-\frac{2\eta \sigma_{11}}{B^{2}}-\frac{s^{2}}{2\pi c^{4}}(\frac{1}{\lambda^{2}}-1/4
)$$\
Thus equations (39) gives the total mass inside the boundary surface $\Sigma$ while equation (40) shows the linear relationship between the $C$ energy for the cylindrically symmetric space-time with the bounding mass over $\Sigma$. Further, equation (41) shows a linear relationship among the fluid parameters $(p_{r}, \eta, q)$ on the bounding surface $\Sigma$. Hence radial pressure is in general non zero on the bounding surface due to dissipative nature of the fluid and the charge on the bounding surface. But when dissipative components of the fluid are switch off then the above result (uncharged) agrees with the results of Herrera etal \[8\]. Also it should be noted that the radial pressure on the boundary does not depend on the charge bounded by $\Sigma$, it depends only on the charge on the surface $\Sigma$.\
Analysis of Dynamical equations:
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From the conservation of energy-momentum i.e. $(T^{\alpha\beta}+E^{\alpha\beta})_{;\beta}=0$ we can have two zero scalars namely $(T^{\alpha\beta}+E^{\alpha\beta});_{\beta}v_{\alpha}~~~and~~~(T^{\alpha\beta}+E^{\alpha\beta});_{\beta}\chi_{\alpha}$\
Using equations (2) and(8) the explicit expressions for these two scalars are $$\frac{\dot{\rho}}{A}+\frac{\dot{B}}{A}(\frac{\rho}{B}+\frac{p_{r}}{B}-2\eta\sigma^{11})+\frac{\dot{C}}{A}(\frac{p_{\bot}}{C}+\frac{\rho}{C}-2\eta\sigma^{22})+\frac{\dot{D}}{A}(\frac{\rho}{D}+\frac{p_{\bot}}{D}-2\eta\sigma^{33})+\frac{q^{'}}{B}+\frac{q}{B}(2\frac{A^{'}}{A}+\frac{C^{'}}{C}+\frac{D^{'}}{D})=0$$ and $$\begin{aligned}
(\frac{p_{r}}{B^{2}}-2\eta\sigma^{11})^{'}+\frac{\dot{q}}{AB}+\frac{q}{AB}(\frac{\dot{C}}{C}+\frac{\dot{D}}{D})+\frac{A^{'}}{A}(\frac{\rho}{B^{2}}+\frac{p_{r}}{B^{2}}-2\eta\sigma^{11})+\frac{B^{'}}{B}(\frac{p_{r}}{B^{2}}-2\eta\sigma^{11})+\frac{C^{'}}{C}(\frac{p_{r}}{B^{2}}
\nonumber
\\
-\frac{p_{\bot}}{B^{2}}-2\eta\sigma^{11}-2\eta\sigma^{22}\frac{C^{2}}{B^{2}})+\frac{D^{'}}{D}(\frac{p_{r}}{B^{2}}-\frac{p_{\bot}}{B^{2}}-2\eta\sigma^{11}+2\eta\sigma^{33}\frac{D^{2}}{B^{2}})-\frac{ss^{'}}{\pi C^{2}D^{2}B}=0 \end{aligned}$$\
Now following the formulation of Misner and Sharp \[2\], we introduce the proper time derivative and proper radial derivative as $$D_{T}=\frac{1}{A}\frac{\delta}{\delta t}~~~~~~and~~~~ D_{R}=\frac{1}{R^{'}}\frac{\delta}{\delta r}$$
so that the fluid velocity in the collapsing situation, can be defined as \[35\] $$U=D_{T}(R)=D_{T}(C) <0
~~~and~~~ V=D_{T}(Rr)=D_{T}(D) <0$$ Using equations (17)-(22) and(45), we can obtain the acceleration of a collapsing matter inside $\Sigma$ as
$$D_{T}(U)=-4\pi R(p_{r}-\frac{4\eta \sigma}{\sqrt{3}})+\tilde{E}\frac{A^{'}}{AB}+\frac{s^{2}}{R^{3}}(2+\frac{1}{2\lambda})-\frac{1}{R^{2}\lambda}(E^{'}-\frac{l}{8})$$
Now combining (43) and (46) we obtain $$\begin{aligned}
(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})D_{T}(U)&=&(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})[\frac{1}{R^{2}\lambda}(E^{'}-\frac{l}{8})+4\pi R(p_{r}-\frac{4\eta \sigma}{\sqrt{3}})-\frac{s^{2}}{R^{3}}(2+\frac{1}{2\lambda})]-\tilde{E}^{2}[D_{R}(p_{r}
\nonumber
\\
&-&\frac{4\eta\sigma}{\sqrt{3}})+\frac{2}{R}(p_{r}-p_{\bot}-2\sqrt{3}\eta\sigma)-\frac{s}{\pi R^{4}}D_{R}(s)]-\frac{2q\tilde{E}}{A}(\frac{\dot{B}}{B}+\frac{\dot{C}}{C})-\frac{\dot{q}\tilde{E}}{A}\end{aligned}$$\
Using (22) and the junction condition $D\stackrel{\Sigma}{=}\lambda C$, we write \[24\] $$\tilde{E} =\frac{C^{'}}{B}=[U^{2}+\frac{s^{2}}{\lambda c^{2}}-\frac{2}{\lambda c}(E^{'}-\frac{l}{8})]^{\frac{1}{2}}$$
Hence using the field equations for the interior manifold we obtain the time rate of change of C-energy as $$D_{T}E^{'}=-4\pi R^{2}\lambda[(p_{r}-\frac{4\eta\sigma}{\sqrt{3}})U +q\tilde{E}]+\frac{s^{2}\dot{C}}{R^{2}A}(2\lambda-\frac{1}{2})$$ Also the above equation can be interpreted as the variation of the total energy inside the collapsing cylinder. Note that due to negativity, of the fluid velocity $v$ the first term on the r.h.s will contribute to the energy of the system provided the radial pressure is restricted as $p_{r}>\frac{4\eta\sigma}{\sqrt{3}}$. Due to negativity, the second term indicates an outflow of energy in the form of radiation during the collapsing process. The third term is coulomb-like force term and it will increase the energy of the system provided $\lambda~>~\frac{1}{4}$.\
Further, using the Einstein field equations (16), (20) and the expression for C- energy in equation (22), the radial derivative of the C energy takes the form $$D_{R}E'=4\pi\rho R^{2}\lambda+\frac{s^{2}}{R^{2}}(2\lambda-\frac{1}{2})+\frac{s}{R}D_{R}(s)+\frac{4\pi q BR^{2}\lambda}{R'}D_{T}(C)+\frac{1}{8\rho R'}$$
This radial derivative can be interpreted as the energy variation between the adjacent cylindrical surfaces within the matter distribution. The first term on the r.h.s. is the usual energy density of the fluid element while the second term and third terms are the conditions due to the electromagnetic field. The fourth term represents contribution due to the dissipative heat flux and the last term will increase or decrease the energy of the system during the collapse of the cylinder provided $R^{'}> ~or~<~0$\
Finally, the collapse dynamics is completely characterized by the equation of motion in equation (47). Normally, for collapsing situation $D_{T}U$ should be negative, i.e, indicating an inward radial flow of the system. Consequently, terms on the r.h.s (of eq.(47)) contributing negatively favours the collapse and positive terms oppose the collapsing process. In an extreme situation the system will be in hydrostatic equilibrium if terms of both signs balance each other. Further, from dimensional analysis the factor $(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})$ can be considered as an inertial mass density, independent of heat flux contribution. The first term on the r.h.s. of eq. (47) can be identified as the gravitational force, indicating the effects of specific length and electric charge in the gravitational contribution. The second term has three contributing components- the pressure gradient (which is negative), local anisotropy of the fluid and electromagnetic field term. The remaining terms represent the heat flux contribution and due to negativity they seem to leave the system along the radial outward streamlines.\
Causal Thermodynamics: The Transport equation
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In causal thermodynamics due to Miller-Israel-Stewart, the transport equation for heat flow is given by \[21\]
$$\tau h^{ab}V^{c}q_{b;c}+q^{a}=-\kappa h^{ab}(T_{,b}+a_{b}T)-\frac{1}{2}\kappa T^{2}(\frac{\tau V^{b}}{\kappa T^{2}})_{;b}q^{a}$$
where $h^{ab}=g^{ab}+V^{a}V^{b}$ is the projection tensor of the $3-$surface orthogonal to the unit time-like vector $V^{a}$, $\kappa$ represents the thermal conductivity, $T$ is the temperature, $\tau$ denotes the relaxation time and $a_{b}T$ is the inertial term due to Tolman. Now due to cylindrical symmetry, the above transport equation (51)simplifies to
$$\tau\dot{q}=-\frac{1}{2A}\kappa \frac{qT^{2}}{\tau}(\frac{\tau}{\kappa T^{2}})-q[\frac{3U}{2R}+G+\frac{1}{\tau}]-\frac{\kappa \tilde{E}D_{R}T}{\tau}-\frac{\kappa TD_{T}U}{\tau\tilde{E}}-\frac{\kappa T}{\tau\tilde{E}R^{2}}[\frac{1}{\lambda}(E^{'}-\frac{l}{8})+4\pi R^{3}(p_{r}-\frac{4\eta\sigma}{\sqrt{3}})-\frac{S^{2}}{R}(2+\frac{1}{2\lambda})]$$
with $G=\frac{1}{A}(\frac{\dot{B}}{B}-\frac{\dot{C}}{C})$\
Now considering proper derivatives in equation(44) of the above equation and using the field velocity (in eq.(45)), and equation of motion (i.e. eq. (47)) one obtains the effects of heat flux or dissipation in the collapsing process as\
$$\begin{aligned}
(1-\alpha)(\rho+ p_{r}-\frac{4\eta\sigma}{\sqrt{3}})D_{T}U =(1-\alpha)F_{grav}+F_{hyd}+ \alpha \tilde{E}^{2}[D_{R}p_{r}+2(p_{r}-p_{\bot}-2\sqrt{3}\eta\sigma)\frac{1}{R}
\nonumber
\\
-\frac{SD_{R}(S)}{\pi R^{4}\lambda^{2}}]-\tilde{E}[D_{T}q+2qG+\frac{4qU}{R}]+\alpha \tilde{E}[D_{T}q+\frac{4qU}{R}+2qG]\end{aligned}$$\
with $$\alpha=\frac{\kappa T}{\tau}(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})^{-1}$$
$$F_{grav}=-(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})[(E^{'}-\frac{l}{8})\frac{1}{\lambda}+4\pi p_{r}R^{3}-(2+\frac{1}{2\lambda})\frac{S^{2}}{R}](\frac{1}{R^{2}})$$
$$F_{hyd}=\tilde{E}^{2}[D_{R}(p_{r}-\frac{4\eta\sigma}{\sqrt{3}})+\frac{2}{R}(p_{r}-p_{\bot}-2\sqrt{3}\eta\sigma)-\frac{S}{\pi R^{4}}D_{R}(S)]$$
The l.h.s. of equation (53) can be interpreted as Newtonian force $F$ with $(\rho+p_{r})(1-\alpha)$ as the inertial mass density. So as $\alpha\rightarrow 1$, $F\rightarrow 0$ i.e. there is no inertial force and collapse will be inevitable due to gravitational attraction. Further, the inertial mass density decreases as long as $0~<~\alpha~<1$ and it increases for $\alpha~>~1$. Moreover, due to equivalence principle the gravitational mass also decrease or increase according as $\alpha~<~or~>~1$ and gives a clear distinction between the expanding and collapsing process due to dynamics of dissipative system. Note that although the gravitational force is affected by the same factor $(1-\alpha)$ but the hydrodynamical force is free from it. Further, combination of all these terms on the r.h.s of equation (53) results the l.h.s i.e. $(1-\alpha)(\rho+p_{r}-\frac{4\pi\sigma}{\sqrt{3}})D_{T}U~<0$, there will be gravitational collapse while there will be expansion if the l.h.s to be positive. Interestingly, if $\alpha$ continuously decreases from a value larger than unity to one less than unity, then there will be a phase transition (collapse to expansion) and bounce will occur. As a result, there will be loss of energy of the system and collapsing cylinder with non-adiabatic source causes emission of gravitational radiation. Therefore, there will be radiation outside the collapsing cylinder and hence the choice of the exterior metric (in eq.(23)) is justified.\
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| ArXiv |
---
abstract: 'Graph representation learning resurges as a trending research subject owing to the widespread use of deep learning for Euclidean data, which inspire various creative designs of neural networks in the non-Euclidean domain, particularly graphs. With the success of these graph neural networks (GNN) in the static setting, we approach further practical scenarios where the graph dynamically evolves. For this case, combining the GNN with a recurrent neural network (RNN, broadly speaking) is a natural idea. Existing approaches typically learn one single graph model for all the graphs, by using the RNN to capture the dynamism of the output node embeddings and to implicitly regulate the graph model. In this work, we propose a different approach, coined EvolveGCN, that uses the RNN to evolve the graph model itself over time. This model adaptation approach is model oriented rather than node oriented, and hence is advantageous in the flexibility on the input. For example, in the extreme case, the model can handle at a new time step, a completely new set of nodes whose historical information is unknown, because the dynamism has been carried over to the GNN parameters. We evaluate the proposed approach on tasks including node classification, edge classification, and link prediction. The experimental results indicate a generally higher performance of EvolveGCN compared with related approaches.'
author:
- 'Aldo Pareja$^{1,2}$[^1]'
- 'Giacomo Domeniconi$^{1,2}$'
- 'Jie Chen$^{1,2}$[^2]'
- 'Tengfei Ma$^{1,2}$'
- 'Toyotaro Suzumura$^{1,2}$'
- |
\
Hiroki Kanezashi$^{1,2}$
- |
Tim Kaler$^{1,3}$Charles E. Leisersen$^{1,3}$ $^1$MIT-IBM Watson AI Lab, $^2$IBM Research, $^3$MIT CSAIL\
{Aldo.Pareja, Giacomo.Domeniconi1}@ibm.com, [email protected]\
[email protected], {tsuzumura, hirokik}@us.ibm.com, {tfk, cel}@mit.edu
bibliography:
- 'reference.bib'
title: 'EvolveGCN: Evolving Graph Convolutional Networks for Dynamic Graphs'
---
Introduction
============
Graphs are ubiquitous data structures that model the pairwise interactions between entities. Learning with graphs encounters unique challenges, including their combinatorial nature and the scalability bottleneck, compared with Euclidean data (e.g., images, videos, speech signals, and natural languages). With the remarkable success of deep learning for the latter data types, there exist renewed interests in the learning of graph representations [@Perozzi2014; @Tang2015; @Cao2015; @Ou2016; @Grover2016] on both the node and the graph level, now parameterized by deep neural networks [@Bruna2014; @Duvenaud2015; @Defferrard2016; @Li2016; @Gilmer2017; @Kipf2017; @Hamilton2017; @Jin2017; @Chen2018; @Velickovic2018].
These neural network models generally focus on a given, static graph. In real-life applications, however, often one encounters a dynamically evolving graph. For example, users of a social network develop friendship over time; hence, the vectorial representation of the users should be updated accordingly to reflect the temporal evolution of their social relationship. Similarly, a citation network of scientific articles is constantly enriched due to frequent publications of new work citing prior art. Thus, the influence, and even sometimes the categorization, of an article varies along time. Update of the node embeddings to reflect this variation is desired. In financial networks, transactions naturally come with time stamps. The nature of a user account may change owing to the characteristics of the involved transactions (e.g., an account participates money laundering or a user becomes a victim of credit card fraud). Early detection of the change is crucial to the effectiveness of law enforcement and the minimization of loss to a financial institute. These examples urge the development of dynamic graph methods that encode the temporal evolution of relational data.
Built on the recent success of graph neural networks (GNN) for static graphs, in this work we extend them to the dynamic setting through introducing a recurrent mechanism to update the network parameters, for capturing the dynamism of the graphs. A plethora of GNNs effectively perform information fusion through aggregating node embeddings from one-hop neighborhoods recursively. A majority of the parameters of the networks is the linear transformation of the node embeddings in each layer. We specifically focus on the graph convolutional network (GCN) [@Kipf2017] because of its simplicity and effectiveness. Then, we propose to use a recurrent neural network to inject the dynamism into the parameters of the GCN, which forms an evolving sequence.
It is worthwhile to distinguish here, on a conceptual level, the proposed method from those [@Seo2016; @Manessia2017; @Narayan2018] also based on a combination of GNNs and recurrent neural networks (RNN, typically an LSTM), with technical details elaborated in the next section. The referenced architectures use GNNs as a feature extractor and RNNs for sequence learning from the extracted features (node embeddings). As a result, one single GNN model is learned for all graphs on the temporal axis. On the other hand, we propose to use an RNN to update the GNN model (i.e., network parameters) at every time step. This approach effectively performs model adaptation, which focuses on the model itself rather than the node embeddings. Therefore, the referenced approaches require the knowledge of the nodes over the whole time span and can hardly expect the performance on new nodes in the future. In contrast, our approach evolves the GNN based on the dynamism of the graphs. Hence, for future graphs with new nodes without historical information, the evolved GNN is still sensible for them.
Related Work
============
Methods for dynamic graphs are often extensions of those for a static one, with an additional focus on the temporal dimension and update schemes. For example, in matrix factorization-based approaches [@Roweis2000; @Belkin2002], node embeddings come from the (generalized) eigenvectors of the graph Laplacian matrix. Hence, DANE [@Li2017] updates the eigenvectors efficiently based on the prior ones, rather than computing them from scratch for each new graph. The dominant advantage of such methods is the computational efficiency.
For random walk-based approaches [@Perozzi2014; @Grover2016], transition probabilities conditioned on history are modeled as the normalized inner products of the corresponding node embeddings. These approaches maximize the probabilities of the sampled random walks. CTDANE [@Nguyen2018] extends this idea by requiring the walks to obey the temporal order. Another work, NetWalk [@Yu2018a], does not use the probability as the objective function; rather, it observes that if the graph does not undergo substantial changes, one only needs to resample a few walks in the successive time step. Hence, this approach incrementally retrains the model with warm starts, substantially reducing the computational cost.
The wave of deep learning introduces a flourish of unsupervised and supervised approaches for parameterizing the quantities of interest with neural networks. DynGEM [@Goyal2017] is an autoencoding approach that minimizes the reconstruction loss, together with the distance between connected nodes in the embedding space. A feature of DynGEM is that the depth of the architecture is adaptive to the size of the graph; and the autoencoder learned from the past time step is used to initialize the training of the one in the following time.
A popular category of approaches for dynamic graphs is point processes that are continuous in time. Know-Evolve [@Trivedi2017] and DyRep [@Trivedi2018] model the occurrence of an edge as a point process and parameterize the intensity function by using a neural network, taking node embeddings as the input. DynamicTriad [@Zhou2018] uses a point process to model a more complex phenomenon—triadic closure—where a triad with three nodes is developed from an open one (a pair of nodes are not connected) to a closed one (all three pairs are connected). HTNE [@Zuo2018] similarly models the dynamism by using the Hawkes process, with additionally an attention mechanism to determine the influence of historical neighbors on the current neighbors of a node. These methods are advantageous for event time prediction because of the continuous nature of the process.
A set of approaches most relevant to this work is combinations of GNNs and recurrent architectures (e.g., LSTM), whereby the former digest graph information and the latter handle dynamism. The most explored GNNs in this context are of the convolutional style and we call them graph convolutional networks (GCN), following the terminology of the related work, although in other settings GCN specifically refers to the architecture proposed by [@Kipf2017]. GCRN [@Seo2016] offers two combinations. The first one uses a GCN to obtain node embeddings, which are then fed into the LSTM that learns the dynamism. The second one is a modified LSTM that takes node features as input but replaces the fully connected layers therein by graph convolutions. The first idea is similarly explored in WD-GCN/CD-GCN [@Manessia2017] and RgCNN [@Narayan2018]. WD-GCN/CD-GCN modifies the graph convolution layers, most notably by adding a skip connection. In addition to such simple combinations, STGCN [@Yu2018] proposes a complex architecture that consists of so-called ST-Conv blocks. In this model, the node features must be evolving over time, since inside each ST-Conv block, a 1D convolution of the node features is first performed along the temporal dimension, followed by a graph convolution and another 1D convolution. This architecture was demonstrated for spatiotemporal traffic data (hence the names STGCN and ST-Conv), where the spatial information is handled by using graph convolutions.
Method
======
In this section we present a novel method, coined *evolving graph convolutional network* (EvolveGCN), that captures the dynamism underlying a graph sequence by using a recurrent model to evolve the graph convolutional network. Throughout we will use subscript $t$ to denote the time index and superscript $l$ to denote the network layer index. Without loss of generality, we assume that all the graphs are built on a common node set of cardinality $n$. A nonexistent node is treated as a dangling node with zero degree, because it does not affect the information flow in graph convolutions. Then, at time step $t$, the input data consists of the pair $(A_t\in{\mathbb{R}}^{n\times n}, X_t\in{\mathbb{R}}^{n\times d})$, where the former is the graph (weighted) adjacency matrix and the latter is the matrix of input node features. Specifically, each row of $X_t$ is a $d$-dimensional feature vector of the corresponding node.
Graph Convolutional Network (GCN) {#sec:gconv}
---------------------------------
A GCN [@Kipf2017] consists of multiple layers of graph convolution, which is similar to a perceptron but additionally has a neighborhood aggregation step motivated by spectral convolution. At time $t$, the $l$-th layer takes the adjacency matrix $A_t$ and the node embedding matrix $H_t^{(l)}$ as input, and uses a weight matrix $W_t^{(l)}$ to update the node embedding matrix to $H_t^{(l+1)}$ as output. Mathematically, we write $$\begin{aligned}
H_t^{(l+1)} &= \operatorname{GCONV}(A_t, H_t^{(l)}, W_t^{(l)}) \nonumber \\
&:= \sigma(\widehat{A}_t H_t^{(l)} W_t^{(l)}), \label{eqn:gcn}\end{aligned}$$ where $\widehat{A}_t$ is a normalization of $A_t$ defined as (omitting time index for clarity): $$\widehat{A}=\widetilde{D}^{-\frac{1}{2}}\widetilde{A}\widetilde{D}^{-\frac{1}{2}},
\quad \widetilde{A}=A+I,
\quad \widetilde{D}=\operatorname{diag}\Bigg(\sum_j\widetilde{A}_{ij}\Bigg),$$ and $\sigma$ is the activation function (typically ReLU) for all but the output layer. The initial embedding matrix comes from the node features; i.e., $H_t^{(0)}=X_t$. Let there be $L$ layers of graph convolutions. For the output layer, the function $\sigma$ may be considered the identity, in which case $H_t^{(L)}$ contains high-level representations of the graph nodes transformed from the initial features; or it may be the softmax for node classification, in which case $H_t^{(L)}$ consists of prediction probabilities.
One sees that the $\operatorname{GCONV}$ layer is similar to a perceptron (a fully connected layer), except for the multiplication with $\widehat{A}_t$ in the front. This matrix is motivated by spectral graph filtering on the graph Laplacian matrix and it results from a linear functional of the Laplacian. On the other hand, one may also interpret the multiplication with $\widehat{A}_t$ as an aggregation of the transformed embeddings of the neighboring nodes.
The parameters of the GCN are the weight matrices $W_t^{(l)}$, for different time steps $t$ and layers $l$. Graph convolutions occur for a particular time but generate new information (i.e., from $H_t^{(l)}$ to $H_t^{(l+1)}$) along the layers. The relationship between $H_t^{(l)}$, $W_t^{(l)}$, and $H_t^{(l+1)}$ is schematically illustrated in the middle part of Figure \[fig:egcu\].
Weight Evolution {#sec:gru}
----------------
At the heart of the proposed method is the update of the weight matrix $W_t^{(l)}$ at time $t$ based on current, as well as historical, information. This requirement can be naturally fulfilled by an RNN, which treats $W_t^{(l)}$ as the hidden state of the system. We focus on the gated recurrent unit (GRU), although other types of RNN, such as LSTM, are also applicable. They require a particularly named “input” argument, which is the information injected to the recurrent system at every time. This information is the node embeddings $H_t^{(l)}$ in our context.
The mathematics is a bit involved, because different from the standard GRU which maps vectors to vectors, we now have matrices. Moreover, the matrix dimensions do not match on the surface. To proceed, let us begin with the definition of a function $g$ in the following pseudocode format, whereby variables are defined only locally and they do not belong to the system of notations we have been using so far:
$Z_t = \operatorname{sigmoid}(W_ZX_t + U_ZH_{t-1} + B_Z)$ $R_t = \operatorname{sigmoid}(W_RX_t + U_RH_{t-1} + B_R)$ $\widetilde{H}_t = \tanh(W_HX_t + U_H(R_t\circ H_{t-1})+B_H)$ $H_t = (1-Z_t) \circ H_{t-1} + Z_t \circ \widetilde{H}_t$
One recognizes that the function $g$ is nothing but the standard GRU modified from the vector version to the matrix version. The involved matrices have the same number of columns; in other words, one may treat the function $g$ as applying the standard GRU on each column of the involved matrices independently. In this context, $X_t$ is the input, $H_{t-1}$ is the past hidden state, and $H_t$ is the new hidden state. Moreover, $Z_t$, $R_t$, and $\widetilde{H}_t$ are the update gate output, the reset gate output, and the pre-output, respectively. We use these notations to familiarize the reader with the standard formulas of GRU.
As hinted earlier, we want to treat the node embeddings $H_t^{(l)}$ as the GRU input and the weight matrix $W_t^{(l)}$ as the (new) GRU hidden state. However, these matrices have different numbers of columns and they are not readily applicable to the function $g$. To this end, we summarize the embedding matrix $H_t^{(l)}$ into one with only $k$ nodes, where $k$ is the number of columns of $W_t^{(l)}$. Then, the function $g$ is applied.
The following pseudocode gives one popular approach for summarizing a matrix $X_t$ with many rows into a matrix $Z_t$ with only $k$ rows (see, e.g., [@Cangea2018]). Note again that in this pseudocode the variables are defined only locally, even though the same names have been used elsewhere. The summarization requires a parameter vector $p$ that is independent of the time index $t$ (but may vary for different $\operatorname{GCONV}$ layers). This vector is used to compute weights for the rows, among which the ones corresponding to the top $k$ weights are selected and weighted for output.
$y_t = X_tp/\|p\|$ $i_t = \text{top-indices}(y_t, k)$ $Z_t = [X_t \circ \tanh(y_t)]_{i_t}$
With the above functions $g$ and $summarize$, we now are ready to evolve the weight matrices: $$\begin{aligned}
W_t^{(l)} &= \operatorname{GRU}(H_t^{(l)}, W_{t-1}^{(l)}) \\
&:= g(summarize(H_t^{(l)}, \#col(W_{t-1}^{(l)}))^T, W_{t-1}^{(l)}),\end{aligned}$$ where $\#col$ denotes the number of columns of a matrix and the superscript $T$ denotes matrix transpose. We reuse the name “GRU” to denote this evolution function. Effectively, it summarizes the node embedding matrix $H_t^{(l)}$ into one with appropriate dimensions and then evolves the weight matrix $W_{t-1}^{(l)}$ in the past time step to $W_t^{(l)}$ for the current time. As opposed to the $\operatorname{GCONV}$ layer which occurs for one particular time step but progresses along the layers, the $\operatorname{GRU}$ occurs for one particular layer but progresses along time.
The relationship between $W_{t-1}^{(l)}$, $H_t^{(l)}$, and $W_t^{(l)}$ is schematically illustrated in the left part of Figure \[fig:egcu\].
Evolving Graph Convolution Unit (EGCU) and Network (EvolveGCN)
--------------------------------------------------------------
Combining $\operatorname{GCONV}$ presented in Section \[sec:gconv\] and $\operatorname{GRU}$ in Section \[sec:gru\], we reach the *evolving graph convolution unit* (EGCU), defined by the following pseudocode:
$W_t^{(l)} = \operatorname{GRU}(H_t^{(l)}, W_{t-1}^{(l)})$ $H_t^{(l+1)} = \operatorname{GCONV}(A_t, H_t^{(l)}, W_t^{(l)})$
Different from the two pseudocodes presented in the preceding subsection, here we do not have notational clash with conventions. For readability, the variables therein need not be considered local. This unit performs graph convolutions along layers and meanwhile evolves the weight matrices over time. It is illustrated in the right part of Figure \[fig:egcu\].
Chaining the units bottom-up, we obtain a GCN with multiple layers for one time step. Then, unrolling over time horizontally, the units form a lattice on which information ($H_t^{(l)}$ and $W_t^{(l)}$) flows. We call the overall model *evolving graph convolutional network* (EvolveGCN).
Task-Specific Training {#sec:training}
----------------------
So far, we have presented the proposed EvolveGCN model from the angle of representation learning, but the real power comes from pairing it with the prediction model for a downstream task and training them together end-to-end. We consider three example tasks here: node classification, edge classification, and link prediction.
For an EvolveGCN with $L$ layers, the output node embeddings are contained in the matrix $H_t^{(L)}$. For notational clarity, in what follows we omit the superscript $L$ and write the output embedding of a node $u$ at time $t$ as $h_t^u$, which is a column vector.
#### Node Classification
Predicting the label of a node $u$ at time $t$ follows the same practice of a standard GCN. One may assume that the node label may change over time so that the problem is more interesting. We let the activation function $\sigma$ of the output $\operatorname{GCONV}$ layer (see with $l=L-1$) be the softmax, so that $h_t^u$ is a probability vector. Then, with the ground-truth one-hot label vector $y_t^u$, the training loss is the weighted cross entropy $$L=-\sum_t\sum_u \alpha_u \sum_i (y_t^u)_i \log (h_t^u)_i,$$ where nonuniform weights $\alpha_u$ (treated as hyperparameters) sometimes help mitigate class imbalance problems.
#### Edge Classification
Predicting the label of an edge $(u,v)$ at time $t$ requires the embeddings of both nodes. We let the activation function of the output $\operatorname{GCONV}$ layer be ReLU, so that $h_t^u$ and $h_t^v$ are interpreted as node embeddings. Then, we introduce an additional parameter matrix $U$, which is independent of time, to form the predictive model $$p_t^{uv}=\operatorname{softmax}(U[h_t^u; h_t^v]).$$ Then, with the ground-truth one-hot label vector $y_t^{uv}$, the training loss is the weighted cross entropy $$L=-\sum_t\sum_{(u,v)} \alpha_{uv} \sum_i (y_t^{uv})_i \log (p_t^{uv})_i,$$ where nonuniform weights $\alpha_{uv}$ sometimes help mitigate class imbalance problems.
#### Link Prediction
The task of link prediction is to leverage information up to time $t$ and predict the existence of an edge $(u,v)$ at time $t+1$. Since historical information has been encoded in the GCN parameters, we base the prediction on $h_t^u$ and $h_t^v$. In order to make them embedding vectors, similar to edge classification, we let the activation function of the output $\operatorname{GCONV}$ layer be ReLU. Then, we introduce a column vector $q$, which is independent of time, to form the predictive model $$p_{t+1}^{uv}=\operatorname{sigmoid}(q^T[h_t^u; h_t^v]),$$ whose output $p_{t+1}^{uv}$ indicates the probability that the edge $(u,v)$ exists at time $t+1$.
The training is not scalable if we use all existent and nonexistent edges, because the number of node pairs scales quadratically with the number of nodes. Typically, existent edges are substantially fewer than nonexistent ones, possibly differing in count by orders of magnitude. Hence, we appeal to negative sampling. To this end, let $P_t$ and $\overline{P}_t$ be the collection of existent and nonexistent edges at time $t$, respectively. For some sampling ratio $\beta$, we sample $\beta|P_t|$ nonexistent edges from $\overline{P}_t$ and form a set $Q_t$. Additionally, we let $y_{t+1}^{uv}$ be the binary label and $C_{t+1}^{uv}$ be the sample cross entropy: $$C_{t+1}^{uv} = -y_{t+1}^{uv}\log p_{t+1}^{uv} -(1-y_{t+1}^{uv})\log (1-p_{t+1}^{uv}).$$ Then, the training loss is $$L=\sum_t\sum_{(u,v)\in P_t} C_{t+1}^{uv}
+\alpha\sum_t\frac{|\overline{P}_t|}{|Q_t|}\sum_{(u,v)\in Q_t} C_{t+1}^{uv},$$ where $\alpha$ is a tunable weighting factor.
Experiments
===========
In this section, we present a comprehensive set of experiments to demonstrate the effectiveness of EvolveGCN. In particular, we experiment with several combinations of data sets and tasks. In addition to existing benchmark data sets, we also work on one generated from a financial fraud simulator. Table \[tab:dataset\] summarizes the information of each data set. Details and processing are presented in individual subsections. Hyperparameters are tuned by using the validation set and test results are reported at the best validation epoch.
--------------- ---------- ---------- ----------------------
\# Nodes \# Edges \# Time Steps
(Train / Val / Test)
Bitcoin-OTC 5,881 35,592 96 / 21 / 21
Bitcoin-Alpha 3,783 24,186 96 / 20 / 20
AMLSim 20,000 117,805 104 / 22 / 23
--------------- ---------- ---------- ----------------------
: Data sets.[]{data-label="tab:dataset"}
Baseline Methods
----------------
We compare EvolveGCN with two baselines.
#### GCN
The first one is GCN without any temporal modeling. We use one single GCN model for all time steps and the loss is accumulated along the time axis.
#### GCN-GRU
The second one is also a single GCN model, but it is co-trained with a recurrent model (GRU) that takes node embeddings as input. We call this approach GCN-GRU, which is conceptually the same as Method 1 of [@Seo2016], except that their GNN is the ChebNet [@Defferrard2016] and their recurrent model is the LSTM.
Data Set: Bitcoin-OTC
---------------------
Bitcoin-OTC[^3] is a who-trusts-whom network of bitcoin users trading on the platform <http://www.bitcoin-otc.com>. Since bitcoin users are anonymous, there emerge ratings of reputation among users for the purpose of reducing transaction risks and avoiding fraud. Members of this network rate the level of trust they have on the other ones in a scale from $-10$ (distrust) to $+10$ (trust).
In this data set, the time of rating spans approximately five years and we use a granularity of approximately two weeks to form a sequence of graphs with 138 time steps. One may easily calculate, according to Table \[tab:dataset\], that on average each graph has only 258 edges (compared this number with the number of nodes, 5,881) and thus is overly sparse, if each edge appears only once in the sequence. The over-sparsity may adversarially affect the usefulness of any node embedding technique assuming graph connectivity. Hence, we maintain a window of size $T$ for the existence of an edge in the graph sequence, since its initial creation. Through extensive experimentation, we find that setting $T=10$ results in good performance for at least the baseline GCN. We use in/out node degree (two-hot) as input features.
#### Edge Classification
We predict the polarity of each rating (binary classification). The hypothesis is that the rating given by a user to another one is based on life knowledge that accumulates over time (e.g., knowledge of existing ratings and identified fraudulent accounts). Hence, the dynamism of the network is a helpful signal for trustfulness prediction, at which a model that captures temporal evolution may excel.
We summarize the performance of EvolveGCN and those of the baselines in Table \[tab:performance\]. The classes are relatively skewed: $89\%$ of the ratings are positive. Because negative ratings (the minority class) are more important (as they reveal distrust and transaction risks), we make the minority class as the positive class and report precision/recall/F1 that focuses on this class. One sees that EvolveGCN achieves the highest recall and F1 score, which means that negative ratings are much more likely to be captured in predictions, promoting safer trading. For completeness, we also include the micro-average F1 score. If we dilute the focus on negative ratings to all ratings, EvolveGCN performs less competatively.
Figure \[fig:otc.f1.over.time\] shows the F1 score over time for the test set.
![F1 score over time, Bitcoin-OTC, edge classification.[]{data-label="fig:otc.f1.over.time"}](image2.png){width=".7\linewidth"}
#### Link Prediction
The task of link prediction is to forecast whether a bitcoin user will rate another one in the next time step. Link prediction shares similarities with a general binary classification, but a distinct characteristic is that the negative class (nonexistent edges) typically dominates the other class. For training, we have discussed in Section \[sec:training\] that negative sampling is necessary in the formulation of the loss function. Similarly, for testing, we also sample a substantial amount of nonexistent edges. The negative sampling ratio for training is $\beta=20$ and that for testing is $\beta=100$.
The results are reported in Table \[tab:performance\]. Naturally, the class imbalance problem is substantially more severe here than that of rating prediction. However, EvolveGCN achieves the best F1 and micro F1 scores simultaneous.
Data Set: Bitcoin-Alpha
-----------------------
The data set Bitcoin-Alpha[^4] is created in the same manner as is Bitcoin-OTC, except that the users and ratings come from a different trading platform, <http://www.btc-alpha.com>. The positive rating ratio is $93\%$. We follow the same experimentation setting as in Bitcoin-OTC and report the results in Table \[tab:performance\]. In both edge classification and link prediction tasks, EvolveGCN achieves the best F1 and micro F1 scores.
[ccccc]{}\
& Precision & Recall & F1 & Micro F1\
GCN & 0.278 & 0.437 & 0.340 & 0.863\
GCN-GRU & 0.301 & 0.437 & 0.357 & 0.872\
EvolveGCN & 0.287 & 0.618 & 0.392 & 0.845\
[ccccc]{}\
& Precision & Recall & F1 & Micro F1\
GCN & 0.341 & 0.394 & 0.365 & 0.9989\
GCN-GRU & 0.310 & 0.318 & 0.314 & 0.9989\
EvolveGCN & 0.487 & 0.369 & 0.420 & 0.9992\
[ccccc]{}\
& Precision & Recall & F1 & Micro F1\
GCN & 0.159 & 0.646 & 0.255 & 0.609\
GCN-GRU & 0.163 & 0.664 & 0.262 & 0.613\
EvolveGCN & 0.168 & 0.611 & 0.263 & 0.646\
[ccccc]{}\
& Precision & Recall & F1 & Micro F1\
GCN & 0.541 & 0.347 & 0.423 & 0.9991\
GCN-GRU & 0.316 & 0.422 & 0.361 & 0.9986\
EvolveGCN & 0.520 & 0.358 & 0.424 & 0.9991\
[ccccc]{}\
& Precision & Recall & F1 & Micro F1\
GCN & 0.068 & 0.263 & 0.108 & 0.959\
GCN-GRU & 0.203 & 0.291 & 0.239 & 0.982\
EvolveGCN & 0.081 & 0.234 & 0.121 & 0.968\
Data Set: AMLSim
----------------
Money laundering is one of the most important crime problems in the finance domain. Since banking transaction data is highly sensitive and rarely publicly available, we have developed a multi-agent simulator called AMLSim (anti-money laundering simulator)[^5], aiming at simulating banking transaction behaviors that include typical money laundering activities. With AMLSim, we generate a benchmark transaction data set[^6].
The data set consists of a temporal graph sequence. A node represents a bank account and an edge represents a time-stamped transaction between two accounts. The data set also comes with node labels that indicate which accounts are suspicious and since when. Basic statistics of the graph sequence is given in Table \[tab:dataset\]. Among the 20,000 accounts, 493 of them are labeled as suspicious.
The task is node classification: predict at a time whether an account becomes suspicious. As one sees, the class distribution is also highly skewed. We report the results in Table \[tab:performance\]. EvolveGCN outperforms GCN but is less competative than GCN-GRU. One possible cause is that suspicious nodes become so only after a few time steps and hence the lack of historical information for a new node is less problematic. Nevertheless, the fact that GCN results in the worst performance signifies the importance of temporal modeling.
Conclusions and Future Work
===========================
A plethora of neural network architectures were proposed recently for graph structured data and their effectiveness have been widely confirmed. In practical scenarios, however, we are often faced with graphs that are constantly evolving, rather than being conveniently static for a once-for-all investigation. The question is how neural networks handle such a dynamism. Combining GNN with RNN is a natural idea. We are not the first to explore this route. Typical approaches use the GNN as a feature extractor and use an RNN to capture the dynamism of the node embeddings. We, on the other hand, use the RNN to evolve the GNN, so that the dynamism is captured in the evolving network parameters. Our method is a model adaptation approach. One advantage is that it handles more flexibly dynamic data, because a node does not need to be present all time around. Experimental results confirm that the proposed approach generally outperforms related ones for a variety of tasks, including node classification, edge classification, and link prediction.
One of the pressing needs is to scale the method to industrial-level graphs, such as transaction graphs that may contain millions or even billions of accounts and transactions. The graph model and the recurrent model have their own limitations regarding scalability. The graph model, particularly in the convolutional style, suffers repeated expansion of the neighborhood. Much of existing work tackles this problem through restricting the neighborhood size (see, e.g., [@Hamilton2017; @Chen2018]). On the other hand, the recurrent model is notoriously sequential and hence good parallelism is difficult to achieve. Emergent architectures using the attention mechanism (see, e.g., [@Vaswani2017]) appear to be a successful remedy.
Another avenue of future research is to tackle the class imbalance problem, which appears common in practical scenarios. For example, in financial activities, frauds are rare compared with normal operations. Prediction of the minority class is like finding a needle in a haystack, wherein outlier prediction approaches appear to be more natural than a classification approach. How one applies the proposed neural network architecture in an outlier prediction framework is the next step of our investigation.
[^1]: Equal contribution
[^2]: Contact author
[^3]: <http://snap.stanford.edu/data/soc-sign-bitcoin-otc.html>
[^4]: <http://snap.stanford.edu/data/soc-sign-bitcoin-alpha.html>
[^5]: <https://github.com/IBM/AMLSim>
[^6]: <https://github.com/IBM/AMLSim/tree/master/sample/20K_cycle200.tgz>
| ArXiv |
---
abstract: 'The advent of modern computers has added an increased emphasis on channeling computational power and statistical methods into digital humanities. Including increased statistical rigor in history poses unique challenges due to the inherent uncertainties of word-of-mouth and poorly recorded data. African genealogies form an important such example, both in terms of individual ancestries and broader historical context in the absence of written records. Our project aims to bridge the lack of accurate maps of Africa during the trans-Atlantic slave trade with the personalized question of where *within* Africa an individual slave may have hailed. We approach this question with a two part mathematical model informed by two primary sets of data. We begin with a conflict intensity surface which can generate capture locations of theoretical slaves, and accompany this with a Markov decision process which models the transport of these slaves through existing cities to the coastal areas. Ultimately, we can use this two-step approach of providing capture locations to a historical trade network in a simulative fashion to generate and visualize the conditional probability of a slave coming from a certain spatial region given they were sold at a certain port. This is a data-driven visual answer to the research question of where the slaves departing these ports originated. [Keywords: Kriging; Markov decision process; Gaussian process; Kernel Density Estimation; Oyo; African Diaspora; Translatlantic slave trade; digital humanities;]{}'
bibliography:
- 'Oyobibliography.bib'
---
[ **Mapping the uncertainty of 19th century West African slave origins using a Markov decision process model.** ]{}
[ **Zachary Mullen and Ashton Wiens and Eric Vance and Henry Lovejoy** ]{}
[****]{}
Introduction
============
The study of colonial empires is dominated by incomplete data and missing records. Despite this difficulty, there is significant interest in tracing the forced diaspora of African peoples via slavery [@lovejoywebsite]. Many black organizations in the modern Americas can trace their origins to the cultural unity required in overcoming the struggles of their subjugation [@chambers2012].
To date, much of the work in understanding this cultural genesis has focused on genealogy and literary interpretation, but the forced relocation of a predominately illiterate population leads to significant shortcomings in availability of written history. As a result, modern understanding of the exodus often lacks comprehensive regional descriptions of the socio-political climate within Africa that enabled the internal slave trade then exploited by colonial European powers. The growing field of digital humanities attempts to expand upon logocentric analyses of African history with modern methods in text mining, linguistic analysis, and machine learning [@lovejoytalk2017]. These GIS and geospatial methods [@knowles2008placing] have been employed heavily in World War II and Holocaust studies [@knowles2014geographies].
One West African slave-trading state was the Oyo empire, which peaked in the late 18th century, culminating in a rapid decline over a series of crises and invasions around the 1820s. During these conflicts, slavers regularly departed from the coast of the Oyo empire and bordering West African states, and many of these voyages are well documented by the slave traders. In addition to ship logs, a handful anecdotal accounts of individual slave movements from the collapse of the Oyo empire have been reconstructed from written and oral records [@kelley2016origins; @kelley2016voyage] (maybe cite slavebiographies.org). Recent work has emphasized integrating the collapse of the Oyo empire into the digital humanities, including the creation of detailed maps on the shifting borders of the collapsing empire [@lovejoy2013redrawing].
One current question regarding the collapse of the Oyo empire is exploring the logistics and detailed movements of the internal slave trade and how those systems actually filled the ships leaving the West African coast. In many cases, the state-controlled ships have accurate passenger counts, ports of arrival, and ports of origin, and modern genealogical explorations can often trace ancestries to those specific ships. However, little historical evidence explicitly connects the passenger logs - where available - and the movements of the ships to the politics of inland Africa at the time. Questions of ancestry often dead-end at these transit points despite the work and literature documenting the internal conflicts during the Oyo collapse.
We attempt to expand on the understanding of the internal slave trade of the Oyo Empire by synthesizing spatial mathematical models onto conflict maps and conjoining them with models for decision processes governing inland slave movements. The first question is one of using discrete events such as recorded dates of battles or towns destroyed to create a model for the location and intensities of conflict. We use spatial smoothing on recorded conflict events to create a continuous density map of the warring regions, augmenting the existing maps of shifting borders by an accompanying picture of which cities and regions in the empire were most likely locations for slavers to capture individuals. We couple this map of conflict regions with a Markov decision process for the Oyo region’s internal slave trading network. We view adjacent or nearby cities as a connected network, and the Markov decision process attempts to ask: “what are the likely movement paths” of slaves captured until their eventual sales and departures via ship or into the trans-Sahara region.
The goal is to provide a functional and descriptive model for the most likely inland origin locations of slaves given a known year and port of origin. As a result, the conflict map and slaver decision process models combine to answer this: we use the conflict map to generate annual maps of likely locations slaves were captured, then pass them into the trading network to determine where slaves captured at those locations would be most likely to leave the region. The resulting counts allow for the inverse question as well; e.g. “for all slaves leaving Lagos in 1824, from which conflict regions did they originate?” This allows our analysis to bridge the process-focused models that stay true to historical narrative with the ends-oriented goals of a genealogist, who may wish to reverse-engineer the historical origin stories. We hope for our exploration to be applicable and available to historians in both other regions of the African diaspora and to studies of other instances of forced transit, such as the Holocaust or the relocation of American indigenous peoples.
Data {#S:Data}
====
**Section in Progress pending collaboration: Describe the conflict data - what historical accounts were used?**
We have several geopolitical data sets, describing the trade routes and conflicts that were we think were present during the collapse of the kingdom of Oyo from approximately 1816-1836 near modern day Togo, Benin, and western Nigeria. The data are shown in \[f:1\]. For each year, we also have approximations of the total number of slaves departing the region as a whole and specific trading ports. The data were collected in \[Mapping the Collapse of Oyo\].
The conflict data is a table where each row describes a 2D spatial location where a conflict occurred. There are variables describing the start year and end year, as well as the intensity of the conflict. The intensity was encoded as a categorical variable with four levels: 0 means a city is founded, 1 means a city is rebuilt, 5 means a city is attacked, and 10 means a city is destroyed. We did not use the founded/rebuilt city data.
Similarly, we have a list of cities with spatial coordinates and the years the city existed (dependent on being destroyed or rebuilt). To infer the trade network among these places, we relied on the map \[fig:1816TradeMap\]. This map has been informed by both available historical records from the time (**CITATION MISSING**) and geographic ease of transit between cities. This representation adds a layer of detail and geopolitical information on top of those published in prior works, such as [@lovejoy2013redrawing].
We encoded the relationships (edges) between the nodes of this graph into an adjacency matrix, describing which cities are connected. An adjacency matrix $A$ for a set of locations (nodes) $s_1, \cdots, s_n$ is of dimension $n \times n$. An entry $A_{ij}$ is nonzero (usually 1) if there is a connection starting at $s_i$ and ending at $s_j$. This formulation describes a directed graphical structure. If the edges are undirected, then $A_{ij} = A_{ji}$ and so $A$ is symmetric. We use this adjacency matrix to construct the probability transition matrix needed in the Markov Decision Process, described in Section \[SS:MDP\].
The third data set we have is the port total data: for each year, the total number of slaves leaving each port was estimated using digitally transcribed hand-written ship logs. Some of the estimates are assigned to an unknown port. This data was not used in formulation of any models we develop in this paper, but we used it as validation data to tune parameters in the model.
Finally, we have shapefile data with prominent geographical features that existed in the region during the historical period. In particular, we include bodies of water in plots which are relevant to identifying the boundaries of the various states. **The data was downloaded from ....**. Several bodies of water which were created since the historical period were removed from the data set.
![Map of Trade in Oyo, 1816[]{data-label="fig:1816TradeMap"}](figures/1816TradeMap.png){width="0.95\linewidth"}
Model
=====
Mapping Conflicts {#SS:krig}
-----------------
The historical narrative surrounding the fall of the Oyo empire is one of borders collapsing inwards from the independence of Dahomey and lost conflicts to Ilorin and Ijebu. While the borders of the resulting countries are relatively well known, the question of slave origins requires unpacking the conflicts themselves to determine the regions within the greater Oyo area most impacted by each conflict. The data available provides conflicts and a measure for their intensity, with battles marked as less intense than the complete destruction of towns. However, a conflict between nations does not unfold over just the sites of the major battles and events: armies mobilize, raid, and occupy villages throughout contested regions. To account for this behavior not having explicit historical accounts, we decided to create a continuous spatial map of the conflict borders given the discrete time and place data available.
### Gaussian Process for Origin Locations
We model our conflicts as arising from a spatially correlated Gaussian Process with an underlying M’tern covariance. Notationally, if we have conflict intensity measures $\boldsymbol{Y}$ observed at 2-D spatial locations $\boldsymbol{S}=\{s_1, s_2, \dots s_n\}$, the Gaussian process considers $\boldsymbol{Y}$ to be a single draw from a multivariate normal on $\mathbb{R}^n$. This corresponds to a log-density of
$$f(\boldsymbol{Y}) \propto -\log\det\left(\Sigma+\tau^2I\right)-\boldsymbol{Y}^T \left(\Sigma+\tau^2I\right)^{-1} \boldsymbol{Y}$$ where $\Sigma_{i,j}$ is given by the Mátern covariance: $
\Sigma_{i,j}=\sigma^2 k(s_i,s_j)+\mathbbm{1}_{\{i=j\}}(\tau^2)$ for $k(s_i,s_j)=\frac{2^{1-\nu}}{\Gamma(\nu)}
\left(a\|s_1-s_2\|\right)^\nu K_\nu(a\|s_1-s_2\|)$ with $K_\nu$ a modified Bessel function of the second kind of order $\nu$ and $\|\cdot\|$ denoting Euclidean distance.
The formal kriging estimator fills in a map of the Oyo region at chosen resolution by taking each desired locations $s_0$ on the fine grid and computing $$\hat{\boldsymbol{Y}}(s_0)=\sigma^2 k(s_0,\vec{s}) \left(\Sigma+\tau^2I\right)^{-1} \boldsymbol{Y}$$
### Alternatives Considered
There are a variety of alternative mathematical options available to bridge a set of discrete spatial observations (sites of conflict) into a smoother continuous map. These include smoothing over observed conflicts onto other locations via splines or the above kriging estimators, treating conflicts as draws from an underlying density and using a kernel density estimator (KDE) to recover that density, or viewing conflict sites as actualizations of an inhomogeneous Poisson point process and estimating the associated intensity function. Of these options, we chose to use a classical Kriging estimator of an underlying Gaussian process for a few reasons. These include:
- A typical Kriging formulation views the data as part of a demeaned autocorrelated process, where the assumption of zero mean rapidly pushes the surface to zero-valued when far from observations [@cressie1992statistics]. This corresponds to the idea that the conflicts and attacked towns themselves were the predominant sources of slaves at the time.
- The parameters available to Kriging covariance models are both flexible and can be interpreted in the units of the data. Specifically, the variance parameter (or sill) is a scaling of the relative importance of minor/major conflicts and the range parameter measures the distances from observed conflicts at which the overall region of conflict exists.
- Where a classical kernel density estimate is symmetric due to its equal weighting of observations-as-kernels, the choice of surface smoothness parameter in a correlated Gaussian Process allows for ridge-like structures that approximate the shifting borders of a conflict region.
Figure \[fig:krigmodel\] shows the flexibility of the Matérn in capturing a shifting border of conflict and contrasts against a kernel density estimate. The leftmost plot shows a kernel density estimate, where the greater *count* of conflicts in the northeastern area generates a corresponding increased conflict intensity. The kriging maps on the other hand show the shapes of the conflicts: the parameters $\nu$ and $a$ have some interaction and different combinations can lead to very similar shapes of conflict borders, with differences largely captures in the magnitude of the range parameters. A larger range - or smaller $a$ - leads to maps with a larger region of uncertainty and non-zero conflict, as shown in the breadth of the yellow region in the middle map compared to the right-most map.
![Left to Right: 1825 conflict map via a kernel density estimate with $h=2$; krig with $\nu=.5$ and $a=5/2$; krig with $\nu=3$ and $a=6$[]{data-label="fig:krigmodel"}](figures/1825conflictkde.pdf "fig:"){width="0.3\linewidth"} ![Left to Right: 1825 conflict map via a kernel density estimate with $h=2$; krig with $\nu=.5$ and $a=5/2$; krig with $\nu=3$ and $a=6$[]{data-label="fig:krigmodel"}](figures/1825conflictnotsmooth.pdf "fig:"){width="0.3\linewidth"} ![Left to Right: 1825 conflict map via a kernel density estimate with $h=2$; krig with $\nu=.5$ and $a=5/2$; krig with $\nu=3$ and $a=6$[]{data-label="fig:krigmodel"}](figures/1825conflictsmooth.pdf "fig:"){width="0.3\linewidth"}
### Estimation
Rather than provide for fully unconstrained estimation of the 4 spatial covariance parameters, we chose to tune the spatial smoothness according to the heuristics of the underlying problem. In general, we want to force ranges to be small enough that regions of high correlation are trapped within the same topographical and population areas as the observed conflicts: we found that a $10km$ range accomplished this. Smoothness was similarly fixed at a 4.5 times differentiable Matern. While this is fairly smooth in the context of dense spatial data, our observations are quite sparse, and higher smoothness helps ensure the ridge-like structure that mimics a shifting border to reflect the ebb and flow of borders in a region of conventional warfare. Lower smoothness would enforce a rapid decay to zero away from observations, and would push our model closer to one found from kernel density estimation, as it would result in conflicts being modeled as small, radially symmetric, disconnected, additive kernels about the observed locations. For the variance parameters of sill and nugget, we fit them via variogram with Cressie weights [@cressie1985fitting] to the 1828 data set, then used those parameters to fill in the remaining years. Using these fixed Matérn covariance parameters are all that is required to perform spatial kriging at any desired location.
In the classical kriging sense, this surface would exist in the units of the $\boldsymbol{Y}$: the 2-valued marker for intensity of battle at a village. However, we can also view the resulting surface as an implied probability density function, where the higher points of the ridge near conflict locations represent regions of increased probability of slave capture. By using the kriging estimator to fill in a high-resolution surface over the Oyo region, we then create an annual empirical cumulative density function by dividing the surface by its overall integral. For each such surface we can simulate sample slave origin locations via direct inversion.
Trading Network for Slave Transport {#SS:MDP}
-----------------------------------
The map in figure \[fig:1816TradeMap\] displays cities within and around Oyo connected by the most probable trade routes at the time. Such a depiction naturally translates to a graph-based approach to capturing the economics of the region. From the map, we construct a transition matrix of valid city-to-city movements, and classify a handful of cities where slaves could depart the region: Lagos, Porto Novo, and Ouidah for Atlantic departure; Abomey and Benin City for departure to the neighboring coastal states; and Djougou, Kalama, Bussa, Ogudu, Tsaragi, and Ogodo for departure into the trans-Saharan slave trade. Rather than directly assign probabilities to the flow of slaves in the region, we turn to a finite horizon Markov decision process to mirror the calculus of slavers at the time. Markov decision processes (MDPs) can be used for sequential decision making but have been underutilized in the social sciences [@boucherie2017markov].
### A Markov Decision Process
A Markov decision process describes the partially deterministic and partially stochastic movement of an agent through a network in discrete time. The agent’s actions at each state are chosen based on the rewards and costs associated with reaching a state in the network, but the actual event that takes place is probabilistic. Formally, a Markov decision process consists of a 5-tuple $(S, A, P_a, R_a, \gamma)$. $S$ is a finite set of states in a network, often spatially located. $A$ are the actions an agent can take from any given state $s \in S$. $S$ and $A$ can also be thought of as the nodes and edges in a network, respectively. In our case, $S$ corresponds to the cities in the trade network, and $A$ are the valid routes in the trade network a slaver can take; the same set of trade nodes discussed in section \[S:Data\]. For an action $a \in A$ taken in state $s$, we must define the probability of actually reaching state $s'$ for all states in $S$. Thus, for each action $a$, we must define $P(s_{t+1}' \, | \, s_{t}, a)$. Similarly, we must provide the expected immediate reward/cost incurred after moving from $s$ to $s'$ via action $a$, which we write as $R(s_{t+1}' \, | \, s_{t}, a)$. The idea is that the MDP will designate a best possible route, but a slaver might choose a slightly different one. The difference between the optimal route and the chosen route is reflected in the probabilities $P$, whereas the rewards that determined the best route are saved in $R$. For our purposes, $R$ includes both negative values that represent cost of movement and positive values that correspond to getting a slave to a point-of-sale. The end result is a “best route” determined by the model for a slaver to reach a point-of-sale, but some chances of deviations along that route to account for the slavers’ personal preferences and/or their imperfect information.
The MDP solves the problem of finding an optimal policy $\pi(s)$ whose value specifies the action $a$ to take by the agent at state $s$. The function $\pi$ is found by maximizing some function of the random rewards. Most often this is the expected discounted rewards: $$\sum^{\infty}_{t=0} {\gamma^t R_{a_t} (s_t, s_{t+1})}$$ The discount factor $\gamma \in [0, 1)$ allows the rewards incurred in later time steps to be downweighted. The discount factor is fixed at $\gamma=1$ because we found it unnecessary in modeling the historical application. Many algorithms have been developed to solve this optimization problem, e.g. using linear or dynamic programming. We use the policy iteration algorithm implemented in the `R` package `MDPtoolbox` [@MDPtoolbox].
Prior to running the MDP, we augment the adjacency matrix $A$ implied by \[fig:1816TradeMap\] by adding an additional row or state corresponding to each point-of-sale city. Movement into these added states represents a sale, and holds the postive contents of the rewards $R$. This flexibility allows some amount of transit between our sale locations - in particular to reflect canoe and naval traffic along the coastal lakes - in order to not view arrival at a port as an inherent end state of the process. Instead, caravans are free to move until the optimal reward is reached, which may include moving along coasts in the presence of unequal sale rewards.
### MDP Heuristics
Similar to the choice of kriging, some *ad hoc* decision-making between modeling options is merited. We harbor multiple criteria for the model for slave transit. For one, we require sufficient noise or stochasticity to allow for slaves captured in similar locations to deviate in port of departure simply by chance, thought of historically as the personal preferences and knowledge of slavers. In addition, we require a model with the ability to downweight probabilities of transit based on conflict intensities: in general, slavers would be incentivized to avoid areas of conflict, and as much of the slave transit in the Oyo is state-sanctioned, avoiding regions of conflict also serves as a proxy for slavers tending to avoid the shifting international borders and stay within their preferred home countries. Finally, it would be ideal for a method to allow for some sale locations to be preferred to others *a priori*, whether that preference is informed by volume of trade or the gradual West-to-East blockade of West African slave ports by the British Navy during the period in question.
This MDP formulation allows for considerable flexibility in meeting our criteria for a transit model.
- In an MDP, an optimal choice is calculated, but the underlying Markov chain allows for pseudo-random (non-“optimal”) movement from step-to-step. With a chosen probability, slavers may choose to traverse a sub-optimal route $a$ out of their city/state $s$, possibly including remaining at state $s$ for an additional time-step.
- Flexibility in rewards can account for both individual slaver preference and the broader temporal shift from the Western Oyo ports and Dahomey to the Eastern ports and Benin. Setting all point-of-sale rewards to be equal asks the question “what the least resistance route to *any* point-of-sale,” whereas varying the reward vector allows for individual slavers to balance preferred or higher revenue sale locations with the implied costs of a longer journey or a journey through regions of conflict.
- The cost-benefits formulation of MDP reward maximization can also easily be adjusted to downweight specific movements. In our case, we explicitly make movement through conflict regions less desirable. More generally, this formulation could be expanded to include disincentives to cross borders, venture through certain terrains, etc. Each of these are in addition to the distance-based cost terms we initialize the model with.
### MDP Parameter Tuning
Because we have no information on the price of slaves as a function of point of sale, we choose as a baseline a variant of the Markov decision process with equal expected rewards for each absorbing state. Transitions along each edge incurred a cost proportional to $D*(1+C)$, where D is the length of that edge and $C$ was the maximum of the conflict kriging predictor along that edge, scaled to an annual maximum of $C=3$. This allows both the transition chains and the slave origin locations to vary with conflict. To illustrate this effect, consider the example of a simple MDP seen in figure \[fig:ToyMDP\]. In this case, we observe the shortest route being taken in the absence of conflict, but a longer route circumventing the conflict when the intense area of conflict would have overlapped one of the routes taken.
![Example of MDP decision chain for a start in S3 with an absorbing state in S1 under no conflict (yellow) or conflict (green)[]{data-label="fig:ToyMDP"}](figures/mdp.pdf){width="0.45\linewidth"}
Figure \[fig:DecisionMaps\] depicts the decision processes for 1825 and 1826 as sets of arrows connecting each city in the trading network. In particular, note the difference in decisions made in in the regions around Abeokuta and the label for Ibadan (founded shortly after this conflict). In general, more trade is flowing northbound in 1826, but we see less traffic through Oyo and Ogodo, instead seeing increases in paths through Ilorin and Kaiama. Some of this traffic is also deflected away from an eventual port of departure in Porto Novo.
![Example of MDP decision chains for 1825 (left) and 1826 (right)[]{data-label="fig:DecisionMaps"}](figures/1825arrow.pdf "fig:"){width="0.45\linewidth"} ![Example of MDP decision chains for 1825 (left) and 1826 (right)[]{data-label="fig:DecisionMaps"}](figures/1826arrow.pdf "fig:"){width="0.45\linewidth"}
We can combine the kriging conflict estimator with the MDP to simulate the capture of a slave at a specific location and the resulting movement of the slave through the trade network to a point-of-sale. Use of unequal rewards to vary these simulative results is discussed further in sections \[SS:FinalMaps\] and \[S:Validation\].
Maps of Location Given Point-of-Sale {#SS:FinalMaps}
------------------------------------
Once the smoothed conflict estimator and MDP process are implemented, repeated simulation of slaves can be passed to the MDP with either varying or identical reward vectors. This allows for us to create a large sample of slaves and their eventual points-of-sale. These can be used to describe the ultimate goals: what were the eventual points-of-sale of slaves and also from what conflict and locations did slaves who departed from specific ports originate?
### Large-Scale Simulation
To gain origin and departure information from the models in \[SS:krig\] and \[SS:MDP\], we generate many slaves from direct inversion of the conflict estimator cumulative density functions. Then, for each such slave, we generate a random reward vector from a distribution specifying the end reward for selling a slave at any given absorbing state in the network. Then, we fit an MDP for each individual slave and reward vector pair, resulting in an optimal policy and eventual path of motion for each individual slave. Ultimately, the routing suggested by an MDP with equal rewards at each of point-of-sale is deterministic: every slave caravan of an identical origin location would choose the same optimal route given the annual conflict map. This doesn’t correspond well to the underlying historical narrative: many routes are nearly identical in terms of distance and slavers may have personal connections and preferences leading them to prefer certain routes to others. Incorporating a random reward vector represents each slaver’s knowledge of the conflicts and rewards present, which allows for a non-deterministic result to the question of where a slave will be sold *given* their capture location. Figure \[fig:rewardvar\] depicts simulated slaves from the 1832 version of the model with different randomness in the rewards. With no randomness the boundaries between colors and eventual points-of-sale are strict, whereas the figures shows increasing uncertainty when routes are nearly equivalent in terms of base cost-to-rewards. Due to the ability to simulate both capture locations and decision processes for any desired number of points, this allows us to generate these mappings to arbitrary probabilistic precision.
![Simulated Slave Origins colored by their points-of-sale. Left to Right: increasing variance in rewards[]{data-label="fig:rewardvar"}](figures/rr1.pdf "fig:"){width="0.3\linewidth"} ![Simulated Slave Origins colored by their points-of-sale. Left to Right: increasing variance in rewards[]{data-label="fig:rewardvar"}](figures/rr3.pdf "fig:"){width="0.3\linewidth"} ![Simulated Slave Origins colored by their points-of-sale. Left to Right: increasing variance in rewards[]{data-label="fig:rewardvar"}](figures/rr5.pdf "fig:"){width="0.3\linewidth"}
### Kernel Density Smoothing For Maps
One question posed by historians is how to integrate this information - at this point a large collection of origin points encoded by their point-of-departure - into more cleanly interpreted spatial maps. As historians and genealogists often know the port of departure of slaves, we can use the repeated samples of simulated data to figure out such intensity maps for the slaves that left a given port on a given year. Because the maps will be a set of points corresponds to individual slaves, we are tasked with a similar question as in creating the conflict estimator: how can we provide a continuous image or heat map for slave origin locations given the simulated slaves? For this we use a simple kernel density estimator, which creates a small, radially decaying kernel function at each simulated slave leaving from a specific port. The addition of each such kernel function for every slave at a given port gives a heat map for slave origin given port of departure.
Formally, the kernel density estimate takes a radially symmetric function $K(r)$ and estimates the regional heat map $\hat{f}$ via the weighted sum
$$\hat{f}(x)=\frac{1}{nh}\sum_{i=1}^n K\left(\frac{|x-x_i|}{h}\right)$$
where $x_1, x_2, \dots x_n$ are the $n$ slave locations for the slaves departing from the port in question. We choose the multivariate normal as the radial function $K$, as is often convention. In general, a kernel density estimate requires only tuning one parameter: the bandwidth $h$ that determines the distance/width of the kernel function centered on each simulated slave. The `R` function `kde2d` in package `MASS` implements the multivariate normal kernel density as its default, and is employed here [@MASS]. While a kernel density function can be sensitive to the number $n$ of points employed, our simulated-based model allows us to simulate any arbitrary amount of spatial samples and construct the resulting kernel density estimator to desired precision. In the applet mentioned in section \[S:App\] we allow $h$ to vary from $.5-2$km for a sample of 10,000 simulated slaves and find this provides an appealing map; a larger simulated sample would in turn allow for smaller bandwidths.
### End-Result Parameter Tuning {#SS:Validation}
As laid out so far, our model includes a considerable amount of parameters with no mathematical optimization strategy. These include the spatial covariance parameters of the kriging estimator, the relative increase in cost of movement to pass through conflict, and the variance in the point-of-sale rewards.
There exist two sources of data that allow for tuning the model to optimize these selections: ship total estimates for ships leaving the southern coast of Oyo and ship ledgers of *names* of slaves as recorded by the colonizing states.
As presently available to us, the passenger counts on each known ship are considered accurate. However, as figure \[fig:MissingShips\] denotes, a considerable amount of slave traffic was not recorded. Many ships that were recorded in our data set also have no specified ports of departure. As a result, a considerable amount of geospatial data is missing, and if that data was biased in any way - whether by the British blockades or some other administrative correlation - drawing geospatial conclusions about the within-Africa geospatial data would inherit these biases.
![Estimated Versus Recorded Trans-Atlantic Slave Departures from the Bight of Benin[]{data-label="fig:MissingShips"}](figures/MissingShips.png){width="0.6\linewidth"}
A more recent second stream of data comes from parsing the transcriptions of slave names from known ship logs. For many ships leaving the area, the Portuguese slave ships recorded each slaves name and attempted to transliterate it into Portuguese. Recent efforts by historians have begun to translate these names, and placed them into the native tongues of the Bight of Benin. This allows us to take a few of the ships and have a new kind of mapping: one of general cultural and linguistic origins. If we take our simulated maps of slave origins, these can be graded and scored against the linguistic data by observing the exact proportions of our simulated data coming from each region. Such a score specifically suggests a $\chi^2$ optimization, where for each set of parameters we can generate a goodness-of-fit measure that best ensures our simulated departures on each ship most closely match the observed data. To date, very few ships have been fully transliterated, but a couple such examples coming from 1832 are shown in Figure \[fig:Linguistic\]. Here, linguistic distributions are shown as categorical data which must be scored against the plotted distributions. One advantage of the $\chi^2$ is that each ship added to the linguistic data records can be independently scored as a $\chi^2$, and their additive score is $\chi^2$ as well.
![Top: Model and Linguistic Data for a ship leaving Lagos, 1832; Bottom: for Ouidah[]{data-label="fig:Linguistic"}](figures/1832LagosModel.png "fig:"){width="0.45\linewidth"} ![Top: Model and Linguistic Data for a ship leaving Lagos, 1832; Bottom: for Ouidah[]{data-label="fig:Linguistic"}](figures/1832Lagos.png "fig:"){width="0.45\linewidth"}\
![Top: Model and Linguistic Data for a ship leaving Lagos, 1832; Bottom: for Ouidah[]{data-label="fig:Linguistic"}](figures/1832OuidahModel.png "fig:"){width="0.45\linewidth"} ![Top: Model and Linguistic Data for a ship leaving Lagos, 1832; Bottom: for Ouidah[]{data-label="fig:Linguistic"}](figures/1832Ouidah.png "fig:"){width="0.45\linewidth"}
Model Summary
-------------
A summary of the model:
1. Take space-time locations for conflict and, for each year, create an estimate for the conflict intensity map that represents the shifting border of the wars involved. This includes some parameters that are viewed as *fixed*. Simulate slave capture locations out of this heat map.
2. Create a trade map designating roads, cities, and common caravan routes at the time. Specify a few cities to be valid points-of-sale in this map.
3. For each and every slave in step 1, create a Markov Decision Process out of the matrix in step 2 by pairing it with a (randomized) reward vector. Record each slaves point of capture and point of sale. This includes at least two flexible parameters: the cost-of-movement through conflict $C$ and the variance of the rewards for each sale location.
4. Aggregate the origin points and points of sale and score the model - either by historians’ debate or a $\chi^2$ as linguistic data become available. Optimize all flexible parameters.
Interactive Web Application {#S:App}
===========================
To make this research more widely available to a general audience, we created an interactive web application using the `Shiny` package in the `R` programming language. The user can select a year and one or more points-of-sale, and the application generates and displays a conditional probability map showing the most likely region of capture based on our simplified model. The app can also display the yearly conflict data as discrete points, a heatmap of the estimated intensity surface, or a contour plot. Furthermore, the annual approximate state borders [@lovejoy2013redrawing] and trade network informing the MDP can be overlayed.
We have run our model independently for each year from 1816-1836 with the annual trade network and reward vectors changing over time, reflecting the historical narrative. For each year, we generate 10,000 capture locations and record the spatial coordinates, the initial location in the network, and the point-of-sale. We use Kriging on these annual data using the methods in this paper to produce an annual conditional probability surface. For each year, we save the conditional probability Kriging surface, the conflict point data, the KDE conflict intensity surface, the trade network, and the state border shapefiles, which are all the data sets required to host the app.
Our web application is easy to use and freely hosted at `website.com`. It allows a general audience to interactively explore the history of West African slave trade by visualizing the data and models used in this paper. Note that we do not claim that these maps display the historical truth, but rather the results from a model which provide an approximation of the truth.
Conclusions and Future Work
===========================
The Markov decision process framework allows for considerable tuning. While we choose rewards for each slave point of sale that are on average equal between locations, scaling the rewards according to the port departure totals to account for the West-to-East blockading of the region by the British Empire would shift the decision processes of the slavers accordingly.
Our model could also be adapted to account for the time variance in the process and the lack or precision in observed conflict dates. One option would be adding positive spatio-temporal correlation from one year’s conflict map to the next. Another option adds a time delay to each step along the Markov decision process, allowing for recalculations as the conflict shifts each year.
A couple additional sources of validation for potential future work merit discussion. First, genetic databases and genealogical tracking have become much more powerful in recent years, and we look forward to an increase in the availability of such data. In particular, if descendents of passengers of any known ships where to compare their genetics to the current genetic mapping of the Bight of Benin areas, we could begin to rapidly improve on the model validation. A second broader issue our model begins to cover is the difference between traffic through the trans-Saharan and Niger areas. To date, historians have little understanding or discussion of the amount of slave traffic that moved north out of the coastal regions, and if our model is able to withstand critique for its treatment of the coastal areas, the estimates the MDP-based model gives for northward flow could help provide initial estimates for this movement.
A final source of tuning and validation would be to fit similar models to similar historical situations with better availability of data. Mapping continuous conflict borders from discrete city observations could be done for nearly any conventional war fought in the $18^{th}$ or $19^{th}$ centuries. Forced transit situations in the Holocaust did not originate from conventional battles as in Oyo, but have considerably better data due to the relative recency, and could be used to better tune the decision process and exit location models.
We look forward to seeing the expansion of mathematical models in creating both maps in the presence of uncertainty and making those tools available in a non-proprietary form to the public and academic genealogists.
| ArXiv |
---
abstract: 'In the thermal dark matter (DM) paradigm, primordial interactions between DM and Standard Model particles are responsible for the observed DM relic density. In [@boehm:2014MNRAS], we showed that weak-strength interactions between DM and radiation (photons or neutrinos) can erase small-scale density fluctuations, leading to a suppression of the matter power spectrum compared to the collisionless cold DM (CDM) model. This results in fewer DM subhaloes within Milky Way-like DM haloes, implying a reduction in the abundance of satellite galaxies. Here we use very high resolution $N$-body simulations to measure the dynamics of these subhaloes. We find that when interactions are included, the largest subhaloes are less concentrated than their counterparts in the collisionless CDM model and have rotation curves that match observational data, providing a new solution to the “too big to fail” problem.'
author:
- |
J. A. Schewtschenko,$^{1,2}$[^1] C. M. Baugh,$^{1}$ R. J. Wilkinson,$^{2}$ C. Bœhm,$^{2,3}$ S. Pascoli,$^{2}$ T. Sawala$^{4}$\
$^1$Institute for Computational Cosmology, Durham University, Durham DH1 3LE, UK\
$^2$Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, UK\
$^3$LAPTH, U. de Savoie, CNRS, BP 110, 74941 Annecy-Le-Vieux, France\
$^4$Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2a, FI-00014 Helsinki, Finland
bibliography:
- 'IDM\_TBTF.bib'
title: 'Dark matter–radiation interactions: the structure of Milky Way satellite galaxies'
---
\[firstpage\]
astroparticle physics – dark matter – galaxies: haloes – large-scale structure of Universe.
Introduction {#sec:intro}
============
The cold dark matter (CDM) model has been remarkably successful at explaining measurements of the cosmic microwave background radiation and the large-scale structure of the Universe. However, in its simplest form, the model faces challenges on small scales; the most pressing of which are the “missing satellite” (@moore_dark_1999 [@Klypin:1999uc]) and “too big to fail” (@BoylanKolchin:2011de) problems. These discrepancies may indicate the need to consider a richer physics phenomenology in the dark sector, although they were first stated without the inclusion of baryonic physics.
The “missing satellite” problem refers to the overabundance of DM subhaloes in Milky Way (MW)-like DM haloes, compared to the observed number of MW satellite galaxies. This comparison between theory and observation requires a connection to be made between subhaloes and galaxies; in the absence of a good model for galaxy formation, this is most readily done using the halo circular velocity. Subsequent simulations that have taken into account baryonic physics suggest that a reduction in the efficiency of galaxy formation in low-mass DM haloes results in many of the excess subhaloes containing either no galaxy at all or a galaxy that is too faint to be observed (@Benson:2002 [@Somerville:2002; @Sawala:2014; @Sawala:2015]).
As the resolution of $N$-body simulations continued to improve, the “too big to fail” problem emerged (@BoylanKolchin:2011de). This concerns the largest subhaloes, which should be sufficiently massive that their ability to form a galaxy is not hampered by heating of the intergalactic medium by photo-ionising photons or heating of the interstellar medium by supernovae. Simulations of vanilla CDM showed that the largest subhaloes are more massive and denser than is inferred from measurements of the MW satellite rotation curves.
The severity of the small-scale problems can be reduced if one considers the mass of the MW, which impacts the selection of MW-like haloes in the simulations but remains difficult to determine (@Wang:2012 [@Cautun:2014dda; @Piffl:2014; @Wang:2015]). A range of alternatives to vanilla CDM have also been proposed e.g. warm DM (@schaeffer_silk), interacting DM (@Boehm:2000gq [@Boehm:2004th; @CyrRacine:2012fz; @Chu:2014lja]), self-interacting DM (@Spergel:1999mh [@Rocha:2012jg; @Vogelsberger:2014pda; @Buckley:2014PhRvD]), decaying DM (@Wang:2014ina) and late-forming DM (@Agarwal:2015). These “beyond CDM” models generally exhibit a cut-off in the linear matter power spectrum at small scales (high wavenumbers) that translates into a reduced number of low-mass DM haloes compared to collisionless CDM at late times.
Most numerical efforts so far to check whether such models could solve the small-scale problems have focussed on either warm DM or self-interacting DM. However, some works have studied the impact of DM scattering elastically with Standard Model particles in the early Universe; for example, with photons ($\boldsymbol{\gamma}$**CDM**) (@Boehm:2000gq [@boehm_interacting_2001; @Sigurdson:2004zp; @Boehm:2004th; @Dolgov:2013una; @Wilkinson:2013kia]), neutrinos ($\boldsymbol{\nu}$**CDM**) (@Boehm:2000gq [@boehm_interacting_2001; @Boehm:2004th; @Mangano:2006mp; @Serra:2009uu; @Wilkinson:2014ksa; @Escudero:2015yka]) and baryons (@Chen:2002yh [@Dvorkin:2013cea; @Aviles:2011ak]).
Such elastic scattering processes are intimately related to the DM annihilation mechanism in the early Universe and are thus directly connected to the DM relic abundance in scenarios where DM is a thermal weakly-interacting massive particle (WIMP). Therefore, rather than being viewed as exotica, interactions between DM and Standard Model particles should be considered as a more realistic realisation of the CDM model. Indeed, instead of assuming that CDM has no interactions beyond gravity, one can actually test this assumption by determining their impact on the linear matter power spectrum and ruling out values of the cross section that are in contradiction with observations. However, it should be noted that the strength of the scattering and annihilation cross sections can differ by several orders of magnitude, depending on the particle physics model.
The $\gamma$CDM and $\nu$CDM scenarios are characterised by the collisional damping of primordial fluctuations, which can lead to a suppression of small-scale power at late times. The collisional damping scale is determined by a single model-independent parameter: the ratio of the scattering cross section to the DM mass. The larger the ratio, the larger the suppression of the matter power spectrum. For simplicity, we assume that the scattering cross section is constant (i.e. temperature-independent), bearing in mind that temperature-dependence would give rise to the same effect but with a different value of the cross section today (@Wilkinson:2013kia [@Wilkinson:2014ksa]). In [@boehm:2014MNRAS], we confirmed that such models can provide an alternative solution to the missing satellite problem in the MW. Here we show that interacting DM could also solve the too big to fail problem[^2].
The paper is organised as follows. In Section \[sec:idmssp:sim\], we describe the setup of the $N$-body simulations that we use to study small structures. In Section \[sec:tbtf\], we investigate whether interacting DM can alleviate the too big to fail problem, using MW observations. Finally, we give our conclusions in Section \[sec:conc\].
Simulations {#sec:idmssp:sim}
===========
{width="90.00000%"}
While the CDM matter power spectrum predicts the existence of structures at all scales (down to earth mass haloes (@Diemand:2005Nature [@Springel:2008cc; @Angulo:2009hf])), interacting DM models predict a suppression of power below a characteristic damping scale that is determined by the ratio of the DM interaction cross section to the DM mass (@boehm_interacting_2001). For allowed $\gamma$CDM and $\nu$CDM models (@boehm:2014MNRAS), the suppression occurs for haloes with masses below $10^8-10^9~M_{\odot}$. Therefore, to study the distribution and properties of structures beyond the linear regime, it is essential to carry out high-resolution $N$-body simulations.
To reach the resolution required to model the dynamics of DM subhaloes within MW-mass DM haloes, we first identify Local Group (LG) candidates in an $N$-body simulation of a large cosmological volume, and then resimulate the region containing these haloes at much higher mass resolution in a “zoom” resimulation. We use the `DOVE` cosmological simulation to identify haloes for resimulation (the criteria used to select the haloes are listed below) [@Sawala:2014arXiv]. The `DOVE` simulation follows the hierarchical clustering of the mass within a periodic cube of side length $100~$Mpc, using particles of mass $8.8 \times 10^{6}~M_{\odot}$ and assuming a WMAP7 cosmology.
Following the `APOSTLE` project [@Fattahi:2015arXiv], which also uses the `DOVE` CDM simulation to identify LG candidates for study at higher resolution, we impose the following three criteria to select candidates for resimulation:
1. [**Mass:**]{} there should be a pair of MW and Andromeda mass host haloes, with masses in the range $(0.5 - 2.5) \times 10^{12}~{M}_\odot$.\
2. [**Environment:**]{} there should be no other large structures nearby, i.e. an environment with an unperturbed Hubble flow out to 4 Mpc.\
3. [**Dynamics:**]{} the separation between the two haloes should be $800 \pm 200$ kpc, with relative radial and tangential velocities below $250$ km $\mathrm{s}^{-1}$ and $100$ km $\mathrm{s}^{-1}$ respectively.
These criteria are more restrictive than those employed in our earlier work on the structure of haloes (@Schewtschenko:2015MNRAS) as they also take into account the internal kinematics of the LG. We obtain four LG candidates and therefore, eight MW-like haloes. If we assume that the gravitational interaction between the LG haloes is limited, the mass, environment and dynamics[^3] of the haloes would not be significantly different if we had run a $\gamma$CDM or $\nu$CDM version of the `DOVE` simulation.
------ ----------------------- ------------------------------------- -------------------------------------------- --
$M_{\rm vir}$ $V_{\max}$ $\sigma_{\mathrm{DM}-\gamma}$
$[10^{12}~M_{\odot}]$ \[$\mathrm{km}$ $\mathrm{s}^{-1}$\] $[\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})]$
AP-1 1.916 200.3
AP-2 1.273 151.5
AP-3 0.987 157.9
AP-4 0.991 163.0
AP-5 2.010 167.5
AP-6 1.934 165.1
AP-7 1.716 163.7 $0,~10^{-10},~10^{-9},$
AP-8 1.558 193.3 $2 \times 10^{-9},~10^{-8}$
------ ----------------------- ------------------------------------- -------------------------------------------- --
: Key properties of the MW-like haloes in the zoom resimulations (Section \[sec:idmssp:sim\]). The first column specifies the `APOSTLE` identifier (ID) for each MW-like halo, while the second and third columns list the virial mass, $M_{\rm vir}$, and maximum circular velocity, $V_{\rm max}$, respectively (for CDM). The fourth column lists the different DM–photon interaction cross sections, $\sigma_{\mathrm{DM}-\gamma}$, used in the zoom resimulations for each LG candidate, where $\sigma_{\rm Th}$ is the Thomson cross section and $\sigma_{\mathrm{DM}-\gamma} = 0$ corresponds to CDM.[]{data-label="tab:idmssp:LGs"}
We perform resimulations with the `P-Gadget3` $N$-body simulation code [@gadget2] assuming the $\gamma$CDM model, bearing in mind that the results for $\nu$CDM would be very similar (see @Schewtschenko:2015MNRAS). We use the same cosmology (WMAP7)[^4], random phases and second-order LPT method [@Jenkins:2010MNRAS] as [@Sawala:2014arXiv]. We resimulate the four LG candidates with a particle mass $m_{\rm part}=7.2\times10^5~M_\odot$ and a comoving softening length $l_{\rm soft}=216$ pc. This corresponds to a mass resolution that is intermediate between levels 4 and 5 in the Aquarius simulations of [@Springel:2008cc] (level 1 being the highest resolution). We also resimulate the two host haloes in one of our LG Candidates (AP-7/AP-8) at an even higher resolution ($m_{\rm part}=6\times10^4~M_\odot$, $l_{\rm soft}=94$ pc; which is comparable to Aquarius level 3). These simulations (denoted with the suffix `-HR`) are used to confirm that our results have converged[^5] and allow us to obtain more reliable predictions for the innermost parts of the halo. Substructures within the host haloes are located using the `AMIGA` halo finder [@ahf_refs].
We run resimulations for zero interaction cross section, which corresponds to collisionless CDM, and for a selection of DM–photon interaction cross sections, as listed in Tab. \[tab:idmssp:LGs\]. We note that the DM–photon interaction cross section value of $\sigma_{\mathrm{DM}-\gamma} = 2\times 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ was shown to solve the missing satellite problem in [@boehm:2014MNRAS], in the absence of baryonic physics effects.
Fig. \[fig:idmssp:apostles\] shows the projected matter density in the `DOVE` simulation box [@Sawala:2014arXiv] (central panel) along with renderings of the four LG candidates from the resimulations. The eight MW-like haloes are listed in Tab. \[tab:idmssp:LGs\] with their respective properties for CDM. Halo properties for $\gamma$CDM vary only slightly (within a few percent) from those in CDM for the cross sections considered here.
Results {#sec:tbtf}
=======
![Top: the circular velocity, $V_{\rm circ}$, versus radius, $r$, for the eleven most massive subhaloes in AP-7-HR for CDM (grey lines) and for $\gamma$CDM with $\sigma_{\mathrm{DM}-\gamma} = 2 \times 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ (red lines). The dashed lines indicate where $V_{\rm circ}$ can still be measured from the simulation but convergence cannot be guaranteed, according to the criteria set out by @power:2003MNRAS. The data points correspond to the observed MW satellites with 1$\sigma$ error bars [@Wolf:2010MNRAS]. Bottom: the $V_{\rm max}$ versus $R_{\rm max}$ results for all eight MW-like haloes, with the same scattering cross sections as in the top panel. The hatched region marks the 2$\sigma$ confidence interval for the observed MW satellites. $V_{\rm max}$ is derived from the observed stellar line-of-sight velocity dispersion, $\sigma_{\star}$, using the assumption that $V_{\rm max} = \sqrt{3} \sigma_{\star}$ [@Klypin:1999uc].[]{data-label="fig:idmssp:tbtf_gcdm"}](pics/tbtf_profile_combined "fig:"){width=".5\textwidth"} ![Top: the circular velocity, $V_{\rm circ}$, versus radius, $r$, for the eleven most massive subhaloes in AP-7-HR for CDM (grey lines) and for $\gamma$CDM with $\sigma_{\mathrm{DM}-\gamma} = 2 \times 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ (red lines). The dashed lines indicate where $V_{\rm circ}$ can still be measured from the simulation but convergence cannot be guaranteed, according to the criteria set out by @power:2003MNRAS. The data points correspond to the observed MW satellites with 1$\sigma$ error bars [@Wolf:2010MNRAS]. Bottom: the $V_{\rm max}$ versus $R_{\rm max}$ results for all eight MW-like haloes, with the same scattering cross sections as in the top panel. The hatched region marks the 2$\sigma$ confidence interval for the observed MW satellites. $V_{\rm max}$ is derived from the observed stellar line-of-sight velocity dispersion, $\sigma_{\star}$, using the assumption that $V_{\rm max} = \sqrt{3} \sigma_{\star}$ [@Klypin:1999uc].[]{data-label="fig:idmssp:tbtf_gcdm"}](pics/tbtf_allLGs "fig:"){width=".5\textwidth"}
We now explore the too big to fail problem and show how the theoretical predictions and observations can be reconciled by including DM interactions beyond gravity.
The too big to fail problem is illustrated in the top panel of Fig. \[fig:idmssp:tbtf\_gcdm\]. Here the rotation curves of the 11 most massive subhaloes[^6] in the CDM resimulation of the halo AP-7-HR clearly lie above the measurements for the “classical” dwarf spheroidal satellites in the MW taken from [@Wolf:2010MNRAS]. In general, one can see that the largest subhaloes in CDM simulations have a higher circular velocity, $V_{\rm circ}$, and therefore more enclosed (dark) matter, than is observed for a given radius.
In the case of $\gamma$CDM, the rotation curves of the most massive satellites are shifted to lower circular velocities, indicating that there is less (dark) matter enclosed within a given radius. Alternatively, one can interpret this as a lower central density or concentration for the haloes in $\gamma$CDM, as seen in [@Schewtschenko:2015MNRAS].
The circular velocity profiles shown in the top panel of Fig. \[fig:idmssp:tbtf\_gcdm\] are plotted using different line styles. The transition occurs at the scale determined by the convergence criteria devised by [@power:2003MNRAS]. At smaller radii (dashed lines), the velocity profiles are not guaranteed to have converged. However, the key point here is that the CDM and $\gamma$CDM resimulations have the same resolution and yet show a clear difference at all radii plotted, with a shift to lower circular velocities for the haloes in $\gamma$CDM.
The bottom panel of Fig. \[fig:idmssp:tbtf\_gcdm\] presents a related view of the too big to fail problem; this time showing the peak velocity in the rotation curve, $V_{\rm max}$, and the radius at which this occurs, $R_{\rm max}$. The hatched region indicates the $2\sigma$ range inferred for the observed MW satellites, assuming that these are DM-dominated and fitting NFW profiles (@Navarro:1997ApJ) to the rotation curve measurements. We allow the halo concentration parameter to vary, following the same technique and assumptions as described in [@Klypin:1999uc][^7].
Again, the collisionless CDM model predicts satellites that lie outside the $2\sigma$ range compatible with observations. Additionally, for CDM, there are many more subhaloes within the range of $V_{\rm max}$–$R_{\rm max}$ plotted than there are observed satellites. The abundance of massive, concentrated subhaloes varies depending on the mass and formation history of the host halo; however, for all the MW-like candidates studied, CDM exhibits a too big to fail problem, which is reduced in the case of $\gamma$CDM.
In Fig. \[fig:idmssp:tbtf\_gcdm2\], we present the results for AP-7 and AP-8 to show the impact of varying the DM–photon interaction cross section. As the cross section is increased, the predicted $V_{\rm max}$ values fall and shift to larger $R_{\rm max}$. This brings the model predictions well within the region compatible with the observational results and also reduces the number of satellites with such rotation curves. Therefore, one can clearly see that interacting DM can alleviate the too big to fail problem for a cross section $\sigma_{\mathrm{DM}-\gamma} \simeq 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ that also solves the missing satellite problem (@boehm:2014MNRAS).
![The $V_{\rm max}$ versus $R_{\rm max}$ results for a range of DM–photon interaction cross sections using the AP-7 and AP-8 haloes. As in Fig. \[fig:idmssp:tbtf\_gcdm\], the hatched region marks the 2$\sigma$ confidence interval for the observed MW satellites, following the methodology of @Klypin:1999uc.[]{data-label="fig:idmssp:tbtf_gcdm2"}](pics/tbtf_LG4){width=".5\textwidth"}
Conclusion {#sec:conc}
==========
There are a multitude of solutions proposed to overcome the small-scale “failures” of cold dark matter (CDM); namely, the “missing satellite” and “too big to fail” problems. Within the collisionless CDM model, these explanations fall into two camps: i) invoking baryonic physics to reduce the efficiency of galaxy formation in low-mass DM haloes [@Sawala:2014arXiv; @2015arXiv151101098S], and ii) exploiting the uncertainty in the mass of the Milky Way (MW) DM halo [@Wang:2012]. Both problems can be diminished using one or both of the above.
Solutions in which the properties of the DM are varied have also been explored. [@Lovell:2013ola] showed that replacing CDM by a warm DM particle of mass $1.5$ keV leads to a reduced abundance of subhaloes in MW-like haloes, and massive subhaloes that are less concentrated than their CDM counterparts, matching observations of the internal dynamics of the MW satellites. [@Vogelsberger:2014pda] investigated the impact of self-interacting DM on the properties of satellite galaxies, finding little change in the global properties of the galaxies but variation in their structure.
Here we have investigated the impact of interactions between DM and radiation on the structure of the MW satellites. Such interactions are well-motivated and may have helped to set the abundance of DM inferred in the Universe today [@Boehm:2003hm; @Peter:2012]; sometimes called the WIMP miracle. As well as its physical basis, this model has the attraction that it is as simple to simulate as CDM. The interactions took place in the early Universe when the densities of DM and radiation were much higher, and are negligible over the time period covered by the simulation. The DM particles are still cold, so there are no issues relating to particle velocity distributions, as would arise in high-resolution simulations of warm DM, particularly for lighter candidates. The only change compared to a CDM simulation is the modification to the matter power spectrum in linear perturbation theory; the DM–radiation interactions result in a damping of the matter power spectrum on small scales.
We have used high resolution $N$-body simulations of DM haloes, which have passed a set of Local Group selection criteria, to show the impact of DM–radiation interactions on the structure of massive subhaloes. Increasing the interaction cross section reduces the mass enclosed within a given radius in the subhaloes, compared to their CDM counterparts, as suggested by our results for a wider population of DM haloes (@Schewtschenko:2015MNRAS). When combined with our earlier work showing that stronger interactions also lead to a reduction in the number of MW subhaloes (@boehm:2014MNRAS), we find that a model with an elastic scattering cross section of $\simeq 10^{-9}~\sigma_{\rm Th}~(m_{\rm DM}/{\rm GeV})$ can solve both of these small-scale problems of CDM. The next step will be to include baryonic physics. This will not alter the qualitative conclusions of our papers, but will relax the constraints on the DM–radiation scattering cross section.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank V. Springel for providing access to the `P-Gadget3` code and the `bPic` rendering code, A. Jenkins for sharing his IC generator code for the “zoom” resimulations with us and J. Halley for helpful discussions. JAS is supported by a Durham University Alumnus Scholarship and RJW is supported by the STFC Quota grant ST/K501979/1. This work was supported by the STFC (grant numbers ST/F001166/1, ST/G000905/1 and ST/L00075X/1) and the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442). This work was additionally supported by the European Research Council under ERC Grant “NuMass” (FP7-IDEAS-ERC ERC-CG 617143). It made use of the DiRAC Data Centric system at Durham University, operated by the ICC on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. SP also thanks the Spanish MINECO (Centro de excelencia Severo Ochoa Program) under grant SEV-2012-0249. CMB acknowledges a research fellowship from the Leverhulme Trust.
Augmented content {#augmented-content .unnumbered}
=================
[l]{}[.14]{} {width=".16\textwidth"}
This paper provides additional multimedia and interactive content embedded in an augmented reality using the open DARO framework for mobile devices. In order to access the data, you need to use a DARO-compatible browser ([`http://icc.dur.ac.uk/\simdaro`](http://icc.dur.ac.uk/~daro)) and scan the DARO QR code printed here.\
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: Recently, it was also demonstrated that one can simultaneously alleviate the small-scale problems of CDM by including interactions between DM and dark radiation on the linear matter power spectrum and DM self-interactions during non-linear structure formation (@Cyr-Racine:2015ihg [@Vogelsberger:2015gpr]).
[^3]: The formation process of structures is slightly delayed by the presence of DM interactions. Therefore, both the separation and the relative velocities may actually lie outside the bound set by the “Dynamics” criterion as the haloes are at a different point in their orbit around each other for $\gamma$CDM. However, as long as this delay between CDM and $\gamma$CDM is not too large, we essentially have the same dynamical system in both cases and the substructures within the host haloes will be unaffected.
[^4]: The fact that we are using the older WMAP7 cosmology instead of the most recent data is not a concern since we are only interested in the effects of DM interactions on a selected local environment.
[^5]: While the properties of some small subhaloes may vary from one resolution level to another (due to minor changes in their formation and accretion histories), the overall scatter for all subhaloes remains the same and thus can be considered to have converged.
[^6]: We have included three more simulated subhaloes than the observed number of dwarf satellites as the most massive subhaloes are considered statistical outliers like the Magellanic clouds, which have been omitted in this study.
[^7]: A plot with the confidence bands for each of the MW satellites can be found in the provided online material and the augmented content of this paper.
| ArXiv |
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abstract: |
Let $X/S $ be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes $H/S$ of $X/S$ with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in $X/S$ containing a given closed subscheme $C$, and intersecting properly a closed set $F$.
Assume now that the base $S$ is the spectrum of a ring $R$ such that for any finite morphism $Z \to S$, ${\mbox{\rm Pic}\kern 1pt}(Z)$ is a torsion group. This condition is satisfied if $R $ is the ring of integers of a number field, or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal $1$-cycles on a regular scheme $X$ quasi-projective and flat over $S $. We also show the existence of a finite surjective $S$-morphism to $\mathbb P_S^d$ for any scheme $X$ projective over $S$ when $X/S$ has all its fibers of a fixed dimension $d$.
KEYWORDS.
Avoidance lemma, Bertini-type theorem, Hypersurface, Moving lemma, Multisection, $1$-cycle, Pictorsion, Quasi-section, Rational equivalence, Zero locus of a section, Noether normalization.
MATHEMATICS SUBJECT CLASSIFICATION: 14A15, 14C25, 14D06, 14D10, 14G40.
address:
- 'IHÉS, 35 route de Chartres, 91440 Bures-sur-Yvette, France'
- 'Université de Bordeaux 1, Institut de Mathématiques de Bordeaux, CNRS UMR 5251, 33405 Talence, France'
- 'Department of mathematics, University of Georgia, Athens, GA 30602, USA'
author:
- Ofer Gabber
- Qing Liu
- Dino Lorenzini
title: Hypersurfaces in projective schemes and a moving lemma
---
[^1]
Let $S={\operatorname{Spec}}R$ be an affine scheme, and let $X/S$ be a quasi-projective scheme. The core of this article is a method, summarized in \[sum\] below, for proving the existence of closed subschemes of $X$ with various favorable properties. As the technical details can be somewhat complicated, we start this introduction by discussing the applications of the method that the reader will find in this article.
Recall (\[zerolocusinvertible\]) that a global section $f$ of an invertible sheaf ${\mathcal L}$ on any scheme $X$ defines a closed subset $H_f$ of $X$, consisting of all points $x \in X$ where the stalk $f_x$ does not generate ${\mathcal L}_x$. Since ${{\mathcal O}}_X f\subseteq {\mathcal L}$, the ideal sheaf ${\mathcal I}:=
{{\mathcal O}}_X f\otimes {\mathcal L}^{-1}$ endows $H_f$ with the structure of closed subscheme of $X$. Let $X \to S$ be any morphism. We call the closed subscheme $H_f$ of $X$ a *hypersurface* (relative to $X \to S$) when no irreducible component of positive dimension of $X_s$ is contained in $H_f$, for all $s\in S$. If, moreover, the ideal sheaf ${\mathcal I}$ is invertible, we say that the hypersurface $H_f$ is *locally principal*. We remark that when a fiber $X_s$ contains isolated points, it is possible for $H_f$ (resp. $(H_f)_s)$ to have codimension $0$ in $X$ (resp. in $X_s$), instead of the expected codimension $1$.
[**A. An Avoidance Lemma for Families.**]{} It is classical that if $X/k$ is a quasi-projective scheme over a field, $C \subsetneq X$ is a closed subset of positive codimension, and $\xi_1,\dots, \xi_r$ are points of $X$ not contained in $C$, then there exists a hypersurface $H$ in $X$ such that $C \subseteq H$ and $\xi_1, \dots, \xi_r \notin H$. Such a statement is commonly referred to as an Avoidance Lemma (see, e.g., \[avoid\]). Our next theorem establishes an Avoidance Lemma for Families. As usual, when $X$ is noetherian, ${\operatorname{Ass}}(X)$ denotes the finite [*set of associated points*]{} of $X$.
[**Theorem \[bertini-type-0\].**]{}
*Let $S$ be an affine scheme, and let $X\to S$ be a quasi-projective and finitely presented morphism. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $X \to S$. Let*
1. $C$ be a closed subscheme of $X$, finitely presented over $S$;
2. $F_1, \dots, F_m$ be subschemes[^2] of $X$ of finite presentation over $S$;
3. $A$ be a finite subset of $X$ such that $A\cap C=\emptyset$.
Assume that for all $s \in S$, $C$ does not contain any irreducible component of positive dimension of $(F_i)_s$ and of $X_s$. Then there exists $n_0>0$ such that for all $n\ge n_0$, there exists a global section $f$ of ${{\mathcal O}}_X(n)$ such that:
1. the closed subscheme $H_f$ of $X$ is a hypersurface that contains $C$ as a closed subscheme;
2. for all $s \in S$ and for all $i\le m$, $H_f$ does not contain any irreducible component of positive dimension of $(F_i)_s$; and
3. $H_f\cap A=\emptyset$.
Assume in addition that $S$ is noetherian, and that $C\cap{\operatorname{Ass}}(X)=\emptyset$. Then there exists such a hypersurface $H_f$ which is locally principal.
When $H_f$ is locally principal, $H_f$ is the support of an effective ample Cartier divisor on $X$. This divisor is ‘horizontal’ in the sense that it does not contain in its support any irreducible component of fibers of $X \to S$ of positive dimension. In some instances, such as in \[bertini-cor1\] and \[generic-S1\], we can show that $H_f$ is a relative effective Cartier divisor, i.e., that $H_f \to S$ is flat. Corollary \[bertini-cor1\] also includes a Bertini-type statement for $X \to S$ with Cohen-Macaulay fibers. We use Theorem \[bertini-type-0\] to establish in \[quasisections\] the existence of finite quasi-sections in certain projective morphisms $X/S$, as we now discuss.
[**B. Existence of finite quasi-sections.**]{} Let $X\to S$ be a surjective morphism. Following EGA [@EGA], IV, §14, p. 200, we define:
\[def.finite-qs\] We call a closed subscheme $C$ of $X$ a *finite quasi-section* when $C \to S$ is finite and surjective. Some authors call *multisection* a finite quasi-section $C \to S$ which is also flat, with $C$ irreducible (see e.g., [@HT], p. [12]{} and [4.7]{}).
When $S$ is integral noetherian of dimension $1$ and $X\to S$ is proper and surjective, the existence of a finite quasi-section $C$ is well-known and easy to establish. It suffices to take $C$ to be the Zariski closure of a closed point of the generic fiber of $X\to S$. When $\dim S>1$, the process of taking the closure of any closed point of the generic fiber does not always produce a closed subset [*finite*]{} over $S$ (see \[easy\]).
[**Theorem \[quasisections\].**]{}
*Let $S$ be an affine scheme and let $X\to S$ be a projective, finitely presented morphism. Suppose that all fibers of $X\to S$ are of the same dimension $d\ge 0$. Let $C$ be a finitely presented closed subscheme of $X$, with $C \to S$ finite but not necessarily surjective. Then there exists a finite quasi-section $T$ of finite presentation which contains $C$. Moreover:*
1. Assume that $S$ is noetherian. If $C$ and $X$ are both irreducible, then there exists such a quasi-section with $T$ irreducible.
2. If $X\to S$ is flat with Cohen-Macaulay fibers (e.g., if $S$ is regular and $X$ is Cohen-Macaulay), then there exists such a quasi-section with $T\to S$ flat.
3. If $X \to S$ is flat and a local complete intersection morphism[^3], then there exists such a quasi-section with $T\to S$ flat and a local complete intersection morphism.
4. Assume that $S$ is noetherian. Suppose that $\pi:X \to S$ has fibers pure of the same dimension, and that $C \to S$ is unramified. Let $Z$ be a finite subset of $S$ (such as the set of generic points of $\pi(C)$), and suppose that there exists an open subset $U$ of $S$ containing $Z$ such that $X \times_S U \to U$ is smooth. Then there exists such a quasi-section $T$ of $X \to S$ and an open set $V \subseteq U$ containing $Z$ such that $T \times_S V \to V$ is étale.
As an application of Theorem \[quasisections\], we obtain a strengthening in the affine case of the classical splitting lemma for vector bundles.
[**Proposition \[splitting\].**]{} [*Let $A$ be a commutative ring. Let $M$ be a projective $A$-module of finite presentation with constant rank $r> 1$. Then there exists an $A$-algebra $B$, finite and faithfully flat over $A$, with $B$ a local complete intersection over $A$, such that $M\otimes_A B $ is isomorphic to a direct sum of projective $B$-modules of rank $1$.*]{}
Another application of Theorem \[quasisections\], to the problem of extending a given family of stable curves ${D} \to Z$ after a finite surjective base change, is found in \[extension.stable.curve\]. It is natural to wonder whether Theorems \[bertini-type-0\] and \[quasisections\] hold for more general bases $S$ which are not affine, such as a noetherian base $S$ having an ample invertible sheaf. It is also natural to wonder if the existence of finite quasi-sections in Theorem \[quasisections\] holds for proper morphisms.
[**C. Existence of Integral Points.**]{} Let $R$ be a Dedekind domain[^4] and let $S= {\operatorname{Spec}}R$. When $X\to S$ is quasi-projective, an integral finite quasi-section is also called an *integral point* in [@MB1], 1.4. The existence of a finite quasi-section in the quasi-projective case over $S={\operatorname{Spec}}{\mathbb Z}$ when the generic fiber is geometrically irreducible is Rumely’s famous Local-Global Principle [@Rum]. This existence result was extended in [@MB1], 1.6, as follows. As in [@MB1], [1.5]{}, we make the following definition.
\[ConditionT\]
We say that a Dedekind scheme $S$ satisfies *Condition* [(T)]{} if:
1. For any finite extension $L$ of the field of fractions $K$ of $R$, the normalization $S'$ of $S$ in ${\operatorname{Spec}}L$ has torsion Picard group ${\mbox{\rm Pic}\kern 1pt}(S')$, and
2. The residue fields at all closed points of $S$ are algebraic extensions of finite fields.
For example, $S$ satisfies Condition (T) if $S$ is an affine integral smooth curve over a finite field, or if $S$ is the spectrum of the ring of $P$-integers in a number field $K$, where $P$ is a finite set of finite places of $K$.
Our next theorem is only a mild sharpening of the Local-Global Principle in [@MB1], 1.7: We show in \[RumelyLG\] that the hypothesis in [@MB1], 1.7, that the base scheme $S$ is excellent, can be removed.
[**Theorem \[RumelyLG\].**]{} [*Let $S$ be a Dedekind scheme satisfying Condition [(T)]{}. Let $X \to S$ be a separated surjective morphism of finite type. Assume that $X$ is irreducible and that the generic fiber of $X \to S$ is geometrically irreducible. Then $X \to S$ has a finite quasi-section.*]{}
Condition (T)(a) is necessary in the Local-Global Principle, but it is not sufficient, as shown by an example of Raynaud over $S={\operatorname{Spec}}\overline{{\mathbb Q}}[x]_{(x)}$ ([@MB0], 3.2, and [@vDM], 5.5). The following weaker condition is needed for our next two theorems.
\[ConditionT\*\] [Let $R$ be any commutative ring and let $S= {\operatorname{Spec}}R$. We say that $R$ or $S$ is *pictorsion* if ${\mbox{\rm Pic}\kern 1pt}(Z)$ is a torsion group for any finite morphism $Z \to S$.]{}
Any semi-local ring $R$ is ${\mbox{pictorsion}}$. A Dedekind domain satisfying Condition (T) is ${\mbox{pictorsion}}$ ([@MB1], 2.3, see also \[lem.torsiondegreed\] (2)). Rings which satisfy the primitive criterion (see \[localglobal\]) are ${\mbox{pictorsion}}$ and only have infinite residue fields.
[**D. A Moving Lemma.**]{} Let $S$ be a Dedekind scheme and let $X$ be a noetherian scheme over $S$. An integral closed subscheme $C$ of $X$ finite and surjective over $S$ is called an *irreducible horizontal $1$-cycle* on $X$. A *horizontal $1$-cycle* on $X$ is an element of the free abelian group generated by the irreducible horizontal $1$-cycles. Our next application of the method developed in this article is a Moving Lemma for horizontal $1$-cycles.
[**Theorem \[mv-1-cycle-local\].**]{}
*Let $R$ be a Dedekind domain, and let $S:={\operatorname{Spec}}R$. Let $X \to S$ be a flat and quasi-projective morphism, with $X$ integral. Let $C$ be a horizontal $1$-cycle on $X$. Let $F$ be a closed subset of $X$. Assume that for all $s \in S$, $F\cap X_s$ and ${\operatorname{Supp}}(C) \cap X_s$ have positive codimension[^5] in $X_s$. Assume in addition that either*
1. $R$ is pictorsion and the support of $C$ is contained in the regular locus of $X$, or
2. $R$ satisfies Condition *(T)*.
Then some positive multiple $mC$ of $C$ is rationally equivalent to a horizontal $1$-cycle $C'$ on $X$ whose support does not meet $F$. Under the assumption [(a)]{}, if furthermore $R$ is semi-local, then we can take $m=1$.
Moreover, if $Y \to S$ is any separated morphism of finite type and $h: X \to Y$ is any $S$-morphism, then $h_*(mC)$ is rationally equivalent to $h_*(C')$ on $Y$.
Example \[ex.extrahyp\] shows that the Condition (T)(a) is necessary for Theorem \[mv-1-cycle-local\] to hold. A different proof of Theorem \[mv-1-cycle-local\] when $S$ is semi-local, $X$ is regular, and $X\to S$ is quasi-projective, is given in [@GLL1], 2.3, where the result is then used to prove a formula for the index of an algebraic variety over a Henselian field ([@GLL1], 8.2).
It follows from [@GLL1], 6.5, that for each $s \in S$, a multiple $m_s C_s$ of the $0$-cycle $C_s$ is rationally equivalent on $X_s$ to a $0$-cycle whose support is disjoint from $F_s$. Theorem 6.5 in [@GLL1] expresses such an integer $m_s$ in terms of Hilbert-Samuel multiplicities. The $1$-cycle $C$ in $X$ can be thought of as a family of $0$-cycles, and Theorem \[mv-1-cycle-local\] may be considered as a Moving Lemma for $0$-cycles in families.
Even for schemes of finite type over a finite field, Theorem \[mv-1-cycle-local\] is not a consequence of the classical Chow’s Moving Lemma. Indeed, let $X$ be a smooth quasi-projective variety over a field $k$. The classical Chow’s Moving Lemma [@Rob] immediately implies the following statement:
\[Chow\] [*Let $Z$ be a $1$-cycle on $X$. Let $F$ be a closed subset of $X$ of codimension at least $2$ in $X$. Then there exists a $1$-cycle $Z'$ on $X$, rationally equivalent to $Z$, and such that ${\operatorname{Supp}}(Z') \cap F
= \emptyset$.* ]{}
Consider a morphism $X \to S$ as in Theorem \[mv-1-cycle-local\], and assume in addition that $S$ is a smooth affine curve over a finite field $k$. Let $F$ be a closed subset as in \[mv-1-cycle-local\]. Such a subset may be of codimension $1$ in $X$. Thus, Theorem \[mv-1-cycle-local\] is not a consequence of Chow’s Moving Lemma for $1$-cycles just recalled, since \[Chow\] can only be applied to $X\to S$ when $F$ is a closed subset of codimension at least $2$ in $X$.
[**E. Existence of finite morphisms to ${\mathbb P}^d_S$.**]{} Let $k$ be a field. A strong form of the Normalization Theorem of E. Noether that applies to graded rings (see, e.g., [@Eis], 13.3) implies that every projective variety $X/k$ of dimension $d$ admits a finite $k$-morphism $X \to {\mathbb P}_k^d$. Our next theorem guarantees the existence of a finite $S$-morphism $X \to {\mathbb P}_S^d$ when $X\to S$ is projective with $R$ ${\mbox{pictorsion}}$, and $d:= \max\{\dim X_s, s\in S\}$.
[**Theorem \[theorem.finiteP\^n\].**]{} [ *Let $R$ be a ${\mbox{pictorsion}}$ ring, and let $S:={\operatorname{Spec}}R$. Let $X \to S$ be a projective morphism, and set $d:= \max\{\dim X_s, s\in S\}$. Then there exists a finite $S$-morphism $ X \to {\mathbb P}^d_S$. If we assume in addition that $\dim X_s= d$ for all $s\in S$, then there exists a finite surjective $S$-morphism $ X \to {\mathbb P}^d_S$.* ]{}
The above theorem generalizes to schemes $X/S$ of any dimension the results of [@Gre2], Theorem 2, and [@CPT], Theorem 1.2, which apply to morphisms of relative dimension $1$. After this article was written, we became aware of the preprint [@CMPT], where the general case is also discussed. We also prove the converse of this theorem:
[**Proposition (see \[conversepictorsion\]).**]{} [ *Let $R$ be any commutative ring and let $S:={\operatorname{Spec}}R$. Suppose that for any $d\ge 0$, and for any projective morphism $X
\to S$ such that $\dim X_s=d$ for all $s\in S$, there exists a finite surjective $S$-morphism $ X \to {\mathbb P}^d_S$. Then $R$ is pictorsion.* ]{}
[**F. Method of proof.**]{} Now that the main applications of our method for proving the existence of hypersurfaces $H_f$ in projective schemes $X/S$ with certain desired properties have been discussed, let us summarize the method.
\[sum\] Let $X\to S$ be a projective morphism with $S={\operatorname{Spec}}R$ affine and noetherian. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $X\to S$. Let $C \subseteq X$ be a closed subscheme defined by an ideal ${\mathcal I}$, and set ${\mathcal I}(n) := {\mathcal I} \otimes {\mathcal O}_X(n)$. Our goal is to show the existence, for all $n$ large enough, of a global section $f $ of ${\mathcal I}(n)$ such that the associated subscheme $H_f$ has the desired properties.
To do so, we fix a system of generators $f_1,\dots, f_N$ of $H^0(X,{\mathcal I}(n)) $, and we consider for each $s\in S$ a subset $\Sigma(s) \subset {\mathbb A}^N(k(s))$ consisting of all the vectors $(\alpha_1,\dots,\alpha_N)$ such that $\sum_i \alpha_i f_{i|{X_s}}$ does not have the desired properties. We show then that all these subsets $\Sigma(s)$, $s\in S$, are the rational points of a single pro-constructible subset $T$ of ${\mathbb A}^N_S$ (which depends on $n$). To find a desired global section $f:=\sum_i a_i f_i$ with $a_i \in R$ which avoids the subset $T$ of ‘bad’ sections, we show that for some $n$ large enough the pro-constructible subset $T$ satisfies the hypotheses of the following theorem. The section $\sigma$ whose existence follows from \[globalize\] provides the desired vector $(a_1,\dots, a_N) \in R^N$.
[**Theorem (see \[globalize\]).**]{}
*Let $S$ be a noetherian affine scheme. Let $T:=T_1\cup \ldots \cup T_m$ be a finite union of pro-constructible subsets of $\mathbb A^N_S$. Suppose that:*
1. For all $i\le m$, $\dim T_i<N$, and $(T_i)_s$ is constructible in $\mathbb A^N_{k(s)}$ for all $s\in S$.
2. For all $s\in S$, there exists a $k(s)$-rational point in ${\mathbb A}^N_{k(s)}$ which does not belong to $T_s$.
Then there exists a section $\sigma$ of $\pi: {\mathbb A}^N_S\to S$ such that $\sigma(S) \cap T = \emptyset$.
To explain the phrasing of (1) in the above theorem, note that the union $T_1\cup \ldots \cup T_m=:T$ is pro-constructible since each $T_i$ is. However, it may happen that $\dim T > \max_i(\dim T_i)$. This can be seen already on the spectrum $T$ of a discrete valuation ring, which is the union of two (constructible) points, each of dimension $0$. The proof of Theorem \[globalize\] is given in section \[Constructible subsets\]. The construction alluded to in \[sum\] of the pro-constructible subset $T$ whose rational points are in bijection with $\Sigma(s)$ is done in Proposition \[constructible-conditions\].
We present our next theorem as a final illustration of the strength of the method. This theorem, stated in a slightly stronger form in section \[Main application\], is the key to the proof of Theorem \[mv-1-cycle-local\] (a), as it allows for a reduction to the case of relative dimension $1$. Note that in this theorem, $S$ is not assumed to be ${\mbox{pictorsion}}$.
[**Theorem (see \[pro.reductiondimension2\]).**]{}
*Let $S$ be an affine noetherian scheme of finite dimension, and let $X\to S$ be a quasi-projective morphism. Let $C$ be a closed irreducible subscheme of $X$, of codimension $d>\dim S$ in $X$. Assume that $C\to S$ is finite and surjective, and that $C \to X$ is a regular immersion. Let $F$ be a closed subset of $X$. Fix a very ample sheaf ${{\mathcal O}}_X(1)$ relative to $X\to S$. Then there exists $n_0>0$ such that for all $n \geq n_0$, there exists a global section $f$ of ${{\mathcal O}}_X(n)$ such that:*
1. $C$ is a closed subscheme of codimension $d-1$ in $H_f$, and $C \to H_f$ is a regular immersion;
2. For all $s\in S$, $H_f$ does not contain any irreducible component of positive dimension of $F_s$.
The proof of Theorem \[pro.reductiondimension2\] is quite subtle and spans sections \[Main application\] and \[compute-coh\]. In Theorem \[mv-1-cycle-local\] (a), we start with the hypothesis that $C$ is contained in the regular locus of $X$. It is not possible in general to expect that a hypersurface $H_f$ containing $C$ can be chosen so that $C$ is again contained in the regular locus of $H_f$. Thus, when no regularity conditions can be expected on the total space, we impose regularity conditions by assuming that $C$ is regularly immersed in $X$. Great care is then needed in the proof of \[pro.reductiondimension2\] to insure that a hypersurface $H_f$ can be found with the property that $C$ is regularly immersed in $H_f$.
Section \[Main application\] contains most of the proof of Theorem \[pro.reductiondimension2\]. Several lemmas needed in the proof of Theorem \[pro.reductiondimension2\] are discussed separately in section \[compute-coh\]. Sections \[hyperf\], \[finite-qs\], \[mv-1c\] and \[finite-pd\] contain the proofs of the applications of our method.
It is our pleasure to thank Max Lieblich and Damiano Testa for the second example in \[ex.avoidance\], Angelo Vistoli for helpful clarifications regarding \[extension.stable.curve\] and \[smooth\_cover\_mg\], and Robert Varley for Example \[Varley\]. We also warmly thank the referees, for a meticulous reading of the article, for several corrections and strengthenings, and many other useful suggestions which led to improvements in the exposition.
[Zero locus of sections of a quasi-coherent sheaf]{}
We start this section by reviewing basic facts on constructible subsets, a concept introduced by Chevalley in [@Che]. We follow the exposition in [@EGA]. We introduce the zero-locus ${\mathcal Z}(\mathcal F,f)$ of a global section $f$ of a finitely presented ${{\mathcal O}}_X$-module $\mathcal F$ on a scheme $X$, and show in \[zero-locus-f\] that this subset is locally constructible in $X$. Given a finitely presented morphism $\pi: X \to Y$, we further define a subset $T_{\mathcal F, f, \pi}$ in $Y$, and show in \[constructible-univ\] that it is locally constructible in $Y$. The main result in this section is Proposition \[constructible-conditions\], which is a key ingredient in the proofs of Theorems \[exist-hyp\] and \[pro.reductiondimension2\].
Let $X$ be a topological space. A subset $T$ of $X$ is *constructible*[^6] if it is a finite union of subsets of the form $U\cap (X\setminus V)$, where $U $ and $ V$ are open and retro-compact[^7] in $X$. A subset $T$ of $X$ is *locally constructible* if for any point $t\in T$, there exists an open neighborhood $V$ of $t$ in $X$ such that $T\cap V$ is constructible in $V$ ([@EGA], Chap. 0, 9.1.3 and 9.1.11). When $X$ is a quasi-compact and quasi-separated scheme[^8] (e.g., if $X$ is noetherian, or affine), then any quasi-compact open subset is retro-compact and any locally constructible subset is constructible ([@EGA], IV.1.8.1).
When $T$ is a subset of a topological space $X$, we endow $T$ with the induced topology, and define the *dimension of $T$* to be the Krull dimension of the topological space $T$. As usual, $\dim T<0$ if and only if $T = \emptyset$. Let $\pi: X\to S$ be a morphism and let $T\subseteq X$ be any subset. For any $s \in S$, we will denote by $T_s$ the subset $\pi^{-1}(s) \cap T$.
\[pro-constr\] Recall that a *pro-constructible* subset in a *noetherian* scheme $X$ is a (possibly infinite) intersection of constructible subsets of $X$ ([@EGA], IV.1.9.4). Clearly, the constructible subsets of $X$ are pro-constructible in $X$, and so are the finite subsets of $X$ ([@EGA], IV.1.9.6), and the constructible subsets of any fiber of a morphism of schemes $X \to Y$ ([@EGA], IV.1.9.5 (vi)). The complement in $X$ of a pro-constructible subset is called an *ind-constructible* of $X$. Equivalently, an ind-constructible subset of $X$ is any union of constructible subsets of $X$.
We are very much indebted to a referee for pointing out that our original hypothesis in Theorem \[globalize\] that $T$ be the union of a constructible subset and finitely many closed strict subsets of fibers $\mathbb A^N_{k(s)}$ could be generalized to the hypothesis that $T$ be pro-constructible. We also thank this referee for suggestions which greatly improved the exposition of the proof of our original Proposition \[constructible-conditions\].
\[lem.construct\]
1. Let $X/k$ be a scheme of finite type over a field $k$, and let $T\subseteq X$ be a constructible subset, with closure $\overline{T} $ in $X$. Then $\dim T = \dim \overline{T}$.
2. Let $X/k$ be as in (a). Let $k'/k$ be a finite extension, and denote by $T_{k'}$ the preimage of $T$ under the map $X \times_k k' \to X$. Then $\dim T_{k'} = \dim T$.
3. Let $Y$ be any noetherian scheme and $\pi: X \to Y$ be a morphism of finite type. Let $T \subseteq X$ be constructible. Assume that for each $y\in Y$, $\dim T_y\leq d$. Then $\dim T \leq \dim Y + d$.
4. Suppose $X$ is a noetherian scheme. Let $T$ be a pro-constructible subset of $X$. Then $T$ has finitely many irreducible components and each of them has a generic point.
(a)-(b) Let $\Gamma$ be an irreducible component of $\overline{T}$ of dimension $\dim \overline{T}$. As $T$ is dense in $\overline{T}$, $T\cap \Gamma$ is dense in $\Gamma$. As $T\cap \Gamma$ is constructible and dense in $\Gamma$, it contains a dense open subset $U$ of $\Gamma$. Therefore, because $\Gamma$ is integral of finite type over $k$, $\dim \Gamma=\dim U \le \dim T \le \dim \overline{T}=\dim \Gamma$ and $\dim T=\dim \overline{T}$. This proves (a).
We also have $$\dim\Gamma=\dim\Gamma_{k'}=\dim U_{k'}\le \dim T_{k'}\le \dim
\overline{T}_{k'}=\dim \overline{T}=\dim\Gamma.$$ This proves (b).
\(c) Let $\{\Gamma_i\}_i$ be the irreducible components of $T$. They are closed in $T$, thus constructible in $X$. As $\dim T=\max_i \{ \dim \Gamma_i\}$ and the fibers of $\Gamma_i\to Y$ all have dimension bounded by $d$, it is enough to prove the statement when $T$ itself is irreducible. Replacing $X$ with the Zariski closure of $T$ in $X$ with reduced scheme structure, we can suppose $X$ is integral and $T$ is dense in $X$. Let $\xi$ be the generic point of $X$ and let $\eta=\pi(\xi)$. As $T$ is constructible and dense in $X$, it contains a dense open subset $U$ of $X$. Then $U_\eta$ is dense in $X_\eta$. Hence $\dim X_\eta=\dim U_\eta\le \dim T_\eta\le d$. Therefore $$\dim T\le \dim X\le \dim \overline{\pi(X)}+d\le \dim Y + d,$$ where the middle inequality is given by [@EGA], IV.5.6.5.
\(d) The subspace $T$ of $X$ is noetherian and, hence, it has finitely many irreducible components ([@Bou], II, § 4.2, Prop. 8 (i), and Prop. 10). Let $\Gamma$ be an irreducible component of $T$. Let $\overline{\Gamma}$ be its closure in $X$. Since $\overline{\Gamma}$ is also irreducible, it has a generic point $\xi\in X$. We claim that $\xi \in \Gamma$, so that $ \xi$ is also the generic point of $\Gamma$. Indeed, suppose that $\Gamma$ is contained in a constructible $W:=\cup_{i=1}^m U_i \cap F_i$, with $U_i$ open and $F_i$ closed in $X$, and such that $(U_i \cap F_i) \cap \Gamma \neq \emptyset$ for all $i=1,\dots, m$. Then there exists $j$ such that $\overline{\Gamma} \subset F_j$. Since $U_j$ contains an element of $\Gamma$ by hypothesis, we find that it must also contain $\xi$, so that $\xi \in W$. The subset $\Gamma$ is pro-constructible in $X$ since it is closed in the pro-constructible $T$. Hence, by definition, $\Gamma $ is the intersection of constructible subsets, which all contain $\xi$. Hence, $\xi \in \Gamma$.
\[zero.locus\] Let $X$ be a scheme. Let ${{\mathcal F}}$ be a quasi-coherent ${{\mathcal O}}_X$-module, such as a finitely presented ${{\mathcal O}}_X$-module ([@EGA], Chap. 0, (5.2.5)). Fix a section $f\in H^0(X, {{\mathcal F}})$. For $x \in X$, denote by $f(x) $ the canonical image of $f$ in the fiber ${{\mathcal F}}(x) := {{\mathcal F}}_x \otimes_{{{\mathcal O}}_{X,x}} k(x)$. We say that [*$f$ vanishes at $x$ if $f(x)=0$*]{} (in ${{\mathcal F}}(x)$). Define $$Z({{\mathcal F}}, f):=\{ x\in X \mid f(x)=0 \}$$ to be the *zero-locus of $f$*.
Let $q: X'\to X$ be any morphism of schemes. Let ${{\mathcal F}}':=q^*{{\mathcal F}}$, and let $f'\in H^0(X', {{\mathcal F}}')$ be the canonical image of $f$. Then $$Z({{\mathcal F}}',f')= q^{-1}(Z({{\mathcal F}},f)).$$ Indeed, for any $x'\in q^{-1}(x)$, the natural morphism ${{\mathcal F}}(x) \to {{\mathcal F}}'(x')={{\mathcal F}}(x)\otimes_{k(x)} k(x')$ is injective.
When ${{\mathcal F}}$ is invertible or, more generally, locally free, then $Z({{\mathcal F}},f)$ is closed in $X$. As our next lemma shows, $Z({{\mathcal F}},f)$ in general is locally constructible. When $X$ is noetherian, this is proved for instance in [@Pearlstein-Schnell], Proposition 5.3. We give here a different proof.
\[zero-locus-f\] Let $X$ be a scheme and let ${{\mathcal F}}$ be a finitely presented ${{\mathcal O}}_X$-module. Then the set $Z({{\mathcal F}}, f)$ is locally constructible in $X$.
Since the statement is local on $X$, it suffices to prove the lemma when $X={\operatorname{Spec}}A$ is affine. We can use the stratification $X=\cup_{1\le i\le n} X_i$ of $X$ described in [@EGA], IV.8.9.5: each $X_i$ is a quasi-compact subscheme of $X$, and ${{\mathcal F}}_i:={{\mathcal F}}\otimes_{{{\mathcal O}}_X} {{\mathcal O}}_{X_i}$ is flat on $X_i$. Let $f_i$ be the canonical image of $f$ in $H^0(X_i, {{\mathcal F}}_i)$. Then $$Z({{\mathcal F}}, f)=\cup_{1\le i\le n} Z({{\mathcal F}}_i, f_i).$$ Since ${{\mathcal F}}_i$ is finitely presented and flat, it is projective ([@Laz], 1.4) and, hence, locally free. So $Z({{\mathcal F}}_i, f_i)$ is closed in $X_i$.
\[prelimSigma\] Let $\pi: X\to Y$ be a finitely presented[^9] morphism of schemes. Let $T$ be a locally constructible subset of $X$. Set $$T_\pi:=\{ y \in Y \mid T_y \mbox{ contains a generic
point of }
X_y \},$$ where a *generic point* of a scheme $X$ is the generic point of an irreducible component of $X$. Such a point is called a maximal point in [@EGA], just before IV.1.1.5. Let ${{\mathcal F}}$ be a finitely presented ${{\mathcal O}}_X$-module and fix a global section $f \in {{\mathcal F}}(X)$. Set $$T_{{{\mathcal F}}, f, \pi}:=\{ y\in Y \mid f \ \text{vanishes at a generic point of}
\ X_y\}.$$ For future use, let us note the following equivalent expression for $T_{{{\mathcal F}}, f, \pi}$. For any $y\in Y$, let $${{\mathcal F}}_y:={{\mathcal F}}\otimes_{{{\mathcal O}}_X} {{\mathcal O}}_{X_y}={{\mathcal F}}\otimes_{{{\mathcal O}}_Y} k(y),$$ and let $f_y$ be the canonical image of $f$ in $H^0(X_y, {{\mathcal F}}_y)$. Let $x \in X_y$. Since the canonical map ${{\mathcal F}}(x)\to
{{\mathcal F}}_y(x)$ of fibers at $x$ is an isomorphism, $f_y$ vanishes at $x$ if and only if $f$ vanishes at $x$. Thus $$T_{{{\mathcal F}}, f, \pi}=\{ y\in Y \mid f_y \ \text{vanishes at a generic point of}
\ X_y\}.$$ When the morphism $\pi$ is understood, we may denote $T_{{{\mathcal F}}, f, \pi}$ simply by $T_{{{\mathcal F}}, f}$. Note that when $\pi= {\rm id}: X \to X$, the set $T_{{{\mathcal F}}, f, {\rm id}}$ is equal to the zero locus $Z({{\mathcal F}},f)$ of $f$ introduced in \[zero.locus\].
\[constructible-univ\] Let $\pi:X\to Y$ be a finitely presented morphism of schemes. Let $T$ be a locally constructible subset of $X$. Let ${{\mathcal F}}$ be a finitely presented ${{\mathcal O}}_X$-module, and fix a section $f\in H^0(X, {{\mathcal F}})$. Then the subsets $T_{\pi}$ and $T_{{{\mathcal F}},f,\pi}$ are both locally constructible in $Y$.
Let us start by showing that $T_\pi$ is locally constructible in $Y$. By definition of locally constructible, the statement is local on $Y$, and it suffices to prove the statement when $Y$ is affine. Assume then from now on that $Y$ is affine. Since $\pi $ is quasi-compact and quasi-separated and $Y$ is affine, $X$ is also quasi-compact and quasi-separated ([@EGA], IV.1.2.6). Hence, $T$ is constructible and we can write it as a finite union of locally closed subsets $T_i:=U_i\cap (X\setminus V_i)$ with $U_i$ and $ V_i$ open and retro-compact. Then $T_y$ contains a generic point of $X_y$ if and only if $(T_i)_y$ contains a generic point of $X_y$ for some $i$. Therefore, it suffices to prove the statement when $T=U\cap (X\setminus V)$ with $U $ and $ V$ open and retro-compact.
We therefore assume now that $T= U \cap Z$, with $U $ and $ X\setminus Z$ open and retro-compact. Fix $y\in Y$. We claim that [*$T_y$ contains a generic point of $X_y$ if and only if there exists $x \in T_y$ such that ${{\operatorname{codim}}}_x(Z_y, X_y)=0$*]{}.
To justify this claim, let us recall the following. Let $\Gamma_1, \dots, \Gamma_n$ be the irreducible components of $Z_y$ passing through $x$ ($X_y$ is noetherian). Then $${{\operatorname{codim}}}_x(Z_y, X_y)=\min_{1\le i\le n} \{ {{\operatorname{codim}}}(\Gamma_i, X_y)\}$$ ([@EGA], 0.14.2.6(i)). So ${{\operatorname{codim}}}_x(Z_y, X_y)=0$ if and only if $Z_y$ contains an irreducible component of $X_y$ passing through $x$. Now, if $T_y$ contains a generic point $\xi$ of $X_y$, then $Z_y$ contains the irreducible component $\overline{\{ \xi\}}\ni \xi$ of $X_y$ and ${{\operatorname{codim}}}_{\xi}(Z_y, X_y)=0$. Conversely, if ${{\operatorname{codim}}}_x(Z_y, X_y)=0$ for some $x\in T_y$, then $Z_y$ contains an irreducible component $\Gamma$ of $X_y$ passing through $x$. As $T_y$ is open in $Z_y$, $T_y\cap \Gamma$ is open in $\Gamma$ and non-empty, so $T_y$ contains the generic point of $\Gamma$.
Since $Z$ is closed, we can apply [@EGA], IV.9.9.1(ii), and find that the set $$X_0:= \{ x \in X \ | \ {{\operatorname{codim}}}_x(Z_{\pi(x)}, X_{\pi(x)})=0\}$$ is locally constructible in $X$. It is easy to check that $$T_\pi = \pi( T \cap X_0).$$ Since $T \cap X_0$ is locally constructible, it follows then from Chevalley’s theorem ([@EGA], IV.1.8.4) that $T_\pi$ is locally constructible in $Y$.
Let us now show that $T_{{{\mathcal F}}, f,\pi}$ is locally constructible in $Y$. Set $T$ to be the zero locus $Z({{\mathcal F}},f)$ of $f$ in $X$, which is locally constructible in $X$ by \[zero-locus-f\]. Then $T_{{{\mathcal F}}, f,\pi}$ is nothing but the associated subset $T_\pi$ which was shown to be locally constructible in $Y$ in the first part of the proposition.
The formation of $T_{{{\mathcal F}}, f, \pi}$ is compatible with base changes $Y'\to Y$, as our next lemma shows.
\[(i)\] Let $\pi:X\to Y$ be a finitely presented morphism of schemes. Let $q: Y' \to Y$ be any morphism of schemes. Let $X':=X\times_Y Y'$ and $\pi':X'\to Y'$. Let ${{\mathcal F}}$ be a finitely presented ${{\mathcal O}}_X$-module, and fix a section $f\in H^0(X, {{\mathcal F}})$. Let ${{\mathcal F}}':={{\mathcal F}}\otimes_{{{\mathcal O}}_Y} {{\mathcal O}}_{Y'}$ and let $f'$ be the image of $f$ in $H^0(X', {{\mathcal F}}')$. Then $T_{{{\mathcal F}}', f',\pi'}=q^{-1}(T_{{{\mathcal F}}, f,\pi})$.
For any $y'\in Y'$, we have a natural $k(y')$-isomorphism $X'_{y'}\to (X_y)_{k(y')}$. Any generic point $\xi'$ of $X'_{y'}$ maps to a generic point $\xi$ of $X_y$, and any generic point of $X_y$ is the image of a generic point of $X'_{y'}$. Moreover, $f'(\xi')$ is identified with the image of $f(\xi)$ under the natural injection ${{\mathcal F}}_y(\xi) \to {{\mathcal F}}'_{y'}(\xi')={{\mathcal F}}_y(\xi)\otimes k(\xi')$.
\[sig\] Let $X\to Y$ be a finitely presented morphism of schemes, and let ${{\mathcal F}}$ be a finitely presented ${{\mathcal O}}_X$-module. Let $N\ge 1$ and let $f_1, \dots, f_N\in H^0(X, {{\mathcal F}})$. For each $y\in Y$, define $$\Sigma(y):=\left\{ (\alpha_1, \ldots, \alpha_N)\in k(y)^N \ \Big| \
\sum_i \alpha_i f_{i,y} \ \text{\rm vanishes at some generic point of}
\ X_y\right\}.$$ When $X_y = \emptyset$, we set $\Sigma(y):= \emptyset$. The subset $\Sigma(y) $ depends on the data $X\to Y$, ${{\mathcal F}}$, and $\{f_1, \dots, f_N\}$.
\[specialcase\] Consider the special case in \[sig\] where $Y={\operatorname{Spec}}k= \{ y\}$, with $k$ a field. For each generic point $\xi $ of $X=X_y$, consider the $k$-linear map $$k^N \longrightarrow {{\mathcal F}}\otimes k(\xi), \quad (\alpha_1,\dots, \alpha_N) \longmapsto \sum \alpha_i f_i(\xi).$$ The kernel $K(\xi)$ of this map is a linear subspace of $k^N$ and, hence, can be defined by a system of homogeneous polynomials of degree $1$. The same equations define a closed subscheme $T({\xi})$ of $\mathbb A^N_y$. Then the set $\Sigma(y)$ is the union of the sets $K(\xi)$, where the union is taken over all the generic points of $X$, and $\Sigma(y)$ is the subset of $k(y)$-rational points of the closed scheme $T:= \cup_\xi T({\xi})$ of $\mathbb A^N_Y$. This latter statement is generalized to any base $Y$ in our next proposition.
\[constructible-conditions\] Let $X\to Y$, ${{\mathcal F}}$, and $\{f_1, \dots, f_N\} \subset H^0(X, {{\mathcal F}})$, be as in [\[sig\]]{}. Then there exists a locally constructible subset $T$ of $\mathbb A^N_Y$ such that for all $y\in Y$, the set of $k(y)$-rational points of $\mathbb A^N_{k(y)}$ contained in $T_y$ is equal to $\Sigma(y)$. Moreover:
1. The set $T$ satisfies the following natural compatibility with respect to base change. Let $Y' \to Y$ be any morphism of schemes, and denote by $q: {\mathbb A}^N_{Y'} \to {\mathbb A}^N_Y$ the associated morphism. Let $X':=X\times_Y Y'\to Y'$ and let ${{\mathcal F}}':={{\mathcal F}}\otimes_{{{\mathcal O}}_Y} {{\mathcal O}}_{Y'}$. Let $f_1', \dots, f_N'$ be the images of $f_1, \dots, f_N$ in $H^0(X', {{\mathcal F}}')$. Then the constructible set $T'$ associated with the data $X'\to Y'$, ${{\mathcal F}}'$, and $f'_1, \dots, f'_N$, is equal to $q^{-1}(T)$. In particular, for all $y'\in Y'$, the set of $k(y')$-rational points of $\mathbb A^N_{k(y')}$ contained in $(q^{-1}(T))_{y'}$ is equal to the set $\Sigma(y')$ associated with $f_1',\dots, f'_N\in H^0(X', {{\mathcal F}}')$.
2. We have $\dim T \leq \dim Y + \sup_{y \in Y} \dim T_y$ when $Y$ is noetherian. In general for each $y \in Y$, $\dim T_y$ is the maximum of the dimension over $k(y)$ of the kernels of the $k(y)$-linear maps $$k(y)^N \longrightarrow {{\mathcal F}}\otimes k(\xi), \quad
(\alpha_1, \dots, \alpha_N)\longmapsto \sum_i \alpha_i f_i(\xi),$$ for each generic point $\xi$ of $X_y$.
Let $\pi: \mathbb A^N_X\to \mathbb A^N_Y$ be the finitely presented morphism induced by the given morphism $X \to Y$. Let $p: \mathbb A^N_X\to X$ be the natural projection, and consider the finitely presented sheaf $p^*{{\mathcal F}}$ on $\mathbb A^N_X$ induced by ${{\mathcal F}}$.
Write $\mathbb A^N_{\mathbb Z}= {\operatorname{Spec}}{\mathbb Z}[u_1, \dots, u_N]$, and identify $H^0(\mathbb A^N_X, p^*{{\mathcal F}})$ with $H^0(X, {{\mathcal F}})\otimes_\mathbb Z \mathbb Z[u_1, \dots, u_N].$ Using this identification, let $f \in H^0(\mathbb A^N_X, p^*{{\mathcal F}})$ denote the section corresponding to $\sum_{1\le i\le N} f_i\otimes u_i$. Apply now Proposition \[constructible-univ\] to the data $\pi: \mathbb A^N_X\to \mathbb A^N_Y$, $p^*{{\mathcal F}}$, and $f$, to obtain the locally constructible subset $T:=T_{p^*{{\mathcal F}}, f}$ of $\mathbb A^N_Y$.
Fix $y\in Y$, and let $z$ be a $k(y)$-rational point in $ \mathbb A^N_Y$ above $y$. We may write $ z=(\alpha_1, \dots, \alpha_N)\in k(y)^N$. The fiber of $\pi$ above $z$ is isomorphic to $X_y$, and the section $ f_z\in (p^*{{\mathcal F}})_z$ is identified with $\sum_i \alpha_i f_i \in H^0(X_y, {{\mathcal F}}_y)$. Therefore, it follows from the definitions that $z\in T_y$ if and only if $z\in \Sigma(y)$, and the first part of the proposition is proved.
\(a) The compatibility of $T$ with respect to a base change $Y'\to Y$ results from Lemma \[(i)\].
\(b) The inequality on the dimensions follows from \[lem.construct\] (c). By the compatibility described in (a), we are immediately reduced to the case $Y={\operatorname{Spec}}k$ for a field $k$, which is discussed in \[specialcase\].
[Sections in an affine space avoiding pro-constructible subsets]{} \[Constructible subsets\]
The following theorem is an essential part of our method for producing interesting closed subschemes of a scheme $X$ when $X \to S$ is projective and $S$ is affine.
\[globalize\] Let $S={\operatorname{Spec}}R$ be a noetherian affine scheme. Let $T:=T_1\cup \ldots \cup T_m$ be a finite union of pro-constructible subsets of $\mathbb A^N_S$. Suppose that:
1. There exists an open subset $V\subseteq S$ with zero-dimensional complement such that for all $i\le m$, $\dim (T_i\cap \mathbb A^N_{V})<N$ and $(T_i)_s$ is constructible in $\mathbb A^N_{k(s)}$ for all $s\in V$.
2. For all $s\in S$, there exists a $k(s)$-rational point in ${\mathbb A}^N_{k(s)}$ which does not belong to $T_s$.
Then there exists a section $\sigma$ of $\pi: {\mathbb A}^N_S\to S$ such that $\sigma(S) \cap T = \emptyset$.
We proceed by induction on $N$, using Claims (a) and (b) below.
[**Claim (a).**]{} [*There exists $\delta\ge 1$ such that for all $s\in V$, $T_s$ is contained in a hypersurface in $\mathbb A^N_{k(s)}$ of degree at most $ \delta$*]{}.
It is enough to prove the claim for each $T_i$. So to lighten the notation we set $T:=T_i$ in this proof. Thus, by hypothesis, $\dim T\cap \mathbb A^N_V<N$. We start by proving that for each $s \in V$, there exist a positive integer $\delta_s$ and an ind-constructible subset (see \[pro-constr\]) $W_s$ of $V$ containing $s$, such that for each $s' \in W_s$, $T_{s'} $ is contained in a hypersurface of degree $\delta_s$ in ${\mathbb A}^N_{k(s')}$. Indeed, let $s \in V$. As $\dim T_s\le \dim T\cap \mathbb A^N_V<N$, and $T_s$ is constructible, $T_s$ is not dense in ${\mathbb A}^N_{k(s)}$ (Lemma \[lem.construct\](a)). Thus, there exists some polynomial $f_s$ of degree $\delta_s>0$ whose zero locus contains $T_s$. Hence, for some affine open neighborhood $V_s$ of $s$, we can find a polynomial $f \in {{\mathcal O}}_V(V_s)[t_1,\dots, t_N]$ of degree $\delta_s$, lifting $f_s$ and defining a closed subscheme $V(f)$ of ${\mathbb A}^N_{V_s}$.
Let $W_1:= \pi({\mathbb A}^N_{V_s} \setminus V(f))$, which is constructible in $V$ by Chevalley’s Theorem. Let $W_2$ be the complement in $V$ of $ \pi(T \cap ({\mathbb A}^N_{V_s} \setminus V(f)))$, which is ind-constructible in $V$ since $\pi(T \cap ({\mathbb A}^N_{V_s} \setminus V(f)))$ is pro-constructible in $V$ ([@EGA], IV.1.9.5 (vii)). Hence, both $W_1$ and $W_2$ are ind-constructible and contain $s$. The intersection $W_s:= W_1 \cap W_2$ is the desired ind-constructible subset containing $s$. Since $V$ is quasi-compact because it is noetherian, and since each $W_s$ is ind-constructible, it follows from [@EGA], IV.1.9.9, that there exist finitely many points $s_1, \dots, s_n $ of $V$ such that $V = W_{s_1} \cup \ldots \cup W_{s_n}$. We can take $\delta:= \max_i\{ \delta_{s_i} \}$, and Claim (a) is proved.
A proof of the following lemma in the affine case is given in [@Samuel], Proposition 13. We provide here an alternate proof.
\[finiteresidue\] Let S be any scheme. Let $c \in \mathbb N$. Then the subset $\{ s\in S \mid {\operatorname{Card}}(k(s))
\le c\}$ is closed in $S$ and has dimension at most $ 0$. When $S$ is noetherian, this subset is then finite.
It is enough to prove that when $S$ is a scheme over a finite prime field $\mathbb F_p$, and $q$ is a power of $p$, the set $\{ s\in S \mid {\operatorname{Card}}(k(s))=q \}$ is closed of dimension $\le 0$.
Let ${\mathbb F}_q$ be a field with $q$ elements. Then any point $s\in S$ with ${\operatorname{Card}}(k(s))=q$ is the image by the projection $S_{{\mathbb F}_q} \to S$ of a rational point of $S_{{\mathbb F}_q}$. Therefore we can suppose that $S$ is a ${\mathbb F}_q$-scheme and we have to show that $S({\mathbb F}_q)$ is closed of dimension $0$. Let $Z$ be the Zariski closure of $S({\mathbb F}_q)$ in $S$, endowed with the reduced structure. Let $U$ be an affine open subset of $Z$. Let $f\in {{\mathcal O}}_Z(U)$. For any $x\in U({\mathbb F}_q)$, $(f^q-f)(x)=0$ in $k(x)$, hence $x\in V(f^q-f)$. As $U({\mathbb F}_q)$ is dense in $U$ and $U$ is reduced, we have $f^q-f=0$. For any irreducible component $\Gamma$ of $U$, this identity then holds on ${{\mathcal O}}(\Gamma)$, so $\Gamma$ is just a rational point. Hence $U=U({\mathbb F}_q)$ and $\dim U=0$. Consequently, $Z=S({\mathbb F}_q)$ is closed and has dimension $0$.
The key to the proof of Theorem \[globalize\] is the following assertion:
[**Claim (b).**]{}
*Suppose $N\ge 1$. Then there exist $t:=t_1 +a_1 \in R[t_1,\dots, t_N]$ with $a_1\in R$, and an open subset $U\subseteq S$ with zero-dimensional complement, such that $H:=V(t)$ is $S$-isomorphic to $\mathbb A^{N-1}_S$ and the pro-constructible subsets $T_1\cap H$, $\dots$, $T_m\cap H$ of $H$ satisfy:*
1. For all $i\le m$, $\dim (T_i\cap H_U)<N-1$, and $(T_i \cap H)_s$ is constructible in $H_s$ for all $s\in U$.
2. For all $s\in S$, there exists a $k(s)$-rational point in $H_s$ which does not belong to $T_s \cap H_s$.
Using Claim (b), we conclude the proof of Theorem \[globalize\] as follows. First, note that when $N=0$, Condition \[globalize\] (2) implies that $T=\emptyset$ and the theorem trivially holds true. When $N\ge 1$, we apply Claim (b) repeatedly to obtain a sequence of closed sets $$\mathbb A^N_S \supset V(t_1+a_1) \supset \dots \supset
V(t_1+a_1, t_2+a_2, \dots, t_N+a_N).$$ The latter set is the image of the desired section, as we saw in the case $N=0$.
[*Proof of*]{} Claim (b): Let $\{\xi_1,\dots, \xi_{\rho}\}$ be the set of generic points of all the irreducible components of the pro-constructible sets $T_i\cap {\mathbb A}^N_V$, $i=1,\dots, m$ (see \[lem.construct\] (d) for the existence of generic points). Upon renumbering these points if necessary, we can assume that for some $r\le \rho$, the image of $\xi_i$ under $\pi: {\mathbb A}^N_S\to S$ has finite residue field if and only if $i> r$. Let $\delta>0$. Let $Z$ be the union of $S\setminus V$ with $\{\pi(\xi_{r+1}),\ldots, \pi(\xi_{\rho})\}$ and with the finite subset of the closed points $s$ of $S$ satisfying ${\operatorname{Card}}(k(s)) \le \delta$ (that this set is finite follows from \[finiteresidue\]). We will later set $\delta$ appropriately to be able to use Claim (a). For each $s \in Z$, we can use \[globalize\] (2) and fix a $k(s)$-rational point $x_s \in {\mathbb A}_{k(s)}^N \setminus {T_s}$.
We now construct a closed subset $V(t) \subset \mathbb A^N_S$ which contains $x_s$ for all $s\in Z$, and does not contain any $\xi_i$ with $i\le r$. Since every point of $Z$ is closed in $S$, the Chinese Remainder Theorem implies that the canonical map $R\to \prod_{s\in Z} k(s)$ is surjective. Let $a \in R$ be such that $a \equiv t_1(x_s)$ in $k(s)$, for all $s\in Z$. Replacing $t_1$ by $t_1-a$, we can assume that $t_1(x_s)=0$ for all $s\in Z$. Let ${{\mathfrak p}}_{j} \subset R[t_1,\dots, t_N]$ be the prime ideal corresponding to $\xi_j$. Let ${{\mathfrak m}}_s \subset R$ denote the maximal ideal of $R$ corresponding to $s \in Z$. Let $I :=\cap_{s \in Z} {{\mathfrak m}}_s$, and in case $Z=\emptyset$, we let $I:=R$. For $t \in R[t_1,\dots, t_N]$, let $I+t:=\{a+t \mid a\in I\}$. [*We claim that:*]{} $$I+t_1 \not\subseteq \cup_{1\le j\le r}{{\mathfrak p}}_j.$$ Indeed, the intersection $(I+t_1)\cap{{\mathfrak p}}_j$ is either empty, or contains $a_j + t_1$ for some $a_j\in I$. In the latter case, $(I+t_1)\cap{{\mathfrak p}}_j=t_1+a_j+ ({{\mathfrak p}}_j\cap I)$. If $ I+t_1 \subseteq \cup_{1\le j\le r}{{\mathfrak p}}_j$, then every $t_1+a$ with $a \in I$ belongs to some $t_1+a_j+ ({{\mathfrak p}}_j\cap I)$. Let ${{\mathfrak q}}_j:=R\cap {{\mathfrak p}}_j$. It follows that $$I\subseteq \cup_j (a_j+{{\mathfrak q}}_j)$$ where the union runs over a subset of $\{1, \ldots, r\}$. Since the domains $R/{{\mathfrak q}}_j$ are all infinite when $j \leq r$, Lemma \[trans-pal-2\] below implies that $I$ is contained in some ${{\mathfrak q}}_{j_0}$ for $1 \leq j_0\le r$. As $I =\cap_{s \in Z} {{\mathfrak m}}_s$, we find that ${{\mathfrak q}}_{j_0}= {{\mathfrak m}}_s$ for some $s\in Z$. This is a contradiction, since for $j\leq r$, $\pi(\xi_j)$ does not belong to $Z$ because the residue field of $\pi(\xi_j)$ is infinite and $\pi(\xi_j) \notin S\setminus V$. This proves our claim.
Now that the claim is proved, we can choose $t\in (I+t_1)\setminus \cup_{1\le j\le r}{{\mathfrak p}}_j$. Clearly, the closed subset $H:=V(t) \subset \mathbb A^N_S$ does not contain any $\xi_i$ with $i\le r$. Since $t$ has the form $t=t_1+a_1$ for some $a_1 \in I = \cap_{s \in Z} {{\mathfrak m}}_s$, we find that $V(t)$ contains $x_s$ for all $s \in Z$. Let $U:=S \setminus Z$. The complement of $U$ in $S$ is a finite set of closed points of $S$. It is clear that $H:=V(t)$ is $S$-isomorphic to $ \mathbb A^{N-1}_S$, and that for each $i$, the fibers of $T_i\cap H \to S$ are constructible.
Let us now prove (i), i.e., that $\dim(T_i\cap H_{U}) < N-1$ for all $i\le m$. Let $\Gamma$ be an irreducible component of some $T_i\cap \mathbb A^N_V$, with generic point $ \xi_j$ for some $j$. If $j>r$, then $\pi(\xi_j)\in Z$ and $\Gamma\cap
\mathbb A^N_U=\emptyset$. Suppose now that $j\le r$. Then $\Gamma\cap \mathbb A^N_U$ is non-empty and open in $\Gamma$, hence irreducible. By construction, $H $ does not contain $\xi_j$ since $j \leq r$. So $\Gamma\cap H_U$ is a proper closed subset of the irreducible space $\Gamma\cap \mathbb A^N_U$. Thus $$\dim (\Gamma\cap H_U)< \dim (\Gamma\cap \mathbb A^N_U) \le \dim \Gamma<N.$$ As $T_i\cap H_U$ is the finite union of its various closed subsets $\Gamma\cap H_U$, this implies that $\dim (T_i\cap H_U)< N-1$.
Let us now prove (ii), i.e., that for all $s \in S$, $H_s$ contains a $k(s)$-rational point that does not belong to $T_s$. When $s\in Z$, $H$ contains the $k(s)$-rational point $x_s$ and this point does not belong to $T_s$. Let now $s \notin Z$. Then $|k(s)|\geq \delta +1$ by construction. Choose now $\delta$ so that the conclusion of Claim (a) holds: for all $s\in V$, $T_s$ is contained in a hypersurface in $\mathbb A^N_{k(s)}$ of degree at most $ \delta$. Then, since $t$ has degree $1$, we find that $H_s\cap T$ is contained in a hypersurface $V(f)$ of $H_s$ with $\deg(f)\leq \delta$.
We conclude that $H_s$ contains a $k(s)$-rational point that does not belong to $T_s$ using the following claim: [*Assume that $k$ is either an infinite field or that $|k|=q \ge \delta+1$. Let $f\in k[T_1,\ldots, T_\ell]$ with $\deg(f) \le \delta$, $f\neq 0$. Then $V(f)(k)\varsubsetneq \mathbb A^\ell(k)$.*]{} When $k$ is a finite field, we use the bound $|V(f)(k)| \leq \delta q^{\ell-1} + (q^{\ell-1} -1)/(q-1)<q^\ell$ found in [@Ser2]. When $k$ is infinite, we can use induction on $\ell$ to prove the claim.
Our next lemma follows from [@McAdam], Theorem 5. We provide here a more direct proof using the earlier reference [@Neumann].
\[trans-pal-2\] Let $R$ be a commutative ring, and let ${{\mathfrak q}}_1,\ldots, {{\mathfrak q}}_r$ be (not necessarily distinct) prime ideals of $R$ with infinite quotients $R/{{\mathfrak q}}_i$ for all $i=1,
\dots, r$. Let $I$ be an ideal of $R$ and suppose that there exist $a_1,\ldots, a_r\in R$ such that $$I\subseteq \cup_{1\le i\le r} (a_i+{{\mathfrak q}}_i).$$ Then $I$ is contained in the union of those $a_i+{{\mathfrak q}}_i$ with $I\subseteq {{\mathfrak q}}_i$. In particular, $I$ is contained in at least one ${{\mathfrak q}}_i$.
We have $I=\cup_{i} ((a_i+{{\mathfrak q}}_i)\cap I)$. If $(a_i+{{\mathfrak q}}_i)\cap I\ne\emptyset$, then it is equal to $\alpha_i + ({{\mathfrak q}}_i\cap I)$ for some $\alpha_i\in I$. Hence $$I=\cup_i (\alpha_i + ({{\mathfrak q}}_i\cap I))$$ where the union runs on part of $\{1,\ldots, r\}$. By [@Neumann], 4.4, $I$ is the union of those $\alpha_i+({{\mathfrak q}}_i\cap I)$ with $I/({{\mathfrak q}}_i\cap I)$ finite. For any such $i$, the ideal $(I+{{\mathfrak q}}_i)/{{\mathfrak q}}_i$ of $R/{{\mathfrak q}}_i$ is finite and, hence, equal to $(0)$ because $R/{{\mathfrak q}}_i$ is an infinite domain.
One can show that the conclusion of Theorem \[globalize\] holds without assuming in \[globalize\] (1) that $T_s$ is constructible in ${\mathbb A}_{k(s)}^N$ for all $s \in V$. Since we will not need this statement, let us only note that [*when $S$ has only finitely many points with finite residue field, then the conclusion of Theorem [\[globalize\]]{} holds if in [\[globalize\] (1)]{} we remove the hypothesis that $T_s$ is constructible for all $s \in V$.*]{} Indeed, with this hypothesis, we do not need to use Claim (a). First shrink $V$ so that $k(s)$ is infinite for all $s\in V$, and then proceed to construct the closed subset $H=V(t)$ discussed in Claim (b). The use of Claim (a) in the proof of (ii) in Claim (b) can be avoided using our next lemma.
Let $k$ be an infinite field, and let $V \subset {\mathbb A}^N_k$ be a closed subset. The property that [*if $\dim(V)<N$, then $V(k) \neq {\mathbb A}^N_k(k)$*]{}, can be generalized as follows.
\[proconstructible.gen\] Let $k$ be an infinite field. Let $T \subset {\mathbb A}^N_k$ be a pro-constructible subset with $\dim(T) <N$. Then $T$ does not contain all $k$-rational points of ${\mathbb A}^N_k $.
Assume that $T$ contains all $k$-rational points of ${\mathbb A}^N_k $. We claim first that $T$ is irreducible. Indeed, if $T=(V(f)\cap T) \cup
(V(g)\cap T)$, then $V(fg)=V(f)\cup V(g)$ contains all $k$-rational points of ${\mathbb A}^N_k$. Thus $V(fg)= {\mathbb A}^N_k$ and either $V(f) \cap T = T$ or $V(g) \cap T = T$. Since $T$ is irreducible, it has a generic point $\xi$ (see \[lem.construct\] (d)), and the closure $F$ of $\xi$ in ${\mathbb A}^N_k$ contains all $k$-rational points of ${\mathbb A}^N_k$. Hence, $F= {\mathbb A}^N_k$, so that $T$ then contains the generic point of ${\mathbb A}^N_k$. Consider now an increasing sequence of closed linear subspaces $F_0 \subset F_1 \subset \dots \subset F_{N}$ contained in ${\mathbb A}^N_k$, with $F_i \cong {\mathbb A}^i_k$. Then $T \cap F_i$ contains all $k$-rational points of $F_i$ by hypothesis, and the discussion above shows that it contains then the generic point of $F_i$. It follows that $\dim(T) =N$.
The hypothesis in \[globalize\] (1) on the dimension of $T$ is needed. Indeed, let $S= {\operatorname{Spec}}{\mathbb Z}$, and $N=1$. Consider the closed subset $V(t^3-t)$ of ${\operatorname{Spec}}{\mathbb Z}[t] = {\mathbb A}^1_{\mathbb Z}$. Let $T$ be the constructible subset of ${\mathbb A}^1_{\mathbb Z}$ obtained by removing from $V(t^3-t)$ the maximal ideals $(2,t-1)$ and $(3,t-1)$. Then, for all $s \in S$, the fiber $T_s$ is distinct from ${\mathbb A}^1_{k(s)}(k(s))$, and $\dim T_s =0$. However, $\dim T=1$, and we note now that there exists no section of ${\mathbb A}^1_{\mathbb Z}$ disjoint from $T$. Indeed, let $V(t-a)$ be a section. If it is disjoint from $T$, then $a \neq 0,1,-1$, and $6 \mid a-1$. So there exists a prime $p>3$ with $p \mid a$, and $V(t-a)$ meets $T$ at the point $(p, t)$.
For a more geometric example, let $k$ be any infinite field. Let $S={\operatorname{Spec}}k[u]$ and ${\mathbb A}^1_S= {\operatorname{Spec}}k[u,t]$. When $T:=V(t^2-u) \subset {\mathbb A}^1_S $, then ${\mathbb A}^1_S\setminus T$ does not contain any section $V(t-g(u))$ of ${\mathbb A}^1_S$. Indeed, otherwise $(t^2-u, t-g(u))=(1)$, and $g(u)^2-u$ would be an element of $k^*$.
[Existence of hypersurfaces]{} \[Main application\]
Let us start by introducing the terminology needed to state the main results of this section.
\[zerolocusinvertible\] Let $X$ be any scheme. A global section $f$ of an invertible sheaf ${\mathcal L}$ on $X$ defines a closed subset $H_f$ of $X$, consisting of all points $x \in X$ where the stalk $f_x$ does not generate ${\mathcal L}_x$. Since ${{\mathcal O}}_X f\subseteq {\mathcal L}$, the ideal sheaf ${\mathcal I}:=
({{\mathcal O}}_X f)\otimes {\mathcal L}^{-1}$ endows $H_f$ with the structure of a closed subscheme of $X$. When $X$ is noetherian and $H_f \neq \emptyset$, it follows from Krull’s Principal Ideal Theorem that any irreducible component $\Gamma$ of $H_f$ has codimension at most $1$ in $X$.
Assume now that $X \to {\operatorname{Spec}}R$ is a projective morphism, and write $X={\operatorname{Proj}}A$, where $A$ is the quotient of a polynomial ring $R[T_0,\dots, T_N]$ by a homogeneous ideal $I$. Let ${{\mathcal O}}_X(1)$ denote the very ample sheaf arising from this presentation of $X$. Let $f\in A$ be a homogeneous element of degree $n$. Then $f$ can be identified with a global section $f\in H^0(X, {{\mathcal O}}_X(n))$, and $H_f$ is the closed subscheme $V_+(f)$ of $X$ defined by the homogeneous ideal $fA$. When $X \to S$ is a quasi-projective morphism and $f$ is a global section of a very ample invertible sheaf ${\mathcal L}$ relative to $X \to S$, we may also sometimes denote the closed subset $H_f$ of $X$ by $V_+(f)$.
Let now $S$ be any affine scheme and $X \to S$ any morphism. We call the closed subscheme $H_f$ of $X$ a *hypersurface* (relative to $X \to S$) when no irreducible component of *positive dimension* of $X_s$ is contained in $H_f$, for all $s\in S$. If, moreover, the ideal sheaf ${\mathcal I}$ is invertible, we say that the hypersurface $H_f$ is *locally principal*. Note that in this case, $H_f$ is the support of an effective Cartier divisor on $ X$. Hypersurfaces satisfy the following elementary properties.
\[hypersurfaces-properties\] Let $S$ be affine. Let $X \to S$ be a finitely presented morphism. Let ${\mathcal L}$ be an invertible sheaf on $X$ and let $f \in H^0(X,
{\mathcal L})$ be such that $H:= H_f$ is a hypersurface on $X$ relative to $X \to S$.
1. If $\dim X_s\ge 1$, then $\dim H_s\leq \dim X_s -1$. If, moreover, $X \to S$ is projective, ${\mathcal L}$ is ample, and $H \neq \emptyset$, then $H_s$ meets every irreducible component of positive dimension of $X_s$, and in particular $\dim H_s=\dim X_s -1$.
2. The morphism $H\to S$ is finitely presented.
3. Assume that $X\to S$ is flat of finite presentation. Then $H$ is locally principal and flat over $S$ if and only if for all $s\in S$, $H$ does not contain any associated point of $X_s$.
4. Assume $S$ noetherian. If $H$ does not contain any associated point of $X$, then $H$ is locally principal.
\(1) Recall that by convention, if $H_s$ is empty, then $\dim H_s <0$, and the inequality is satisfied. Assume now that $H_s$ is not empty. By hypothesis, $H_s$ does not contain any irreducible component of $X_s$ of positive dimension. Since $H_s$ is locally defined by one equation, we obtain that $\dim H_s\leq \dim X_s-1$. The strict inequality may occur for instance in case $\dim X_s \geq 2$, and $H_s$ does not meet any component of $X_s$ of maximal dimension.
Consider now the open set $X_f:= X \setminus H$. Under our additional hypotheses, for any $s\in S$, $X_f \cap X_s=(X_s)_{f_s}$ is affine and, thus, can only contain irreducible components of dimension $0$ of the projective scheme $X_s$.
\(2) Results from the fact that $H$ is locally defined by a single equation in $X$.
\(3) See [@EGA], IV.11.3.8, c) $\Leftrightarrow$ a). Each fiber $X_s$ is noetherian. Use the fact that in a noetherian ring, an element is regular if and only if it is not contained in any associated prime.
\(4) The property is local on $X$, so we can suppose $X={\operatorname{Spec}}A$ is affine and $\mathcal L={{\mathcal O}}_X\cdot e$ is free. So $f=he$ for some $h\in A$. The hypothesis $H\cap {\operatorname{Ass}}(X)=\emptyset$ implies that $h$ is a regular element of $A$. So the ideal sheaf $\mathcal I=({{\mathcal O}}_X f)\otimes
\mathcal L^{-1}$ is invertible.
We can now state the main results of this section.
\[exist-hyp\] Let $S$ be an affine noetherian scheme of finite dimension, and let $X\to S$ be a quasi-projective morphism with a given very ample invertible sheaf ${{\mathcal O}}_X(1)$.
1. Let $C$ be a closed subscheme of $X$;
2. Let $F_1, \dots, F_m$ be locally closed subsets[^10] of $X$ such that for all $s\in S$ and for all $i\le m$, $C_s$ does not contain any irreducible component of positive dimension of $(F_i)_s$;
3. Let $A$ be a finite subset of $X$ such that $A\cap C=\emptyset$.
Then there exists $n_0>0$ such that for all $n\ge n_0$, there exists a global section $f$ of ${{\mathcal O}}_X(n)$ such that:
1. $C$ is a closed subscheme of $H_f$,
2. for all $s \in S$ and for all $i\le m$, $H_f$ does not contain any irreducible component of positive dimension of $F_i\cap X_s$, and
3. $H_f\cap A=\emptyset$. Moreover,
4. if, for all $s \in S$, $C$ does not contain any irreducible component of positive dimension of $X_s$, then there exists $f$ as above such that $H_f$ is a hypersurface relative to $X \to S$. If in addition $C\cap {\operatorname{Ass}}(X)=\emptyset$, then there exists $f$ as above such that $H_f$ is a locally principal hypersurface.
We will first give a complete proof of Theorem \[exist-hyp\] in the case where $X \to S$ is projective in \[Proofreductiondimension2\], after a series of technical lemmas. The proof of \[exist-hyp\] when $X\to S $ is only assumed to be quasi-projective is given in \[quasiproj\]. Theorem \[exist-hyp\] will be generalized to the case where $S$ is not noetherian in \[bertini-type-0\].
Theorem \[pro.reductiondimension2\] below is the key to reducing the proof of the Moving Lemma \[mv-1-cycle-local\] (a) to the case of relative dimension 1. This theorem is stated in a slightly different form in the introduction, and we note in \[lem.remcodim\] (3) that the two versions are compatible.
\[pro.reductiondimension2\] Let $S$ be an affine noetherian scheme of finite dimension, and let $X\to S$ be a quasi-projective morphism with a given very ample invertible sheaf ${{\mathcal O}}_X(1)$ relative to $X \to S$. Assume that the hypotheses [(i)]{}, [(ii)]{}, and [(iii)]{} in [\[exist-hyp\]]{} hold. Suppose further that
1. $C\to S$ is finite,
2. $C\to X$ is a regular immersion and $C$ has pure codimension[^11] $d> \dim S$ in $X$, and
3. for all $s\in S$, ${{\operatorname{codim}}}(C_s, X_s)\ge d$.
Then there exists $n_0>0$ such that for all $n\ge n_0$ there exists a global section $f$ of ${{\mathcal O}}_X(n)$ such that $H_f$ satisfies [(1)]{}, [(2)]{}, and [(3)]{} in [\[exist-hyp\]]{}, and such that $H_f$ is a locally principal hypersurface, $C \to H_f$ is a regular immersion, and $C$ has pure codimension $d-1$ in $H_f$.
Suppose now that $\dim S= 1$. Then there exists a closed subscheme $Y$ of $X$ such that $C$ is a closed subscheme of $Y$ defined by an invertible sheaf of ideals of $Y$ (i.e., $C$ corresponds to an effective Cartier divisor on $Y$). Moreover, for all $s\in S$ and all $i\le m$, any irreducible component $\Gamma$ of $F_i \cap X_s$ is such that $\dim(\Gamma \cap Y_s) \leq \max(\dim(\Gamma)-(d-1),0)$. In particular, if $(F_i)_s$ has positive codimension in $X_s$ in a neighborhood of $C_s$, then $F_i \cap Y_s$ has dimension at most $0$ in a neighborhood of $C_s$.
The main part of Theorem \[pro.reductiondimension2\] is given a complete proof in the case where $X \to S$ is projective in \[Proofreductiondimension2-a\]. The proof when $X\to S $ is only assumed to be quasi-projective is given in \[quasiproj\]. We prove here the end of the statement of Theorem \[pro.reductiondimension2\], where we assume that $\dim S = 1$. Apply Theorem \[pro.reductiondimension2\] $(d-1)$ times, starting with $X':=V_+(f)$, $F'_i:=F_i\cap V_+(f)$, and $C \subseteq X'$. Note that at each step Condition \[pro.reductiondimension2\] (c) holds by Lemma \[lem.remcodim\] (2).
\[lem.remcodim\] Let $S$ be a noetherian scheme, and let $\pi: X\to S$ be a morphism of finite type. Let $C$ be a closed subset of $X$, with $C \to S$ finite.
1. Let $s \in S$ be such that $C_s$ is not empty. Then the following are equivalent:
1. ${{\operatorname{codim}}}(C_s, X_s)\ge d$.
2. Every point $x$ of $C_s$ is contained in an irreducible component of $X_s$ of dimension at least equal to $d$ (equivalently, $\dim_x X_s\ge d$ for all $x \in C_s$)[^12]
2. Let ${\mathcal L}$ be a line bundle on $X$ with a global section $f$ defining a closed subscheme $H_f$ which contains $C$. Let $s\in S$. Suppose that ${{\operatorname{codim}}}(C_s, X_s)\ge d$. Then ${{\operatorname{codim}}}(C_s, (H_f)_s)\ge d-1$.
3. Assume that $C$ has codimension $d\ge 0$ in $X$ and that each irreducible component of $C$ dominates an irreducible component of $S$ (e.g., when $C\to S$ is flat). Then for all $s\in S$, ${{\operatorname{codim}}}(C_s, X_s)\ge d$. In particular, if $X/S $ and $C$ satisfy the hypotheses of the version of [\[pro.reductiondimension2\]]{} given in the introduction, then they satisfy the hypotheses of Theorem [\[pro.reductiondimension2\]]{} as stated above.
\(1) This is immediate since, $X_s$ being of finite type over $k(s)$, $C_s$ is the union of finitely many closed points of $X_s$.
\(2) We can suppose $C_s$ is not empty. Let $x \in C_s$. Then $x$ is contained in an irreducible component $\Gamma'$ of $X_s$ of dimension at least equal to $d$. Consider an irreducible component $\Gamma $ of $\Gamma' \cap (H_f)_s$ which contains $x$. Since $\Gamma' \cap (H_f)_s$ is defined in $\Gamma'$ by a single equation, we find that $\dim(\Gamma) \ge \dim(\Gamma')- 1 \ge d - 1$, as desired.
\(3) Let $\xi$ be a generic point of $C$. By hypothesis, $\pi(\xi)$ is a generic point of $S$ and $\xi$ is closed in $X_{\pi(\xi)}$. So $$\dim_{\xi} X_{\pi(\xi)}=\dim {{\mathcal O}}_{X_{\pi(\xi)}, \xi}=\dim {{\mathcal O}}_{X, \xi}\ge{{\operatorname{codim}}}_{\xi}(C, X)\ge d.$$ The set $\{ x\in X | \dim_x X_{\pi(x)} \ge d\}$ is closed ([@EGA], IV.13.1.3). Since this set contains the generic points of $C$, it contains $C$. Hence, when $C_s $ is not empty, ${{\operatorname{codim}}}(C_s, X_s)\ge d$ by (1)(b). When $C_s=\emptyset$, ${{\operatorname{codim}}}(C_s,X_s)=+\infty$ by definition and the statement of (3) also holds.
In the version of [\[pro.reductiondimension2\]]{} given in the introduction, we assume that $C$ is irreducible, that $C \to S$ is finite and surjective, and that $C$ has codimension $d > \dim S$ in $X$. It follows then from (3) that (c) in Theorem [\[pro.reductiondimension2\]]{} is automatically satisfied.
\[emp.notation\] \[emp.proofpro.reductiondimension2\] We fix here some notation needed in the proofs of \[exist-hyp\] and \[pro.reductiondimension2\]. Let $S = {\operatorname{Spec}}R $ be a noetherian affine scheme. Consider a projective morphism $X\to S$. Fix a very ample sheaf ${{\mathcal O}}_X(1)$ on $X$ relative to $S$. As usual, if ${{\mathcal F}}$ is any quasi-coherent sheaf on $X$ and $s \in S$, let ${{\mathcal F}}_s$ denote the pull-back of ${{\mathcal F}}$ to the fiber $X_s$ and, if $x\in X$, ${{\mathcal F}}(x):={{\mathcal F}}_x\otimes k(x)$ (see \[zero.locus\]).
Let $C \subseteq X $ be a closed subscheme defined by an ideal sheaf ${{\mathcal J}}$. For $n\ge 1$, set ${{\mathcal J}}(n):= {{\mathcal J}}\otimes {{\mathcal O}}_X(n)$, and for $s \in S$, let $ {{\mathcal J}}_s(n):= {{\mathcal J}}_s \otimes {{\mathcal O}}_{X_s}(n)={{\mathcal J}}(n)_s$. Let ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s$ denote the image of ${{\mathcal J}}_s\to {{\mathcal O}}_{X_s}$. When $x \in C \cap X_s$, we note the following natural isomorphisms of $k(x)$-vector spaces: $$({{\mathcal J}}(n)|_C)_s(x) \longrightarrow {{\mathcal J}}_s(n)/{{\mathcal J}}_s^2(n) \otimes k(x) \longrightarrow {{\mathcal J}}_s(n) \otimes k(x) \longrightarrow {{\mathcal J}}(n) \otimes k(x)$$ and $${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)/{ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s^2(n) \otimes k(x) \longrightarrow { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n) \otimes k(x).$$ To prove Theorem \[exist-hyp\] and Theorem \[pro.reductiondimension2\], we will show the existence of $f \in H^0(X, {{\mathcal J}}(n))$, for all $n$ sufficiently large, such that the associated closed subscheme $ H_f \subset X$ satisfies the conclusions of the theorems. To enable us to use the results of the previous section to produce the desired $f$, we define the following sets.
Let $n$ be big enough such that ${{\mathcal J}}(n)$ is generated by its global sections. Fix a system of generators $f_1, \dots, f_N$ of $H^0(X, {{\mathcal J}}(n))$. Let $s\in S$. Denote by $\overline{f}_{i,s}$ the image of $f_i$ in $H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n))$. Let $F$ be a locally closed subset of $X$.
> $\bullet$ Let $\Sigma_F(s)$ denote the set of $(\alpha_1,\ldots, \alpha_N)
> \in k(s)^N$ such that the closed subset $V_+(\sum_{i=1}^N \alpha_i \overline{f}_{i,s})$ in $X_s$, defined by the global section $\sum_{i=1}^N \alpha_i \overline{f}_{i,s}$ of ${{\mathcal O}}_{X_s}(n)$, contains at least one irreducible component of $F_s$ of positive dimension.
For the purpose of \[pro.reductiondimension2\], we will also consider the following set.
> $\bullet$ Let $\Sigma_C(s)$ denote the set of $(\alpha_1,\ldots, \alpha_N)
> \in k(s)^N$ for which there exists $x\in C \cap X_s$ such that the image of $\sum_{i=1}^N \alpha_i (f_{i}|_{X_s})\in H^0(X_s, {{\mathcal J}}_s(n))$ vanishes in ${{\mathcal J}}_s(n) \otimes k(x)$.
To lighten the notation, we will not always explicitly use symbols to make it clear that indeed the sets $\Sigma_C(s)$ and $\Sigma_F(s)$ depend on $n$ and on $f_1, \dots, f_N$. We will use the fact that if $f \in H^0(X, {{\mathcal J}}(n))$ and $\overline{f}_{s}$ is its image in ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)$, then $V_+(f) \cap X_s = V_+(\overline{f}_{s})$.
\[remove-isolated-pts\] Let $S$ be an affine noetherian scheme, and let $X\to S$ be a morphism of finite type. Let $F$ be a locally closed subset of $X$. Let ${\bf F}$ be the union of the irreducible components of positive dimension of $F_s$, when $s$ runs over all points of $S$. Then ${\bf F}$ is closed in $F$.
Assume now that $X\to S$, $n$, ${{\mathcal J}}(n)$, and $\{f_1,\dots, f_N\}$ are as above in [\[emp.notation\]]{}. Then
1. $\Sigma_{F}(s)=\Sigma_{\bf{F}}(s)$ for all $s \in S$.
2. There exists a natural constructible subset $T_F$ of $\mathbb A^N_S$ such that for all $s\in S $, $\Sigma_F(s)$ is exactly the set of $k(s)$-rational points of $\mathbb A^N_{k(s)}$ contained in $(T_F)_s$,
Endow $F$ with the structure of a reduced subscheme of $X$ and consider the induced morphism $g: F \to S$. Then the set of $x \in F$ such that $x$ is isolated in $g^{-1}(g(x))$ is open in $F$ ([@EGA], IV.13.1.4). Thus, ${\bf F}$ is closed in $F$.
\(1) By construction, for all $s\in S$, $F_s$ and ${\bf F}_s$ have the same irreducible components of positive dimension, so $\Sigma_F(s)=\Sigma_{\bf F}(s)$ for all $s \in
S$.
\(2) By (1) we can replace $F$ by ${\bf F}$ and suppose that for all $s\in S$, $F_s$ contains no isolated point. Endow $F$ with the structure of a reduced subscheme of $X$. Let ${{\mathcal O}}_F(1):={{\mathcal O}}_X(1)|_F$. Consider the following data: the morphism of finite type $F \to S$, the sheaf $ {{\mathcal O}}_F(n)$, and the sections $h_1,\dots, h_N$ in $H^0(F,
{{\mathcal O}}_F(n))$, with $h_i:= f_i|_F$. We associate to this data, for each $s \in S$, the subset $\Sigma(s) $ as in \[sig\]. We claim that for each $s \in S$, we have $\Sigma_F(s)=\Sigma(s)$. For convenience, recall that $$\begin{array}{c}
\Sigma(s)=\left\{ (\alpha_1, \ldots, \alpha_N)\in k(s)^N \ \Big| \
\sum_{i=1}^N \alpha_i h_{i,s} \ \text{\rm vanishes at some generic point of}
\ F_s\right\}.
\end{array}$$ Let $f\in H^0(X, {{\mathcal J}}(n))\subseteq H^0(X, {{\mathcal O}}_X(n))$ and let $h=f|_F\in H^0(F, {{\mathcal O}}_F(n))$. Recall that $\bar{f}_s$ denotes the image of $f_s$ under the natural map ${{\mathcal J}}_s(n) \to {{\mathcal O}}_{X_s}(n)$. Thus, $\bar{f}_s$ is nothing but the image of $f\in H^0(X, {{\mathcal O}}_X(n))$ under the natural map $H^0(X, {{\mathcal O}}_X(n)) \to H^0(X_s, {{\mathcal O}}_{X_s}(n))$. For any $s\in S$ and for any $x\in F_s$, we have $$x\in V_+(\bar{f}_s) \Longleftrightarrow \bar{f}_s(x)=0\in {{\mathcal O}}_X(n)\otimes k(x)
\Longleftrightarrow h_s(x)=0 \in{{\mathcal O}}_F(n)\otimes k(x).$$ Since we are assuming that $F_s$ does not have any irreducible component of dimension $0$, $\Sigma_F(s)$ is equal to $$\begin{array}{c}
\left\{ (\alpha_1, \ldots, \alpha_N)\in k(s)^N \ \Big| \
\sum_{i=1}^N \alpha_i \overline{f}_{i,s} \ \text{\rm vanishes at some generic point of } F_s \right\}.
\end{array}$$ Therefore, $\Sigma_F(s)=\Sigma(s)$. We can thus apply Proposition \[constructible-conditions\] to the above data $F \to S$, $ {{\mathcal O}}_F(n)$, and the sections $h_1,\dots, h_N$, to obtain a natural constructible subset $T_F$ of $\mathbb A^N_S$ such that for all $s\in S$, $\Sigma_F(s)$ is exactly the set of $k(s)$-rational points of $\mathbb A^N_{k(s)}$ contained in $(T_F)_s$.
Our goal now is to bound the dimension of $(T_F)_s$ so that Theorem \[globalize\] can be used to produce the desired $f \in H^0(X, {{\mathcal J}}(n))$. Let $V/k$ be a projective variety over a field $k$, endowed with a very ample invertible sheaf ${{\mathcal O}}_V(1)$. Recall that the [*Hilbert polynomial*]{} $P_V(t)\in \mathbb Q[t]$ is the unique polynomial such that $P_V(n)=\chi({{\mathcal O}}_V(n))$ for all integers $n$ (where $\chi({{\mathcal G}})$ denotes as usual the Euler-Poincaré characteristic of a coherent sheaf ${{\mathcal G}}$). A finiteness result for the Hilbert polynomials of the fibers of a projective morphism, needed in the final step of the proof of our next lemma, is recalled in \[Mumford\].
\[cor.constructible2\] Let $S={\operatorname{Spec}}R$ be an affine noetherian scheme and let $X\to S$ be a projective morphism. Let ${{\mathcal O}}_X(1)$ be a very ample invertible sheaf relative to $X \to S$. Let $C$ be a closed subscheme of $X$ with ideal sheaf ${{\mathcal J}}$, and let $F$ be a locally closed subset of $X$. Assume that for all $s \in S$, no irreducible component of $F_s$ of positive dimension is contained in $C_s$.
Let $c\in {\mathbb N}$. Then there exists $n_0\in {\mathbb N}$ such that for all $n\ge n_0$ and for any choice $\{f_1,\ldots, f_N\}$ of generators of $H^0(X, {{\mathcal J}}(n))$, the constructible subset $T_F\subseteq \mathbb A^N_S$ introduced in [\[remove-isolated-pts\] (2)]{} satisfies $\dim (T_F)_s \leq N-c$ for all $s \in S$.
Lemma \[remove-isolated-pts\] (1) shows that we can suppose that the locally closed subset $F$ is such that for all $s \in S$, $F_s$ has no isolated point. We now further reduce to the case where $F$ is open and dense in $X$.
Let $Z$ be the Zariski closure of $F$ in $X$. Then $F$ is open and dense in $Z$. Endow $Z$ with the induced structure of reduced closed subscheme. Denote by ${{\mathcal J}}{{\mathcal O}}_Z$ the image of ${{\mathcal J}}$ under the natural homomorphism ${{\mathcal O}}_X \to {{\mathcal O}}_Z$. This sheaf is the sheaf of ideals associated with the image of the closed immersion $C \times_X Z \to Z$. The morphism of ${{\mathcal O}}_X$-modules ${{\mathcal J}}\to {{\mathcal J}}{{\mathcal O}}_Z$ is surjective with kernel ${\mathcal K}$. Since ${{\mathcal O}}_X(1)$ is very ample, we find that there exists $n_0>0$ such that for all $n \geq n_0$, $H^1(X, {\mathcal K}(n)) =(0)$, so that the natural map $$H^0(X, {{\mathcal J}}(n)) \longrightarrow H^0(Z, {{\mathcal J}}{{\mathcal O}}_Z(n))$$ is surjective. Fix $n \geq n_0$, and fix a system of generators $\{f_1,\ldots, f_N\}$ of $H^0(X,{{\mathcal J}}(n))$. It follows that the images of $f_1, \dots, f_N$ generate the $R$-module $H^0(Z, {{\mathcal J}}{{\mathcal O}}_Z(n))$. Note that $C\times_X Z$ does not contain any irreducible component of $F_s$ for all $s$. It follows that it suffices to prove the bound on the dimension of $(T_F)_s$ when $Z=X$, that is, when $F$ is open and dense in $X$. We need the following fact:
\[use.it\] Let $S$ be a noetherian scheme. Let $X\to S$ be a morphism of finite type. Then there exist an affine scheme $S'$ and a quasi-finite surjective morphism of finite type $S' \to S$ with the following properties:
1. $S'$ is the disjoint union of its irreducible components.
2. Let $X':=X\times_S S'$, and let $\Gamma_1, \dots, \Gamma_m$ be the irreducible components of $X'$ endowed with the induced structure of reduced closed schemes. Then for $i=1,\dots, m$, the fibers of $\Gamma_{i}\to S'$ are either empty or geometrically integral.
3. For each $s'\in S'$, the irreducible components of $X'_{s'}$ are exactly the irreducible components of the non-empty $(\Gamma_i)_{s'}$, $i=1, \dots, m$.
We proceed by noetherian induction on $S$. We can suppose $S$ is reduced and $X\to S$ is dominant. First consider the case $S={\operatorname{Spec}}K$ for some field $K$. Then there exists a finite extension $L/K$ such that each irreducible component of $X_L$, endowed with the structure of reduced closed subscheme, is geometrically integral (see [@EGA], IV.4.5.11 and IV.4.6.6). The lemma is proved with $S'={\operatorname{Spec}}L$.
Suppose now that the property is true for any strict closed subscheme $Z$ of $S$ and for the scheme of finite type $X\times_S Z\to Z$. If $S$ is reducible with irreducible components $S_1,\dots, S_\ell$, then by the induction hypothesis we can find $S'_i\to S_i$ with the desired properties (a)-(c). Then it is enough to take $S'$ equal to the disjoint union of the $S'_i$. Now we are reduced to the case $S$ is integral. Let $\eta$ be the generic point of $S$ and let $K=k(\eta)$. Let $L/K$ be a finite extension defined as in the zero-dimensional case above. Restricting $S$ to a dense open subset $V$ if necessary, we can find a finite surjective morphism $\pi: U\to V$ with $U$ integral that extends ${\operatorname{Spec}}L\to {\operatorname{Spec}}K$. Let $X_1, \dots, X_r$ be the (integral) irreducible components of $X\times_V U$. Their generic fibers over $U$ are geometrically integral.
It follows from [@EGA], IV.9.7.7, that there exists a dense open subset $U'$ of $U$ such that $X_i \times_{V} U' \to U'$ has geometrically integral fibers for all $i=1,\dots,r$. Restricting $U'$ further if necessary, we can suppose that the number of geometric irreducible components in the fibers of $X \times_{V} U' \to U'$ is constant ([@EGA], IV.9.7.8). Note now that for each $y \in U'$, the irreducible components of $(X\times_V U')_y$ are exactly the fibers $(X_i)_y$, $i=1, \dots, r$. As $S\setminus \pi(U\setminus U')$ is open and dense in $S$, it contains a dense affine open subset $V'$ of $S$. By induction hypothesis, there exists $T'\to (S\setminus V')_{\mathrm{red}}$ with the desired properties (a)-(c). Let $S'$ be the disjoint union of $\pi^{-1}(V')$ with $T'$. It is clear that $S'$ satisfies the properties (a)-(c).
Let us now return to the proof of Lemma \[cor.constructible2\]. We now proceed to prove that [*it suffices to bound the dimension of $(T_F)_s$ for all $s\in S$ when all fibers of $X \to S$ are integral*]{}. To prove this reduction, we use the fact that the formation of $T_F$ is compatible with any base change $S'\to S$ as in \[constructible-conditions\] (a), and the fact that the dimension of a fiber $(T_F)_s$ is invariant by finite field extensions in the sense of \[lem.construct\] (b). Finally, the conditions that $C_s$ does not contain any irreducible component of $F_s$ is also preserved by base change. While making these reductions, care will be needed to keep track of the hypothesis that $f_1,\dots, f_N$ generate $H^0(X, {{\mathcal J}}(n))$.
Let $g: S' \to S$ be as in Lemma \[use.it\] with natural morphism $g': X \times_S S' \to X$. Let $F'$ be the pre-image of $F$ in $X\times_S S'$. For any $s\in S$ and $s'\in S'$ lying over $s$, $\dim (T_F)_s=\dim (T_F)_{s'}$ and $(T_F)_{s'}=(T_{F'})_{s'}$ is the finite union of the $(T_{\Gamma_i\cap F'})_{s'}$. Increasing $n_0$ if necessary, we find using Fact \[Mumford\] (i) that the natural map $$H^0(X, {{\mathcal J}}(n))\otimes {{\mathcal O}}(S')\longrightarrow H^0(X', g'^*{{\mathcal J}}(n))$$ is an isomorphism. Denote now by ${{\mathcal J}}{{\mathcal O}}_{\Gamma_i}$ the image of $g'^{*}{{\mathcal J}}$ under the natural map $g'^{*}{{\mathcal J}}\to {{\mathcal O}}_{X'} \to {{\mathcal O}}_{\Gamma_i}$. The morphism $g'^{*}{{\mathcal J}}\to {{\mathcal J}}{{\mathcal O}}_{\Gamma_i}$ of ${{\mathcal O}}_{X'}$-modules is surjective. Increasing $n_0 $ further if necessary, we find that $$H^0(X', g'^*{{\mathcal J}}(n)) \longrightarrow H^0(\Gamma_i, {{\mathcal J}}{{\mathcal O}}_{\Gamma_i}(n))$$ is surjective for all $i=1,\dots, m$, where the twisting is done with the very ample sheaf ${{\mathcal O}}_{X'}(1):= g'^* {{\mathcal O}}_X(1)$ relative to $X' \to S'$. It follows that the images of $f_1,\dots, f_N$ in $H^0(\Gamma_i, {{\mathcal J}}{{\mathcal O}}_{\Gamma_i}(n))$ also form a system of generators of $H^0(\Gamma_i, {{\mathcal J}}{{\mathcal O}}_{\Gamma_i}(n))$. Therefore, we can replace $X\to S$ with $\Gamma\to S'$ for $\Gamma$ equal to some $\Gamma_i$. Now we are in the situation where all fibers of $X\to S$ are integral.
By \[constructible-conditions\] (b), if $F_s\ne\emptyset$, $\dim (T_F)_s$ is the dimension of the kernel of the natural map $$k(s)^N \to {{\mathcal O}}_{F_s}(n)\otimes k(\xi)={{\mathcal O}}_{X_s}(n)\otimes k(\xi)$$ defined by the $\bar{f}_{i,s}$ and where $\xi$ is the generic point of $X_s$. This map is given by sections in $H^0(X, {{\mathcal J}}(n))$, so it factorizes into a sequence of linear maps $$k(s)^N\to H^0(X, {{\mathcal J}}(n))\otimes k(s) \to H^0(X_s, {{\mathcal J}}_s(n))
\to H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n))\to {{\mathcal O}}_{X_s}(n)\otimes k(\xi),$$ where the first one is surjective because $f_1,\dots, f_N$ generate $H^0(X, {{\mathcal J}}(n))$, the composition of the second and the third is surjective (independently of $s$) by \[Jn-barJn\] (a) (after increasing $n_0$ if necessary so that \[Jn-barJn\] (a) can be applied), and the last one is injective because $X_s$ is integral. If $F_s=\emptyset$, then $(T_{F})_s=\emptyset$. Therefore, in any case $$\dim (T_F)_s \le N - \dim_{k(s)} H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)).$$
We now end the proof of Lemma \[cor.constructible2\] by showing that after increasing $n_0$ if necessary, we have $\dim_{k(s)} H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)) \geq c$ for all $s\in S_F:=\{ s\in S \mid F_s\ne\emptyset\}$. We note that for all $s\in S_F$, $\dim F_s>0$ (since $F_s$ has no isolated point) so $C_s$ does not contain $F_s$ and, thus, $C_s\ne X_s$. As $X_s$ is irreducible, we have $\dim C_{s}<\dim X_{s}$. It follows that the Hilbert polynomial $P_{C_{s}}(t)$ of $C_{s}$ satisfies $\deg P_{C_{s}}(t)<\deg P_{X_{s}}(t)$. Since the set of all Hilbert polynomials $P_{X_{s}}(t)$ and $ P_{C_{s}}(t)$ with $s\in S$ is finite (\[Mumford\] (iii)), and since such polynomials have positive leading coefficient ([@Har], III.9.10), we can assume, increasing $n_0$ if necessary, that $$P_{X_{s}}(n)-P_{C_{s}}(n)\ge c$$ for all $s\in S_F$. Using \[Mumford\] (ii), and increasing $n_0$ further if necessary, we find that $$H^i(X_s, {{\mathcal O}}_{X_{s}}(n)) =(0)
=H^i(C_{s}, {{\mathcal O}}_{C_{s}}(n))$$ for all $i\ge 1$ and for all $s\in S$. We have $P_{X_{s}}(n) = \chi({{\mathcal O}}_{X_{s}}(n))$, and $P_{C_{s}}(n) = \chi({{\mathcal O}}_{C_{s}}(n))$ for all $n \geq 1$. Therefore, using the above vanishings for $i>0$, we find that for all $s\in S$, $$P_{X_s}(n)-P_{C_s}(n)= \dim H^0(X_{s}, {{\mathcal O}}_{X_s}(n))-
\dim H^0(C_{s}, {{\mathcal O}}_{C_s}(n)).$$ Hence, for all $s \in S_F$, $$\dim H^0(X_{s}, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)) \geq \dim H^0(X_{s}, {{\mathcal O}}_{X_{s}}(n))
-\dim H^0(C_{s}, {{\mathcal O}}_{C_{s}}(n))\ge c$$ and the lemma is proved.
Assume now that $C \to S$ is as in Theorem \[pro.reductiondimension2\], and let ${{\mathcal J}}$ denote the ideal sheaf of $C$ in $X$, as in \[emp.notation\]. In particular, $C \to S$ is finite, $C \to X$ is a regular immersion, $C$ has pure codimension $d$ in $X$, and for all $s \in S$, ${{\operatorname{codim}}}(C_s,X_s) \geq d$. This latter hypothesis and \[lem.remcodim\] (1.b) imply that $C_s$ does not contain any isolated point of $X_s$. Therefore, for any $x\in C_s$, $({ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s)_x\ne 0$ and, hence, both ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)/{ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s^2(n)
\otimes k(x)$ and ${{\mathcal J}}_s(n)/{{\mathcal J}}_s^2(n) \otimes k(x) $, are non-zero. In fact, as $C\to X$ is a regular immersion, ${{\mathcal J}}(n)/{{\mathcal J}}^2(n)$ is a rank $d$ vector bundle on $C$.
\[cor.constructible\] Assume that $C \to S$ is as in Theorem [\[pro.reductiondimension2\]]{}, with $C$ of pure codimension $d$ in $X$ and suppose $X\to S$ is projective. Keep the notation in [\[emp.notation\]]{}. Let $n_0>0$ be an integer such that for all $n \geq n_0$, ${{\mathcal J}}(n)$ is generated by its global sections, and $H^1(X,{{\mathcal J}}^2(n)) =(0)$. Then for all $n \geq n_0$, and for any system of generators $f_1, \dots, f_N$ of $H^0(X, {{\mathcal J}}(n))$, there exists a constructible subset $T_C $ of $\mathbb A^N_S$ such that
1. for all $s\in S$, $\Sigma_C(s)$ is exactly the set of $k(s)$-rational points of $\mathbb A^N_{k(s)}$ contained in $(T_C)_s$, and
2. $\dim (T_C)_s \le N-d$.
Consider the data consisting of the morphism $C\to S$, the sheaf ${{\mathcal F}}:={{\mathcal J}}(n)|_C$ on $C$, and the images of $f_1,\dots, f_N$ under the natural map $H^0(X,{{\mathcal J}}(n) ) \to H^0(C, {{\mathcal F}})$. To this data is associated in \[sig\] a set $\Sigma(s)$ for any $s \in S$. As $C_s$ is finite, $\Sigma_C(s)$ is nothing but the set $\Sigma(s)$. We thus apply Proposition \[constructible-conditions\] to the above data to obtain a constructible set $T_C$ of $\mathbb A^N_S$ such that for all $s\in S$, $\Sigma_C(s)$ is exactly the set of $k(s)$-rational points of $\mathbb A^N_{k(s)}$ contained in $(T_C)_s$.
Our additional hypothesis implies that the images of the sections $f_1,\dots, f_N$ generate $H^0(X,{{\mathcal J}}(n)/{{\mathcal J}}^2(n))$, which we identify with $ H^0(C, {{\mathcal F}})$. Since $C \to S$ and $S$ are affine, and $C_s$ is finite for each $s \in S$, we have isomorphisms $$H^0(C, {{\mathcal F}}) \otimes k(s) \simeq
H^0(C_s,{{\mathcal F}}_s) \simeq \oplus_{x \in C_s} ({{\mathcal F}}_s)_x.$$ It follows that for each $x \in C_s$, the natural map $H^0(C_s,{{\mathcal F}}_s) \to ({{\mathcal F}}_s)_x$ is surjective. As $({{\mathcal F}}_s)_x$ is free of rank $d$ and the images of $f_1, \dots, f_N$ generate $H^0(C_s, {{\mathcal F}}_s)$, we find that the linear maps $k(s)^N \to {{\mathcal F}}(x)$ in \[constructible-conditions\] (b) are surjective for all $s \in
S$. It follows that $\dim (T_C)_s \le N-d$ (the equality holds if $C_s\ne\emptyset$).
\[Proofreductiondimension2\] [*Proof of Theorem [\[exist-hyp\]]{} when $\pi: X \to S$ is projective.* ]{} Let $\{F_1, \dots, F_m\}$ be the locally closed subsets of $X$ given in (ii) of the statement of the theorem. When $C$ does not contain any irreducible component of positive dimension of $X_s$ for all $s \in S$, we set $F_0:=X$ and argue below using the set $\{F_0, F_1, \dots, F_m\}$. Let $A$ denote the finite set given in (iii). When $C\cap {\operatorname{Ass}}(X)=\emptyset$, we enlarge $A$ if necessary by adjoining to it the finite set ${\operatorname{Ass}}(X)$.
Let $A_0\subset X$ be the union of $A$ with the set of the generic points of the irreducible components of positive dimension of $(F_{1})_s, \dots, (F_{m})_s$, for all $s\in S$. When relevant, we also add to $A_0$ the generic points of the irreducible components of positive dimension of $(F_{0})_s$, for all $s\in S$. Using [@EGA], IV.9.7.8, we see that the number of points in $A_0\cap X_s$ is bounded when $s$ varies in $S$. We are thus in a position to apply Lemma \[existence-fn\] (a) with the set $A_0$. Let $c:=1+\dim S$. Let $n_0$ be an integer satisfying simultaneously the conclusion of Lemma \[existence-fn\] (a) for $A_0$, and of Lemma \[cor.constructible2\] for $c$ and for each locally closed subset $F=F_i$, with $i=1,\dots, m$, and $i=0$ when relevant.
Fix now $n\ge n_0$, and fix $f_1, \dots, f_N$ a system of generators of $H^0(X, {{\mathcal J}}(n))$. Increasing $n_0$ if necessary, we can assume using Lemma \[Jn-barJn\] that for all $s \in S$, the composition of the canonical maps $$H^0(X, {{\mathcal J}}(n)) \otimes k(s) \longrightarrow H^0(X_s, {{\mathcal J}}_s(n))
\longrightarrow H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n))$$ is surjective. Let $T_{F_1}, \dots, T_{F_m}$ be the constructible subsets of ${\mathbb A}_S^N$ pertaining to $\Sigma_{F_1}(s), \dots,
\Sigma_{F_m}(s)$ and whose existence is proved in \[remove-isolated-pts\] (2). When relevant, we also consider $T_{F_0}$ and $\Sigma_{F_0}(s)$. Since Lemma \[cor.constructible2\] is applicable for $c$ and for each $F=F_i$, we find that for all $s\in S$ we have $$\dim (T_{F_i})_s\le N-c=N-1-\dim S.$$ It follows from \[lem.construct\] (c) that $\dim T_{F_i}\le N-1$.
Let $\pi(A) := \{s_1, \dots, s_r\} \subseteq S$. Fix $s_j \in \pi(A)$, and for each $x\in A\cap X_{s_j}$, consider the hyperplane of $\mathbb A^N_{k(s_j)}$ defined by $\sum_i \alpha_i f_i(x)=0$. This is indeed a hyperplane because otherwise $f_i(x)=0$ for all $i\le N$ at $x$, which would imply that $x\in C$, but $A\cap C=\emptyset$ by hypothesis. Denote by $T_{A_j}$ the finite union of all such hyperplanes of $\mathbb A^N_{k(s_j)}$, for each $x\in A\cap X_{s_j}$. The subset $T_{A_j}$ is pro-constructible in $\mathbb A^N_S$ (see \[pro-constr\]). It has dimension $N-1$, and its fibers $(T_{A_j})_s$ are constructible for each $s \in S$ (and $(T_{A_j})_s$ is empty if $s \neq s_j$).
We now apply Theorem \[globalize\] to the set of pro-constructible subsets $T_{A_j}$, $j=1,\dots, r$ and $T_{F_i}$, $i=1,\dots, m$, and $i=0$ when relevant. Our discussion so far implies that these pro-constructible subsets all satisfy Condition (1) in \[globalize\] with $V=S$. Let $T=(\cup_j T_{A_j})\cup (\cup_i
T_{F_i})$. For each $s \in S$, the element $f_{s,n} \in H^0(X_s,{{\mathcal J}}_s(n))$ exhibited in Lemma \[existence-fn\] (a) gives rise to a $k(s)$-rational point of $\mathbb A_{k(s)}^N$ not contained in $T_s$. So Condition (2) in \[globalize\] is also satisfied by $T$. We can thus apply Theorem \[globalize\] to find a section $(a_1,\ldots, a_N)\in R^N=\mathbb A_S^N(S)$ such that for all $s \in S$, $(a_1(s), \ldots, a_N(s))$ is a $k(s)$-rational point of $\mathbb A^N_{k(s)}$ that is not contained in $T_s$.
Let $f:=\sum_{i=1}^N a_if_i$ and consider the closed subscheme $H_f \subset X$. As $f\in H^0(X, {{\mathcal J}}(n))$, $C$ is a closed subscheme of $H_f$. By definition of $T_{F_i}$ and $T_{A_j}$, for all $s\in S$ and for all $0 \leq i\le m$, $H_f$ does not contain any irreducible component of $(F_i)_s$ of positive dimension and $H_f\cap A=\emptyset$. This proves the conclusions (1), (2), and (3) of \[exist-hyp\].
When the hypothesis of (4) is satisfied, we have included in our proof above conditions pertaining to $F_0=X$, and we find then that $H_f$ contains no irreducible component of $X_s$. It is thus by definition a hypersurface relative to $X \to S$. If furthermore $C\cap {\operatorname{Ass}}(X)=\emptyset$, as we enlarged $A$ to include ${\operatorname{Ass}}(X)$, we have $H_f\cap {\operatorname{Ass}}(X)=\emptyset$. Hence, it follows from Lemma \[hypersurfaces-properties\] (4) that $H_f$ is locally principal. This proves (4), and completes the proof of Theorem \[exist-hyp\] when $X\to S$ is projective.
\[Proofreductiondimension2-a\] [*Proof of Theorem [\[pro.reductiondimension2\]]{} when $\pi: X \to S$ is projective.* ]{} We assume now that $C \to S$ is finite. Thus $C_s$ is finite for each $s \in S$, and we find that $C_s$ does not contain any irreducible component of positive dimension of $X_s$. Let $\{F_1, \dots, F_m\}$ be the locally closed subsets of $X$ given in (ii) of \[exist-hyp\]. We set $F_0:=X$ and argue as in the proof of \[exist-hyp\] above using the set $\{F_0, F_1, \dots, F_m\}$. Let $A$ denote the finite set given in (iii) of \[exist-hyp\]. We have that $C\cap{\operatorname{Ass}}(X)=\emptyset$: indeed, for all $x\in C$, $\mathrm{depth}({{\mathcal O}}_{X,x})\ge d>0$, so that $x\notin \mathrm{Ass}(X)$. We therefore enlarge $A$ if necessary by adjoining to it the finite set ${\operatorname{Ass}}(X)$. We define $A_0$ and $c:=1+\dim S$ exactly as in the proof of \[exist-hyp\] in \[Proofreductiondimension2\]. Let $n_0$ be an integer satisfying simultaneously the conclusion of Lemma \[existence-fn\] (b) for $A_0$, of Lemma \[cor.constructible\], and of Lemma \[cor.constructible2\] for $c$ and for each locally closed subset $F=F_i$, with $i=0,1,\dots, m$.
Fix now $n\ge n_0$, and fix $f_1, \dots, f_N$ a system of generators of $H^0(X, {{\mathcal J}}(n))$. Increasing $n_0$ if necessary, we can assume using Lemma \[Jn-barJn\] that for all $s \in S$, the composition of the canonical maps $$H^0(X, {{\mathcal J}}(n)) \otimes k(s) \longrightarrow H^0(X_s, {{\mathcal J}}_s(n))
\longrightarrow H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n))$$ is surjective. Let $T_{F_0}, T_{F_1}, \dots, T_{F_m}$ be the constructible subsets of ${\mathbb A}_S^N$ pertaining to $\Sigma_{F_0}(s)$, $\Sigma_{F_1}(s), \dots$, $\Sigma_{F_m}(s)$, and whose existence is proved in \[remove-isolated-pts\] (2). As in the proof \[Proofreductiondimension2\], we find that $\dim T_{F_i}\le N-1$ for each $i=0,\dots, m$. Define now $T_{A_j}$, $j=1,\dots, r$ as in the proof \[Proofreductiondimension2\]. Again, $T_{A_j}$ is pro-constructible in $\mathbb A^N_S$, it has dimension $N-1$, and its fibers $(T_{A_j})_s$ are constructible for each $s \in S$.
Since Lemma \[cor.constructible\] is applicable, we can also consider the constructible subset $T_C$ of $\mathbb A^N_S$ pertaining to $\Sigma_C(s)$. Since we assume that $d > \dim S$, we find from \[cor.constructible\] that $$\dim (T_C)_s\le N-d\le N-(\dim S+1)$$ for all $s\in S$. Thus, it follows from \[lem.construct\] (c) that $\dim T_C\leq N-1$.
As in \[Proofreductiondimension2\], we set $T$ to be the union of the sets $T_{F_i}$, $i=0,\dots, m$, and $T_{A_j}$, $j=1,\dots, r$. Lemma \[existence-fn\] (b) implies that $\mathbb A^N_{k(s)}$ is not contained in $(T_C\cup T)_s$ because $({ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)/{ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}^2_s(n))\otimes
k(x)\ne 0$ for all $x\in C_s$ (see the paragraph before \[cor.constructible\]). Applying Theorem \[globalize\] to the pro-constructible subsets $T_C$, $T_{F_i}$, $i=0,\dots, m$, and $ T_{A_j}$, $j=1,\dots, r$, we find $(a_1, \dots, a_N)\in \mathbb A^N_S(S)$ such that for all $s \in S$, $(a_1(s), \ldots, a_N(s))$ is a $k(s)$-rational point of $\mathbb A^N_{k(s)}$ that is not contained in $(T_C\cup T)_s$.
Let $f:=\sum_{i=1}^N a_if_i$ and consider the closed subscheme $H_f \subset X$. As in \[Proofreductiondimension2\], we find that $H_f$ satisfies the conclusions (1), (2), and (3) of \[exist-hyp\], and that $H_f$ is a locally principal hypersurface.
It remains to use the properties of the set $T_C$ to show that $C$ is regularly immersed in $H_f$, and that $C$ is pure of codimension $d-1$ in $H_f$. Indeed, this is a local question. Fix $x \in C$. Let $I:= {{\mathcal J}}_x \subset {{\mathcal O}}_{X,x}$ and let $g \in I$ correspond to the section $f$. Since the image of $g$ in $I/I^2 \otimes k(x)$ is non-zero by the definition of $T_C$, the image of $g$ in the free ${{\mathcal O}}_{C,x}$-module $I/I^2$ can be completed into a basis of $I/I^2$, and it is then well-known that $g$ belongs to a regular sequence generating $I$. This concludes the proof of Theorem \[pro.reductiondimension2\] when $X\to S$ is projective.
\[quasiproj\] [*Proof of Theorems [\[exist-hyp\]]{} and [\[pro.reductiondimension2\]]{} when $X\to S$ is quasi-projective.* ]{} Since ${{\mathcal O}}_X(1)$ is assumed to be very ample relative to $X \to S$, there exists a projective morphism $\overline{X} \to S$ with an open immersion $X \to \overline{X}$ of $S$-schemes, and a very ample sheaf ${{\mathcal O}}_{\overline{X}}(1)$ relative to $\overline{X} \to S$ which restricts on $X$ to the given sheaf ${{\mathcal O}}_X(1)$.
Let us first prove Theorem [\[exist-hyp\]]{}. We are given in [\[exist-hyp\]]{} (i) a closed subscheme $C$ of $X$. Let ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeC\else\ensuremath{C}\fi \kern-0.1em } }}$ be the scheme-theoretical closure of $C$ in ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeX\else\ensuremath{X}\fi \kern-0.1em } }}$. We are given in [\[exist-hyp\]]{} (ii) $m$ locally closed subsets $F_1, \dots, F_m$ of $X$. Since $X$ is open in ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeX\else\ensuremath{X}\fi \kern-0.1em } }}$, each set $F_i$ is again locally closed in ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeX\else\ensuremath{X}\fi \kern-0.1em } }}$. It is clear that the finite subset $A \subset X$ given in [\[exist-hyp\]]{} (iii) which does not intersect $C$ is such that $A \subset { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeX\else\ensuremath{X}\fi \kern-0.1em } }}$ does not intersect ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeC\else\ensuremath{C}\fi \kern-0.1em } }}$.
We are thus in the position to apply Theorem \[exist-hyp\] to the projective morphism $\overline{X} \to S$ with the data $ { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeC\else\ensuremath{C}\fi \kern-0.1em } }}$, $ F_1, \dots, F_m$, and $ A$. When $C$ satisfies the first hypothesis of \[exist-hyp\] (4), we set $F_0 :=X$ and add the locally closed subset $F_0$ to the list $F_1,\dots, F_m$, as in the proof \[Proofreductiondimension2\]. When $C\cap {\operatorname{Ass}}(X)=\emptyset$, we replace $A$ by $A \cup {\operatorname{Ass}}(X)$. We can then conclude that there exists $n_0>0$ such that for any $n\ge n_0$, there exists a global section $f $ of $ {{\mathcal O}}_{\overline{X}}(n)$ such that the closed subscheme $H_f$ in ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeX\else\ensuremath{X}\fi \kern-0.1em } }}$ contains ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeC\else\ensuremath{C}\fi \kern-0.1em } }}$ as a closed subscheme and satisfies the conclusions (2), (3), and, when relevant, (4), of \[exist-hyp\] for $\overline{X} \to S$. The restriction of $f$ to ${{\mathcal O}}_X(n)$ defines the desired closed subscheme $H_f \cap X$ satisfying the conclusions of Theorem \[exist-hyp\] for $X \to S$.
Let us now prove Theorem \[pro.reductiondimension2\], where we assume that $C \to S$ is finite and, hence, proper. It follows that ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeC\else\ensuremath{C}\fi \kern-0.1em } }}=C$. We apply Theorem \[pro.reductiondimension2\] to the projective morphism ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeX\else\ensuremath{X}\fi \kern-0.1em } }} \to S$, and the data $C$, $F_1,\dots, F_m$, and $A$. We can then conclude that there exists $n_0>0$ such that for any $n\ge n_0$, there exists a global section $f $ of $ {{\mathcal O}}_{\overline{X}}(n)$ such that the closed subscheme $H_f$ in ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeX\else\ensuremath{X}\fi \kern-0.1em } }}$ contains ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmodeC\else\ensuremath{C}\fi \kern-0.1em } }}$ as a closed subscheme and satisfies the conclusions (2), (3), and (4), of \[exist-hyp\] for $\overline{X} \to S$. The restriction of $f$ to ${{\mathcal O}}_X(n)$ defines the desired closed subscheme $H_f \cap X$ satisfying the conclusions of Theorem \[pro.reductiondimension2\] for $X \to S$.
[Variations on the classical Avoidance Lemma]{} \[compute-coh\]
In this section, we prove various assertions used in the proofs of Theorem \[exist-hyp\] and Theorem \[pro.reductiondimension2\]. The main result in this section is Lemma \[existence-fn\].
\[Mumford\] Let $S$ be a noetherian scheme, and let $\pi: X \to S$ be a projective morphism. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $\pi$, and let ${\mathcal F}$ be any coherent sheaf on $X$.
1. Let $g:S'\to S$ be a morphism of finite type, and consider the cartesian square $$\xymatrix{
X':=X \times_S S' \ar@{>}[r]^{\hspace*{12mm} g'} \ar[d]^{\pi'} & X \ar[d]^{\pi}
\\
S' \ar@{>}[r]^{g} & S. \\
}$$ Then there exists a positive integer $n_0$ such that for all $n \geq n_0$, the canonical morphism $g^*\pi_*({\mathcal F}(n)) \longrightarrow \pi'_{*}g'^{*}({\mathcal F}(n))$ is an isomorphism.
2. There exists a positive integer $n_0$ such that for all $n \geq n_0$ and for all $s \in S$, $H^i(X_s, {\mathcal F}_s(n))=(0)$ for all $i>0$, and $\pi_*{{\mathcal F}}(n) \otimes k(s) \longrightarrow H^0(X_s, {\mathcal F}_s(n))$ is an isomorphism.
3. The set of Hilbert polynomials $\{ P_{X_s}(t)\in \mathbb Q[t] \mid s
\in S\}$ is finite.
The properties in the statements are local on the base, and we may thus assume that $S$ is affine. In this case, there is no ambiguity in the definition of a projective morphism, as all standard definitions coincide when the target is affine ([@EGA], II.5.5.4 (ii)). The proofs of (i) and (ii) when $X= {\mathbb P}^d_S$ and any coherent sheaf ${{\mathcal F}}$ can be found, for instance, in [@MumL], p. 50, (i), and [@MumL], p. 58, (i) (see also [@Sernesi], step 3 in the proof of Theorem 4.2.11). The statement (iii) follows from [@MumL], p. 58, (ii). The general case follows immediately using the closed $S$-immersion $i: X \to {\mathbb P}^d_S$ defining ${{\mathcal O}}_X(1)$.
\[m-regular\] Let us recall the definition and properties of $m$-regular sheaves needed in our next lemmas. Let $X$ be a projective variety over a field $k$, with a fixed very ample sheaf ${{\mathcal O}}_X(1)$. Let ${{\mathcal F}}$ be a coherent sheaf on $X$, and let ${{\mathcal F}}(n) := {{\mathcal F}}\otimes {{\mathcal O}}_X(n)$. Let $m\in \mathbb Z$. Recall ([@MumL], Lecture 14, p. 99) that ${{\mathcal F}}$ is called *$m$-regular* if $H^i(X, {{\mathcal F}}(m-i))=0$ for all $i\ge 1$.
Assume that ${{\mathcal F}}$ is *$m$*-regular. Then it is known (see, e.g., [@Sernesi], Proposition 4.1.1) that for all $n\ge m$,
1. ${{\mathcal F}}$ is $n$-regular,
2. $H^i(X, {{\mathcal F}}(n))=0$ for all $i\ge 1$,
3. ${{\mathcal F}}(n)$ is generated by its global sections, and
4. The canonical homomorphism $$H^0(X, {{\mathcal F}}(n))\otimes H^0(X, {{\mathcal O}}_X(1)) \longrightarrow H^0(X, {{\mathcal F}}(n+1))$$ is surjective.
\[Mumford2\] Let $S$ be a noetherian scheme, and let $\pi: X \to S$ be a projective morphism. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $\pi$, and let ${\mathcal F}$ be any coherent sheaf on $X$. Then there exists a positive integer $n_0$ such that for all $n \geq n_0$ and all $s\in S$, the sheaf ${\mathcal F}_s$ is $n$-regular on $X_s$.
Let $r$ denote the maximum of $\dim X_s$, $s \in S$. This maximum is finite ([@EGA], IV.13.1.7). Then $H^i(X_s, {\mathcal F}_s(n))=(0)$ for all $i \geq r+1 $ and for any $n$. Using \[Mumford\] (ii), there exists $n_1>0$ such that $H^i(X_s, {\mathcal F}_s(n))=(0)$ for all $s \in S$, for all $n \geq n_1$, and for all $i>0$. It follows that $ {\mathcal F}_s $ is $n$-regular for all $s \in S$ and for all $n\geq n_0:=r+n_1$.
We now discuss a series of lemmas needed in the proof of \[existence-fn\].
\[Jn-barJn\] Let $\pi : X\to S$ be a projective scheme over a noetherian scheme $S$. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $\pi$. Let $C$ be a closed subscheme of $X$, with sheaf of ideals ${{\mathcal J}}$ in ${{\mathcal O}}_X$. Let ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s $ denote the image of ${{\mathcal J}}_s$ in ${{\mathcal O}}_{X_s} $. Then there exists $n_0 \in {\mathbb N}$ such that for all $n\ge n_0$ and for all $s\in S$,
1. The canonical map $\pi_*{{\mathcal J}}(n)\otimes k(s)\longrightarrow H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n))$ is surjective.
2. The sheaf ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s$ is $n$-regular.
By the generic flatness theorem [@EGA], IV.6.9.3, there exist finitely many locally closed subsets $U_i$ of $S$ such that $S = \cup_i U_i$ (set theoretically), and such that when each $U_i$ is endowed with the structure of reduced subscheme of $S$, then $C_{U_i}:=C\times_S U_i\to U_i$ is flat. Refining each $U_i$ by an affine covering, we can suppose $U_i$ affine.
Denote by $U$ one of these affine schemes $U_i$. Denote by $\mathcal K$ and ${{\mathcal J}}'$ the kernel and image of the natural morphism ${{\mathcal J}}\otimes_{{{\mathcal O}}_X} {{\mathcal O}}_{X_U} \to {{\mathcal O}}_{X_{U}}$, with associated exact sequence of sheaves on $X_{U}$ $$0 \longrightarrow \mathcal K \longrightarrow {{\mathcal J}}\otimes_{{{\mathcal O}}_X} {{\mathcal O}}_{X_U} \longrightarrow {{\mathcal J}}'
\longrightarrow 0.$$ For all $n\in \mathbb Z$, we then have the exact sequence $$0 \to \mathcal K(n) \to {{\mathcal J}}(n)\otimes_{{{\mathcal O}}_X} {{\mathcal O}}_{X_U} \to
{{\mathcal J}}'(n)\to 0.$$ Since $X_{U} \to U$ is projective, we can find $n_1$ such that $H^1(X_{U}, \mathcal K(n))=(0) $ for all $n \geq n_1$ (Serre Vanishing). Using \[Mumford\] (ii), we find that by increasing $n_1$ if necessary, we can assume that for all $n\ge n_1$ and for all $s\in U$, $$\label{Jns}
H^0(X_{U}, {{\mathcal J}}'(n))\otimes k(s)\longrightarrow H^0(X_s, {{\mathcal J}}'(n)_s)$$ is an isomorphism. The exact sequence $0\to {{\mathcal J}}' \to {{\mathcal O}}_{X_U} \to {{\mathcal O}}_{C_U} \to 0$ induces an exact sequence $0\to {{\mathcal J}}'_s \to {{\mathcal O}}_{X_s} \to {{\mathcal O}}_{C_s} \to 0$ for all $s \in U$ because $C\times_S U\to U$ is flat. It follows that ${{\mathcal J}}'_s= { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s$. We can thus apply \[Mumford2\] to the morphism $X_{U} \to U$ and the sheaf ${{\mathcal J}}'$ to obtain that ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s$ is $n$-regular for all $n \geq n_1$ and all $s \in U$(after increasing $n_1$ further if necessary.)
For any $s \in U$ and for $n \geq n_1$, consider the commutative diagram: $$\xymatrix{
H^0(X_{U}, {{\mathcal J}}(n)\otimes{{\mathcal O}}_{U})\otimes k(s)
\ar@{>>}[r] \ar[d] \ar[dr] & H^0(X_{U}, {{\mathcal J}}'(n))\otimes k(s) \ar[d] \\
H^0(X_s, {{\mathcal J}}(n)_s) \ar[r] & H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)). \\
}$$ The top horizontal map is surjective because $H^1(X_{U}, \mathcal K(n))=(0)$, and the right vertical arrow is an isomorphism by the isomorphism above. Thus, the bottom arrow $H^0(X_s, {{\mathcal J}}(n)_s) \to H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n))$ is surjective for all $n\ge n_1$ and all $s\in U$.
To complete the proof of (b), it suffices to choose $n_0$ to be the maximum in the set of integers $n_1$ associated with each $U_i $ in the stratification. To complete the proof of (a), we further increase $n_0$ if necessary to be able to use the isomorphism in \[Mumford\] (ii) applied to ${\mathcal F}={{\mathcal J}}$ on $X\to S$.
\[fn-bis\] Let $X$ be a projective variety over a field $k$ with a fixed very ample sheaf ${{\mathcal O}}_X(1)$. Let $C$ be a closed subscheme of $X$. Let ${{\mathcal J}}$ denote the ideal sheaf of $C$ in $X$, and assume that ${{\mathcal J}}$ is $m_0$-regular for some $m_0 \geq 0$. Let $D$ be a finite set of closed points of $C$. Let $\{\xi_1, \dots, \xi_r\}$ be a finite subset of $X$ disjoint from $C$.
1. If ${\mathrm{Card}}(k)\geq r+{\mathrm{Card}}(D)$, then for all $n\ge m_0$, there exists a section $f_n\in H^0(X, {{\mathcal J}}(n))$ such that $V_+(f_n)$ does not contain any $\xi_i$, and such that, for all $x\in D$ such that $({{\mathcal J}}(n)/{{\mathcal J}}^2(n))\otimes k(x) \neq (0)$, the image of $f_n$ in $({{\mathcal J}}(n)/{{\mathcal J}}^2(n))\otimes k(x)$ is non-zero.
2. There exists an integer $n_0>0$ such that for all $n\ge n_0$, there exists a section $f_n\in H^0(X, {{\mathcal J}}(n))$ as in [(a)]{}.
It suffices to prove the lemma for the subset of $D$ obtained by removing from $D$ all points $x$ such that $({{\mathcal J}}(n)/{{\mathcal J}}(n)^2)\otimes k(x)=0$. We thus suppose now that $({{\mathcal J}}(n)/{{\mathcal J}}(n)^2)\otimes k(x)\ne 0$ for all $x\in D$. Note also that the natural map $({{\mathcal J}}(n)/{{\mathcal J}}(n)^2)\otimes k(x) \to {{\mathcal J}}(n)\otimes k(x)$ is an isomorphism for all $x\in D$, and we will use the latter expression.
\(a) Let $x \in D$ and $n \geq m_0$. Consider the $k$-linear map $$H^0(X, {{\mathcal J}}(n))\longrightarrow {{\mathcal J}}(n)\otimes k(x)$$ and denote by $H_x$ its kernel. Since ${{\mathcal J}}(n)$ is generated by its global sections (\[m-regular\] (c)), this map is non-zero and $H_x \neq H^0(X, {{\mathcal J}}(n))$.
Let $B= \oplus_{j \geq 0} H^0(X, {{\mathcal O}}_X(j))$. This is a graded $k$-algebra and $X\simeq {\operatorname{Proj}}B$. Let ${{\mathfrak p}}_1,\dots, {{\mathfrak p}}_r$ be the homogeneous prime ideals of $B$ defining $\xi_1, \dots, \xi_r$. Let $J$ be the homogeneous ideal $\oplus_{j \geq 0} H^0(X, {{\mathcal J}}(j))$ of $B$. Then $C$ is the closed subscheme of $X$ defined by $J$. By hypothesis, for each $i \leq r$, ${{\mathfrak p}}_i$ neither contains $J$ nor $B(1)$. Let $J(n):=H^0(X, {{\mathcal J}}(n))$. We claim that for each $i \leq r$, $J(n)\cap {{\mathfrak p}}_i$ is a proper subspace of $J(n)$. Indeed, if $J(n_0)\cap {{\mathfrak p}}_i = J(n_0)$ for some $n_0 \geq m_0$, then the surjectivity of the map in \[m-regular\] (d) implies that $J(n)\cap {{\mathfrak p}}_i = J(n)$ for all $n \geq n_0$. This would imply $C=V_+(J)\supseteq V_+({{\mathfrak p}}_i)\ni \xi_i$.
We have constructed above at most $r+{\mathrm{Card}}(D)$ proper subspaces of $H^0(X, {{\mathcal J}}(n))$. Since $r+{\mathrm{Card}}(D) \leq {\mathrm{Card}}(k)$ by hypothesis, the union of these proper subspaces is not equal to $H^0(X,{{\mathcal J}}(n))$ (\[union\]). Since any element $f_n$ in the complement of the union of these subspaces satisfies the desired properties, (a) follows.
\(b) Let ${{\mathcal J}}_D$ be the ideal sheaf on $X$ defining the structure of reduced closed subscheme on $D$. Choose $m \geq 0$ large enough such that both ${{\mathcal J}}$ and $ {{\mathcal J}}{{\mathcal J}}_D$ are $m$-regular. As $H^1(X, ({{\mathcal J}}{{\mathcal J}}_D)(n))=(0)$ for $n \geq m$ by \[m-regular\] (b), the map $$H^0(X, {{\mathcal J}}(n))\longrightarrow H^0(X, {{\mathcal J}}(n){|_D})
=H^0(X, {{\mathcal J}}(n)/{{\mathcal J}}{{\mathcal J}}_D(n)))$$ is surjective for all $n\ge m$. Note now the isomorphisms $$H^0(X, {{\mathcal J}}(n){|_D})
\longrightarrow \oplus_{x\in D} ({{\mathcal J}}(n){|_D})_x \longrightarrow \oplus_{x\in D} {{\mathcal J}}(n)\otimes k(x).$$ Let then $f\in H^0(X, {{\mathcal J}}(n))$ be a section such that its image in ${{\mathcal J}}(n)\otimes k(x)$ is non-zero for each $x \in D$. Keep the notation introduced in (a). Then $I:=\oplus_{n\ge 0} H^0(X, {{\mathcal J}}^2(n))$ is a homogeneous ideal of $B$ and $J^2\subseteq I\subseteq J$. Hence $I\not\subseteq {{\mathfrak p}}_i$ for all $i\le r$, since otherwise $J\subseteq {{\mathfrak p}}_i$, which contradicts the hypothesis that $\xi_i\notin C$.
Lemma \[avoid\] (a) below implies then the existence of $n_0 \geq 0$ such that for all $n \geq n_0$, there exists $x_n\in I(n)$ such that $f_n:=f+x_n\notin\cup_{1\le i\le r}{{\mathfrak p}}_i$. We have $f_n\in J(n)=H^0(X, {{\mathcal J}}(n))$ and for all $x\in D$, $f_n$ is non-zero in ${{\mathcal J}}(n)\otimes k(x)$.
The following Prime Avoidance Lemma for graded rings is needed in the proof of \[fn-bis\]. This lemma is slightly stronger than 4.11 in [@GLL1]. For related statements, see [@SH], Theorem A.1.2., or [@Bou], III, 1.4, Prop. 8, page 161. We do not use the statement \[avoid\] (b) in this article.
\[avoid\] Let $B=\oplus_{n\ge 0}B(n)$ be a graded ring. Let $I=\oplus_{n\ge 0}I(n)$ be a homogeneous ideal of $B$. Let ${{\mathfrak p}}_1,\dots,{{\mathfrak p}}_r$ be homogeneous prime ideals of $B$ not containing $B(1)$ and not containing $I$.
1. Then there exists an integer $n_0 \geq 0$ such that for all $n\ge n_0$ and for all $f\in B(n)$, we have $$f+I(n)\not\subseteq \cup_{1\le i\le r} {{\mathfrak p}}_i.$$
2. Let $k$ be a field with $\mathrm{Card}(k)>r$, and assume that $B$ is a $k$-algebra. If $I$ can be generated by elements of degree at most $ d$, then in [(a)]{} we can take $n_0=d$.
We can suppose that there are no inclusion relations between ${{\mathfrak p}}_1, \dots, {{\mathfrak p}}_r$.
\(a) Let $i\le r$ and set $I_i:=I\cap (\cap_{j\ne i}{{\mathfrak p}}_j)$. We first observe that there exists $n_i\ge 0$ such that for all $n \ge n_i$, we have $I_i(n)\not\subseteq {{\mathfrak p}}_i$. Indeed, as $I_i\not\subseteq {{\mathfrak p}}_i$ and $I_i$ is homogeneous, we can find a homogeneous element $\alpha$ in $ I_i\setminus {{\mathfrak p}}_i$. Let $t\in B(1)\setminus {{\mathfrak p}}_i$. Set $n_i:=\deg \alpha$. Then for all $n\ge n_i$, we have $t^{n-n_i}\alpha \in I_i(n)\setminus {{\mathfrak p}}_i$.
Let $n_0:=\max_{1\le i\le r} \{ n_i\}$. Let $n\ge n_0$ and let $f\in B(n)$. If $f\notin\cup_i {{\mathfrak p}}_i$, then clearly $f+I(n)\not\subseteq \cup_{1\le i\le r} {{\mathfrak p}}_i$. Assume now that $f \in\cup_i {{\mathfrak p}}_i$, and for each $j$ such that $f \in {{\mathfrak p}}_j$, choose $t_j\in I_j(n)\setminus {{\mathfrak p}}_j$. Then we easily verify that $$f+\sum_{{{\mathfrak p}}_j\ni f} t_j \ \in (f+I(n))\setminus \cup_{1\le i\le r} {{\mathfrak p}}_i.$$
\(b) Let $n\ge d$. For each $j \leq r$, let us show that $I(n)\not\subseteq {{\mathfrak p}}_j$. Suppose by contradiction that $I(n)\subseteq {{\mathfrak p}}_j$, and choose $t\in B(1)\setminus {{\mathfrak p}}_j$. Then $t^{n-e}I(e)\subseteq I(n)\subseteq {{\mathfrak p}}_j$ for all $1 \leq e \leq d$. Hence, $I(e)\subseteq {{\mathfrak p}}_j$, and then $I\subseteq {{\mathfrak p}}_j$ because $I$ can be generated by the union of the $I(e)$, $1 \leq e \leq d$. Contradiction.
Let $f\in B(n)$, and suppose that $f+I(n)\subseteq \cup_{1\le i\le r} {{\mathfrak p}}_i$. Then $$I(n) = \cup_{1\le i \le r} \left((-f+{{\mathfrak p}}_i)\cap I(n)\right).$$ If $(-f+{{\mathfrak p}}_i)\cap I(n)$ is not empty, pick $c_i \in ((-f+{{\mathfrak p}}_i)\cap I(n))$, and let $W_i:= -c_i +((-f+{{\mathfrak p}}_i)\cap I(n))$. The reader will easily check that $W_i$ is a $k$-subspace of the $k$-vector space $I(n)$. Moreover, we claim that $W_i \neq I(n)$. Indeed, if $W_i = I(n)$, then $I(n) = c_i + W_i = (-f+{{\mathfrak p}}_i)\cap I(n)$. But then $f\in {{\mathfrak p}}_i$, which implies that $I(n)\subseteq {{\mathfrak p}}_i$, a contradiction. Therefore, the $k$-vector space $I(n)$ is a finite union of at most $r$ proper $k$-affine subspaces, and this is also a contradiction (\[union\]).
\[union\] Let $V$ be a vector space over a field $k$. For $i=1,\dots, m$, let $v_i\in V$ and let $V_i$ be a proper subspace of $V$. If $\mathrm{Card}(k)\ge m+1$, then $$V\ne (v_1 +V_1) \cup \ldots \cup (v_m +V_m).$$ If $\mathrm{Card}(k)\ge m$, then $V\ne V_1 \cup \ldots \cup V_m.$
Assume that $\mathrm{Card}(k)\ge m+1$, and that $V= (v_1 +V_1) \cup \ldots \cup (v_m +V_m)$. We claim then that $V= V_1 \cup \ldots \cup V_m$. Indeed, fix $x \in V_1$, and let $y \in V \setminus V_1$. Since $\mathrm{Card}(k^*) \geq m$, we can find at least $m$ elements of the form $v_1+(x+\lambda y)$ with $\lambda \in k^*$, and $$v_1+(x+\lambda y)\in V\setminus (v_1+V_1)\subseteq \cup_{2\le i\le m} (v_i+V_i).$$ Thus there exist an index $i$ and distinct $\lambda_1 $, $\lambda_2$ in $k^*$ such that $v_1+x+\lambda_1 y$ and $ v_1+x+\lambda_2 y$ both belong to $v_{i}+V_i$. It follows that $(\lambda_1-\lambda_2)y\in V_i$ and, thus, $y\in V_i$. Hence, $V= V_1 \cup \ldots \cup V_m$. The second statement of the lemma is well-known and can be found for instance in [@BBS], Lemma 2.
Our final lemma in this section is a key ingredient in the proofs of Theorem \[exist-hyp\] and Theorem \[pro.reductiondimension2\] and is used to insure that Condition (2) in Theorem \[globalize\] holds for $n$ big enough uniformly in $s\in S$.
\[existence-fn\] Let $S$ be a noetherian affine scheme, and let $X\to S$ be projective. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf on $X$ relative to $X\to S$. Let $C:=V({{\mathcal J}})$ be a closed subscheme of $X$. Let $A_0$ be a subset of $X$ disjoint from $C$ and such that there exists $c_0\in \mathbb N$ with $\mathrm{Card} (A_0\cap X_s) \le c_0$ for all $ s\in S$. Let ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s$ denote the image of ${{\mathcal J}}_s $ in ${{\mathcal O}}_{X_s} $. Then there exists $n_0 \geq 0$ such that for all $s \in S$ and for all $n \geq n_0$,
1. There exists $f_{s,n}\in H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n))$ whose zero locus $V_+(f_{s,n})$ in $X_s$ does not contain any point of $A_0$.
2. Suppose that $C\to S$ is finite. Then there exists $f_{s,n}\in H^0(X_s, { \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n))$ as in [(a)]{} such that the image of $f_{s,n}$ in $({ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)/{ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s^2(n))\otimes k(x)$ is non-zero for all $x\in C$ with $({ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n)/{ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}^2_s(n))\otimes k(x)\neq (0)$.
When $C\to S$ is finite, we increase $c_0$ if necessary so we can assume that $\mathrm{Card}(C_s)\le c_0$ for all $s\in S$. Let $$Z_0:=\{ s\in S \mid {\mathrm{Card}}(k(s)) \le 2c_0\}.$$ Lemma \[finiteresidue\] shows that $Z_0$ is a finite set.
Let $n_0$ be such that Lemma \[Jn-barJn\] applies. Fix $n>n_0$. It follows from \[Jn-barJn\] (b) that ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s$ is $n$-regular for all $s \in S$. Let $s \in S \setminus Z_0$. Then ${\mathrm{Card}}(k(s)) > {\mathrm{Card}}(A_0) + {\mathrm{Card}}(C_s)$. Parts (a) and (b) both follow from Lemma \[fn-bis\] (a) applied to ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s$, with $D$ empty in the proof of (a), and $D = C_s$ in the proof of (b). For the remaining finitely many points $s\in Z_0$, we increase $n_0$ if necessary so that we can use Lemma \[fn-bis\] (b) for each $s \in Z_0$.
[Avoidance lemma for families]{} \[hyperf\]
We present in this section further applications of our method. Our first result below is a generalization of Theorem \[exist-hyp\], where the noetherian hypothesis on the base has been removed.
\[bertini-type-0\] Let $S$ be an affine scheme, and let $X\to S$ be a quasi-projective and finitely presented morphism. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $X \to S$. Let
1. $C$ be a closed subscheme of $X$, finitely presented over $S$;
2. $F_1, \dots, F_m$ be subschemes of $X$ of finite presentation over $S$;
3. $A$ be a finite subset of $X$ such that $A\cap C=\emptyset$.
Assume that for all $s \in S$, $C$ does not contain any irreducible component of positive dimension of $(F_i)_s$ and of $X_s$. Then there exists $n_0>0$ such that for all $n\ge n_0$, there exists a global section $f$ of ${{\mathcal O}}_X(n)$ such that:
1. The closed subscheme $H_f$ of $X$ is a hypersurface that contains $C$ as a closed subscheme;
2. \[HF\] For all $s \in S$ and for all $i\le m$, $H_f$ does not contain any irreducible component of positive dimension of $(F_i)_s$; and
3. \[add-A\] $H_f\cap A=\emptyset$.
Assume in addition that $S$ is noetherian, and that $C\cap{\operatorname{Ass}}(X)=\emptyset$. Then there exists such a hypersurface $H_f$ which is locally principal.
The last statement when $S$ is noetherian is immediate from the main statement of the theorem: simply apply the main statement of the theorem with $A$ replaced by $A\cup{\operatorname{Ass}}(X)$. The fact that $H_f$ is locally principal when $H_f \cap {\operatorname{Ass}}(X) = \emptyset$ is noted in \[hypersurfaces-properties\] (4).
Let us now prove the main statement of the theorem. First we add $X$ to the set of subschemes $F_i$. Then the property of $H_f$ being a hypersurface results from \[bertini-type-0\] (2). Our main task is to reduce to the case $S$ is noetherian and of finite dimension, in order to then apply Theorem \[exist-hyp\]. Using [@EGA], IV.8.9.1 and IV.8.10.5, we find the existence of an affine scheme $S_0$ of finite type over $\mathbb Z$, and of a morphism $S\to S_0$ such that all the objects of Theorem \[bertini-type-0\] descend to $S_0$. More precisely, there exists a quasi-projective scheme $X_0\to S_0$ such that $X$ is isomorphic to $X_0\times_{S_0} S$. We will denote by $p : X\to X_0$ the associated ‘first projection’ morphism. There also exists a very ample sheaf ${{\mathcal O}}_{X_0}(1)$ relative to $X_0 \to S_0$ whose pull-back to $X$ is ${{\mathcal O}}_{X}(1)$. There exists a closed subscheme $C_0$ of $X_0$ such that $C$ is isomorphic to $C_0 \times_{S_0} S$. Finally, there exists subschemes $F_{1,0}, \dots, F_{m,0}$ of $X_0$ such that $F_i$ is isomorphic to $F_{i,0} \times_{S_0} S$. Let $A_0:=p(A)$.
Since $S_0$ is of finite type over $\mathbb Z$, $S_0$ is noetherian and of finite dimension. The data $X_0, C_0, \{ (F_{1,0}\setminus C_0),
\dots, (F_{m,0}\setminus C_0) \}, A_0$ satisfy the hypothesis of Theorem \[exist-hyp\]. Let $n_0> 0$ and, for all $n\ge n_0$, an $f_0\in H^0(X_0, {{\mathcal O}}_{X_0}(n))$ be given by Theorem \[exist-hyp\] with respect to these data. Let $H_{f}$ be the closed subscheme of $X$ define by the canonical image of $f_0$ in $H^0(X, {{\mathcal O}}_X(n))$. Then $H_f=H_{f_0}\times_{S_0} S$ contains $C$ as a closed subscheme and $H_f\cap A=\emptyset$. It remains to check Condition (2) of \[bertini-type-0\]. Let $\xi$ be the generic point of an irreducible component of positive dimension of $F_{i,s}$. Let $s=p(s_0)$ and let $\xi_0=p(\xi)$. Then an open neighborhood of $\xi$ in $(F_i)_s$ has empty intersection with $C$. As $C=C_0\times_{S_0} S$, this implies that the same is true for $\xi_0$ in $(F_{i,0})_s$. Hence $\xi_0$ is the generic point of an irreducible component of $(F_{i,0}\setminus C_0)_s$ of positive dimension. Thus $\xi_0\notin H_{f_0}$ and $\xi\notin H_f$.
The classical Avoidance Lemma states that if $X/k$ is a quasi-projective scheme over a field, $C \subsetneq X$ is a closed subset of positive codimension, and $\xi_1,\dots, \xi_r$ are points of $X$ not contained in $C$, then there exists a hypersurface $H$ in $X$ such that $C \subseteq H$ and $\xi_1, \dots, \xi_r \notin H$.
Let $S$ be a noetherian scheme, and let $X/S$ be a quasi-projective scheme. One may wonder whether it is possible to strengthen Theorem \[bertini-type-0\], the Avoidance Lemma for Families, by strengthening its Condition (\[HF\]). The following example shows that Theorem \[bertini-type-0\] does not hold if Condition (\[HF\]) is replaced by the stronger Condition (\[HF\]’): For all $s \in S$, $H_f$ does not contain any irreducible component of $F_s$.
Let $S={\operatorname{Spec}}R$ be a Dedekind scheme such that ${\mbox{\rm Pic}\kern 1pt}(S)$ is not a torsion group (see, e.g., [@Gol], Cor. 2). Let ${\mathcal L}$ be an invertible sheaf on $S$ of infinite order. Consider as in \[threesections\] the scheme $X={\mathbb P}({{\mathcal O}}_S \oplus {\mathcal L})$ with its natural projective morphism $X \to S$. Let $F$ be the union of the two horizontal sections $C_0$ and $C_{\infty}$. If Theorem \[bertini-type-0\] with Condition (\[HF\]’) holds, then there exists a hypersurface $H_f$ which is a finite quasi-section of $X \to S$ (as defined in \[def.finite-qs\]), and which does not meet $F$. Proposition \[threesections\] shows that this can only happen when ${\mathcal L}$ has finite order.
\[ex.avoidance\] Let $k$ be any field, and let $X/k$ be an irreducible proper scheme over $k$. Let $C \subsetneq X$ be a closed subscheme, and let $\xi_1,\dots, \xi_r$ be points of $X$ not contained in $C$. We may ask whether an Avoidance Lemma holds for $X/k$ in the following senses: (1) Does there exist a line bundle ${\mathcal L}$ on $X$ and a section $f \in {\mathcal L}(X)$ such that the closed subscheme $H_f$ contains $C$ and $\xi_1, \dots, \xi_r \notin H_f$? We may also ask (2) whether there exists a codimension $1$ subscheme $H$ of $X$ such that $H$ contains $C$ and $\xi_1, \dots, \xi_r \notin H$.
The answer to the first question is negative, as there exist proper schemes $X/k$ with ${\mbox{\rm Pic}\kern 1pt}(X)= (0)$. For instance, a normal proper surface $X/k$ over an uncountable field $k$ with ${\mbox{\rm Pic}\kern 1pt}(X)= (0)$ is constructed in [@Sch], section 3.
The answer to the second question is also negative when $X/k$ is not smooth. Recall the example of Nagata-Mumford ([@Art], pp. 32-33). Consider the projective plane ${\mathbb P}^2_k/k$, and fix an elliptic curve $E/k$ in it, with origin $O$. Assume that $E(k)$ contains a point $x$ of infinite order. Fix ten distinct multiples $n_ix$, $i=1,\dots, 10$. Blow up ${\mathbb P}^2_k$ at these ten points to get a scheme $Y/k$. Since $x$ has infinite order, any codimension $1$ closed subset of $Y$ intersects the strict transform $F$ of $E$ in $Y$. Now $F$ has negative self-intersection on $Y$ by construction, and so there exists an algebraic space $Z$ and a morphism $Y \to Z$ which contracts $F$. The algebraic space $Z$ is not a scheme, since the image $z$ of $F$ in $Z$ cannot be contained in an open affine of $Z$. It follows from [@LM], 16.6.2, that there exists a scheme $X$ with a finite surjective morphism $X \to Z$. The finite set consisting of the preimage of $z$ in $X$ meets every codimension $1$ subscheme $H$ of $X$.
Let $S$ be an affine scheme and let $X \to S$ be projective and smooth. Fix a very ample invertible sheaf ${{\mathcal O}}_X(1)$ relative to $X\to S$ as in Theorem \[bertini-type-0\]. It is not possible in general to find $n>0$ and a global section $f \in {{\mathcal O}}_X(n)$ such that $H_f \to S$ is smooth. Examples of N. Fakhruddin illustrating this point can be found in [@Poo], 5.14 and 5.15. M. Nishi ([@Nishi], and [@DH], Remarks, page 80, (b)) gave an example of a nonsingular cubic surface $C$ in ${\mathbb P}^4_k$ which is not contained in any nonsingular hypersurface of ${\mathbb P}^4_k$. Our next two corollaries are examples of weaker ‘theorems of Bertini-type for families’, where for instance smooth is replaced by Cohen-Macaulay.
Recall that a locally noetherian scheme $Z$ is $(\mathrm{S}_\ell)$ for some integer $\ell\ge 0$ if for all $z\in Z$, the depth of ${{\mathcal O}}_{Z,z}$ is at least equal to $\min\{\ell, \dim{{\mathcal O}}_{Z,z}\}$ ([@EGA], IV.5.7.2).
\[bertini-cor1\] Let $S$ be an affine scheme, and let $X\to S$ be a quasi-projective and finitely presented morphism. Let $C$ be a closed subscheme of $X$ finitely presented over $S$. Assume that for all $s \in S$, $C$ does not contain any irreducible component of positive dimension of $X_s$. Suppose that for some $\ell \geq 1$, $X_s$ is $(\mathrm{S}_\ell)$ for all $s \in S$. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $X \to S$.
Then there exists $n_0>0$ such that for all $n\ge n_0$, there exists a global section $f$ of ${{\mathcal O}}_X(n)$ such that $ H_f \subset X$ is a hypersurface containing $C$ as a closed subscheme, and the fibers of $H_f\to S$ are $(\mathrm{S}_{\ell-1})$. In particular, if the fibers of $X\to S$ are Cohen-Macaulay, then the same is true of the fibers of $H_f\to S$. Moreover:
1. Assume that $X_s$ has no isolated point for all $s \in S$. If $X\to S$ is flat, then $H_f\to S$ can be assumed to be flat and locally principal.
2. Assume that $S$ is noetherian and that ${\operatorname{Ass}}(X) \cap C= \emptyset$. Then $H_f \to S$ can be assumed to be locally principal.
We apply Theorem \[bertini-type-0\] to $X\to S$ and $C$, with $F_1=X$, $m=1$ and with $A= \emptyset$. Let $H:= H_f$ be a hypersurface in $X$ as given by \[bertini-type-0\]. For all $s\in S$, $H_s$ does not contain any irreducible component of $X_s$ of positive dimension. Since $X_s$ is $(\mathrm{S}_\ell)$ with $\ell\ge 1$, $X_s$ has no embedded points. At an isolated point of $X_s$ contained in $H$, $H_s$ is trivially $(\mathrm{S}_{k})$ for any $k \geq 0$. At all other points, it follows that $H_s$ is locally generated everywhere by a regular element and, thus, $H_s$ is $(\mathrm{S}_{\ell-1})$.
The statement (a) follows from \[hypersurfaces-properties\] (3). For (b), we apply Theorem \[bertini-type-0\] and Lemma \[hypersurfaces-properties\] (4) to $X\to S$ and $C$, with $F_1=X$ and with $A={\operatorname{Ass}}(X)$.
\[generic-S1\] Let $S$ be an affine irreducible scheme of dimension $1$. Let $X\to S$ be quasi-projective and flat of finite presentation. Assume that its generic fiber is $(\mathrm{S}_1)$, and that it does not contain any isolated point. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $X \to S$. Then there exists $n_0>0$ such that for all $n\ge n_0$, there exists a global section $f$ of ${{\mathcal O}}_X(n)$ such that $H_f \subset X$ is a locally principal hypersurface, flat over $S$.
Consider the set $M$ of all $x \in X$ such that $X_s$ is not $(\mathrm{S}_1)$ at $x$ (or, equivalently, the set of all $x\in X$ such that $x$ is contained in an embedded component of $X_s$). Then this set is constructible ([@EGA], IV.9.9.2 (viii), and even closed since $X\to S$ is flat, [@EGA], IV.12.1.1 (iii)). Since the generic fiber is $(\mathrm{S}_1)$, the image of $M$ in $S$ must be finite because $S$ has dimension $1$. Therefore, there are only finitely many $s \in S$ such that the fiber $X_s$ is not $(\mathrm{S}_1)$. Let $M'$ denote the set of associated points in the fibers that are not $(\mathrm{S}_1)$. This set is finite. Apply now Theorem \[bertini-type-0\] with the very ample sheaf ${{\mathcal O}}_X(1)$ and with $F_i=C=\emptyset$ and $A=M'$, to find a hypersurface $H_f$ which does not intersect $M'$. This hypersurface is locally principal and flat over $S$ by \[hypersurfaces-properties\] (3). Indeed, by construction, $(H_f)_s$ does not contain any irreducible component of $X_s$ of positive dimension. Our hypothesis on the generic fiber having no isolated point implies that $X_s$ has no isolated point for all $s$, since the set of all points that are isolated in their fibers is open ([@EGA], IV.13.1.4).
We discuss below one additional application of Theorem \[globalize\].
\[generic\_smoothness\] Let $S={\operatorname{Spec}}R$ be an affine scheme and let $\pi:X\to S$ be projective and finitely presented. Let $C$ be a closed subscheme of $X$, finitely presented over $S$. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $X \to S$. Let $Z \subset S$ be a finite subset. Suppose that
1. $\pi:X\to S$ is smooth at every point of $\pi^{-1}(Z)$;
2. for all $s\in Z$, $C_{s}$ is smooth and ${{\operatorname{codim}}}_x(C_s, X_s)>\frac{1}{2}\dim_x X_s$ for all $x\in C_s$;
3. for all $s\in S$, $C_s$ does not contain any irreducible component of positive dimension of $X_s$, and for all $s \in Z$, $X_s$ has no isolated point.
Then there exists an integer $n_0$ such that for all $n \geq n_0$, there exists a global section $f$ of ${{\mathcal O}}_X(n)$ such that $H_f$ is a hypersurface containing $C$ as a closed subscheme and such that $H_f\to S$ is smooth in an open neighborhood of $\pi^{-1}(Z)\cap H_f$.
By arguing as in the proof of Theorem \[bertini-type-0\] (using also [@EGA], IV.11.2.6, or [@EGA], IV.17.7.8), we find that it suffices to prove the proposition in the case where $S$ is noetherian and has finite dimension.
Let ${{\mathcal J}}$ be the ideal sheaf defining $C$. As in the proof in \[Proofreductiondimension2\] of Theorem \[exist-hyp\], when $m=1$ and $F_1=X$, there exists $n_0$ such that for any $n\ge n_0$ and for any choice of generators $f_1, \dots, f_N$ of $H^0(X, {{\mathcal J}}(n))$, the associated constructible subset $T_X\subseteq \mathbb A^N_S$ (see \[remove-isolated-pts\]) has dimension at most $ N -1$. For each $s\in Z$, we can apply Lemma \[constructible-sm\] to the following data: the morphism $X_s \to {\operatorname{Spec}}k(s)$, $C_s\subseteq X_s$ defined by the ideal sheaf ${ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s \subseteq {{\mathcal O}}_{X_s}$ (notation as in \[emp.notation\]), and the sections $\overline{f}_{i,s}$, image of $f_i$ in $H^0(X_s,{ \vbox{ \hrule height 0.5pt \kern0.25ex \hbox{ \kern-0.1em \ifmmode{{\mathcal J}}\else\ensuremath{{{\mathcal J}}}\fi \kern-0.1em } }}_s(n))$, $i=1,\dots, N$. We denote by $T_{Z,s}$ the constructible subset of $\mathbb A^N_{k(s)}$ associated in Lemma \[constructible-sm\] (1) with this data. Note that $T_{Z,s}$ is then a pro-constructible subset of $\mathbb A^N_S$.
Let $n$ and $ f_1, \dots, f_N$ be as above. Consider the finite union $T$ of $T_X$ and the pro-constructible subsets $T_{Z,s}$, $s\in Z$. It is pro-constructible with constructible fibers over $S$. Lemma \[constructible-sm\] shows that Conditions (1) and (2) in \[globalize\] (with $V=S$) are satisfied, after increasing $n_0$ if necessary. Then by Theorem \[globalize\], there exists a section $(a_1,\ldots, a_N)\in R^N=\mathbb A_S^N(S)$ such that for all $s \in S$, $(a_1(s), \ldots, a_N(s))$ is a $k(s)$-rational point of $\mathbb A^N_{k(s)}$ that is not contained in $T_s$. Let $f:=\sum_{i=1}^N a_if_i$ and consider the closed subscheme $H_f \subset X$. As $f\in H^0(X, {{\mathcal J}}(n))$, $C$ is a closed subscheme of $H_f$. By definition of $T_{X}$, we find that for all $s\in S$, $H_f$ does not contain any irreducible component of $X_s$ of positive dimension, so that $H_f$ is a hypersurface relative to $X \to S$. By definition of $T_{Z,s}$, we find that $(H_f)_s$ is smooth for all $s\in Z$. Since $X \to S$ is flat in a neighborhood $U$ of $\pi^{-1}(Z)$, and since $X_s$ does not contain any isolated point when $s \in Z$, we find that $U \cap H_f \to S $ is flat at every point of $\pi^{-1}(Z)$ (\[hypersurfaces-properties\] (3)). So $H_f\to S$ is smooth in a neighborhood of $\pi^{-1}(Z)$ by the openness of the smooth locus.
\[constructible-sm\] Let $S$ be a noetherian scheme. Let $X\to S$ be a quasi-projective morphism. Let ${{\mathcal O}}_X(1)$ be a very ample sheaf relative to $X \to S$. Let $C$ be a closed subscheme of $X$ defined by a sheaf of ideals ${{\mathcal J}}$. Let $n\ge 1$ and let $f_1, \dots, f_N \in H^0(X, {{\mathcal J}}(n))$.
1. Define for any $s\in S$ $$\Sigma_{{\mathrm{sing}}}(s):=
\left\{ (a_1,\dots, a_N) \in k(s)^N \mid V_+(\sum_i a_i \overline{f}_{i,s})\subseteq X_{s}
\text{\ is not smooth over } k(s)\right\}.$$ Then there exists a constructible subset $T_{{\mathrm{sing}}}$ of ${\mathbb A}^N_S$ such that for all $s\in S$, the set of $k(s)$-rational points of ${\mathbb A}^N_S$ contained in $T$ is equal to $\Sigma_{{\mathrm{sing}}}(s)$. Moreover, $T_{{\mathrm{sing}}}$ is compatible with base changes $S'\to S$ as in Proposition [\[constructible-conditions\] (a)]{}.
2. Let $k$ be a field and assume that $S={\operatorname{Spec}}k$. Suppose also that $C$ and $X$ are projective and smooth over $k$, and that ${{\operatorname{codim}}}_x(C, X)> \frac{1}{2}\dim_x X$ for all $x\in C$.
Then there exists $n_0$ such that for any $n\ge n_0$ and for any choice of a system of generators $f_1, \dots, f_N$ of $H^0(X, {{\mathcal J}}(n))$, we have $\dim T_{\mathrm{sing}}\le N-1$, and there exists $(a_1, \dots, a_N)\in k^N$ such that $V_+(\sum_i a_if_i)$ is smooth and does not contain any irreducible component of $X$.
\(1) Write ${\mathbb A}^N_S={\operatorname{Spec}}{{\mathcal O}}_S[u_1, \dots, u_N]$. Consider the natural projections $$p: {\mathbb A}^N_X\to X, \quad
q: {\mathbb A}^N_X\to \mathbb A^N_S.$$ With the appropriate identifications, consider the global section $\sum_{1\le i\le N} u_i f_i$ of $p^*({{{\mathcal J}}}(n))$, which defines the closed subscheme $Y:=V_+(\sum_{1\le i\le N} u_i {f}_i )$ of ${\mathbb A}^N_X$. Consider $$\mathcal S: =\{ y\in Y \mid Y_{q(y)} \ \text{is not smooth at $y$ over }
k(q(y))\}.$$ Set $T_{{\mathrm{sing}}}:=q(\mathcal S)$. Clearly $T_{{\mathrm{sing}}}$ satisfies all the requirements of the lemma except for the constructibility, which we now prove. By Chevalley’s theorem, it is enough to show that $\mathcal S$ is constructible. The subset $\mathcal S$ is the union for $0\le d\le \max_{z\in {\mathbb A}^N_S} \dim Y_z$ ([@EGA], IV.13.1.7) of the subsets $$\mathcal S_d:=\{ y\in Y \mid \dim_y Y_{q(y)} \le d <
\dim_{k(y)} (\Omega^1_{Y/{\mathbb A}^N_S}\otimes k(y))\}.$$ Thus it is enough to show that $\mathcal S_d$ is constructible. The set $\{ y\in Y \mid \dim_y Y_{q(y)} \le d \}$ is open by Chevalley’s semi-continuity theorem ([@EGA], IV.13.1.3). On the other hand, for any coherent sheaf ${{\mathcal F}}$ on $Y$, the subset $\{ y\in Y \mid \dim_{k(y)}({{\mathcal F}}\otimes k(y)) \ge d+1\}$ is closed in $Y$. So $\mathcal S_d$ is constructible and (1) is proved.
\(2) By the compatibility with base changes, the dimension of $T_{{\mathrm{sing}}}$ can be computed over an algebraic closure of $k$. Now over an infinite field, Theorem (7) of [@KA], p. 787, in the simplest possible case, where the subvariety is smooth and equal to the open subset in Theorem (7), implies that the generic point of ${\mathbb A}^N_k$ is not contained in $T_{\mathrm{sing}}$ for all $n$ big enough and for all systems of generators $\{f_1,\dots, f_N\}$ of $H^0(X, {{\mathcal J}}(n))$. As $T_{\mathrm{sing}}$ is constructible, we have $\dim T_{\mathrm{sing}}\le N-1$.
Let $\Gamma_1,\dots, \Gamma_m$ be the connected components of $X$. They are irreducible and $$H^0(X, {{\mathcal J}}(n))=\oplus_{1\le i\le m} H^0(\Gamma_i, {{\mathcal J}}(n)|_{\Gamma_i}).$$ By hypothesis, $\dim C\cap \Gamma_i < \frac{1}{2} \dim \Gamma_i$. So by [@KA], loc. cit., when $k$ is infinite, and by [@Poo2], Theorem 1.1 (i) when $k$ is finite, increasing $n_0$ if necessary, for any $n\ge n_0$, there exists $g_i\in H^0(\Gamma_i, {{\mathcal J}}(n)|_{\Gamma_i})$ such that $V_+(g_i)\subset \Gamma_i$ is smooth and of dimension $\dim\Gamma_i -1$. Let $f:=g_1\oplus \cdots \oplus g_m\in H^0(X, {{\mathcal J}}(n))$. Then $V_+(f)$ is a smooth subvariety of $X$ not containing any irreducible component of $X$. Let $f_1,\dots, f_N$ be any system of generators of $H^0(X, {{\mathcal J}}(n))$, and write $f=\sum_i a_if_i$ with $a_i\in k$. Then $(a_1,\dots, a_N)\in k^N$ is the desired point.
[Finite quasi-sections]{} \[finite-qs\]
Let $X\to S$ be a surjective morphism. We call a closed subscheme $T$ of $X$ a *finite quasi-section* when $T \to S$ is finite and surjective (\[def.finite-qs\]). We establish in \[quasisections\] the existence of a finite quasi-section for certain types of projective morphisms. The existence of quasi-finite quasi-sections locally on $S$ for flat or smooth morphisms is discussed in [@EGA], IV.17.16.
When $S$ is integral noetherian of dimension $1$ and $X\to S$ is proper, the existence of a finite quasi-section $T$ is well-known and easy to establish. It suffices to take $T$ to be the Zariski closure of a closed point of the generic fiber of $X\to S$. Then $T\to S$ have fibers of dimension $0$ (see, e.g., [@Liubook], 8.2.5), so it is quasi-finite and proper and, hence, finite. When $\dim S>1$, the process of taking the closure of a closed point of the generic fiber does not always produce a closed subset [*finite*]{} over $S$, as the simple example below shows.
\[easy\] Let $S={\operatorname{Spec}}A$ with $A$ a noetherian integral domain, and let $K={\operatorname{Frac}}(A)$. Let $X=\mathbb P^1_A$. Choose coordinates and write $X = {\rm Proj \ } A[t_0,t_1]$. Let $P\in X_K(K)$ be given as $(a:b)$, with $a, b\in A\setminus \{ 0 \}$. When $(bt_0-at_1)$ is a prime ideal in $A[t_0, t_1]$, then $T:=V_+(bt_0-at_1)$ is the Zariski closure of $P$ in $X$. When in addition $aA+bA\ne A$, $T$ is not finite over $S$. For a concrete example with $S$ regular of dimension $2$, take $k$ a field and $A=k[t, s]$, with $a=t$, and $ b=s$. (Note that when $\dim(A)=1$ and $aA+bA\ne A$, the ideal $(bt_0-at_1)$ is never prime in $A[t_0,t_1]$).
More generally, to produce $K$-rational points on the generic fiber of ${\mathbb P}^n_S \to S$ for some $n>1$ whose closure is not finite over $S$, we can proceed as follows. Let $T \to S$ be the blowing-up of $S$ with respect to a coherent sheaf of ideals $I$, and choose $I$ so that $T\to S$ is not finite. Then $T\to S$ is a projective morphism, and we can choose $T\to {\mathbb P}^n_S$ to be a closed immersion over $S$ for some $n>0$. Let $\xi$ denote the generic point of the image of $T$ in $X:={\mathbb P}^n_S$. Then $\xi$ is a closed point of the generic fiber of $X\to S$, and the closure of $\xi$ in $X$ is not finite over $S$.
The composition $\mathbb P^d_T= {\mathbb P}^d_S \times T \to T \to S$ is an example of a projective morphism which does not have any finite quasi-section. In this example, one irreducible fiber has dimension greater than $d$.
Before turning to the main theorem of this section, let us note here an instance of interest in arithmetic geometry where the closure of a rational point of the generic fiber is a section.
\[closurequasisection\] Let $S$ be a noetherian regular integral scheme, with function field $K$. Let $X\to S$ be a proper morphism such that no geometric fiber $X_{\bar{s}}$ contains a rational curve. Then any $K$-rational point of the generic fiber of $X\to S$ extends to a section over $S$.
Let $T$ be the (reduced) Zariski closure of a rational point of the generic fiber of $X\to S$. Consider the proper birational morphism $f: T\to S$. Denote by $E:=E(f)$ the exceptional set of $f$, that is, the set of points $x \in T$ such that $f$ is not a local isomorphism at $x$. Suppose $E\ne\emptyset$. Since $S$ is regular, by van der Waerden’s purity theorem ([@EGA], IV.21.12.12 or [@Liubook], 7.2.22), $E$ has pure codimension $1$ in $T$. Let $\xi$ be a generic point of $E$ and let $s=f(\xi)$. Using the dimension formula ([@EGA], IV.5.5.8, [@Liubook], 8.2.5) and because $S$ is regular hence universally catenary ([@EGA], IV.5.6.4), we find $$\mathrm{trdeg}_{k(s)} k(\xi)=\dim {{\mathcal O}}_{S,s}-1.$$ Let $\overline{T}\to T$ be the normalization of $T$ and let $\eta$ be a point of $\overline{T}$ lying over $\xi$. Then by Krull-Akizuki, ${{\mathcal O}}_v:={{\mathcal O}}_{\overline{T}, \eta}$ is a discrete valuation ring. It has center $s$ in $S$. As $k(\eta)$ is algebraic (even finite) over $k(\xi)$, we have $\mathrm{trdeg}_{k(s)} k(\eta)=\dim
{{\mathcal O}}_{S,s}-1$. So ${{\mathcal O}}_v$ is a prime divisor of $K(S)$ in the sense of [@Abh], Definition 1. It follows from a theorem of Abhyankar ([@Abh], Proposition 3) that $k(\eta)$ is the function field of a ruled variety of positive dimension over $k(s)$. One can also prove this result in a more geometric flavor as in [@Liubook], Exercise 8.3.14 (a)-(b) (the hypothesis that the base scheme is Nagata is not needed in our situation as the local rings which intervene are all regular). So $\overline{T}_{\bar{s}}$ contains a rational curve. As $\overline{T}_{\bar{s}}\to T_{\bar{s}}$ is integral, the image of such a curve is a rational curve in $T_{\bar{s}}$. It follows that $X_{\bar{s}}$ contains a rational curve, and this is a contradiction. So $E$ is empty and $T\to S$ is an isomorphism.
\[quasisections\] Let $S$ be an affine scheme and let $\pi:X\to S$ be a projective, finitely presented morphism. Suppose that all fibers of $X\to S$ are of the same dimension $d\ge 0$. Let $C$ be a finitely presented closed subscheme of $X$,
with $C \to S$ finite but not necessarily surjective. Then there exists a finite quasi-section $T \to S$ of finite presentation which contains $C$. Moreover:
1. Assume that $S$ is noetherian. If $C$ and $X$ are both irreducible, then there exists such a quasi-section with $T$ irreducible.
2. If $X\to S$ is flat with Cohen-Macaulay fibers (e.g., if $S$ is regular and $X$ is Cohen-Macaulay), then there exists such a quasi-section with $T\to S$ flat.
3. If $X \to S$ is flat and a local complete intersection morphism[^13], then there exists such a quasi-section with $T\to S$ flat and a local complete intersection morphism.
4. Assume that $S$ is noetherian. Suppose that $\pi:X \to S$ has fibers pure of the same dimension, and that $C \to S$ is unramified. Let $Z$ be a finite subset of $S$ (such as the set of generic points of $\pi(C)$), and suppose that there exists an open subset $U$ of $S$ containing $Z$ such that $X \times_S U \to U$ is smooth. Then there exists such a quasi-section $T$ of $X \to S$ and an open set $V \subseteq U$ containing $Z$ such that $T \times_S V \to V$ is étale.
To prove the first conclusion of the theorem, it suffices to show that $X/S$ has a finite quasi-section $T$ of finite presentation. Then $T \cup C$ is a finite quasi-section which contains $C$. If $d=0$, then $X\to S$ itself is finite. Suppose $d\ge 1$. It follows from Theorem \[bertini-type-0\], with $ A=\emptyset$ and $F=\emptyset$, that there exists a hypersurface $H$ in $X$. By definition of a hypersurface, for all $s \in S$, $H_s$ does not contain any irreducible component of $X_s$ of positive dimension. Lemma \[hypersurfaces-properties\](1) and our hypotheses show that every fiber $H_s$ has dimension $d-1$. Lemma \[hypersurfaces-properties\](2) shows that $H/S $ is also finitely presented. Repeating this process another $d-1$ times produces the desired quasi-section.
\(1) Since $X$ is assumed irreducible and since the fibers of $X \to S$ are all not empty by hypothesis, we find that $X\to S$ is surjective and that $S$ is irreducible. When $d=0$, $X\to S$ is then an irreducible finite quasi-section, and contains $C$ as a closed subscheme. Assume now that $d \geq 1$. Then we can find a hypersurface $H_f$ which contains $C$ as a closed subscheme (\[bertini-type-0\]). Since $S$ is noetherian, we can use \[components-dom\] below and the assumption that $C$ is irreducible to find an irreducible component $\Gamma$ of $H_f$ which contains (set-theoretically) $C$, dominates $S$, and such that all fibers of $\Gamma \to S$ have dimension $d-1$. Let ${{\mathcal J}}_C$ and ${{\mathcal J}}_{\Gamma}$ denote the sheaves of ideals in ${{\mathcal O}}_X$ defining $C$ and $\Gamma$, respectively. Then some positive power ${{\mathcal J}}_{\Gamma}^m$ is contained in ${{\mathcal J}}_C$, and we endow the irreducible closed set $\Gamma$ with the structure of scheme given by the structure sheaf ${{\mathcal O}}_X/{{\mathcal J}}_{\Gamma}^m$. By construction, the scheme $\Gamma$ is irreducible and contains $C$ as a closed subscheme. If $d-1>0$, we repeat the process with $\Gamma \to S$.
\(2) When $d=0$, the statement is obvious. Assume now that $d >0$. Since $X_s$ has no embedded point for all $s \in S$, we find that for each $i \geq 0$, the set $X_i$ of all $x \in X$ such that every irreducible component of $X_{\pi(x)}$ passing through $x$ has dimension $i$ is open in $X$ ([@EGA], IV.12.1.1 (ii), using here that $X \to S$ is flat). Moreover, since $X_s$ is Cohen-Macaulay for all $s$, the irreducible components of $X_s$ passing through a given point $x$ have the same dimension. We find that $X$ is the disjoint union of the open sets $X_i$. Each $X_i \to S$ is of finite presentation, since each $X_i$ is open and closed in $X$ ([@EGA], IV.1.6.2 (i)).
Consider now $X_0 \to S$, which is clearly quasi-finite of finite presentation and flat. Since $X \to S $ is projective, $X_0 \to S$ is then also finite ([@EGA] IV.8.11.1). We apply Corollary \[bertini-cor1\] (a) to the finitely presented scheme $X':=(X \setminus X_0) \to S$ and the finite quasi-section $C' := C \times_X X'$. We obtain a hypersurface $H'$ containing $C'$, and using the same method as in the proof of the first statement of the theorem, we obtain a finite flat quasi-section $T'$ of $X' \to S$ containing $C'$. Then $T:= T' \cup X_0$ is the desired finite flat quasi-section.
To prove (3), we proceed as in (2), and remark that the hypersurface $H'$ obtained from Corollary \[bertini-cor1\] (a) is flat and locally principal, so that its fiber ${H'}_s$ is l.c.i. over $k(s)$ when $X_s$ is. By hypothesis, $X_0/S$ has only l.c.i fibers, and (3) follows.
\(4) When $d=0$, $X \to S$ is the desired finite quasi-section, since it is étale over the given open subset $U$ of $S$. Assume now that $d>0$. By hypothesis, $C \to S$ is finite and unramified, so that for each $s \in S$, $C_s \to {\operatorname{Spec}}k(s)$ is smooth. Moreover, since we are assuming that the fibers are pure of dimension $d$, Condition (iii) in \[generic\_smoothness\] is satisfied. We can therefore apply Proposition \[generic\_smoothness\] with $Z$, to find a hypersurface $H_f$ of $X\to S$ containing $C$ as a closed subscheme, with $H_f$ smooth over an open neighborhood $W$ of $Z$ in $S$. For all $s\in S$, $X_s$ is pure of dimension $d$ and $(H_f)_s$ is a hypersurface in $X_s$. Thus, $(H_f)_s$ is pure of dimension $d-1$ for all $s\in S$. Therefore, the above discussion can be applied to the morphism $H_f \to S$, which induces a smooth morphism $H_f\times_S W\to W$, to produce a hypersurface $H_{f_2} $ of $H_f\to S$ containing $C$ as a closed subscheme, with $H_{f_2}$ smooth over an open neighborhood $W_2$ of $Z$ in $S$. Thus, we obtain the desired finite quasi-section after $d$ such steps.
\[components-dom\] Let $S$ be affine, noetherian, and irreducible, with generic point $\eta$. Let $\pi: X\to S$ be a morphism of finite type. For each irreducible component $\Delta $ of $X$, suppose that $\Delta \to S$ has generic fiber of positive dimension. Let $\mathcal L$ be an invertible sheaf on $X$ with a global section $f$, and assume that $H:=H_f \subset X$ is a hypersurface relative to $X \to S$. Then:
1. Each irreducible component $\Gamma$ of $H$ dominates $S$.
2. Assume in addition that for some $d\ge 1$, the fibers of each morphism $\Delta \to S$ all have dimension $d$. Then $X\to S$ is equidimensional of dimension $d$ and $H\to S$ is equidimensional of dimension $d-1$.
\(1) Apply [@EGA], IV.13.1.1, to each morphism $\Delta \to S$ to find that for all $s\in S$, all irreducible components of $X_s$ have positive dimension. Let $\Gamma$ be an irreducible component of $H$. Let $Z$ denote the Zariski closure of $\pi(\Gamma)$ in $S$. We need to show that $Z=S$. Let us first show by contradiction that ${{\operatorname{codim}}}(\Gamma, X_Z)>0$. Otherwise, $\Gamma$ contains an irreducible component $T$ of $X_Z$. Let $t$ be the generic point of $T$. Since $T_{\pi(t)}$ is irreducible and dense in $T$, it is an irreducible component of $X_{\pi(t)}$. In particular, $T_{\pi(t)}$ has positive dimension, and is contained in $\Gamma$. This contradicts our hypothesis that $H$ is a hypersurface.
Every irreducible component of $X_Z$ is contained in an irreducible component of $X$, and every irreducible component of $X$ has non-empty generic fiber. Thus, if $Z\ne S$, then ${{\operatorname{codim}}}(X_Z, X)>0$, and $${{\operatorname{codim}}}(\Gamma, X)\ge {{\operatorname{codim}}}(\Gamma, X_Z)+{{\operatorname{codim}}}(X_Z, X)\ge 2.$$ This is a contradiction with the inequality ${{\operatorname{codim}}}(\Gamma, X)\leq 1$, which follows from Krull’s Principal Ideal Theorem. Hence, $Z=S$.
\(2) Let us first show that $X\to S$ is equidimensional of dimension $d\ge 1$. The definition of [*equidimensional*]{} is found in [@EGA], IV.13.3.2. We use [@EGA], IV.13.3.3 to prove our claim. Indeed, our hypotheses imply that the image under $X \to S$ of each irreducible component $\Delta$ of $X$ is $S$, and that the generic fibers of all induced morphisms $\Delta \to S$ have equal dimension $d$. Let $x \in \Delta$, and let $s \in S$ be its image. We have $\dim_x \Delta_s \leq \dim \Delta_s=d$. We find using [@EGA], IV.13.1.6, that $\dim_x \Delta_s \geq \dim \Delta_\eta=d$. Our claim follows immediately.
Let now $\Gamma$ be an irreducible component of $H$. We know from (1) that $\Gamma \to S$ is dominant. Thus, using [@EGA], IV.13.3.3, to show that $H \to S$ is equidimensional of dimension $d-1$, it suffices to show that $\Gamma \to S$ is equidimensional of dimension $d-1$. Since $\Gamma_\eta$ is an irreducible component of the hypersurface $H_\eta\cap \Delta_\eta$ of $\Delta_\eta$, we have $ \dim \Gamma_\eta= \dim \Delta_\eta - 1 = \dim X_\eta -1$.
Let $s \in S$ be such that $\Gamma_s $ is not empty. Our hypothesis that $H$ is a hypersurface implies that $\Gamma_s$ does not contain any irreducible component of $X_s$ of positive dimension. Thus, $\dim\Gamma_s\le \dim X_s-1=d-1$. By [@EGA], IV.13.1.6, $\Gamma_s$ is equidimensional of dimension $d-1$. It follows that $\Gamma\to S$ is equidimensional of dimension $d-1$.
Let $S$ be an affine integral scheme. The scheme $X:= {\mathbb P}^1_S \sqcup S$ is an $S$-scheme in a natural way, and every irreducible component of $X$ dominates $S$. Any proper closed subset of $S$ defined by a principal ideal is a hypersurface $H_0$ of $S$. Thus, there exist hypersurfaces $H:=H_1 \sqcup H_0$ of $X$ such that the irreducible component $H_0$ of $H$ does not dominate $S$. As \[components-dom\] (a) shows, this cannot happen when every irreducible component of $X$ has a generic fiber of positive dimension.
Let $S$ be a noetherian affine scheme. A variant of Theorem \[quasisections\] can be obtained when the morphism $\pi:X\to S$ is only assumed to be quasi-projective, but satisfies the following additional condition: There exists a scheme $\overline{X}$ with a projective morphism $\pi': \overline{X} \to S$ having all fibers of dimension $d > 0$, and an open $S$-immersion $X \to \overline{X}$ with dense image and $\dim (\overline{X}\setminus X)<d$. Keeping all other hypotheses of Theorem \[quasisections\] in place, its conclusions then also hold under the above weaker hypotheses on $\pi:X\to S$. The proof of this variant is similar to the proof of Theorem \[quasisections\], and consists in applying Theorem \[bertini-type-0\] $d$ times, starting with the data $\overline{X}$, $C$, $F:=\overline{X}\setminus X$, and the finite set $A$ containing the generic points of $F$.
Let $S$ be an affine scheme, and let $ X \to S$ be a [*smooth*]{}, projective, and surjective, morphism. We may ask whether $X \to S$ always admits a [*finite étale*]{} quasi-section. (The existence of a quasi-finite étale quasi-section is proved in [@EGA], IV.17.16.3 (ii).) The answer to the above question is known in two cases of arithmetic interest.
First, let $S$ be a smooth affine geometrically irreducible curve over a finite field. Let $X \to S$ be a smooth and surjective morphism, with geometrically irreducible generic fiber. Then $X/S$ has a finite étale quasi-section ([@Tam], Theorem (0.1)).
Let now $S = {\operatorname{Spec}}{\mathbb Z}$. The answer to this question in this case is negative, as examples of K. Buzzard [@Buz] show. Indeed, a positive answer to this question over $S = {\operatorname{Spec}}{\mathbb Z}$ would imply that any smooth, projective, surjective, morphism $X \to {\operatorname{Spec}}{\mathbb Z}$ has a generic fiber which has a ${\mathbb Q}$-rational point. The hypersurface $X/S$ in ${\mathbb P}^7_{\mathbb Z}$ defined by the quadratic form $f(x_1,\dots, x_8)$ associated with the $E_8$-lattice is smooth over $S$ because the determinant of the associated symmetric matrix is $\pm 1$, and the generic fiber of $X/S$ has no ${\mathbb R}$-points because $f$ is positive definite.
Let $S= {\operatorname{Spec}}{{\mathcal O}}_K$, where $K$ is a number field. Let $L/K$ denote the extension maximal with the property that the integral closure ${{\mathcal O}}_L$ of ${{\mathcal O}}_K$ in $L$ is unramified over ${{\mathcal O}}_K$. Does the above question have a positive answer if $L/K$ is infinite? Obviously, if it is possible to find such a $K$ and $L$ where $L \subset {\mathbb R}$, then the example of Buzzard would still show that the answer is negative. We do not know if examples of such $K$ exist.
Some conditions on the dimension of the fibers of a projective morphism $X \to S$ are indeed necessary for a finite quasi-section to exist, as the following proposition shows.
\[fqs-dim\] Let $X$ and $ S$ be irreducible noetherian schemes. Let $\pi : X\to S$ be a proper morphism, and suppose that $\pi $ has a finite quasi-section $T$.
1. Assume that $\pi: X\to S$ is generically finite. Then $\pi$ is finite.
2. Assume that the generic fiber of $X\to S$ has dimension $1$. If $X$ is regular, then for all $s\in S$, $X_s$ has an irreducible component of dimension $1$.
\(a) Since $\pi$ is generically finite and $X$ is irreducible, the generic fiber of $X\to S$ is reduced to one point, namely, the generic point of $X$. Since $T\to S$ is surjective, $T$ meets the generic fiber of $X\to S$, and so it contains the generic point of $X$. Thus, $T=X$ set-theoretically. Since $X_{\mathrm{red}}\subseteq T$, we find that is $X_{\mathrm{red}}$ is finite over $S$. Since $X$ is then quasi-finite and proper, it is finite over $S$.
\(b) Let $\Gamma$ be an irreducible component of $T$ which surjects onto $S$. Let us first show that ${{\operatorname{codim}}}(\Gamma, X)=1$. Let $Y$ be an irreducible closed subset of $X$ of codimension $1$ which contains $\Gamma$. Since the generic fiber of $X\to S$ has dimension $1$, the generic fibers of $\Gamma \to S$ and $Y \to S$ are both irreducible and $0$-dimensional. Hence, these generic fibers are equal. Therefore, $\Gamma=Y$ and $\Gamma$ has codimension $1$ in $X$. Since $X$ is regular, $\Gamma$ is then the support of a Cartier divisor on $X$. By hypothesis, for all $s \in S$, $\Gamma_s$ is not empty, and has dimension $0$. Thus, for all $s\in S$ and all $t\in \Gamma_s$, we have $0=\dim_t \Gamma_s\ge \dim_t X_s -1$. It follows that the irreducible components of $X_s$ which intersect $\Gamma_s$ all have dimension at most $ 1$. Since every irreducible component of $X_s$ has dimension at least $1$ ([@EGA], IV.13.1.1), (b) follows.
As the following example shows, it is not true in general in Proposition \[fqs-dim\] (b) that for all $s \in S$, all irreducible components of $X_s$ have dimension $1$. Let $S$ be regular of dimension $d\ge 2$. Fix a section $T$ of $\mathbb P^1_S\to S$. Let $x_0$ be a closed point of $\mathbb P^1_S$ not contained in $T$ and lying over a point $s\in S$ with $\dim_{s} S=d$. Let $X\to \mathbb P^1_S$ be the blowing-up of $x_0$. Then $X$ is regular, and $X\to S$ has the preimage of $T$ as a section (and thus it has a finite quasi-section). However, $X_{s}$ consists of the union of a projective line and the exceptional divisor $E$ of $X\to \mathbb P^1_S$, which has dimension $d$. So $\dim X_{s}=d\ge 2$.
Our next example shows that some regularity assumption on $X$ is necessary in \[fqs-dim\] (b). Let $k$ be any field, $R:=k[ t_1, t_2]$, and $B:= R[u_0, u_1, u_2]/(t_1u_2-t_2u_1)$. Consider the induced projective morphism $$X:={\operatorname{Proj}}( B) \longrightarrow S:= {\operatorname{Spec}}R = \mathbb A^2_{k}.$$ The scheme $X$ is singular at the point $P$ corresponding to the homogeneous ideal $(t_1,t_2,u_1,u_2)$ of $B$. The fibers of $X \to S$ are isomorphic to $\mathbb P^1_{k(s)}$ if $s \neq (0,0)$. When $s = (0,0)$, then $X_s$ is isomorphic to $\mathbb P^2_{k(s)}$. The morphism $X \to S$ has a finite section $T$, corresponding to the homogeneous ideal $(u_1,u_2)$. As expected in view of the proof of \[fqs-dim\] (b), any section of $X\to S$, and in particular the section $T$, contains the singular point $P$.
We conclude this section with two applications of Theorem \[quasisections\].
\[splitting\] Let $A$ be a commutative ring. Let $M$ be a projective $A$-module of finite presentation with constant rank $r> 1$. Then there exists an $A$-algebra $B$, finite and faithfully flat over $A$, with $B$ a local complete intersection over $A$, such that $M\otimes_A B $ is isomorphic to a direct sum of projective $B$-modules of rank $1$.
Let $S := {\operatorname{Spec}}A$. Let ${\mathcal M}$ denote the locally free ${{\mathcal O}}_S$-module of rank $r$ associated with $M$. Let $X := {\mathbb P}({\mathcal M})$. Then the natural map $X \to S$ is projective, smooth, and its fibers all have dimension $r-1$. We are thus in a position to apply Theorem \[quasisections\] (3) to obtain the existence of a finite flat quasi-section $f:T \to S$ as in \[quasisections\] (3). In particular, $T={\operatorname{Spec}}B$ for some finite and faithfully flat $A$-algebra $B$, with $B$ a local complete intersection over $A$. Moreover, the existence of an $S$-morphism $g:T \to X$ corresponds to the existence of an ${{\mathcal O}}_T$-invertible sheaf ${\mathcal L}_1$ and of a surjective morphism $f^*{\mathcal M} \to {\mathcal L}_1$. Let ${\mathcal M}_1$ denote the kernel of this morphism. The ${{\mathcal O}}_T$-module ${\mathcal M}_1$ is locally free of rank $r-1$, and $f^*{\mathcal M} \cong {\mathcal L}_1 \oplus {\mathcal M}_1$. We may thus proceed as above and use Theorem \[quasisections\] (3) another $r-2$ times to obtain the conclusion of the corollary.
The proposition strengthens, in the affine case, the classical splitting lemma for vector bundles ([@F-L], V.2.7). When $A$ is of finite type over an algebraically closed field $k$ and is regular, it is shown in [@Sum], 3.1, that it is possible to find a finite faithfully flat [*regular*]{} $A$-algebra $B$ over which $M$ splits.
We provide now an example of a commutative ring $A$ with a finitely generated projective module $M$ which is not free and such that it is not possible to find a finite *étale* $A$-algebra $B$ which splits $M$ into a direct sum of rank $1$ projective modules. For this, we exhibit a ring $A$ such that the étale fundamental group of ${\operatorname{Spec}}A$ is trivial and such that ${\mbox{\rm Pic}\kern 1pt}(A) = (0)$. Then, if a projective module $M$ of finite rank is split over a finite *étale* $A$-algebra $B$, it must be split over $A$. Since ${\mbox{\rm Pic}\kern 1pt}(A)=(0)$, we find then that $M$ is a free module. Let $n >2$ and consider the algebra $$A:=\mathbb C[x_1,\dots, x_{2n}]/(x_1^2+ \dots + x_{2n}^2 -1).$$ This ring is regular, and it is well-known that it is a UFD, so that ${\mbox{\rm Pic}\kern 1pt}(A)=(0)$ (see, e.g., [@Swan], Theorem 5). It is shown in [@ST], Theorem 3.1 (use $p=2$), that for each $n>2$, there exists a projective module $M$ of rank $n-1$ which is not free. Let now $X:={\operatorname{Spec}}A$. The étale fundamental group of $X$ is trivial if the topological fundamental group of $X({\mathbb C})$ is trivial (use [@SGA1], XII, Corollaire 5.2). The topological fundamental group of $X({\mathbb C})$ is trivial because there exists a retraction $X({\mathbb C}) \to S^{2n-1}$, where $S^{2n-1}$ is the real sphere in ${\mathbb R}^{2n}$ given by the equation $x_1^2+ \dots + x_{2n}^2 =1$ (see, e.g., [@Wood], section 2). It is well-known that the fundamental group of $S^{2n-1}$ is trivial for all $n \geq 2$. Hence, the module $M$ cannot be split after a finite étale base change.
Let $S$ be a scheme and let $U\subseteq S$ be an open subset. Given a family $C \to U$ of stable curves over $U$, conditions are known (see, e.g., [@dJ-O]) to insure that this family extends to a family of stable curves over $S$. It is natural to consider the analogous problem of extending a given family $D \to Z$ of stable curves over a closed subset $Z$ of $S$. For this, we may use the existence of finite quasi-sections in appropriate moduli spaces, as in the proposition below.
Let $\overline{\mathcal M}:=\overline{\mathcal M}_{g,S}$ be the proper Deligne-Mumford stack of stable curves of genus $g$ over $S$ (see [@D-M], 5.1). Our next proposition uses the following statement: [*Over $S={\operatorname{Spec}}\mathbb Z$, the stack $\overline{\mathcal M}_{g,S}$ admits a coarse moduli space $\overline{M}_{g,\mathbb Z}$ which is a projective scheme over ${\operatorname{Spec}}\mathbb Z$*]{}. Such a statement is found in an appendix in GIT [@Mum], page 228, with a sketch of proof. See also [@Kol], 5.1, for another brief proof.
\[extension.stable.curve\] Let $S$ be a noetherian affine scheme. Let $Z$ be a closed subscheme of $S$, and let $D\to Z$ be a stable curve of genus $g\ge 2$. Then there exist a finite surjective morphism $S'\to S$ mapping each irreducible component of $S'$ onto an irreducible component of $S$, a finite surjective morphism $Z' \to Z$, a closed $S$-immersion $Z' \to S'$, and a stable curve $\mathcal D\to S'$ of genus $g$ with a morphism $D\times_Z Z' \to \mathcal D$ such that the diagram below commutes and the top square in the diagram is cartesian: $$\xymatrix{
D\times_Z Z' \ar[d] \ar@{^{(}->}[r] & {\mathcal D} \ar[d] \\
Z' \ar@{->>}[d] \ar@{^{(}->}[r] & S' \ar@{->>}[d]\\
Z \ar@{^{(}->}[r] & S. \\
}$$
Let $\overline{\mathcal M}:=\overline{\mathcal M}_{g,S}$ be the proper Deligne-Mumford stack of stable curves of genus $g$ over $S$ (see [@D-M], 5.1). We first construct a finite surjective morphism $X\to \overline{\mathcal M}$ such that $X$ is a scheme, projective over $S$ and with constant fiber dimensions over $S$. It is known that over $\mathbb Z$, the coarse moduli space $\overline{M}_{g,\mathbb Z}$ of $\overline{\mathcal M}$ is a projective scheme and that its fibers over ${\operatorname{Spec}}{\mathbb Z}$ are all geometrically irreducible of the same dimension $3g-3$. Let $\overline{M}:=\overline{M}_{g,\mathbb Z}\times_{{\operatorname{Spec}}{\mathbb Z}} S$. Then we have a canonical morphism $\overline{\mathcal M}\to \overline{M}$ which is proper and a universal homeomorphism (hence quasi-finite). By construction, the $S$-scheme $\overline{M}$ is projective with constant fiber dimension.
Since $\overline{\mathcal M}$ is a noetherian separated Deligne-Mumford stack, there exists a (representable) finite surjective morphism from a scheme $X$ to $\overline{\mathcal M}$ ([@LM], 16.6). The composition $X \to\overline{\mathcal M}\to \overline{M}$ is a finite (because proper and quasi-finite) surjective morphism of schemes. Thus $X\to S$ is projective since $S$ is affine and $\overline{M}\to S$ is projective. So $X\to S$ is projective and all its fibers have the same dimension.
The curve $D\to Z$ corresponds to an element in the set $\overline{\mathcal M}(Z)$, which in turn corresponds to a finite morphism $Z\to \overline{\mathcal M}$. So $Z':=Z\times_{\overline{\mathcal M}} X$ is a scheme, finite surjective over $Z$ and finite over $X$. Let $Z_0$ denote the schematic image of $Z'$ in $X$. It is finite over $S$.
To be able to apply Theorem \[quasisections\] (1), we note the following. Let $T$ be the disjoint union of the reduced irreducible components of $S$. Replacing if necessary $S$ with $T$ and $D\to Z$ with $D\times_S T\to Z\times_S T$, we easily reduce the proof of the proposition to the case where $S$ is irreducible. Once $S$ is assumed irreducible, we use the fact that $\overline{M}\to S$ is proper with irreducible fibers to find that $\overline{M}$ is also irreducible. Replacing $X$ by an irreducible component of $X$ which dominates $\overline{M}$, we can suppose that $X$ is irreducible.
Theorem \[quasisections\] (1) can then be applied to the morphism $X \to S$ and to each irreducible component of $Z_0$. We obtain a finite quasi-section $S_0$ of $X/S$ containing (set-theoretically) $Z_0$ and such that each irreducible component of $S_0$ maps onto $S$. Modifying the structure of closed subscheme on $S_0$ as in the proof of \[quasisections\] (1), we can suppose that $Z_0$ is a subscheme of $S_0$.
Because $S_0$ is affine, it is clear that there exists a scheme $S'$, finite and faithfully flat (and even l.c.i.) over $S_0$, and a closed immersion $Z' \to S'$ making the following diagram commute: $$\xymatrix{
Z' \ar@{->>}[d] \ar@{^{(}->}[r] & S' \ar@{->>}[d] & \\
Z_0 \ar@{^{(}->}[r] & S_0 .\\
}$$ As $S'\to S_0$ is flat, each irreducible component of $S'$ maps onto an irreducible component of $S_0$, hence onto $S$.
The stable curve $\mathcal D\to S'$ whose existence is asserted in the statement of Proposition \[extension.stable.curve\] corresponds to the element of $\overline{\mathcal M}(S') $ given by the composition of the finite morphisms $S' \to S_0\to X\to \overline{\mathcal M}$.
\[smooth\_cover\_mg\] Consider the finite surjective $S$-morphism $X \to \overline{\mathcal M}$ introduced at the beginning of the proof of \[extension.stable.curve\] above. If we can find such a cover $X\to \overline{\mathcal M}$ such that $X\to S$ is flat with Cohen-Macaulay fibers (resp., with l.c.i. fibers), then using Theorem \[quasisections\] (2) and (3), we can further require in the statement of Proposition \[extension.stable.curve\] that $S'\to S$ be finite and faithfully flat (resp., l.c.i.).
When some prime number $p$ is invertible in ${{\mathcal O}}_S(S)$, then it is proved in [@dJ-P], 2.3.6.(1) and 2.3.7, that there exists such an $X$ which is even smooth over $S$. Therefore, in this case, we can find a morphism $S'\to S$ which is finite, faithfully flat, and l.c.i.
[Moving lemma for 1-cycles]{} \[mv-1c\]
We review below the basic notation needed to state our moving lemma. Let $X$ be a noetherian scheme. Let ${\mathcal Z}(X)$ denote the free abelian group on the set of closed integral subschemes of $X$. An element of ${\mathcal Z}(X)$ is called a *cycle*, and if $Y$ is an integral closed subscheme of $X$, we denote by $[Y]$ the associated element in ${\mathcal Z}(X)$.
Let ${\mathcal K}_X$ denote the sheaf of meromorphic functions on $X$ (see [@K], top of page 204 or [@Liubook], Definition 7.1.13). Let $f\in \mathcal K_X^*(X)$. Its associated principal Cartier divisor is denoted by ${\operatorname{div}}(f)$ and defines a cycle on $X$: $$[{\operatorname{div}}(f)]=\sum_{x} {\operatorname{ord}}_x(f_x)[\overline{\{x\}}]$$ where $x$ ranges through the points of codimension $1$ in $X$, and ${\operatorname{ord}}_x: \mathcal K_{X,x}^* \to \mathbb Z$ is defined, for a regular element of $g \in {{\mathcal O}}_{X,x}$, to be the length of the ${{\mathcal O}}_{X,x}$-module ${{\mathcal O}}_{X,x}/(g)$.
A cycle $Z$ is *rationally equivalent to $0$* or *rationally trivial*, if there are finitely many integral closed subschemes $Y_i$ and non-zero rational functions $f_i$ on $Y_i$ such that $Z=\sum_i [{\operatorname{div}}(f_i)]$. Two cycles $Z$ and $Z'$ are *rationally equivalent* in $X$ if $Z-Z'$ is rationally equivalent to $0$. We denote by ${\mathcal A}(X)$ the quotient of $\mathcal Z(X)$ by the subgroup of rationally trivial cycles.
A morphism of schemes of finite type $\pi : X\to Y$ induces by [*push forward of cycles*]{} a group homomorphism $\pi_*: {\mathcal Z}(X) \to
{\mathcal Z}(Y)$. If $Z $ is any closed integral subscheme of $X$, then $\pi_*([Z]):= [k(Z):k(\overline{\pi(Z)})] [\overline{\pi(Z)}]$, with the convention that $[k(Z):k(\overline{\pi(Z)})]=0$ if the extension $k(Z)/k(\overline{\pi(Z)})$ is not finite.
\[emp.proper\] Let $S$ be a noetherian scheme which is universally catenary and equidimensional at every point (for instance, $S$ is regular). Assume that both $X\to S$ and $Y\to S$ are morphisms of finite type, and let $\pi:X\to Y$ be a [*proper*]{} morphism of $S$-schemes. Let $C$ and $C'$ be two cycles on $X$ which are rationally equivalent. Then $\pi_*(C)$ and $\pi_*(C')$ are rationally equivalent on $Y$ ([@Th], Note 6.7, or Proposition 6.5 and 3.11). We denote by $\pi_*: {\mathcal A}(X) \to {\mathcal A}(Y)$ the induced morphism. For an example showing that the hypotheses on $S$ are needed for $\pi_*: {\mathcal A}(X) \to {\mathcal A}(Y)$ to be well-defined, see [@GLL1], 1.3.
We are now ready to state the main theorem of this section. Recall that the support of a horizontal $1$-cycle $C$ in a scheme $X$ over a Dedekind scheme $S$ is a finite quasi-section (\[ConditionT\*\]). The definitions of [*Condition*]{} (T) and of [*pictorsion*]{} are given in [(\[ConditionT\])]{} and [(\[ConditionT\*\])]{}, respectively.
\[mv-1-cycle-local\] Let $R$ be a Dedekind domain, and let $S:={\operatorname{Spec}}R$. Let $X \to S$ be a flat and quasi-projective morphism, with $X$ integral. Let $C$ be a horizontal $1$-cycle on $X$. Let $F$ be a closed subset of $X$. Assume that for all $s \in S$, $F\cap X_s$ and ${\operatorname{Supp}}(C) \cap X_s$ have positive codimension in $X_s$. Assume in addition that either
1. $R$ is pictorsion and the support of $C$ is contained in the regular locus of $X$, or
2. $R$ satisfies Condition *(T)*.
Then some positive multiple $mC$ of $C$ is rationally equivalent to a horizontal $1$-cycle $C'$ on $X$ whose support does not meet $F$. Under the assumption [(a)]{}, if furthermore $R$ is semi-local, then we can take $m=1$.
Moreover, if $Y \to S$ is any separated morphism of finite type and $h: X \to Y$ is any $S$-morphism, then $h_*(mC)$ is rationally equivalent to $h_*(C')$ on $Y$.
The proof of Theorem \[mv-1-cycle-local\] is postponed to \[mv-1-cycle-local.Proof\]. We first briefly introduce below needed facts about contraction morphisms. We then discuss several statements needed in the proof of \[mv-1-cycle-local\] (b) when $S$ is not excellent.
\[contraction\] Let $R$ be a Dedekind domain, and $S:={\operatorname{Spec}}R$. Let $X\to S$ be a projective morphism of relative dimension $1$, with $X$ integral. Let $C$ be an effective Cartier divisor on $X$, flat over $S$. Then
1. There exists $m_0 \geq 0$ such that the invertible sheaf ${{\mathcal O}}_X(mC)$ is generated by its global sections for all $m\geq m_0 $.
2. The morphism $X':={\operatorname{Proj}}\left(\oplus_{m\ge 0} H^0(X, {{\mathcal O}}_X(mC))\right) \longrightarrow S$ is projective, with $X'$ integral, and the canonical morphism $u : X\to X'$ is projective, with $u_*{{\mathcal O}}_X={{\mathcal O}}_{X'}$ and connected fibers.
3. For any vertical prime divisor $\Gamma$ on $X$, $u|_{\Gamma}$ is constant if $\Gamma\cap {\operatorname{Supp}}C=\emptyset$, and is finite otherwise.
4. Let $Z$ be the union of the vertical prime divisors of $X$ disjoint from ${\operatorname{Supp}}C$. Then $u$ induces an isomorphism $ X\setminus Z\to X'\setminus u(Z)$.
In [@BLR], Theorem 1 in 6.7, a similar statement is proved, with $R$ local, and $X$ normal. (The normality is not assumed in [@Em] and [@Pie]. A global base is considered in [@Liubook], 8.3.30.) We leave it to the reader to check that the proof of [@BLR], 6.7/1, can be used [*mutatis mutandis*]{} to prove \[contraction\]. Part (a) follows from the first part of the proof of 6.7/1. Part (b) follows from 6.7/2. Part (c) follows from the second part of the proof of 6.7/1. We now give a proof of (d). The morphism $u$ is birational because it induces an isomorphism $X_{\eta} \to X'_{\eta}$ over the generic point $\eta$ of $S$, since $C_\eta$ is ample, being effective of positive degree. It follows that $Z$ is the union of finitely many prime divisors of $X$. As $u$ has connected fibers, it follows from (c) that $Z=u^{-1}(u(Z))$. The restriction $v : X\setminus Z\to X'\setminus u(Z)$ of $u$ is thus projective and quasi-finite. Therefore, $v$ is finite and, hence, affine. As ${{\mathcal O}}_{X'\setminus \pi(Z)}=v_*{{\mathcal O}}_{X\setminus Z}$, $v$ is an isomorphism.
Let $K$ be a field of characteristic $p>0$. Let $K':=K^{p^{-\infty}}$ be the perfect closure of $K$. Let $n \geq 0$ and set $q:= p^n$. Let $K^{1/q}$ denote the extension of $K$ in $K'$ generated by the $q$-th roots of all elements of $K$. Let $i: K \to K^{1/q}$ denote the natural inclusion, and let $\rho: K^{1/q}\to K$ be defined by $\lambda\mapsto \lambda^{q}$. The composition $F:= \rho \circ i: K \to K$ is the $q$-th Frobenius morphism of $K$. By definition, given a morphism $Y \to {\operatorname{Spec}}K$, the morphism $Y^{(q)} \to {\operatorname{Spec}}K$ is the base change $(Y \times_{{\operatorname{Spec}}K, F^*} {\operatorname{Spec}}K) \to {\operatorname{Spec}}K$. It follows that we have a natural isomorphism of $K$-schemes: $$\label{Frob}
Y^{(q)}\simeq (Y\times_{{\operatorname{Spec}}K, i^*} {{\operatorname{Spec}}(K^{1/q})})\times_{{\operatorname{Spec}}(K^{1/q}), \rho^*}{\operatorname{Spec}}K.$$
\[normalization-smooth\] Let $K$ be a field of characteristic $p>0$. Let $Y \to {\operatorname{Spec}}K$ be a morphism of finite type, with $Y$ integral of dimension $1$. Then there exists $n\geq 0$ such that the normalization of $(Y^{(p^n)})_{\mathrm{red}}$ is smooth over $K$.
The normalization $Z$ of $(Y_{K'})_{\mathrm{red}}$ is regular and, hence, smooth over the perfect closure $K'$. There is a finite sub-extension $L/K$ of $K'$ such that the curve $Z$ and the morphism $Z\to (Y_{K'})_{\mathrm{red}}$ are defined over $L$. This implies that the normalization of $(Y_L)_{\mathrm{red}}$ is $Z_{/L}$, hence smooth over $L$. Let $q=p^n$ be such that $L\subseteq
K^{1/q}$. As $Z_{/L}\to Y_L$ is finite and induces an isomorphism on the residue fields at the generic points, the same is true for $$Z_{K^{1/q}}\longrightarrow (Y_L)_{K^{1/q}}=Y_{K^{1/q}}.$$ Using $\rho: K^{1/q}\to K$ and , $$(Z_{K^{1/q}})_K \longrightarrow (Y_{K^{1/q}})_K\simeq Y^{(q)}$$ is finite and induces an isomorphism on the residue fields at the generic points. As the left-hand side is smooth, this morphism is the normalization of $(Y^{(q)})_{\mathrm{red}}$.
\[Chow-radicial\] Let $S$ be a universally catenary noetherian scheme which is equidimensional at every point. Let $\pi: X\to X_0$ be a finite surjective morphism of $S$-schemes of finite type, with induced homomorphism of Chow groups $\pi_* : {\mathcal A}(X)\to {\mathcal A}(X_0)$. Then
1. The cokernel of $\pi_*$ is a torsion group.
2. If $\pi$ is a homeomorphism, then the kernel of $\pi_*$ is also a torsion group.
Our hypotheses on $S$ allow us to use \[emp.proper\], so that the morphism $\pi_* : {\mathcal A}(X)\to {\mathcal A}(X_0)$ is well-defined.
\(1) Let $Z_0$ be an integral closed subscheme on $X_0$, and let $Z$ be an irreducible component of $\pi^{-1}(Z_0)$ whose image in $X_0$ is $Z_0$. When $Z$ is endowed with the reduced induced structure, $Z \to Z_0$ is finite and surjective, and $\pi_*[Z]=[k(Z): k(Z_0)][Z_0]$. Hence, the cokernel of $\mathcal Z(X)\to \mathcal
Z(X_0)$ is torsion, and the same holds for the corresponding homomorphism of Chow groups.
\(2) Let $W_0$ be an integral closed subscheme of $X_0$. Since $\pi$ is a homeomorphism, $W:=\pi^{-1}(W_0)$ is irreducible, and we endow it with the reduced induced structure. The induced morphism $\pi: W \to W_0$ is finite and surjective between integral noetherian schemes. Let $f\in k(W_0)$ be a non-zero rational function. Using for instance [@Liubook], 7.1.38, we find that $$\pi_*([\mathrm{div}_W(\pi^*f)])=[k(W): k(W_0)][\mathrm{div}_{W_0}(f)].$$ This implies that for every integer multiple $r $ of $[k(W): k(W_0)]$, $r[\mathrm{div}_{W_0}(f)]=\pi_*(D_r)$ for some principal cycle $D_r$ on $X$.
Now let $Z$ be any cycle on $X$ such that $\pi_*Z$ is principal on $X_0$. Then for a suitable integer $N$, $N\pi_*Z=\pi_*(D)$ for some principal cycle $D$ on $X$. Since $\pi$ is a homeomorphism, $\pi_* : \mathcal Z(X)\to \mathcal Z(X_0)$ is injective. Therefore, $NZ=D$ in $\mathcal Z(X)$, and the class of $NZ$ is trivial in $\mathcal A(X)$.
For our next proposition, recall that a normal scheme $X$ is called [*${\mathbb Q}$-factorial*]{} if every Weil divisor $D$ on $X$ is such that some positive integer multiple of $D$ is the cycle associated with a Cartier divisor on $X$.
\[normalization-finite\] Let $S$ be a Dedekind scheme with generic point $\eta$. Let $X\to S$ be a dominant morphism of finite type, with $X$ integral. Suppose that the normalization of $X_\eta$ is smooth over $k(\eta)$. Then
1. The normalization morphism $\pi: X'\to X$ is finite.
2. If $X$ is normal, then the following properties are true.
1. The completion $\widehat{{{\mathcal O}}}_{X,x}$ is normal for all $x\in X$.
2. The locus $\mathrm{Reg}(X)$ of regular points of $X$ is open in $X$.
3. If $\dim X_\eta=1$ and $S$ satisfies [Condition (T)]{}, then $X$ is $\mathbb Q$-factorial.
When $S$ is assumed to be excellent, then $X$ is also excellent and most of the statements in the proposition follow from this property. The statement \[normalization-finite\](b)(3) can be found in [@MB1], 3.3. We now give a proof of \[normalization-finite\] without assuming that $S$ is excellent.
\(a) We can and will assume that $X$ is affine. As $\pi_\eta: X'_\eta\to X_\eta$ is finite, there exists a factorization $X'\to X''\to X$ with $X''\to X$ finite and birational, and such that $X'_\eta\to X''_\eta$ is an isomorphism (simply take generators of ${{\mathcal O}}_{X'_{\eta}}(X'_\eta)$ which belong to ${{\mathcal O}}_{X'}(X')$). Replacing $X$ with $X''$, we can suppose that $X_\eta$ is smooth. The smooth locus of $X\to S$ is open and contains $X_\eta$, so it contains an open set of the form $X_V:=X\times_S V$ for some dense open subset $V$ of $S$. So $X'_V=X_V$ and we find that $\pi_*{{\mathcal O}}_{X'}/{{\mathcal O}}_X$ is supported on finitely many closed fibers $X_{s_1},\dots, X_{s_n}$.
To show that $\pi$ is finite, it is enough to show that the normalization morphism of $X\times_S {\operatorname{Spec}}({{\mathcal O}}_{S,s_i})$ is finite for all $s_i$. Therefore, we can suppose that $S={\operatorname{Spec}}R$ for some discrete valuation ring $R$. Let $\widehat{R}$ be the completion of $R$. As $X_\eta$ is smooth, the normalization morphism $\widehat{\pi}: (X_{\widehat{R}})'\to X_{\widehat{R}}$ is an isomorphism on the generic fiber. It is finite because $\widehat{R}$ is excellent. By [@Liubook], 8.3.47 and 8.3.48, $\widehat{\pi}$ descends to a finite morphism $Z\to X$ over $R$. By faithfully flat descent, this implies that $Z$ is normal and, thus, isomorphic to $X'$, and that $X'\to X$ is finite and $X'_{\widehat{R}}=(X_{\widehat{R}})'$ is normal.
\(b) Suppose now that $X$ is normal with smooth generic fiber. To prove (1), let $x\in X$ with image $s\in S$. Then ${{\mathcal O}}_{X,x}$ is also the local ring of $X\times_S{\operatorname{Spec}}{{\mathcal O}}_{S,s}$ at $x$. To prove that its completion is normal, we can thus suppose that $S$ is local. We can even restrict to $s$ closed in $S$ as $X_V$ is regular. Let $R={{\mathcal O}}_{S,s}$. We saw above that $X_{\widehat{R}}$ is normal. As $\widehat{{{\mathcal O}}}_{X,x}$ is also the completion of ${{\mathcal O}}_{X_{\widehat{R}},x}$ (see, e.g., [@Liubook], 8.3.49(b)), it is normal because $X_{\widehat{R}}$ is excellent ([@EGA], IV.7.8.3 (vii)).
\(2) We have $\mathrm{Reg}(X)\supseteq X_V$ and $\mathrm{Reg}(X)\cap
X_s=\mathrm{Reg}(X\times_S {\operatorname{Spec}}{{\mathcal O}}_{S,s})$ for all $s\in S\setminus
V$. As $S\setminus V$ consists of finitely many closed points of $S$, $\mathrm{Reg}(X)$ is open by [@EGA], IV.6.12.6 (ii).
\(3) The statement of (3) is proved in [@MB1], Lemme 3.3, provided that the singular points of $X$ are isolated, and that [@MB1], Théorème 2.8, holds when $A={{\mathcal O}}_{X,x}$. In our case, the singular points of $X$ are isolated by (2). Théorème 2.8 in [@MB1] is proved under the hypothesis that $A$ is excellent, but the proof in [@MB1] only uses the fact that the completion of $A$ is normal (in step 2.10). So in our case, this property is satisfied by (1).
\[mv-1-cycle-local.Proof\] [*Proof of Theorem [\[mv-1-cycle-local\]]{} when [(a)]{} holds.*]{} It suffices to prove the theorem in the case where the given $1$-cycle is the cycle associated with an integral closed subscheme of $X$ finite over $S$. We will denote again by $C $ this integral closed subscheme. As in the proof of Theorem 2.3 in [@GLL1], we reduce the proof of \[mv-1-cycle-local\] to the case where $C \to X$ is a regular immersion[^14] as follows.
Proposition 3.2 in [@GLL1] shows the existence of a finite birational morphism $D\to C$ such that the composition $D\to C \to S$ is an l.c.i. morphism. Since $C$ is affine, there exists for some $N \in {\mathbb N}$ a closed immersion $D \to C\times_S \mathbb P^N_S\subseteq X\times_S \mathbb P^N_S$. Note that since $C$ is contained in the regular locus of $X$, then $D$ is contained in the regular locus of $X\times_S \mathbb P^N_S$. We claim that it suffices to prove the theorem for the $1$-cycle $D$ and the closed subset ${\bf F}:= F \times_S \mathbb P^N_S$ in the scheme $X\times_S \mathbb P^N_S$. Indeed, let $D'$ be a horizontal $1$-cycle whose existence is asserted by the theorem in this case, with $mD$ rationally equivalent to $D'$. In particular, ${\operatorname{Supp}}(D') \cap {\bf F} = \emptyset$. Consider the projection $p : X\times_S \mathbb P^N_S\to X$, which is a projective morphism. Then $p_*(D) = C$ because $D\to C$ is birational. It follows from \[emp.proper\] that $mC=p_*(mD)$ is rationally equivalent to the horizontal $1$-cycle $C':= p_*(D')$ on $X$. Moreover, ${\operatorname{Supp}}(C') \cap F = \emptyset$. Since $D/S$ is l.c.i., each local ring ${{\mathcal O}}_{D,x}$, $x \in D$, is an absolute complete intersection ring, and the closed immersion $D \to X\times_S \mathbb P^N_S$ is a regular immersion ([@EGA], IV.19.3.2). Finally, consider a morphism $h:X \to Y$ as in the last statement of the theorem. Apply this statement to $mD$, $D'$, and to the associated morphism $h':X\times_S \mathbb P^N_S \to Y\times_S \mathbb P^N_S$. Since the projection $Y\times_S \mathbb P^N_S \to Y $ is proper, we find as desired that $h_*(mC)$ is rationally equivalent to $h_*(C')$ on $Y$.
Let us now assume that $C \to X$ is a regular immersion. Let $d$ denote the codimension of $C$ in $X$. If $d>1$, we can apply Theorem \[pro.reductiondimension2\] (as stated in the introduction since $C$ is integral) and obtain a closed subscheme $Y$ of $X$ such that $C$ is the support of a Cartier divisor on $Y$ and such that $F \cap Y_s$ is finite for all $s\in S$. Clearly, $C$ is also the support of a Cartier divisor on $Y_{{\mathrm{red}}}$, and on any irreducible component of $Y_{{\mathrm{red}}}$ passing through $C$. Thus, we are reduced to proving the theorem when $X$ is integral of dimension $2$ and $F$ is quasi-finite over $S$. Note that after this reduction process, we cannot and do not assume anymore that $C$ is contained in the regular locus of $X$.
When $d=1$, we do not apply \[pro.reductiondimension2\], but we note that in this case too $F \cap X_s$ is finite for all $s\in S$. Indeed, since $C\to S$ is finite, the generic point of $C$ is a closed point in the generic fiber $X_{\eta} $ of $X \to S$. Since the codimension of $C$ in $X$ is $d=1$, and since the generic fiber is a scheme of finite type over a field, we find that one irreducible component of $X_{\eta} $ has dimension $1$. Since $S$ is a Dedekind scheme and $X\to S$ is flat with $X$ integral, we find that all fibers are equidimensional of dimension $\dim X_{\eta}=1$ ([@Liubook], 4.4.16). Hence, our hypothesis on $F$ implies that $F \to S$ is quasi-finite.
\[endofproof\] Since $X/S$ is quasi-projective and $X$ is integral, there exists an integral scheme $\overline{X}$ with a projective morphism $\overline{X} \to S$ and an $S$-morphism $X \to \overline{X}$ which is an open immersion. Let $\overline{F}$ be the Zariski closure of $F$ in $\overline{X}$. The closed subscheme $\overline{F}$ is finite over $S$ because $F \to S$ is quasi-finite and $S$ has dimension $1$. Recall that by definition, a horizontal $1$-cycle on $X$ is finite over $S$. Hence, $C$ is closed in $\overline{X}$. Since $C$ is the support of a Cartier divisor on $X$, we find that $C$ is also the support of a Cartier divisor on $\overline{X}$. We are thus in a situation where we can consider the contraction morphism $u : \overline{X}\to X'$ associated to ${C}$ in \[contraction\]. Let $Z$ denote the union of the irreducible components $E$ of the fibers of $\overline{X}\to S$ such that $E \cap {\operatorname{Supp}}({C}) = \emptyset$. Let $U= X\setminus (Z\cap X)$. Then ${\operatorname{Supp}}C \subseteq U$, and $u|_U$ is an isomorphism onto its image. Let $F'=u(\overline{F}\cup Z\cup (\overline{X}\setminus X))\cup u({\operatorname{Supp}}(C))$. Then $X'\setminus F'\subseteq u(U)$, and $F'$ is finite over $S$. We endow $F'$ with the structure of a reduced closed subscheme of $X'$.
Now suppose that $R$ is ${\mbox{pictorsion}}$. Then ${\mbox{\rm Pic}\kern 1pt}(F')$ is a torsion group by hypothesis. So, fix $n>0$ such that ${{\mathcal O}}_{X'}(nC)|_{F'}$ is trivial. Since $C$ meets every irreducible component of every fiber of $X'\to S$, the sheaf ${{\mathcal O}}_{X'}(C)$ is relatively ample for $X'\to S$ ([@EGA], III.4.7.1). Let $\mathcal I$ denote the ideal sheaf of $F'$ in $X'$. Then there exists a multiple $m$ of $n$ such that $H^1(X', \mathcal I \otimes {{\mathcal O}}_{X'}(mC))=(0) $. It follows that a trivialization of ${{\mathcal O}}_{X'}(mC)|_{F'}$ lifts to a section $f\in H^0(X', {{\mathcal O}}_{X'}(mC))$.
Recall that by definition, ${{\mathcal O}}_{X'}(mC)$ is a subsheaf of $\mathcal K_{X'}$. We thus consider $f\in H^0(X', {{\mathcal O}}_{X'}(mC)) \subseteq \mathcal K_{X'}(X')$ as a rational function. The support of the divisor ${\rm div}_{X'}(f)+mC$ is disjoint from $F'$ by construction. In particular, it is contained in $u(U)$ and is horizontal, and ${\rm div}_{X'}(f)$ has also its support contained in $u(U)$. Considering the pull-back of the divisors under $\overline{X}\to X'$ shows that the divisor $C':={\rm div}_{\overline{X}}(f)+mC$ is contained in $U$, disjoint from $F$, horizontal and linearly equivalent to $mC$ on $\overline{X}$.
When $R$ is semi-local, the set $F' \subset X'$ is a finite set of points. Thus we may apply Proposition 6.2 of [@GLL1] directly to the Cartier divisor whose support is $u(C)$ to find a Cartier divisor $D$ linearly equivalent to $u(C)$ and whose support does not meet $F'$.
It remains to prove the last statement of the theorem, which pertains to the morphism $h: X \to Y$. To summarize, in the situation of \[mv-1-cycle-local\] (a), when $C$ is integral, we found a closed integral subscheme $W$ of $X$ containing $C$, a projective scheme $\overline{W}/S$ containing $W$ as a dense open subset, and $m\ge 1$ (with $m=1$ when $R$ is semi-local) such that $mC$ is rationally equivalent on $\overline{W}$ to some horizontal $1$-cycle $C'$ contained in $W$. The morphism $h : X\to Y$ in the statement of \[mv-1-cycle-local\] induces an $S$-morphism $h : W\to Y$. Our proof now proceeds as in [@GLL1], proof of Proposition 2.4(2). For the convenience of the reader, we recall the main ideas of that proof here.
Let $g$ be the function on $W$ such that $[{\operatorname{div}}_W(g)] = mC - C'$, Let $\Gamma\subseteq \overline{W}\times_S Y$ be the schematic closure of the graph of the rational map $\overline{W}\dasharrow Y$ induced by $h:W \to Y$. Let $p: \Gamma \to \overline{W} $ and $q: \Gamma\to Y$ be the associated projection maps over $S$. Since $\Gamma$ is integral and its generic point maps to the generic point of $W$, the rational function $g$ on $W$ induces a rational function, again denoted by $g$, on $\Gamma$. As $p : p^{-1}(W)\to W$ is an isomorphism, we let $p^*(C)$ and $p^*(C')$ denote the preimages of $C$ and $C'$ in $p^{-1}(W)$; they are closed subschemes of $\Gamma$. Since $g$ is an invertible function in a neighborhood of $\overline{W} \setminus W$, $[{\operatorname{div}}_\Gamma(g)] =mp^*(C)-p^*(C')$, and $p^*(mC)$ and $p^*(C')$ are rationally equivalent on $\Gamma$. Then, as $q$ is proper and $S$ is universally catenary, $q_*p^*(mC)$ and $q_*p^*C'$ are rationally equivalent in $Y$. Since $h_*C =q_*p^*C$ and $h_*C'= q_*p^*C'$, we find that $h_*(mC)$ is rationally equivalent to $h_*(C') $ in $Y$.
[*Proof of Theorem [\[mv-1-cycle-local\]]{} when [(b)]{} holds.*]{} It suffices to prove the theorem in the case where the given $1$-cycle is the cycle associated with an integral closed subscheme of $X$ finite over $S$. We will denote again by $C $ this integral closed subscheme. By hypothesis, $X \to S$ is flat, so all its fibers are of the same dimension $d$.
Since $C$ is not empty, $C_s$ is not empty, and thus has positive codimension in $X_s$ by hypothesis. Therefore, we find that $d \geq 1$. Moreover, $C$ does not contain any irreducible component of positive dimension of $F_s$ and of $X_s$. If $d>1$, we fix a very ample invertible sheaf ${{\mathcal O}}_X(1)$ on $X$ and apply Theorem \[exist-hyp\] to $X \to S$, $C$, and $F$, to find that there exists $n>0$ and a global section $f$ of ${{\mathcal O}}_X(n)$ such that the closed subscheme $H_f$ of $X$ is a hypersurface that contains $C$ as a closed subscheme, and such that for all $s \in S$, $H_f$ does not contain any irreducible component of positive dimension of $F_s$. Using Lemma \[components-dom\] and the assumption that $C$ is integral, we can find an irreducible component $\Gamma$ of $H_f$ which contains $C$, and such that all fibers of $\Gamma \to S$ have dimension $d-1$. If $d-1>1$, we repeat the process with $\Gamma$ endowed with the reduced induced structure, $C$, and $F \cap \Gamma$.
It follows that we are reduced to proving the theorem when $X \to S$ has fibers of dimension $1$ and $X$ is integral. In this case, $F \to S$ is quasi-finite. We now reduce to the case where $X$ is normal and $X\to S$ has a smooth generic fiber. Let $K$ denote the function field of $S$. When $K$ has positive characteristic $p>0$, consider the homeomorphism $\pi: X\to X^{(p^n)}$ with $n$ as in \[normalization-smooth\], so that the normalization of the reduced generic fiber of $X^{(p^n)}$ is smooth over $K$. Applying \[Chow-radicial\] to $\pi_*: {\mathcal A}(X) \to {\mathcal A}(X^{(p^n)})$, we find that it suffices to prove \[mv-1-cycle-local\] for $X^{(p^n)}$, $\pi(C)$, and $\pi(F)$. So we can suppose that the normalization of the reduced generic fiber $X_{K}$ is smooth.
Let $\pi: X'\to X$ be the normalization morphism. By \[normalization-finite\] (a), this morphism is finite. Using \[Chow-radicial\] applied to $\pi_*: {\mathcal A}(X') \to {\mathcal A}(X)$, we see that it is enough to prove \[mv-1-cycle-local\] for $X'$, $\pi^{-1}(C)$, and $\pi^{-1}(F)$. Replacing $X$ with $X'$ if necessary, we can now suppose that $X$ is normal, and that $X_K$ is smooth over $K$.
We can now apply \[normalization-finite\] (3) and we find that $X$ is $\mathbb Q$-factorial. So there exists an integer $n>0$ such that the effective Weil divisor $nC$ is associated to a Cartier divisor on $X$. We are thus reduced to the case where $C$ is a Cartier divisor on $X$, and the statement then follows from the end of the proof \[endofproof\] of Case (a).
We show in our next theorem that in Rumely’s Local-Global Principle as formulated in [@MB1], 1.7, the hypothesis that the base scheme $S$ is excellent can be removed.
\[RumelyLG\] Let $S$ be a Dedekind scheme satisfying Condition [(T)]{}. Let $X \to S$ be a separated surjective morphism of finite type. Assume that $X$ is irreducible and that the generic fiber of $X \to S$ is geometrically irreducible. Then $X \to S$ has a finite quasi-section.
In [@MB1], the hypothesis that $S$ is excellent is only used in 3.3 (which relies on 2.8) and, implicitly, in 2.5. The removal of the hypothesis that $S$ is excellent in 2.5 is addressed in \[lem.torsiondegreed\] (2). To prove the Local-Global Principle, it is enough to prove it for integral quasi-projective schemes of relative dimension $1$ over $S$ ([@MB1], 3.1).
Assume that $S$ is not excellent. Consider a finite $S$-morphism $X \to X^{(p^n)}$ such that the normalization of the reduced generic fiber of $X^{(p^n)} \to S$ is smooth (\[normalization-smooth\]). Clearly, $X^{(p^n)} \to S$ has a finite quasi-section if and only if $X\to S$ has one. Similarly, since $(X^{(p^n)})_{\mathrm{red}} \to X^{(p^n)}$ is a finite $S$-morphism, $(X^{(p^n)})_{\mathrm{red}} \to S$ has a finite quasi-section if and only if $X^{(p^n)}\to S$ has one. We also find from \[normalization-finite\] (a) that the normalization morphism $X' \to (X^{(p^n)})_{\mathrm{red}}$ is finite, and again $(X^{(p^n)})_{\mathrm{red}} \to S$ has a finite quasi-section if and only if $X' \to S$ has one. Thus we are reduced to the case where $X$ is normal and the generic fiber of $X \to S$ is smooth. We now proceed as in the proof of \[normalization-finite\] (3) to remove the ‘excellent’ hypothesis in [@MB1], 2.8, and in [@MB1], 3.3.
The following proposition is needed to produce the examples below which conclude this section.
\[threesections\] Let $S$ be a noetherian irreducible scheme. Let ${\mathcal L}$ be an invertible sheaf over $S$, and consider the scheme $X:= \mathbb P({\mathcal O}_S \oplus {\mathcal L})$, with the associated projective[^15] morphism $\pi: X \to S$. Denote by $C_0$ and $C_\infty$ the images of the two natural sections of $\pi$ obtained from the projections ${\mathcal O}_S \oplus {\mathcal L} \to {\mathcal O}_S$ and ${\mathcal O}_S \oplus {\mathcal L} \to {\mathcal L}$. Suppose that there exists a finite flat quasi-section $g:Y\to S$ of $X \to S$ of degree $d$ which does not meet $F:= C_0 \cup C_\infty$. Then ${\mathcal L}^{\otimes d}$ is trivial in ${\mbox{\rm Pic}\kern 1pt}(S)$.
Let $X':= X \times_S Y$, with projection $\pi':X' \to Y$. Clearly, $\pi'$ corresponds to the natural projection $\mathbb P({\mathcal O}_{Y} \oplus g^*{\mathcal L}) \to Y$. We find that the morphism $\pi'$ has now three pairwise disjoint sections, corresponding to three homomorphisms from ${\mathcal O}_{Y} \oplus g^*{\mathcal L}$ to lines bundles, two of them being the obvious projection maps.
We claim that three such pairwise disjoint sections can exist only if $\mathcal L':=g^*{\mathcal L}$ is the trivial invertible sheaf. Let $\mathcal N\subset {{\mathcal O}}_{Y}\oplus {\mathcal L}'$ be the submodule corresponding to the third section ([@EGA], II.4.2.4). For any $y\in Y$, $\mathcal N\otimes k(y)$ is different from $\mathcal L'\otimes k(y)$ (viewed as a submodule of $({{\mathcal O}}_{Y}\oplus {\mathcal L}')\otimes k(y)$) because in the fiber above $y$, the section defined by $\mathcal N$ is disjoint from the section defined by the projection to ${{\mathcal O}}_{Y}$, so the image of $\mathcal N\otimes k(y)$ in the quotient $k(y)$ is non zero. Therefore the canonical map $\mathcal N\to {{\mathcal O}}_{Y}\oplus {\mathcal L}' \to {{\mathcal O}}_{Y}$ is surjective and, hence, it is an isomorphism. Similarly, the canonical map $\mathcal N\to \mathcal L'$ is an isomorphism. Therefore $\mathcal L'\simeq {{\mathcal O}}_{Y}$. It is known (see, e.g., [@Gur], 2.1) that since $Y \to S$ is finite and flat, the kernel of the induced map ${\mbox{\rm Pic}\kern 1pt}(S) \to {\mbox{\rm Pic}\kern 1pt}(Y)$ is killed by $d$.
\[ex.extrahyp\] Let $R$ be any Dedekind domain and let $S= {\operatorname{Spec}}R$. Our next example shows that Theorem \[mv-1-cycle-local\] can hold only if $R$ has the property that ${\mbox{\rm Pic}\kern 1pt}(R')$ is a torsion group for all Dedekind domains $R'$ finite over $R$.
Indeed, choose an invertible sheaf ${\mathcal L}$ over $S$, and consider the scheme $X:= \mathbb P({\mathcal O}_S \oplus {\mathcal L})$, with the associated smooth projective morphism $\pi: X \to S$. Let $C_0$ and $C_\infty$ be as in \[threesections\], and let $C:= C_0 + C_\infty$. Let $F:= {\operatorname{Supp}}(C)$.
If Theorem \[mv-1-cycle-local\] holds, then a multiple of $C$ can be moved, and there exists a horizontal $1$-cycle $C'$ of $X$ such that ${\operatorname{Supp}}(C') \cap F = \emptyset$. Hence, we find the existence of an integral subscheme $Y$ of $X$, finite and flat over $S$, and disjoint from $F$. Thus, \[threesections\] implies that ${\mathcal L}$ is a torsion element in ${\mbox{\rm Pic}\kern 1pt}(S)$, and for Theorem \[mv-1-cycle-local\] to hold, it is necessary that ${\mbox{\rm Pic}\kern 1pt}(S)$ be a torsion group. Repeating the same argument starting with any invertible sheaf ${\mathcal L}'$ over any $S'$ (which is regular, and finite and flat over $S$) and considering the map $\mathbb P({\mathcal O}_{S'} \oplus {\mathcal L}') \to S' \to S$, we find that for Theorem \[mv-1-cycle-local\] to hold, it is necessary that ${\mbox{\rm Pic}\kern 1pt}(S')$ be a torsion group.
An analogue of Theorem \[mv-1-cycle-local\] cannot be expected to hold when $S$ is assumed to be a smooth [*proper*]{} curve over a field $k$, even when $k$ is a finite field. Indeed, suppose that $X \to S$ is given as in \[mv-1-cycle-local\] with both $X/k$ and $S/k$ smooth and proper. Then any ample divisor $C$ on $X$ will have positive intersection number $(C \cdot D)_X$ with any curve $D$ on $X$. Such an ample divisor then cannot be contained in a fiber of $X \to S$, and thus must be finite over $S$. Set $F={\operatorname{Supp}}(C)$. The conclusion of Theorem \[mv-1-cycle-local\] cannot hold in this case: it is not possible to find on $X$ a divisor rationally equivalent to $C$ which does not meet the closed set $F={\operatorname{Supp}}(C)$.
\[ex.extrahyp2\] Keep the notation introduced in Example \[ex.extrahyp\], and choose a non-trivial line bundle $\mathcal L$ of finite order $d>1$ in ${\mbox{\rm Pic}\kern 1pt}(S)$. Let $X:=\mathbb P({{\mathcal O}}_S\oplus \mathcal L)$. Let $F:= C_0 \cup C_\infty$. Theorem \[mv-1-cycle-local\] (with the appropriate hypotheses on $S$) implies that a multiple $mC_0$ of $C_0$ can be moved away from $F$. We claim that $C_0$ itself cannot be moved away from $F$. Indeed, otherwise, there exist finitely many finite quasi-sections $Y_i \to S$ in $X \setminus F$ such that the greatest common divisor of the degrees $d_i$ of $Y_i \to S$ is $1$ (because $C_0\to S$ has degree $1$). Hence, as in \[ex.extrahyp\], we find that the order of $\mathcal L$ divides $d_i$ for all $i$. Since $\mathcal L$ has order $d>1$ by construction, we have obtained a contradiction. In fact, we find that $dC_0$ is the smallest positive multiple of $C_0$ that could possibly be moved away from $F$.
[Finite morphisms to [${\mathbb P}^d_S$]{}.]{} \[finite-pd\]
Let $X \to S$ be an affine morphism of finite type, with $S={\operatorname{Spec}}R$. Assume that $S$ is irreducible with generic point $\eta$, and let $d:= \dim X_\eta$. When $R=k$ is a field, the Normalization Theorem of E. Noether states that there exists a finite $k$-morphism $X \to {\mathbb A}_k^d$. When $R$ is not a field, no [*finite*]{} $S$-morphism $X \to {\mathbb
A}_S^d$ may exists in general, even when $X\to S$ is surjective and $S$ is noetherian.
When $R=k$ is a field, a stronger form of the Normalization Theorem that applies to graded rings (see, e.g., [@Eis], 13.3) implies that every projective variety $X/k$ of dimension $d$ admits a finite $k$-morphism $X \to {\mathbb P}_k^d$. Our main theorem in this section, Theorem \[theorem.finiteP\^n\] below, guarantees the existence of a finite $S$-morphism $X \to {\mathbb P}_S^d$ when $X\to S$ is projective with $R$ ${\mbox{pictorsion}}$ (\[ConditionT\*\]), and $d:= \max\{\dim X_s, s\in S\}$. A converse to this statement is given in \[conversepictorsion\]. We end this section with some remarks and examples of pictorsion rings.
\[theorem.finiteP\^n\] \[morphism-to-Pd\] Let $R$ be a ${\mbox{pictorsion}}$ ring, and let $S:={\operatorname{Spec}}R$. Let $X \to S$ be a projective morphism, and set $d:= \max\{\dim X_s, s\in S\}$. Then there exists a finite $S$-morphism $r: X \to {\mathbb P}^d_S$. If we assume in addition that $\dim X_s= d$ for all $s\in S$, then $r$ is surjective.
Identify $X$ with a closed subscheme of a projective space $P:=\mathbb P^N_S$. Assume first that $X \to S$ is of finite presentation. We first apply Theorem \[bertini-type-0\] to the projective scheme $P\to S$ with ${{\mathcal O}}_P(1)$, $C=\emptyset$, $m=1$, and $F_1=X$, to find $n_0>0$ and $f_0\in H^0(P, {{\mathcal O}}_P(n_0))$ such that $X\cap H_{f_0}\to S$ has all its fibers of dimension $\le d-1$ (use \[hypersurfaces-properties\] (1)). We apply again Theorem \[bertini-type-0\], this time to $P\to S$ and ${{\mathcal O}}_P(n_0)$, $C=\emptyset$, $m=1$, and $F_1=X\cap H_{f_0}$. We find an integer $n_1$ and a section $f_1\in H^0(P, {{\mathcal O}}_P(n_0n_1))$ such that $(X\cap H_{f_0})\cap H_{f_1}$ has fibers over $S$ of maximal dimension $d-2$. We continue this process $d-2$ additional times, to find a sequence of homogeneous polynomials $f_0,\dots, f_{d-1}$ such that the closed subscheme $Y:=X\cap H_{f_{d-1}}\cap \ldots \cap H_{f_0}$ has all its fibers of dimension at most $0$ and, hence, is finite over $S$ since it is projective (see [@EGA], IV.8.11.1). Note that replacing $f_i$ by a positive power of $f_i$ does not change the topological properties of the closed set $H_{f_i}$. So we can suppose $f_1, \dots, f_{d-1}\in H^0(P, {{\mathcal O}}_P(n))$ for some $n>0$.
Since $S$ is ${\mbox{pictorsion}}$, ${\mbox{\rm Pic}\kern 1pt}(Y)$ is a torsion group. So there exists $j\ge 1$ such that ${{\mathcal O}}_P(nj)|_Y\simeq {{\mathcal O}}_Y$. Let $e\in H^0(Y, {{\mathcal O}}_P(nj)|_Y)$ be a basis. As $Y$ is finitely presented over $S$, both $Y$ and $e$ can be defined on some noetherian subring $R_0$ of $R$ ([@EGA], IV.8.9.1(iii)). By Serre’s Vanishing Theorem on $\mathbb P^N_{R_0}$ applied with the very ample sheaf ${{\mathcal O}}_{\mathbb P^N_{R_0}}(nj)$, we can find $k>0$ such that $e^{\otimes k}$ lifts to a section $f_d \in H^0(P, {{\mathcal O}}_P(njk))$. It follows that $H_{f_d}\cap Y=\emptyset$.
We have constructed $d+1$ sections $f_0^{jk}, \dots, f_{d-1}^{jk}, f_d$ in $H^0(P, {{\mathcal O}}_P(njk))$, whose zero loci on $X$ have empty intersection. The restrictions to $X$ of these sections define a morphism $r : X\to \mathbb P^d_S$. Since $X\to S$ is of finite presentation and $\mathbb P^d_S\to S$ is separated of finite type, the morphism $r : X\to \mathbb P^d_S$ is also of finite presentation ([@EGA] IV.1.6.2 (v), or [@EGA1], I.6.3.8 (v)). By a standard argument (see e.g. [@Ked], Lemma 3), the morphism $r : X\to \mathbb P^d_S$ is finite. When $\dim X_s=d$, as $X_s\to \mathbb P^d_{k(s)}$ is finite, it is also surjective.
Let us consider now the general case where $X \to S$ is not assumed to be of finite presentation. The scheme $X $, as a closed subscheme of $\mathbb P^N_S$, corresponds to a graded ideal $J \subset R[T_0,\dots, T_N]$. Then $X$ is a filtered intersection of subschemes $X_\lambda \subset\mathbb P^N_S$ defined by finitely generated graded subideals of $J$. Thus each natural morphism $f_\lambda: X_\lambda \to S$ is of finite presentation. The points $x$ of $X_\lambda$ where the fiber dimension dim$_x(f_\lambda^{-1}(f_\lambda(x))$ is greater than $d$ form a closed subset $E_\lambda$ of $X_\lambda$ ([@EGA], IV.13.1.3). Since the fibers of $Z \to S$ are of finite type over a field, the corresponding set for $X \to S$ is nothing but $\cap_\lambda E_\lambda$, and by hypothesis, the former set is empty. Since $\cap_\lambda E_\lambda$ is a filtered intersection of closed subsets in the quasi-compact space $\mathbb P^N_S$, we find that $E_{\lambda_0} $ is empty for some $\lambda_0$. We can thus apply the statement of the theorem to the morphism of finite presentation $f_{\lambda_0}: X_{\lambda_0} \to S$ and find a finite $S$-morphism $X_{\lambda_0} \to \mathbb P^d_S$. Composing with the closed immersion $X \to X_{\lambda_0}$ produces the desired finite morphism $X \to \mathbb P^d_S$.
\[rem.flat\] Assume that the morphism $r: X \to {\mathbb P}^d_S$ obtained in the above theorem is finite and surjective. When $S$ is a noetherian regular scheme and $X$ is Cohen-Macaulay and irreducible, then $r$ is also flat. Indeed, $\mathbb P^d_S$ is regular since $S$ is. Since $r$ is finite and surjective, $X $ is irreducible, and $ \mathbb P^d_S$ is universally catenary, we find that $\dim {{\mathcal O}}_{X,x}= \dim {{\mathcal O}}_{\mathbb P^d_S,r(x)}$ for all $x \in X$ ([@Liubook], 8.2.6). We can then use [@A-K], V.3.5, to show that $r$ is flat.
Let us note here one class of projective morphisms $X \to {\operatorname{Spec}}R$ which satisfy the conclusion of \[theorem.finiteP\^n\] without a pictorsion hypothesis on $R$.
\[NoPictorsionHyp\] Let $R$ be a noetherian ring of dimension $1$ with $S:={\operatorname{Spec}}R$ connected. Let $\mathcal E$ be any locally free ${{\mathcal O}}_S$-module of rank $r \geq 2$. Then there exists a finite $S$-morphism ${\mathbb P}(\mathcal E) \to {\mathbb P}^{r-1}_S$ of degree $r^{r-1}$.
Let $S$ be any scheme. Recall that given any locally free sheaf of rank $r$ of the form $\mathcal L_1 \oplus \dots \oplus \mathcal L_r$, with $\mathcal L_i$ invertible for $i=1,\dots, r$, there exists a finite $S$-morphism $${\mathbb P}(\mathcal L_1 \oplus \dots \oplus \mathcal L_r)
\longrightarrow {\mathbb P}(\mathcal L_1^{\otimes d} \oplus \dots
\oplus \mathcal L_r^{\otimes d})$$ of degree $d^{r-1}$, defined on local trivializations by raising the coordinates to the $d$-th tensor power.
Assume now that $S= {\operatorname{Spec}}R$ is connected, and recall that when $R$ is noetherian of dimension $1$, any locally free ${{\mathcal O}}_S$-module $\mathcal E$ of rank $r$ is isomorphic to a locally free ${{\mathcal O}}_S$-module of the form ${{\mathcal O}}_S^{r-1} \oplus \mathcal L$, where $\mathcal L$ is some invertible sheaf on $S$ ([@Ser], Proposition 7)[^16], and $\wedge^r \mathcal E \cong \mathcal L$. Consider the morphism ${\mathbb P}(\mathcal E) \to {\mathbb
P}({{\mathcal O}}_S^{r-1} \oplus \mathcal L^{\otimes r})$ of degree $r^{r-1}$ described above. We claim that ${\mathbb P}({{\mathcal O}}_S^{r-1}\oplus \mathcal
L^{\otimes r})$ is isomorphic to $ \mathbb P^{r-1}_S$. Indeed, we find that $ \mathcal
L^{\oplus r}$ is isomorphic to $\mathcal {{\mathcal O}}_S^{r-1}\oplus \mathcal L^{\otimes r}$ using the result quoted above, and ${\mathbb P}(\mathcal L^{\oplus r})$ is $S$-isomorphic to ${\mathbb P}({{\mathcal O}}_S^{r}) = \mathbb P^{r-1}_S$ ([@EGA], II.4.1.4).
To prove a converse to Theorem \[theorem.finiteP\^n\] in \[conversepictorsion\], we will need the following proposition.
\[ProjSpace\] Let $S$ be a connected noetherian scheme. Let $\mathcal E$ be a locally free sheaf of rank $n+1$. Consider the natural projection morphism $\pi: {\mathbb P}(\mathcal E) \to S$.
1. Any invertible sheaf on ${\mathbb P}(\mathcal E)$ is isomorphic to a sheaf of the form ${{\mathcal O}}_{{\mathbb P}(\mathcal E)}(m) \otimes \pi^*(\mathcal L)$, where $m \in {\mathbb Z}$ and $\mathcal L$ is an invertible sheaf on $S $.
2. Assume that $S={\operatorname{Spec}}R$ is affine, and let $f: {\mathbb P}^n_S \to {\mathbb P}^n_S$ be a finite morphism. Then $f^*({{\mathcal O}}_{\mathbb P^n_S}(1))$ is isomorphic to a sheaf of the form ${{\mathcal O}}_{\mathbb P^n_S}(m) \otimes \pi^*(\mathcal L)$, where $\mathcal L$ is an invertible sheaf on $S $ of finite order and $m>0$.
\(a) This statement is well-known (it is found for instance in [@Mum], page 20, or in [@EGA], II.4.2.7). Since we did not find a complete proof in the literature, let us sketch a proof here. Let ${\mathcal M}$ be an invertible sheaf on ${\mathbb P}(\mathcal E)$. For each $s \in S$, the pull-back ${\mathcal M}_s $ of ${\mathcal M}$ to the fiber over $s$ is of the form ${{\mathcal O}}_{{\mathbb P}_{k(s)}^n}(m_s)$ for some integer $m_s$. Using the fact that the Euler characteristic of ${\mathcal M}_s $ is locally constant on $S$ ([@MilAV], 4.2 (b)), we find that $m_s$ is locally constant on $S$. Since $S$ is connected, $m_s=m$ for all $s\in S$. The conclusion follows as in [@MilAV], 5.1.
\(b) Let ${{\mathcal O}}(1):= {{\mathcal O}}_{{\mathbb P}_S^n}(1)$. Using (a), we find that $f^*({{\mathcal O}}(1))$ is isomorphic to a sheaf of the form ${{\mathcal O}}(m) \otimes \pi^*(\mathcal L)$, where $\mathcal L$ is an invertible sheaf on $S $. We have $m>0$ because over each point $s$, $f^*({{\mathcal O}}(1))_s$ is ample, being the pull-back by a finite morphism of the ample sheaf ${{\mathcal O}}(1)_s$, and is isomorphic to ${{\mathcal O}}(m)_s$.
Write $M:= H^0(S,\mathcal L)$, and identify $H^0(\mathbb P^n_S, (\pi^*\mathcal L) (m))$ with $M \otimes_R R[x_0, x_1, ..., x_n]_m$, where $R[x_0, x_1, ..., x_n]_m$ denotes the set of homogeneous polynomials of degree $m$. The section of ${{\mathcal O}}(1)$ corresponding to $x_i \in R[x_0, x_1, ..., x_n]_1$ pulls back to a section of $f^*({{\mathcal O}}(1))$ which we identify with an element $F_i \in M \otimes_R R[x_0, x_1, ..., x_n]_m$.
Since $M$ is locally free of rank $1$, there is a cover $\cup_{j=1}^t D(s_j) $ of $ S$ by special affine open subsets such that $M \otimes_R R[1/s_j]$ has a basis $t_j$. Hence, for each $i\le n$, we can write $F_i = t_j \otimes G_{ij}$ with $G_{ij} \in R[1/s_j][x_0,\dots, x_n]_m$. Denote the resultant of $G_{0j}, \dots, G_{nj}$ by $\text{Res}(G_{0j}, \dots, G_{nj})$ (see [@Jou], 2.3). We claim that $\text{Res}(G_{0j},\dots, G_{nj}) $ is a unit in $ R[1/s_j]$. Indeed, over $D(s_j) := {\operatorname{Spec}}R[1/s_j]$, the restricted morphism $f_{S_j} : {\mathbb P}^n_{S_j} \to {\mathbb P}^n_{S_j}$ is given by the global sections of ${{\mathcal O}}(m)_{\mid D(s_j)}$ corresponding to $G_{0j}, \dots, G_{jn} \in R[1/s_j][x_0,\dots, x_n]_m$. Since these global sections generate the sheaf ${{\mathcal O}}(m)_{\mid D(s_j)}$, we find that the hypersurfaces $G_{0j}=0$, $\dots$, $G_{nj}=0$ cannot have a common point and, thus, that $\text{Res}(G_{0j},\dots, G_{nj}) \in R[1/s_j]^*$.
For $j=1,\dots, t$, consider now $$r_j:= \text{Res}(G_{0j}, \dots, G_{nj})t_j^{\otimes (n+1)m^{n}} \in M^{\otimes (n+1)m^{n}} \otimes R[1/s_j].$$ Since $\text{Res}(G_{0j},\dots, G_{nj}) \in R[1/s_j]^*$, the element $r_j$ is a basis for $M^{\otimes (n+1)m^{n}} \otimes R[1/s_j]$. We show now that $M^{\otimes (n+1)m^{n}}$ is a free $R$-module of rank $1$ by showing that the elements $r_j$ can be glued to produce a basis $r$ of $M^{\otimes (n+1)m^{n}}$ over $R$. Indeed, over $D(s_j) \cap D(s_k)$, we note that there exists $a \in R[1/s_j, 1/s_k]$ such that $at_j=t_k$. Then from $F_i= t_j \otimes G_{ij} = t_k \otimes G_{ik}$, we conclude that $G_{ij}=aG_{ik}$, so that $$\text{Res}(G_{0j},\dots, G_{nj})= a^{(n+1)m^{n}}\text{Res}(G_{0k},\dots, G_{nk})$$ ([@Jou], 5.11.2). We thus find that $r_j$ is equal to $r_k$ when restricted to $D(s_j) \cap D(s_k)$, as desired.
\[example.infiniteorder\] Let $S$ be a connected noetherian affine scheme. Assume that ${\mbox{\rm Pic}\kern 1pt}(S)$ contains an element $\mathcal L$ of infinite order. Suppose that $\mathcal L$ can be generated by $d+1$ sections for some $d\ge 0$. We construct in this example a projective morphism $X_{\mathcal L} \to S$, with fibers of dimension $d$, and such that there exists no finite $S$-morphism $X_{\mathcal L} \to {\mathbb P}^d_S$. Let ${{\mathcal O}}(1):= {{\mathcal O}}_{{\mathbb P}_S^d}(1)$.
Using $d+1$ global sections in $\mathcal L(S)$ which generate $\mathcal L$, define a closed $S$-immersion $i_1: S \to {\mathbb P}^d_S$, with $i^*_1({\mathcal O}(1)) = \mathcal L$. Consider also the closed $S$-immersion $i_0: S \to {\mathbb P}^d_S$ given by $(1:0:\ldots:
0)\in \mathbb P^d_S(S)$, so that $i^*_0({\mathcal O}(1) ) = {{\mathcal O}}_S$. Consider now the scheme $X_{\mathcal L}$ obtained by gluing two copies of ${\mathbb P}^d_S$ over the closed subschemes $\text{Im}(i_0)$ and $\text{Im}(i_1)$ ([@Ana], 1.1.1). It is noted in [@Ana], 1.1.5, that under our hypotheses, the resulting gluing is endowed with a natural morphism $\pi: X_{\mathcal L} \to S$ which is separated and of finite type. Recall that the scheme $X_{\mathcal L}$ is endowed with two natural closed immersions $\varphi_0: X_0= {\mathbb P}^d_S \to X_{\mathcal L}$ and $\varphi_1: X_1= {\mathbb P}^d_S \to X_{\mathcal L}$ such that $\varphi_0 \circ i_0 = \varphi_1 \circ i_1$. Moreover, the $S$-morphism $(\varphi_0,\varphi_1): {\mathbb P}^d_S \sqcup {\mathbb P}^d_S \to X_{\mathcal L}$ is finite and surjective. Since ${\mathbb P}^d_S \to S$ is proper, we find that $X_{\mathcal L} \to S$ is also proper.
Suppose that there exists a finite $S$-morphism $f: X_{\mathcal L} \to
{\mathbb P}^d_S$. Then, using \[ProjSpace\] (b), we find that there exist two torsion invertible sheaves ${{\mathcal G}}_0$ and ${{\mathcal G}}_1$ on $S$ and $m
\geq 0$ such that $(f \circ \varphi_0 \circ i_0)^*({{\mathcal O}}_{\mathbb
P^d_S}(1))= {{\mathcal G}}_0$, and $(f \circ \varphi_1 \circ i_1)^*({{\mathcal O}}_{\mathbb
P^d_S}(1))= \mathcal L^{\otimes m} \otimes {{\mathcal G}}_1$. Since we must have then ${{\mathcal G}}_0$ isomorphic to $\mathcal L^{\otimes m} \otimes {{\mathcal G}}_1$ and since ${\mathcal L}$ is not torsion, we find that such a morphism $f$ cannot exist.
To conclude this example, it remains to show that $\pi: X_{\mathcal L} \to S$ is a projective[^17] morphism. For this, we exhibit an ample sheaf on $X_{\mathcal L}$ as follows. Consider the sheaf ${\mathcal F}_0:= (\pi \circ \varphi_0)^*({\mathcal L})(1)$ on $X_0= {\mathbb P}^d_S$ and the sheaf ${\mathcal F}_1:={{\mathcal O}}(1)$ on $X_1= {\mathbb P}^d_S$. We clearly have a natural isomorphism of sheaves $i_0^*({\mathcal F}_0) \rightarrow i_1^*({\mathcal F}_1)$ on $S$. Thus, we can glue the sheaves ${\mathcal F}_0$ and ${\mathcal F}_1$ to obtain a sheaf ${\mathcal F}$ on $X_{\mathcal L}$. Since both ${\mathcal F}_0$ and ${\mathcal F}_1$ are invertible, we find that ${\mathcal F}$ is also invertible on $X_{\mathcal L}$ (such a statement in the affine case can be found in [@Fer], 2.2). Under the finite $S$-morphism $(\varphi_0,\varphi_1): {\mathbb P}^d_S \sqcup {\mathbb P}^d_S \to X_{\mathcal L}$, the sheaf ${\mathcal F}$ pulls back to the sheaf restricting to ${\mathcal F}_0$ on $X_0$ and ${\mathcal F}_1$ on $X_1$. In particular, the pull-back is ample (since $\mathcal L$ is generated by its global sections), and since $X_{\mathcal L}\to S$ is proper, we can apply [@EGA], III.2.6.2, to find that ${\mathcal F} $ is also ample.
Let $R$ be a Dedekind domain and $S:= {\operatorname{Spec}}R$. Let $X \to S$ be a projective morphism with fibers of dimension $1$. When $R$ is pictorsion, Theorem \[theorem.finiteP\^n\] shows that there exists a finite $S$-morphism $X \to {\mathbb P}^1_S$. It is natural to wonder, when $R$ is not assumed to be pictorsion, whether it would still be possible to find a locally free ${{\mathcal O}}_S$-module $\mathcal E$ of rank $2$ and a finite $S$-morphism $X \to {\mathbb P}(\mathcal E)$. The answer to this question is negative, as the following example shows.
Assume that ${\mbox{\rm Pic}\kern 1pt}(S)$ contains an element $\mathcal L$ of infinite order. Then $\mathcal L$ can be generated by $2$ sections. Consider the projective morphism $X_{\mathcal L} \to S$ constructed in Example \[example.infiniteorder\]. Suppose that there exists a locally free ${{\mathcal O}}_S$-module $\mathcal E$ of rank $2$ and a finite $S$-morphism $X_{\mathcal L} \to {\mathbb P}(\mathcal E)$. Proposition \[NoPictorsionHyp\] shows that there exists then a finite $S$-morphism ${\mathbb P}(\mathcal E) \to {\mathbb P}^1_S$. We would then obtain by composition a finite $S$-morphism $X_{\mathcal L} \to {\mathbb P}^1_S$, which is a contradiction. We thank Pascal Autissier for bringing this question to our attention.
We are now ready to prove a converse to Theorem \[theorem.finiteP\^n\].
\[conversepictorsion\] Let $R$ be any commutative ring and let $S:={\operatorname{Spec}}R$. Suppose that for any $d \geq 0$, and for any projective morphism $X \to S$ such that $\dim X_s=d$ for all $s\in S$, there exists a finite surjective $S$-morphism $ X \to {\mathbb P}^d_S$. Then $R$ is pictorsion.
When $R$ is noetherian of finite Krull dimension $\dim R$, then $R$ is pictorsion if for all projective morphisms $X\to S$ such that $\dim X_s \le \dim R$ for all $s\in S$, there exists a finite $S$-morphism $X \to {\mathbb P}^{\dim R}_S$.
Let $R'$ be a finite extension of $R$, and let $\mathcal L' \in {\mbox{\rm Pic}\kern 1pt}({\operatorname{Spec}}R')$. The sheaf $\mathcal L'$ descends to an element ${\mathcal L}$ of ${\mbox{\rm Pic}\kern 1pt}({\operatorname{Spec}}R_0)$ for some noetherian subring $R_0$ of $R'$. For each connected component $S_i$ of ${\operatorname{Spec}}R_0$, let $d_i $ be such that ${\mathcal L}_{|S_i}$ can be generated by $d_i+1$ global sections. Let $d:= {\rm max}(d_i)$. When $R$ is noetherian we take $R=R_0$, and when in addition $\dim(R) < \infty$, we can always choose $d \leq \dim(R)$.
Assume now that $\mathcal L'$ is of infinite order. It follows that ${\mathcal L}$ is of infinite order on some connected component of ${\operatorname{Spec}}R_0$. Apply the construction of Example \[example.infiniteorder\] to each connected component $S_i$ of ${\operatorname{Spec}}R_0$ where ${\mathcal L}$ has infinite order, with a choice of $d$ global sections of ${\mathcal L}_{|S_i}$ which generate ${\mathcal L}_{|S_i}$. We obtain a projective scheme $X_i \to S_i$ with fibers of dimension $d$ and which does not admit a finite morphism to $ X_i \to {\mathbb P}^{d}_{S_i}$. If $S_j$ is a connected component of ${\operatorname{Spec}}R_0$ such that ${\mathcal L}$ has finite order, we set $X_j\to S_j$ to be ${\mathbb P}^{d}_{S_j} \to S_j$. We let $X$ denote the disjoint union of the schemes $X_i$. The natural morphism $X\to {\operatorname{Spec}}R_0$ has fibers of dimension $d$, and does not admit a finite morphism $ X \to {\mathbb P}^{d}_{R_0}$.
Let $R_1$ be any noetherian ring such that $R_0 \subset R_1 \subset R'$. By construction, the pullback of $\mathcal L $ to ${\operatorname{Spec}}R_1$ has infinite order on some connected component of ${\operatorname{Spec}}R_1$. Since the construction in Example \[example.infiniteorder\] is compatible with pullbacks, we conclude that $
X \times_{R_0} R_1$ does not admit a finite morphism to ${\mathbb P}^{d}_{R_1}$. It follows then from [@EGA] IV.8.8.2 and IV.8.10.5 that there is no finite morphism $X \times_{R} R' \to {\mathbb P}^{d}_{R'}$. Hence, there is no finite $R$-morphism $X \times_{R} R' \to {\mathbb P}^{d}_{R}$. Replacing $X \times_{R} R'$ with its disjoint union with ${\mathbb P}^{d}_{R}$ if necessary, we obtain a projective morphism to $S$ with fibers of dimension $d$ and which does not factor through a finite morphism to $ {\mathbb P}^d_S$.
We present below an example of an affine regular scheme $S$ of dimension $3$ with a locally free sheaf $\mathcal E$ of rank $2 $ of the form ${\mathcal E}= {{\mathcal O}}_S \oplus {\mathcal L}$ such that ${\mathbb P}({\mathcal E})$ does not admit a finite $S$-morphism to ${\mathbb P}^1_S$.
Let $V$ be any smooth connected quasi-projective variety over a field $k$. Let $\mathcal E$ be a locally free sheaf of rank $r $ on $V$. Let $p : {\mathbb P}(\mathcal E) \to V$ denote the associated projective bundle. Denote by $A(V)$ the Chow ring of algebraic cycles on $V$ modulo rational equivalence. Let $\xi$ denote the class in $A({\mathbb P}(\mathcal E))$ of the invertible sheaf ${{\mathcal O}}_{\mathbb P(\mathcal E)}(1)$. Then $p$ induces a ring homomorphism $p^*: A(V) \to A({\mathbb P}(\mathcal E))$, and $A({\mathbb P}(\mathcal E))$ is a free $A(V)$-module generated by $1, \xi, \dots, \xi^{r-1}$. For $i=0,1,\dots, r$, one defines (see, e.g., [@Har], page 429) the $i$-th Chern class of ${\mathcal E}$, $c_i(\mathcal E) \in A^i(V)$, and these classes satisfy the requirements that $c_0(\mathcal E)=1$ and $$\sum_{i=0}^r (-1)^i p^*(c_i(\mathcal E)) \xi^{r-i}=0$$ in $A^r({\mathbb P}(\mathcal E))$. When $\mathcal E = {{\mathcal O}}_V \oplus {\mathcal L}$ for some invertible sheaf ${\mathcal L}$, we find that $c_2(\mathcal E)=0$.
Consider now the case where $\mathcal E$ has rank $2$ and suppose that there exists a finite $V$-morphism $f: {\mathbb P}(\mathcal E) \to {\mathbb P}^1_V$. Then $f^*({{\mathcal O}}_{{\mathbb P}^1_V}(1))$ is isomorphic to a locally free sheaf of the form $p^*\mathcal M \otimes {{\mathcal O}}_{{\mathbb P}(\mathcal E)}(m)$ for some $m >0$ and some invertible sheaf $\mathcal M$ on $V$ (\[ProjSpace\] (a)). Consider the ring homomorphism $$f^*: A({\mathbb P}^1_V) =A(V)[h]/(h^2) \longrightarrow A({\mathbb P}(\mathcal E)) = A(V)[\xi]/(\xi^2 - c_1(\mathcal E) \xi + c_2(\mathcal E)) ,$$ where $h$ denote the class in $A({\mathbb P}^1_V)$ of the invertible sheaf ${{\mathcal O}}_{\mathbb P^1_V}(1)$. It follows that in $A({\mathbb P}(\mathcal E))$, $f^*(h)= a + m \xi$ with $a \in A^1(V)$, and $(a + m \xi)^2=0$. Hence, $m^4(c_1(\mathcal E)^2 -4 c_2(\mathcal E)) = 0$ in $A^2(V)$. Thus, in $A^2(V)_{\mathbb Q}$, $c_1(\mathcal E)^2 =4 c_2(\mathcal E)$. Choose now $\mathcal E = {{\mathcal O}}_V \oplus {\mathcal L}$ for some invertible sheaf ${\mathcal L}$. Then $0=c_2(\mathcal E)=c_1(\mathcal E)^2 = c_1(\mathcal L)^2$ in $A^2(V)_{\mathbb Q}$.
We are now ready to construct our example. Recall that under our hypotheses on $V$, there exists an affine variety $S$ and a surjective morphism $\pi: S \to V$ such that $\pi $ is a torsor under a vector bundle (Jouanolou’s device, [@JouDevice], 1.5). We will use only the simplest case of this construction, when $V= {\mathbb P}^2_k$. In this case, $S$ is the affine variety formed by all $3 \times 3$-matrices which are idempotent and have rank $1$. We claim that we have an isomorphism $$\pi^*: A(V)_{\mathbb Q} \to A(S)_{\mathbb Q}.$$ Indeed, this statement with the Chow rings replaced by $K$-groups is proved in [@Jou2], 1.1. Then we use the fact that the Chern character determines an isomorphism of ${\mathbb Q}$-algebras $ch : K^0(X)_{\mathbb Q} \longrightarrow A(X)_{\mathbb Q} $ where $K^0(X)$ denotes the Grothendieck group of algebraic vector bundles on a smooth quasi-projective variety $X$ over a field ([@Ful], 15.2.16 (b)).
Choose on $S$ the line bundle $\mathcal L:= \pi^*{{\mathcal O}}_{\mathbb P^2}(1)$. Then since $A(V)_{\mathbb Q}= {\mathbb Q}[h]/(h^3)$, we find that $h^2 \neq 0$, so that $c_1(\mathcal L)^2 \neq 0$ in $A(S)_{\mathbb Q}$. Hence, we have produced a smooth affine variety $S$ of dimension $3$, and a locally free sheaf $\mathcal E:={{\mathcal O}}_S \oplus \mathcal L$ such that $\mathbb P(\mathcal E)$ does not admit a finite surjective $S$-morphism to $\mathbb P^1_S$.
We conclude this section with some remarks and examples of pictorsion rings (\[ConditionT\*\]). We first note the following.
Let $R$ be any commutative ring. Denote by $R^{\mathrm{red}}$ the quotient of $R$ by its nilradical. Then $R$ is pictorsion if and only if $R^{\mathrm{red}}$ is pictorsion.
Since $R\to R^{\mathrm{red}}$ is a finite homomorphism, it is clear that if $R$ is pictorsion, then so is $R^{\mathrm{red}}$. Assume now that $R^{\mathrm{red}}$ is pictorsion and let $R\to R'$ be a finite homomorphism. Then $R^{\mathrm{red}}\to (R')^{\mathrm{red}}$ is a finite homomorphism. Thus ${\mbox{\rm Pic}\kern 1pt}((R')^{\mathrm{red}})$ is a torsion group. As we can see using Nakayama’s lemma, ${\mbox{\rm Pic}\kern 1pt}(R')\to
{\mbox{\rm Pic}\kern 1pt}((R')^{\mathrm{red}})$ is injective, so ${\mbox{\rm Pic}\kern 1pt}(R') $ is a torsion group.
Recall that a pictorsion Dedekind domain $R$ satisfies Condition [(T) (a)]{} in \[ConditionT\] by definition (that is, ${\mbox{\rm Pic}\kern 1pt}(R_L)$ is a torsion group for any Dedekind domain $R_L$ obtained as the integral closure of $R$ in a finite extension $L/K$). The statement of (2) below is found in [@MB1], 2.3, when $R$ is excellent. We follow the proof given in [@MB1], modifying it only in 2.5 to also treat the case where $R$ is not excellent. We do not know of an example of a Dedekind domain which satisfies Condition [(T) (a)]{} and which is not pictorsion.
\[lem.torsiondegreed\] Let $R$ be a Dedekind domain with field of fractions $K$.
1. Let $L/K$ be a finite extension of degree $d$, and let $R'$ denote a sub-$R$-algebra of $L$, integral over $R$. Then the kernel of ${\mbox{\rm Pic}\kern 1pt}(R) \to {\mbox{\rm Pic}\kern 1pt}(R')$ is killed by $d$.
2. If $R$ satisfies Condition [(T)]{} in [\[ConditionT\]]{}, then $R$ is ${\mbox{pictorsion}}$.
\(1) When $R'$ is finite and flat over $R$, this is well-known (see, e.g., [@Gur], 2.1). (The hypothesis that $R$ is Dedekind is used here to insure that the ring $R'$ is flat over $R$.) In general, let $M$ be a locally free $R$-module of rank $1$ such that $M \otimes_R R'$ is isomorphic as $R'$-module to $R'$. Then there exist a finite $R$-algebra $A$ contained in $R'$ such that $M \otimes_R A$ is isomorphic as $A$-module to $A$. It follows that $M^{d}$ is trivial in ${\mbox{\rm Pic}\kern 1pt}(R)$, since $A/R$ is finite.
\(2) Let $S={\operatorname{Spec}}R$. Let $Z$ be a finite $S$-scheme. We need to show that ${\mbox{\rm Pic}\kern 1pt}(Z)$ is torsion. The proof in [@MB1], 2.3 - 2.6, is complete when $R$ is excellent. When $R$ is not necessarily excellent, only 2.5 needs to be modified as follows. Assume that $Z$ is reduced. Let $Z'\to Z$ be the normalization morphism, which need not be finite. Then $Z'$ is a finite disjoint union of Dedekind schemes, and the hypothesis that $R$ satisfies Condition (T)(a) implies that ${\mbox{\rm Pic}\kern 1pt}(Z')$ is a torsion group. Let $\mathcal L\in{\mbox{\rm Pic}\kern 1pt}(Z)$. Then there exists $n\ge 1$ such that $\mathcal L^{\otimes n}\otimes {{\mathcal O}}_{Z'}\simeq {{\mathcal O}}_{Z'}$. This isomorphism descends to some $Z$-scheme $Z_{\alpha}$ with $\pi: Z_{\alpha}\to Z$ finite and birational. We now use the proof of 2.5 in [@MB1], applied to $Z_1=Z_{\alpha}$ (instead of the normalization which is not necessarily finite), to find that the kernel of ${\mbox{\rm Pic}\kern 1pt}(Z)\to {\mbox{\rm Pic}\kern 1pt}(Z_{\alpha})$ is torsion. Hence, $\mathcal L$ is torsion.
\[Bezout\] Let $R$ be a Dedekind domain with field of fractions $K$. Let $\overline{R}$ denote the integral closure of $R$ in an algebraic closure $\overline{K}$ of $K$. The following are equivalent:
1. Condition [(T)(a)]{} in [\[ConditionT\]]{} holds.
2. $\overline{R}$ is a Bézout domain (i.e., all finitely generated ideals of $\overline{R}$ are principal).
That (1) implies (2) is the content of Theorem 102 in [@Ka]. Assume that (2) holds, and let $R_L$ be the integral closure of $R$ in a finite extension $L/K$. Let $I$ be a non-zero ideal in $R_L$. Then $I\overline{R}$ is principal. Hence, there exists a finite extension $F/L$ such that in the integral closure $R_F$ of $R$ in $F$, $IR_F$ is principal. Since the kernel of ${\mbox{\rm Pic}\kern 1pt}(R_L) \to {\mbox{\rm Pic}\kern 1pt}(R_F)$ is killed by $[F:L]$ (\[lem.torsiondegreed\]), we find that $I$ has finite order in ${\mbox{\rm Pic}\kern 1pt}(R_L)$.
Keep the notation of \[Bezout\], and denote by $R_F$ the integral closure of $R$ in any algebraic extension $F/K$. Then Condition (1) in \[Bezout\] implies that ${\mbox{\rm Pic}\kern 1pt}(R_F)$ is a torsion group. Indeed, one finds that ${\mbox{\rm Pic}\kern 1pt}(R_F) = \varinjlim {\mbox{\rm Pic}\kern 1pt}(R_L)$, with the direct limit taken over all finite extensions $L/K$ contained in $F$.
Condition (2) in \[Bezout\] is equivalent to ${\mbox{\rm Pic}\kern 1pt}(\overline{R})= (0)$. Indeed, the ring $\overline{R}$ is a Prüfer domain ([@Ka], Thm. 101), and a Prüfer domain $D$ is a Bézout domain if and only if ${\mbox{\rm Pic}\kern 1pt}(D)= (0)$.
We now recall two properties of commutative rings and relate them to the notion of pictorsion introduced in this article. A [*local-global*]{} ring $R$ is a commutative ring where the following property holds: whenever $f \in R[x_1,\dots, x_n]$ is such that the ideal of values $(f(r), r \in R^n)$ is equal to the full ring $R$, then there exists $r \in R^n$ such that $f(r) \in R^*$. A commutative ring $R$ [*satisfies the primitive criterion*]{} if, whenever $f(x) = a_nx^n+\dots + a_0$ is such that $(a_n,\dots, a_0)=R$ (such $f$ is called [*primitive*]{}), then there exists $r \in R$ such that $f(r) \in R^*$. A ring $R$ satisfies the primitive criterion if and only if it is local-global and for each maximal ideal $M$ of $R$, the residue field $R/M$ is infinite ([@McW], Proposition, or [@EG], 3.5).
\[localglobal\] Let $R$ be a local-global commutative ring. Then every finite $R$-algebra $R'$ has ${\mbox{\rm Pic}\kern 1pt}(R')=(1)$. In particular, $R$ is pictorsion.
The ring $R'$ is also a local-global ring ([@EG], 2.3). In a local-global ring, every finitely generated projective $R$-module of constant rank is free ([@McW], Theorem, or [@EG], 2.10). It follows that ${\mbox{\rm Pic}\kern 1pt}(R') = (1)$.
\[pictrivial\] Rings which satisfy the primitive criterion can be constructed as follows (see, e.g., [@vK], 1.13, and also [@EG], section 5). Let $R$ be any commutative ring, and consider the multiplicative subset $S$ of $R[x]$ consisting of all primitive polynomials. Then the ring $R(x):= S^{-1}R[x]$ satisfies the primitive criterion[^18]. Indeed, suppose that $g(y) \in R(x)[y]$ is primitive. Then write $g(y) = \sum_{i=0}^n f_i(x) y^i$, with $f_i(x) \in R(x)$. It is easy to reduce to the case where $f_i(x) \in R[x]$ for all $i$. Since $g(y) $ is primitive, we find that the ideal generated by the coefficients of the polynomials $f_0(x),\dots,f_n(x)$ is the unit ideal of $R$. Hence, choosing $y:=x^t$ for $t$ large enough, we find that $
g(x^t)$ is a primitive polynomial in $R[x]$ and thus is a unit in $R(x)$.
We have seen already in this article examples of commutative rings $R$ such that for every finite morphism ${\operatorname{Spec}}R' \to {\operatorname{Spec}}R$, ${\mbox{\rm Pic}\kern 1pt}(R')$ is trivial (\[pictrivial\]), or ${\mbox{\rm Pic}\kern 1pt}(R')$ is finite but not necessarily trivial (take $R={\mathbb Z}$ or ${\mathbb F}_p[x]$). When needed, such rings could be called [*pictrivial*]{} and [*picfinite*]{}, respectively.
Let us note in this example a ring $R$ which is pictorsion and such that at least one of the groups ${\mbox{\rm Pic}\kern 1pt}(R')$ is not finite. Consider the algebraic closure $\overline{\mathbb F}_p$ of $\mathbb F_p$, and let $R:= \overline{\mathbb F}_p[x]$. Then $R$ is pictorsion because it satisfies Condition (T) (\[lem.torsiondegreed\] (2)). Indeed, let $R'$ be the integral closure of $R$ in a finite extension of $\overline{\mathbb F}_p(x)$. Then $U:={\operatorname{Spec}}(R')$ is a dense open subset of a smooth connected projective curve $X/ \overline{\mathbb F}_p$. One shows that the natural restriction map ${\mbox{\rm Pic}\kern 1pt}(X) \to {\mbox{\rm Pic}\kern 1pt}(U)$ induces a surjective map ${\mbox{\rm Pic}\kern 1pt}^0(X) \to {\mbox{\rm Pic}\kern 1pt}(U)$ with finite kernel. When the genus of $X$ is bigger than $0$, it is known that ${\mbox{\rm Pic}\kern 1pt}^0(X)$, which is isomorphic to the $\overline{\mathbb F}_p$-points of the Jacobian of $X$, is an infinite torsion group.
We also note that the ring $R:= \overline{\mathbb F}_p[x,y]$ is not pictorsion. Indeed, let $X/\overline{\mathbb F}_p$ be a smooth projective surface over $\overline{\mathbb F}_p$ such that its Néron-Severi group NS$(X)$ has rank greater than one (for instance, $X$ could be the product of two smooth projective curves). Let $D \subset X$ be an irreducible divisor whose complement $V:= X \setminus D$ is affine. Write $V={\operatorname{Spec}}A$, and use Noether’s Normalization Lemma to view $A$ as a finite $\overline{\mathbb F}_p[x,y]$-algebra. We claim that ${\mbox{\rm Pic}\kern 1pt}(V)$ is not a torsion group. Indeed, the natural restriction map ${\mbox{\rm Pic}\kern 1pt}(X) \to {\mbox{\rm Pic}\kern 1pt}(V)$ is surjective, with kernel generated by the class of $D$. If ${\mbox{\rm Pic}\kern 1pt}(V)$ is torsion, then the quotient of $\mathrm{NS}(X)$ by the subgroup generated by image of $D$ is torsion. This contradicts the hypothesis on the rank of $\mathrm{NS}(X)$.
\[Varley\] Robert Varley suggested the following example of a Dedekind domain $A$ which is ${\mbox{pictorsion}}$ with infinitely many maximal ideals, each having residue field which is not an algebraic extension of a finite field, i.e., such that $A$ does not satisfies Condition (T)(b) in \[ConditionT\]. Rather than providing a direct proof that the ring below is pictorsion, we interpret the example in light of the above definitions:
[*Let $Z$ denote a countable subset of ${\mathbb C}$. Consider the polynomial ring ${\mathbb C}[x]$, and let $T$ denote the multiplicative subset of all polynomials which do not vanish on $Z$. Then $A:= T^{-1}({\mathbb C}[x])$ satisfies the primitive criterion, and is thus pictorsion by .* ]{}
Indeed, let $F(y) \in A[y]$ be a primitive polynomial. Up to multiplication by elements of $T$, we can assume that $F(y)=f_n(x) y^n + \dots + f_0(x)$ with $f_i \in {\mathbb C}[x]$ for all $i$, and that $x-z$ does not divide $\gcd(f_i(x), i=0,\dots, n)$ for all $z \in Z$. We claim that there exists $a \in {\mathbb C}$ such that $F(a) \in {\mathbb C}[x]$ is coprime to $x-z$ for all $z \in Z$. This shows that $F(a)$ is invertible in $A:= T^{-1}({\mathbb C}[x])$, and thus $A$ satisfies the primitive criterion. To prove this claim, let us think of $F(y)$ as a polynomial $F(x,y)$ in two variables, and let us first note that the curve $F(x,y)=0$ intersects the line $x-z=0$ in at most $\deg(F)$ places. Thus, there are only countably many points in the plane ${\mathbb C}^2$ of the form $(z,v)$ with $z \in Z$ and $F(z,v)=0$. Therefore, it is possible to choose $a \in {\mathbb C}$ such that $F(x,a)=0$ does not contain any of these countably many points.
Let $\overline{\mathbb Q}$ denote the algebraic closure of ${\mathbb Q}$. In view of the above example, it is natural to wonder whether there exists a multiplicative subset $T$ of $\overline{\mathbb Q}[x]$ such that $R:=T^{-1}(\overline{\mathbb Q}[x])$ is pictorsion and ${\operatorname{Spec}}R$ is infinite. Clearly, the integral closure $\tilde{R}$ of $R$ in the algebraic closure of $\overline{\mathbb Q}(x)$ must be Bézout (\[Bezout\]). A related question is addressed in [@vDM], section 5, and in [@GM].
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[^1]: D.L. was supported by NSF Grant 0902161.
[^2]: Each $F_i$ is a closed subscheme of an open subscheme of $X$ ([@EGA], I.4.1.3).
[^3]: Since the morphism $X\to S$ is flat, it is a local complete intersection morphism if and only if every fiber is a local complete intersection morphism (see, e.g., [@Liubook], 6.3.23).
[^4]: A Dedekind domain in this article has dimension $1$, and a Dedekind scheme is the spectrum of a Dedekind domain.
[^5]: The definition of codimension in [@EGA], Chap. 0, 14.2.1, implies that the codimension of the empty set in $X_s$ is $+\infty$, which we consider to be positive.
[^6]: See [@EGA], Chapter 0, 9.1.2. Beware that in the second edition [@EGA1], Chapter 0, 2.3, a [*globally constructible*]{} subset now refers to what is called a [*constructible*]{} subset in [@EGA].
[^7]: A topological space $X$ is [*quasi-compact*]{} if every open covering of $X$ has a finite refinement. A continuous map $f : X \to Y $ is [*quasi-compact*]{} if the inverse image $f^{-1}(V )$ of every quasi-compact open $V$ of $Y$ is quasi-compact. A subset $Z$ of $X$ is [*retro-compact*]{} if the inclusion map $Z \to X$ is quasi-compact.
[^8]: $X$ is [*quasi-separated*]{} if and only if the intersection of any two quasi-compact open subsets of $X$ is quasi-compact ([@EGA], IV.1.2.7).
[^9]: A morphism $\pi:X \to Y$ is [*finitely presented*]{} (or [*of finite presentation*]{}) if it is locally of finite presentation, quasi-compact, and quasi-separated ([@EGA], IV.1.6.1).
[^10]: Recall that a [*locally closed*]{} subset $F$ of a topological space $X$ is the intersection of an open subset $U$ of $X$ with a closed subset $Z$ of $X$. When $X$ is a scheme, we can endow $F$ with the structure of a [*subscheme*]{} of $X$ by considering $U$ as an open subscheme of $X$ and $F$ as the closed subscheme $Z \cap U$ of $U$ endowed with the reduced induced structure.
[^11]: By [*pure*]{} codimension $d$, we mean that every irreducible component of $C$ has codimension $d$ in $X$.
[^12]: Recall that $\dim_x X_s$ is the infimum of $\dim U$, where $U$ runs through the open neighborhoods of $x$ in $X_s$.
[^13]: Since the morphism $X\to S$ is flat, it is a local complete intersection morphism if and only if every fiber is a local complete intersection morphism (see, e.g., [@Liubook], 6.3.23).
[^14]: The hypothesis that $C\to X$ is a regular immersion is equivalent to the condition that $C \to X$ is a local complete intersection morphism (see, e.g., [@Liubook], 6.3.21).
[^15]: This morphism is projective by definition, see [@EGA], II.5.5.2. It is then also proper ([@EGA], II.5.5.3).
[^16]: When $k$ is an algebraically closed field and $R$ is a finitely generated regular $k$-algebra of dimension $2$, conditions on $R$ are given in [@M-S], Theorem 2, which ensure that such an isomorphism also exists for any locally free sheaf $\mathcal E$ on ${\operatorname{Spec}}R$.
[^17]: For an example where the gluing of two projective spaces over a ‘common’ closed subscheme is not projective, see [@Fer], 6.3.
[^18]: The ring $R(x)$ is considered already in [@Kru], page 535 after Hilfssatz 1. The notation $R(x)$ was introduced by Nagata (see the historical remark in [@Nag], p. 213). When $R$ is a local ring, the extension $R\to R(x)$ is used to reduce some considerations to the case of local rings with infinite residue fields (see, e.g., [@SH], 8.4, p. 159). Let $X$ be any scheme with an ample invertible sheaf. An affine scheme $X'$ with a faithfully flat morphism $X' \to X$ is constructed in [@Fer2], 4.3, in analogy with the purely affine situation ${\operatorname{Spec}}R(x) \to {\operatorname{Spec}}R$.
| ArXiv |
---
abstract: '[Starting from a Skyrme interaction with tensor terms, the $\beta$-decay rates of $^{52}$Ca have been studied within a microscopic model including the $2p-2h$ configuration effects. We observe a redistribution of the strength of Gamow-Teller transitions due to the $2p-2h$ fragmentation. Taking into account this effect results in a satisfactory description of the neutron emission probability of the $\beta$-decay in $^{52}$Ca.]{}'
author:
- ' $^{1),2)}$'
title: 'Strength fragmentation of Gamow-Teller transitions and delayed neutron emission of atomic nuclei'
---
The multi-neutron emission is basically a multistep process consisting of (a) the $\beta$-decay of the parent nucleus (N, Z) which results in feeding the excited states of the daughter nucleus (N - 1, Z + 1) followed by the (b) $\gamma$-deexcitation to the ground state or (c) multi-neutron emissions to the ground state of the final nucleus (N - 1 - X, Z + 1), see e.g., Ref. [@b05]. Predictions of the multi-neutron emission are needed for the analysis of radioactive beam experiments and for modeling of astrophysical r-process. Recent experiments gave an evidence for strong shell effects in exotic calcium isotopes [@w13; @s13]. For this reason, the $\beta$-decay properties of neutron-rich isotope $^{52}$Ca provides valuable information [@h85], with important tests of theoretical calculations.
[ccccccc]{} $\lambda_i^{\pi}=1_i^+$&&\
& Expt. & QRPA & 2PH & Expt. & QRPA & 2PH\
$1_1^+$ &1.64 & 1.5 & 1.3 &4.2$\pm$0.1& 4.3 & 4.3\
$1_2^+$ &2.75 & & 3.9 &4.5$\pm$0.2& & 6.4\
$1_3^+$ &3.46 & & 4.2 &5.3$\pm$0.5& & 9.2\
$1_4^+$ &4.27 & 5.0 & 4.9 &4.0$\pm$0.5& 3.2 & 3.3\
One of the successful tools for studying charge-exchange nuclear modes is the quasiparticle random phase approximation (QRPA) with the self-consistent mean-field derived from a Skyrme energy-density functional (EDF) since these QRPA calculations enable one to describe the properties of the parent ground state and Gamow-Teller (GT) transitions using the same EDF. Making use of the finite rank separable approximation (FRSA) [@gsv98] for the residual interaction, the approach has been generalized for the coupling between one- and two-phonon components of the wave functions [@svg04]. The FRSA in the cases of the charge-exchange excitations and the $\beta$-decay was already introduced in Refs. [@svg12; @ss13] and in Ref. [@svbag14; @e15], respectively. In the case of the $\beta$ decay of $^{52}$Ca, we use the EDF T45 which takes into account the tensor force added with refitting the parameters of the central interaction [@TIJ]. The pairing correlations are generated by a zero-range volume force with a strength of -315 MeVfm$^{3}$ and a smooth cut-off at 10 MeV above the Fermi energies [@svbag14]. This value of the pairing strength has been fitted to reproduce the experimental neutron pairing energy of $^{52}$Ca obtained from binding energies of neighbouring Ca isotopes.
Taking into account the basic ideas of the quasiparticle-phonon model (QPM) [@solo; @ks84], the Hamiltonian is then diagonalized in a space spanned by states composed of one and two QRPA phonons [@svbag14], $$\begin{aligned}
\Psi _\nu (J M) = \left(\sum_iR_i(J \nu )Q_{J M i}^{+}+
\sum_{\lambda _1i_1\lambda _2i_2}P_{\lambda _2i_2}^{\lambda
_1i_1}( J \nu )\left[ Q_{\lambda _1\mu _1i_1}^{+}\bar{Q}_{\lambda
_2\mu _2i_2}^{+}\right] _{J M }\right)|0\rangle, \label{wf}\end{aligned}$$ where $Q_{\lambda \mu i}^{+}\mid0\rangle$ are the wave functions of the one-phonon states of the daughter nucleus (N - 1, Z + 1); $\bar{Q}_{\lambda\mu i}^{+} |0\rangle$ is the one-phonon excitation of the parent nucleus (N, Z). We use only the two-phonon configurations $[1^{+}_{i}\otimes 2^{+}_{i'}]_{QRPA}$. In the allowed GT approximation, the $\beta^{-}$-decay rate is expressed by summing the probabilities (in units of $G_{A}^{2}/4\pi$) of the energetically allowed transitions ($E_{k}^{\mathrm{GT}}\leq Q_{\beta}$) weighted with the integrated Fermi function $$\begin{aligned}
T_{1/2}^{-1}=D^{-1}\left(\frac{G_{A}}{G_{V}}\right)^{2}
\sum\limits_{k}f_{0}(Z+1,A,E_{k}^{\mathrm{GT}})B(GT)_{k},\end{aligned}$$ $$E_{k}^{\mathrm{GT}}=Q_{\beta}-E_{1^+_k},$$ where $G_A/G_V$=1.25 and $D$=6147 s. $E_{1_k^+}$ denotes the excitation energy of the daughter nucleus. As proposed in Ref. [@ebnds99], this energy can be estimated by the following expression: $$E_{1^{+}_{k}}\approx E_{k}-E_{\textrm{2QP},\textrm{lowest}}.$$ $E_{k}$ are the eigenvalues of the wave functions (\[wf\]) and $E_{\textrm{2QP},\textrm{lowest}}$ corresponds the lowest two-quasiparticle energy. The difference in the characteristic time scales of the $\beta$ decay and subsequent particle emission processes justifies an assumption of their statistical independence (see Ref. [@b05] for more details). The $P_{n}$ probability of the delayed neutron emission is defined as the ratio of the integral $\beta$-strength to the excited states above the neutron separation energy of the daughter nucleus.
The spectrum of four low-energy $1^+$ states of $^{52}$Sc is shown in Table 1. The structure peculiarities are reflected in the $\log ft$ values. We find that the dominant contribution in the wave function of the first (fourth) $1^+$ state comes from the configuration $\{\pi1f_{7/2}\nu1f_{5/2}\}$ ($\{\pi1f_{7/2}\nu1f_{7/2}\}$). The inclusion of the four-quasiparticle configurations $\{\pi1f_{7/2}\nu1f_{5/2} \nu2p_{3/2}\nu2p_{1/2}\}$ and $\{\pi1f_{7/2}\nu1f_{5/2} \nu2p_{3/2}\nu2p_{3/2}\}$ plays the key role in our calculations of the states $1_{2}^+$ and $1_{3}^+$, respectively. The inclusion of the two-phonon configurations results in the $P_{n}$ value of 5%, and the quantitative agreement with the experimental data [@h85] is satisfactory. Note that this value is almost three times less than that within the one-phonon approximation.
In summary, by starting from the Skyrme mean-field calculations the GT strength in the $Q_{\beta}$-window has been studied within the model including the $2p-2h$ fragmentation. We analyze this effect on the $\beta$-transition rates in the case of $^{52}$Ca. Including the $2p-2h$ configurations leads to qualitative agreement with existence of four low-energy $1^+$ states of $^{52}$Sc. As a result, the probability of the delayed neutron emission is decreased.
I would like to thank I.N. Borzov, Yu.E. Penionzhkevich, and D. Verney for fruitful collaboration, N.N. Arsenyev and E.O. Sushenok for help. This work is partly supported by CNRS-RFBR Agreement No. 16-52-150003, the IN2P3-JINR agreement, and RFBR Grant No. 16-02-00228.
[99]{} $\beta$-delayed neutron emission in the $^{78}$Ni region // Phys. Rev. C. 2005. V. 71. P. 065801. Masses of exotic calcium isotopes pin down nuclear forces // Nature. 2013. V. 498. P. 346–349. Evidence for a new nuclear ‘magic number’ from the level structure of $^{54}$Ca // Nature. 2013. V. 502. P. 207–210. Beta decay of the new isotopes $^{52}$K, $^{52}$Ca, and $^{52}$Sc; a test of the shell model far from stability // Phys. Rev. C. 1985. V. 31. P. 2226–2237. Finite rank approximation for random phase approximation calculations with Skyrme interactions: an application to Ar isotopes // Phys. Rev. C. 1998. V. 57. P. 1204–1209. Effects of phonon-phonon coupling on low-lying states in neutron-rich Sn isotopes // Eur. Phys. J. A. 2004. V. 22. P. 397–403. Charge-exchange excitations with Skyrme interactions in a separable approximation// Prog. Theor. Phys. 2012. V. 128. P. 489–506. Tensor correlation effects on Gamow-Teller resonances in $^{120}$Sn and $N=80,82$ isotones// Prog. Theor. Exp. Phys. 2013. V. 2013. P. 103D03. Influence of 2p-2h configurations on $\beta$-decay rates// Phys. Rev. C. 2014. V. 90. P. 044320. Low-lying intruder and tensor-driven structures in $^{82}$As revealed by $\beta$-decay at a new movable-tape-based experimental setup// Phys. Rev. C. 2015. V. 91. P. 064317. Tensor part of the Skyrme energy density functional: Spherical nuclei// Phys. Rev. C. 2007. V. 76. P. 014312. Theory of atomic nuclei: quasiparticles and phonons. Bristol and Philadelphia, Institute of Physics, 1992. Fragmentation of the Gamow-Teller resonance in spherical nuclei// J. Phys. G. 1984. V. 10. P. 1507-1522. $\beta$-decay rates of r-process waiting-point nuclei in a self-consistent approach// Phys. Rev. C. 1999. V. 60. P. 014302.
| ArXiv |
---
author:
- |
[Nicolas Weber,]{} [Florian Schmidt,]{} [Mathias Niepert,]{} [Felipe Huici]{}\
NEC Laboratories Europe, Systems and Machine Learning Group
title: 'BrainSlug: Transparent Acceleration of Deep Learning Through Depth-First Parallelism'
---
=1
| ArXiv |
---
abstract: 'A consecutive formalism and analysis of *exactly solvable radial reflectionless potentials with barriers*, which in the spatial semiaxis of radial coordinate $r$ have one hole and one barrier, after which they fall down monotonously to zero with increasing of $r$, is presented. It has shown, that at their shape such potentials look qualitatively like radial scattering potentials in two-partial description of collision between particles and nuclei or radial decay potentials in the two-partial description of decay of compound spherical nuclear systems. An analysis shows, that the particle propagates without the smallest reflection and without change of an angle of motion (or tunneling) during its scattering inside the spherically symmetric field of the nucleus with such radial potential of interaction, i. e. the nuclear system with such interacting potential shows itself as *invisible* for the incident particle with any kinetic energy. An approach for construction of a hierarchy such reflectionless potentials is proposed, wave functions of the first potentials of this hierarchy are found.'
author:
- |
Sergei P. Maydanyuk [^1]\
*Institute for Nuclear Research, National Academy of Sciences of Ukraine*\
*prosp. Nauki, 47, Kiev-28, 03680, Ukraine*
bibliography:
- 'Ref\_IMe.bib'
title: Invisible nuclear system
---
[**PACS numbers:**]{} 11.30.Pb 03.65.-w, 12.60.Jv 03.65.Xp, 03.65.Fd, [**Keywords:**]{} invisible nucleus, supersymmetry, exactly solvable model, reflectionless radial potentials, inverse power potentials, potentials of Gamov’s type, SUSY-hierarchy
Introduction \[sec.1\]
======================
An interest to methods of supersymmetric quantum mechanics (SUSY QM) has been increasing every year. Initially constructed for a description of a symmetry between bosons and fermions in field theories, these methods during their development have formed completely independent section in quantum mechanics [@Cooper.1995.PRPLC].
Today, the methods of SUSY QM are a powerful tool for calculation and analysis of spectral characteristics of quantum systems, they have shown as extremely effective in obtaining of new types of *exactly solvable potentials* and in analysis of their properties, in an evident explanation of such unusual phenomena from the point of view of common sense as a *resonant tunneling*, a *reflectionless penetration* (or an *absolute transparency*) of the potentials (differed from the resonant tunneling by that it exists in a whole energy spectrum, where a coefficient of reflection is not only minimal but equals to zero also), *reinforcement of the barrier permeability* and *breaking of tunneling symmetry in opposite directions during the propagation of multiple of particles*, *absolute reflection for above-barrier energies*, *bound states in continuous energy spectra* of systems [@Zakhariev.1993.PHLTA; @Zakhariev.1994.PEPAN].
A number of papers has been increasing every year. Here, I should like to note a fine review [@Cooper.1995.PRPLC], to note intensively developed methods of *Nonlinear (also Polynomial, $N$-fold) supersymmetric quantum mechanics* in ), methods of *shape invariant potentials* with different types of parameters transformations (for example, see ), methods of a description of *self-similar potentials* studied by *Shabat* [@Shabat.1992.INPEE] and *Spiridonov* [@Spiridonov.1992.PRLTA; @Spiridonov.hep-th/0302046] and concerned with $q$-supersymmetry, methods of other types of potentials deformations and symmetries (for example, see [@Gomez-Ullate.quant-ph/0308062]), non-stationary approaches for a description of properties and behavior of quantum systems [@Samsonov.2002.Proc_IM]. One can note papers unified methods of supersymmetry with methods of inverse problem of quantum mechanics, and I should like to mention to nice monography [@Chadan.1977] and reviews [@Zakhariev.1994.PEPAN; @Zakhariev.1999.PEPAN] (with a literature list there). An essential progress has achieved in development of the methods of SUSY QM in spaces with different geometries [@Samsonov.1997.RusPhysJ], in non-commutative spaces [@Ghosh.2005.EPJC]. Having a powerful and universal apparatus, now the methods of SUSY QM find their application in a number of tasks of field theories, in QCD, in development of different models of quantum gravity, cosmology and other.
However, in this paper I propose to pay attention into the reflectionless phenomenon in some types of spherical symmetric quantum systems (one note in development of SUSY QM formalism for different scattering problems). We find out a new type of radial exactly solvable reflectionless potential, which in its shape has one hole and one barrier, after which it falls down monotonously to zero with increasing of radial coordinate $r$ [@Maydanyuk.2005.APNYA]. Qualitatively, such potential looks like scattering potentials in two-partial description of collision between particle and spherically symmetric nucleus or decay potentials in the two-partial description of decay of compound spherical nuclear system. An analysis has shown that the particle propagates without the smallest reflection and without change of an angle of motion (or tunneling) in its scattering in the spherically symmetric field of the nucleus with such radial potential of interaction, i. e. the nuclear system with such potential shows itself as *invisible* for the incident particle with any kinetic energy. And this paper is devoted to an analysis of such radial reflection potentials.
SUSY-interdependence between spectral characteristics of potentials partners in the radial problem \[sec.2\]
============================================================================================================
Darboux transformations \[sec.2.1\]
-----------------------------------
Let’s consider a formalism of Darboux transformations in a problem about motion of a particle with mass $m$ in the spherically symmetric potential field (also see [@Andrianov.hep-th/9404061; @Bagrov.quant-ph/9804032]). The spherical symmetry of the potential allows to reduce this problem to the one-dimensional problem about the motion of this particle in the radial field $V(r)$, defined on the positive semiaxis of $r$, where wave function of such system looks like: $$\psi(r, \theta, \varphi) =
\displaystyle\frac{\chi_{nl}(r)}{r}
Y_{lm} (\theta, \varphi),
\label{eq.2.1.1}$$ and the radial Schrödinger equation has a form: $$H \chi_{nl}(r) =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2} \chi_{nl}(r)}{dx^{2}} +
\biggl(V_{n}(r) +
\displaystyle\frac{l(l+1) \hbar^{2}}{2mr^{2}} \biggr)
\chi_{nl}(r) =
E_{n} \chi_{nl}(r)
\label{eq.2.1.2}$$ and differs from the one-dimensional Schrödinger equation by a presence of a centrifugal term. One can reduce this equation to one-dimensional one by replacement: $$\bar{V}_{n}(r) =
V_{n}(r) + \displaystyle\frac{l(l+1) \hbar^{2}}{2mr^{2}}.
\label{eq.2.1.3}$$
As in the one-dimensional case, one can introduce operators $A_{1}$ and $A_{1}^{+}$: $$\begin{array}{ll}
A_{1} =
\displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d}{dr}
+ W_{1}(r), &
A_{1}^{+} =
-\displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d}{dr}
+ W_{1}(r),
\end{array}
\label{eq.2.1.4}$$ where $W_{1}(r)$ is a function, defined in the positive semiaxis $0 \le r < +\infty$ and continuous in it with an exception of some possible points of discontinuity. Then one can determine an interdependence between two hamiltonians of the propagation of the particle with mass $m$ in the fields $\bar{V}_{1}(r)$ and $\bar{V}_{2}(r)$: $$\begin{array}{l}
H_{1} = A_{1}^{+} A_{1} + C_{1} =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2}}{dr^{2}}
+ \bar{V}_{1}(r), \\
H_{2} = A_{1} A_{1}^{+} + C_{1} =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2}}{dr^{2}}
+ \bar{V}_{2}(r),
\end{array}
\label{eq.2.1.5}$$ where each potential is expressed through one function $W_{1}(r)$: $$\begin{array}{ll}
\bar{V}_{1}(r) =
W_{1}^{2}(r) - \displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d W_{1}(r)}{dr} + C_{1}, &
\bar{V}_{2}(r) =
W_{1}^{2}(x) + \displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d W_{1}(r)}{dr} + C_{1}.
\end{array}
\label{eq.2.1.6}$$ One can find: $$\bar{V}_{2} (r) - \bar{V}_{1}(r) =
V_{2} (r) - V_{1}(r) =
2\displaystyle\frac{\hbar}{\sqrt{2m}}
\displaystyle\frac{d W_{1}(r)}{dr}.
\label{eq.2.1.7}$$ The determination of the potentials $V_{1}(r)$ and $V_{2}(r)$ of two quantum systems on the basis of one function $W_{1}(r)$ establishes the interdependence between spectral characteristics (spectra of energy, wave functions, S-matrixes) of these systems. We shall consider this interdependence, as the interdependence given by Darboux transformations in the radial problem, and we shall name $W_{1}(r)$ as *superpotential*, potentials $V_{1}(r)$ and $V_{2}(r)$ as *supersymmetric potentials-partners*. Note, that there is a constant $C_{1}$ in the definition (\[eq.2.1.2\]) of the hamiltonians of two quantum systems. If to choose $C_{1}=E^{(1)}_{0}, E^{(2)}_{0} \ne E^{(1)}_{0}$ ($E^{(1)}_{0}$ and $E^{(2)}_{0}$ are the lowest levels of energy spectra of the first and second hamiltonians $H_{1}$ and $H_{2}$), then we obtain the most widely used construction two hamiltonians $H_{1}$ and $H_{2}$ in the one-dimensional case on the basis of the operators $A_{1}$ and $A_{1}^{+}$ (for example, see p. 287–289 in [@Cooper.1995.PRPLC]). However, this case corresponds to bound states in the discrete regions of the energy spectra of two studied quantum systems. For study of scattering, decay or synthesis processes in the radial consideration usually we deal with unbound states with the continuous region of the energy spectra (with the lowest energy levels $C_{1} = E^{(1)}_{0} = E^{(2)}_{0} = 0$) of quantum systems. Therefore, one need to use $C_{1}=0$ for obtaining the interdependence between the spectral characteristics of two systems on the basis of Darboux transformations (and we obtain a construction of hierarchy of potentials as in [@Maydanyuk.2005.APNYA], see p. 443–445): $$\begin{array}{l}
H_{1} = A_{1}^{+} A_{1} =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2}}{dr^{2}}
+ \bar{V}_{1}(r), \\
H_{2} = A_{1} A_{1}^{+} =
-\displaystyle\frac{\hbar^{2}}{2m}
\displaystyle\frac{d^{2}}{dr^{2}}
+ \bar{V}_{2}(r).
\end{array}
\label{eq.2.1.8}$$
The interdependence between wave functions \[sec.2.3\]
------------------------------------------------------
We shall study two quantum systems, in each of which there is the scattering of the particle on the potential $V_{1}(r)$ or $V_{1}(r)$. Further, we shall not consider processes, concerned with loss of complete energy of systems (for example, dissipation, bremsstrahlung etc.). The energy spectra of these systems are *continuous*, and their lowest levels are *zero*. In accordence with (\[eq.2.1.8\]), we write: $$\begin{array}{l}
H_{1} \chi^{(1)}_{k,l} =
A_{1}^{+} A_{1} \chi^{(1)}_{k,l} =
E^{(1)}_{k,l} \chi^{(1)}_{k,l}, \\
H_{2} \chi^{(2)}_{k^{\prime},l^{\prime}} =
A_{1} A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}} =
E^{(2)}_{k^{\prime},l^{\prime}} \chi^{(2)}_{k^{\prime},l^{\prime}},
\end{array}
\label{eq.2.3.1}$$ where $E^{(1)}_{k, l}$ and $E^{(2)}_{k^{\prime}, l^{\prime}}$ are the energy levels of two systems with orbital quantum numbers $l$ and $l^{\prime}$, $\chi^{(1)}_{k,l}(x)$ and $\chi^{(2)}_{k^{\prime}, l^{\prime}}(x)$ are the radial components of wave functions, concerned with these levels, $k = \displaystyle\frac{1}{\hbar}\sqrt{2mE^{(1)}_{k,l}}$ and $k^{\prime} =
\displaystyle\frac{1}{\hbar}\sqrt{2mE^{(2)}_{k^{\prime},l^{\prime}}}$ are wave vectors corresponding to the levels $E^{(1)}_{k,l}$ and $E^{(2)}_{k^{\prime}, l^{\prime}}$. From (\[eq.2.3.1\]) we obtain: $$H_{2} (A_{1} \chi^{(1)}_{k,l}) =
A_{1} A_{1}^{+} (A_{1} \chi^{(1)}_{k,l}) =
A_{1} (A_{1}^{+} A_{1} \chi^{(1)}_{k,l}) =
A_{1} (E^{(1)}_{k,l} \chi^{(1)}_{k,l}) =
E^{(1)}_{k,l} (A_{1} \chi^{(1)}_{k,l}).
\label{eq.2.3.2}$$ We see, that the function $f(r)=A_{1} \chi^{(1)}_{k,l}(r)$ is the eigen-function of the operator $\hat{H}_{2}$ with quantum number $l$ to a constant factor, i. e. it represents the wave function $\chi^{(2)}_{k^{\prime},l}(r)$ of the hamiltonian $H_{2}$. The energy level $E^{(1)}_{k,l}$ must be the eigen-value of this operator exactly, i. e. it represents the energy level $E^{(2)}_{k^{\prime},l}$ of this hamiltonian. Here, new wave function and energy level have the same index ${k^{\prime}}$. One can write: $$\begin{array}{lcr}
A_{1} \chi^{(1)}_{k,l} (r) =
N_{2} \chi^{(2)}_{k^{\prime},l} (r), &
E^{(1)}_{k,l} = E^{(2)}_{k^{\prime},l}, &
N_{2} = const.
\end{array}
\label{eq.2.3.3}$$
Taking into account (\[eq.2.3.1\]), one can write: $$H_{1} (A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}) =
A_{1}^{+} A_{1} (A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}) =
A_{1}^{+} (A_{1} A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}) =
A_{1}^{+} (E^{(2)}_{k^{\prime},l^{\prime}}
\chi^{(2)}_{k^{\prime},l^{\prime}}) =
E^{(2)}_{k^{\prime},l^{\prime}}
(A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}}).
\label{eq.2.3.4}$$ and obtain: $$\begin{array}{lcr}
A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}} (r) =
N_{1} \chi^{(1)}_{k,l^{\prime}} (r), &
E^{(2)}_{k^{\prime},l^{\prime}} = E^{(1)}_{k,l^{\prime}}, &
N_{1} = const.
\end{array}
\label{eq.2.3.5}$$
Thus, we obtain the following interdependences between the wave functions and the levels of the continuous energy spectra of two systems SUSY-partners in the radial problem: $$\begin{array}{cccc}
\chi^{(1)}_{k,l^{\prime}} (r) =
\displaystyle\frac{1}{N_{1}}
A_{1}^{+} \chi^{(2)}_{k^{\prime},l^{\prime}} (r), &
\chi^{(2)}_{k^{\prime},l} (r) =
\displaystyle\frac{1}{N_{2}}
A_{1} \chi^{(1)}_{k,l} (r), &
E^{(1)}_{k,l} = E^{(2)}_{k^{\prime},l}, &
E^{(1)}_{k,l^{\prime}} = E^{(2)}_{k^{\prime},l^{\prime}}.
\end{array}
\label{eq.2.3.6}$$ Darboux transformations establish the interdependence between the wave functions for the same energy levels of two systems. The coefficients $N_{1}$ and $N_{2}$ can be calculated from a normalization conditions for the wave functions (for the continuous energy spectra), and boundary condition are defined by scattering or decay process.
The interdependence between amplitudes of transittion and reflection \[sec.2.4\]
--------------------------------------------------------------------------------
For scattering the radial superpotential $W_{1}(r)$, the potentials $V_{1}(r)$ and $V_{2}(r)$ are finite in the whole spatial region of their definition and in asymptotic they tend to zero: $$\begin{array}{ll}
W_{1} (r \to +\infty) = 0, &
V_{1} (r \to +\infty) = V_{2} (r \to +\infty) = 0.
\end{array}
\label{eq.2.4.1}$$ Let’s find an interdependence between resonant and potential components of S-matrixes of these systems (for example, also see [@Andrianov.hep-th/9404061]).
One can describe the particle motion in the direction to zero inside the fields $V_{1}(r)$ and $V_{2}(r)$ with use of plane waves $e^{-ikr}$ (we assume, that the plane waves of both systems have the same wave vectors $k$). In spatial asymptotic regions we obtain transmitted waves $T_{1}(k)e^{ikx}$ and $T_{2}(k)e^{ikx}$, which are formed in result of total propagation (with possible tunneling) through the potentials and describe the resonant scattering of the particle on the potentials, and reflected waves $R_{1}(k)e^{ikx}$ and $R_{2}(k)e^{ikx}$, which are formed in result of reflection from the potentials and describe the potential scattering of the particle on the potentials. For each process of scattering one can write components of wave functions, which are formed in result of the transmission through the potential and the reflection from it: $$\begin{array}{ll}
\chi_{inc+ref}^{(1)}(k, r \to +\infty) =
\bar{N}_{1} (e^{-ikr} + R_{1} e^{ikr}), &
\chi_{tr}^{(1)}(k, r \to +\infty) \to
\bar{N}_{1} T_{1} e^{ikr}, \\
\chi_{inc+ref}^{(2)}(k, r \to +\infty) =
\bar{N}_{2} (e^{-ikr} + R_{2} e^{ikr}), &
\chi_{tr}^{(2)}(k, r \to +\infty) \to
\bar{N}_{2} T_{2} e^{ikr},
\end{array}
\label{eq.2.4.2}$$ where the coefficients $\bar{N}_{1}$ and $\bar{N}_{2}$ can be found from the normalization conditions.
Using (\[eq.2.4.2\]) for the wave functions in asymptotic region, taking into account the interdependence (\[eq.2.3.6\]) between them and definitions (\[eq.2.1.1\]) for the operators $A_{1}$ and $A_{1}^{+}$, we obtain: $$\begin{array}{l}
\bar{N}_{1} \biggl( e^{-ikr} + R_{1} e^{ikr} \biggr) =
\displaystyle\frac{\bar{N}_{2}}{N_{1}}
\displaystyle\frac{ik\hbar}{\sqrt{2m}}
\biggl(e^{-ikr} - R_{2} e^{ikr} \biggr), \\
\bar{N}_{1} T_{1} e^{ikr} =
-\displaystyle\frac{\bar{N}_{2}}{N_{1}}
\displaystyle\frac{ik\hbar}{\sqrt{2m}} T_{2} e^{ikr}.
\end{array}
\label{eq.2.4.3}$$ These expressions are carried out only, if items with the same exponents are equal between themselves. We find: $$\bar{N}_{1} =
\displaystyle\frac{\bar{N}_{2}}{N_{1}}
\displaystyle\frac{ik\hbar}{\sqrt{2m}}
\label{eq.2.4.4}$$ and $$\begin{array}{cc}
R_{1}(k) = - R_{2}(k), &
T_{1}(k) = - T_{2}(k).
\end{array}
\label{eq.2.4.5}$$
Exp. (\[eq.2.4.5\]) establish the interdependence between the amplitudes of the transmission $T_{1}(k)$, $T_{2}(k)$ and the amplitudes of the reflection $R_{1}(k)$, $R_{2}(k)$ for the particle relatively two potentials. Squares of modules of the transmitted and reflected amplitudes represent the resonant and potential components of the S-matrixes for two systems. We see, that all these values do not depend on the normalized coefficients $N_{1}$, $N_{2}$, $\bar{N}_{1}$, $\bar{N}_{2}$.
One can introduce the matrix of scattering $S_{l}(k)$ for $l$-partial wave: $$\chi_{nl}(r) \sim S_{l}(k) e^{ikr} - (-1)^{l} e^{-ikr}
\label{eq.2.4.6}$$ and determine a phase shift $\delta_{l}(k)$: $$e^{i\delta_{l}(k)} = S_{l}(k).
\label{eq.2.4.7}$$ Then with taking into account (\[eq.2.4.2\]), we find: $$S_{l}(k) = (-1)^{l+1} (R_{l}(k) + T_{l}(k)).
\label{eq.2.4.8}$$ One can see, how these partial components of the S-matrixes and the phases for two systems are interdependent (also see [@Cooper.1995.PRPLC], p. 278–279): $$\begin{array}{cc}
S_{l}^{(1)}(k) = - S_{l}^{(2)}(k), &
\delta_{l}^{(1)}(k) = \delta_{l}^{(2)}(k) + \pi/2.
\end{array}
\label{eq.2.4.9}$$
Let’s consider a spherically symmetric quantum system with the radial potential, to which a zero amplitude of the reflection $R(k)$ of the wave function corresponds. The particle during its scattering in this field propagates into a center without the smallest reflection by the field. In particular, such is a nul radial potential. We shall name such quantum systems and their radial potentials as *reflectionless* or *absolutely transparent*. Then from (\[eq.2.4.5\]) one can see, that the potential-partner for the reflectionless potential is reflectionless also in that region, where it is finite. If such potential is finite on the whole region of its definition, then it is reflectionless completely (i. e. in standard definition of quantum mechanics). A series of the finite potentials of hierarchy, which contains the nul radial potential, should be reflectionless also. Using this simple idea and knowing a form of only one reflectionless potential, one can construct many new exactly solvable radial reflectionless potentials.
Spherically symmetric systems with absolute transparency \[sec.3\]
==================================================================
A radial reflectionless potentials with barriers \[sec.3.1\]
------------------------------------------------------------
In [@Maydanyuk.2005.APNYA] (see sec. 5.3.2, p. 459–462) an one-dimensional superpotential, defining a reflectionless potential which in semiaxis $0 < x < +\infty$ has one hole, one barrier and then with increasing of $x$ falls down monotonously to zero in asymptotic region, had found. As this superpotential is obtained on the basis of interdependence between two one-dimensional hamiltonians with continuous energy spectra, one can use it in the problem about scattering of a particle in the spherically symmetric field with a barrier and with orbital quantum number $l=0$. In such case, we have: $$\begin{array}{ll}
W(r) =
\displaystyle\frac{2\beta - \alpha}{f(\bar{r})} -
\displaystyle\frac{\beta}{\bar{r}}, &
\mbox{при } 2\beta \ne \alpha,
\end{array}
\label{eq.3.1.1}$$ where $$f(\bar{r}) = C(2\beta - \alpha) \bar{r}^{2\beta / \alpha} +
\bar{r}.
\label{eq.3.1.2}$$ Here $\bar{r} = r+r_{0}$, $\beta$ and $C$ are arbitrary real positive constants, $r_{0}$ is a positive number close to zero, and a designation $\alpha = \displaystyle\frac{\hbar}{\sqrt{2m}}$ is introduced. This superpotential is defined on the positive semiaxis of $r$ (at $r > r_{0}$).
Let’s find potentials-partners for the superpotential (\[eq.3.1.1\]). In accordance with (\[eq.2.1.6\]), we obtain: $$\begin{array}{lcl}
V_{1,2}(r) & = &
\displaystyle\frac{(2\beta - \alpha)^{2}} {f^{2}(\bar{r})} -
\displaystyle\frac{2\beta (2\beta - \alpha)}
{\bar{r} f(\bar{r})} +
\displaystyle\frac{\beta^{2}}{\bar{r}^{2}} \pm \\
& \pm &
\Biggl(
\displaystyle\frac{(2\beta - \alpha)^{2}} {f^{2}(\bar{r})} -
\displaystyle\frac{2\beta (2\beta - \alpha)}
{\bar{r} f(\bar{r})} +
\displaystyle\frac{\alpha\beta}{\bar{r}^{2}}
\Biggr)
\end{array}
\label{eq.3.1.3}$$ or $$\begin{array}{lcl}
V_{1}(r) & = &
\displaystyle\frac{\beta (\beta - \alpha)} {\bar{r}^{2}}, \\
V_{2}(r) & = &
2\displaystyle\frac{(2\beta - \alpha)^{2}} {f^{2}(\bar{r})} -
\displaystyle\frac{4\beta (2\beta-\alpha)}
{\bar{r} f(\bar{r})} +
\displaystyle\frac
{\beta (\beta+\alpha)} {\bar{r}^{2}}.
\end{array}
\label{eq.3.1.4}$$
From (\[eq.3.1.4\]) one can see that at $\beta = \alpha$ the first potential $V_{1}(r)$ obtains zero value and, therefore, it becomes reflectionless. Then, according to (\[eq.2.4.5\]), if the second potential $V_{2}(r)$ is finite in a whole region of its definition, then it should be reflectionless also. At $\beta = \alpha$ we obtain: $$V_{2}(r) =
\displaystyle\frac{2\alpha^{2}}
{\biggl(r + r_{0} + \displaystyle\frac{1}{C\alpha}\biggr)^{2}}.
\label{eq.3.1.5}$$ We see, that this potential is finite in the whole region of its definition at any values of the parameters $C>0$ and $r_{0} \ge 0$. Thus, we have obtained the reflectionless potential of the inverse power type with a shift to the left, which is defined on the whole positive semiaxis of $r$ (including $r=0$ and $r_{0}=0$).
In accordance with [@Maydanyuk.2005.APNYA] (see p. 452–455, sec. 5.1.2), one can construct a hierarchy of the inverse power potentials, and a general solution of the potential with arbitrary number $n$ can be written down so: $$\begin{array}{ll}
V_{n}(r) =
\displaystyle\frac{\gamma_{n} \alpha^{2}}{\bar{r}^{2}}, &
\gamma_{n \pm 1} = 1 + \gamma_{n} \pm \sqrt{4\gamma_{n}+1}.
\end{array}
\label{eq.3.1.6}$$ If to require, that the first potential $V_{1}(r)$ in this hierarchy must be constant (i. e. at $\gamma_{1}=0$ and $n=1$), then all hierarchy of the inverse power potentials (\[eq.3.1.6\]) becomes the *hierarchy of the reflectionless inverse power potentials*, and the solution (\[eq.3.1.5\]) becomes the general solution for the reflectionless inverse power potential. Note, that *when the hierarchy of the inverse power potentials becomes reflectionless, then the coefficients $\gamma_{n}$ become integer numbers*. We write its first values: $$\gamma_{n} = 0, 2, 6, 12, 20, 30, 42...
\label{eq.3.1.7}$$
Now, if to calculate $\beta_{n}$ for given $\gamma_{n}$ with number $n$ from (\[eq.3.1.7\]) from the following condition: $$\beta_{n} (\beta_{n}-\alpha) = \gamma_{n} \alpha^{2},
\label{eq.3.1.8}$$ then the first potential $V_{1}(r)$ from (\[eq.3.1.4\]) becomes reflectionless inverse power potential (at $\beta =\beta_{n}$). The second potential $V_{2}(r)$ from (\[eq.3.1.4\]) is finite in the whole region of its definition (including $r=0$) and should be reflectionless also, however it is not inverse power potential. So, substituting the coefficients $\gamma_{n}$ with other numbers $n$ into the second expression (\[eq.3.1.4\]) for the potential $V_{2}(r)$, one can construct the whole hierarchy of the radial reflectionless potentials of this new type.
In Fig. \[fig.1\] the potential $V_{2}(r)$ for the chosen values of the parameters $C$ and $\gamma_{n}$ is shown. From here one can see, that such potential has one hole and one barrier, after which it falls down monotonously to zero with increasing of the radial coordinate $r$.
![ A dependence of the radial potential $V_{2}(r)$ on $C$ and $\gamma_{n}$: (a) the barrier maximum and the hole minimum of this potential are changed along the axis $r$ at the change of $C$ (at $C = 0.01, 0.1, 0.3, 1.0, 2.5$, $\gamma_{n}=6$, $r_{0}=0.5$); (b) the barrier maximum of this potential practically does not changed along the axis $r$ at the change of $\gamma_{n}$ (at $C = 1$, $\gamma_{n}=2, 6, 12, 20 $, $r_{0}=0.5$). \[fig.1\]](f31a.eps "fig:"){width="57mm"} ![ A dependence of the radial potential $V_{2}(r)$ on $C$ and $\gamma_{n}$: (a) the barrier maximum and the hole minimum of this potential are changed along the axis $r$ at the change of $C$ (at $C = 0.01, 0.1, 0.3, 1.0, 2.5$, $\gamma_{n}=6$, $r_{0}=0.5$); (b) the barrier maximum of this potential practically does not changed along the axis $r$ at the change of $\gamma_{n}$ (at $C = 1$, $\gamma_{n}=2, 6, 12, 20 $, $r_{0}=0.5$). \[fig.1\]](f31b.eps "fig:"){width="57mm"}
In its behavior such potential looks qualitatively like radial potentials with barriers used in theory of nuclear collisions for a description of scattering of particles on spherical nuclei, and for a description of decay and synthesis of nuclei of a spherical type also. This potential is reflectionless, if the parameter $\gamma_{n}$ has discrete values from the sequence (\[eq.3.1.7\]). For any reflectionless potential with given $\gamma_{n}$ one can displace continuously its barrier and hole along an axis $r$ by use of the parameter $C$. Such deformation of the shape of the reflectionless potential is shown in Fig. \[fig.2\].
![The reflectionless radial exactly solvable potential $V_{2}(r)$ with the barrier. Continuous change of its shape at variation of $C$ ($\gamma_{n}=6$, $r_{0}=0.5$) \[fig.2\]](f32a.eps "fig:"){width="80mm"} ![The reflectionless radial exactly solvable potential $V_{2}(r)$ with the barrier. Continuous change of its shape at variation of $C$ ($\gamma_{n}=6$, $r_{0}=0.5$) \[fig.2\]](f32b.eps "fig:"){width="80mm"}
An analysis of wave functions \[sec.3.2\]
-----------------------------------------
### Wave functions for the reflectionless inverse power potential \[sec.3.2.1\]
Let’s find a radial wave function describing the scattering of the particle on the inverse power reflectionless potential (\[eq.3.1.5\]) at $l=0$. For the potential $V_{1}(r)$ with zero value of (\[eq.3.1.4\]) one can write its radial wave function (for arbitrary energy level concerned with wave vector $k$) at $l=0$ by such a way (at $\beta=\alpha$): $$\chi_{l=0}^{(1)}(k,r) =
\bar{N}_{1} (e^{-ikr} - S_{l=0}^{(1)} e^{ikr}).
\label{eq.3.2.1.1}$$ Then, one can find a radial wave function at $l=0$ for the reflectionless potential $V_{2}(r)$ of (\[eq.3.1.5\]) (for the energy level corresponding to the wave vector $k$) on the basis of the second expression of (\[eq.2.3.6\]). Taking into account (\[eq.2.1.4\]) and (\[eq.2.4.9\]), we obtain: $$\begin{array}{lcl}
\chi_{l=0}^{(2)}(k,r) & = &
\displaystyle\frac{1}{N_{2}} A_{1} \chi_{l=0}^{(1)}(k,r) =
\displaystyle\frac{\bar{N}_{1}}{N_{2}}
\biggl( \alpha\displaystyle\frac{d}{dr} + W(r) \biggr)
\Bigl(e^{-ikr} - S_{l=0}^{(1)} e^{ikr}\Bigl) = \\
% & = &
% i\displaystyle\frac{\bar{N}_{1}}{k\alpha N_{2}}
% \biggl[
% \biggl( 1 + \displaystyle\frac{iW(r)}{k\alpha} \biggr)
% e^{-ikr} -
% S_{l=0}^{(2)}
% \biggl( 1 - \displaystyle\frac{iW(r)}{k\alpha} \biggr)
% e^{ikr}
% \biggr] = \\
& = &
\chi_{l=0}^{(-)}(k,r) - S_{l=0}^{(2)}\chi_{l=0}^{(+)}(k,r),
\end{array}
\label{eq.3.2.1.2}$$ where $$\chi_{l=0}^{(\pm)}(k,r) =
\bar{N}_{2}
\biggl( 1 \mp \displaystyle\frac{iW(r)}{k\alpha} \biggr)
e^{\pm ikr}
\label{eq.3.2.1.3}$$ and $$\bar{N}_{2} = i\displaystyle\frac{\bar{N}_{1}}{k\alpha N_{2}}
\label{eq.3.2.1.4}$$
In accordance with main statements of quantum mechanics, for applying such form of the radial wave function to the description of scattering of the particle in the field of the potential $V_{2}(r)$, it needs to achieve a boundary requirement $\chi_{l=0}^{(2)}(k,r) \to 0$ at $r \to 0$, which gives a finiteness of the wave function (\[eq.2.1.1\]) at $r=0$ (and $S_{l=0}^{(2)}$ must have finite values and be not zero). One can see from (\[eq.3.2.1.2\]), that it is fulfilled only in case ($W(r)$ is real): $$\begin{array}{lcl}
Re (S_{l=0}^{(2)}) =
\displaystyle\frac{k^{2}\alpha^{2}-W^{2}(0)}
{k^{2}\alpha^{2}+W^{2}(0)}, &
Im (S_{l=0}^{(2)}) =
\displaystyle\frac{2W(0)k\alpha}{k^{2}\alpha^{2}+W^{2}(0)},
\end{array}
\label{eq.3.2.1.5}$$ where $$W(0) =
-\displaystyle\frac{\alpha}{r_{0} + \displaystyle\frac{1}{C\alpha}}.
\label{eq.3.2.1.6}$$ For the partial components of the S-matrix the following property $|S_{l=0}|^{2} = 1$ is fulfilled also. In limit $r \to 0$ we obtain the following expression for the radial wave function: $$\chi_{l=0}^{(2)}(k,r) =
\bar{N}_{2} \biggl(1 + \displaystyle\frac{iW(0)}{k\alpha}\biggr)
\biggl(e^{-ikr} - S_{l=0}^{(2)}
\displaystyle\frac{k\alpha - iW(0)}{k\alpha + iW(0)}
e^{ikr}\biggl),
\label{eq.3.2.1.7}$$ which in its form coincides with Exp. (\[eq.3.2.1.1\]) for the wave function for the potential $V_{1}(r)$ from (\[eq.3.1.4\]) with zero value.
In Fig. \[fig.3\] real and imaginary parts of the wave function near to the point $r=0$ are shown (here the starting formulas (\[eq.3.2.1.2\])–(\[eq.3.2.1.3\]) are taken). From Fig. \[fig.3\] (a, b) one can see a deformation of the imaginary part of this wave function with change of the wave vector $k$ and the parameter $C$. The real part of the wave function in its behavior looks like the imaginary part (see Fig. \[fig.3\] (c)). Here, one can see also that at such choice of the real and imaginary parts of the partial components of the S-matrix the wave function leaves from its zero value at $r=0$.
![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the parameter $C$ (the values of $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$ are chosen): (a) a displacement of peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the wave vector $k$ (at $k = 0.3, 0.5, 0.7 $, $C=1 $); (b) the displacement of the peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the parameter $C$ (at $C = 0.1, 0.5, 1.0$, $k=0.5$); (c) the real part of the wave function is shown (at $k = 0.3, 0.5, 0.7$, $C=1$) \[fig.3\]](f02.eps "fig:"){width="50mm"} ![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the parameter $C$ (the values of $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$ are chosen): (a) a displacement of peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the wave vector $k$ (at $k = 0.3, 0.5, 0.7 $, $C=1 $); (b) the displacement of the peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the parameter $C$ (at $C = 0.1, 0.5, 1.0$, $k=0.5$); (c) the real part of the wave function is shown (at $k = 0.3, 0.5, 0.7$, $C=1$) \[fig.3\]](f03.eps "fig:"){width="50mm"} ![The dependence of the radial wave function (\[eq.3.2.1.2\]) from the wave vector $k$ and the parameter $C$ (the values of $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$ are chosen): (a) a displacement of peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the wave vector $k$ (at $k = 0.3, 0.5, 0.7 $, $C=1 $); (b) the displacement of the peaks of the imaginary part of the wave function along the semiaxis of $r$ is shown with change of the parameter $C$ (at $C = 0.1, 0.5, 1.0$, $k=0.5$); (c) the real part of the wave function is shown (at $k = 0.3, 0.5, 0.7$, $C=1$) \[fig.3\]](f04.eps "fig:"){width="50mm"}
In Fig. \[fig.4\] an evident picture of behavior of the imaginary part of the wave function close to point $r=0$ with continuous change of the wave vector $k$ and the parameter $C$ is shown.
![ Behavior of the imaginary part of the wave function (\[eq.3.2.1.2\]) depending on the wave vector $k$ and the parameter $C$ (the values $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$) are used): (a) dependence on the wave vector $k$ (at $C=1$); (b) dependence on the parameter $C$ (at $k=0.5$) \[fig.4\]](f06.eps "fig:"){width="80mm"} ![ Behavior of the imaginary part of the wave function (\[eq.3.2.1.2\]) depending on the wave vector $k$ and the parameter $C$ (the values $\alpha=1$, $r_{0}=0$, $\bar{N}_{2}=1$) are used): (a) dependence on the wave vector $k$ (at $C=1$); (b) dependence on the parameter $C$ (at $k=0.5$) \[fig.4\]](f07.eps "fig:"){width="80mm"}
Note, that according to (\[eq.2.4.9\]), the condition (\[eq.3.2.1.5\]) can bring to not zero values of the radial wave function $\chi_{l=0}^{(1)}(k,r)$ at $r \to 0$ and can give discontinuity of the total wave function. However, a variation of the phase of $S_{l=0}^{(1)}$ does not change the form of the potential $V_{1}(r)$, which remains zero and reflectionless. In other words, the reflectionless potential $V_{1}(r)$ allows an arbitrariness in a choice of boundary conditions for the wave function at point $r=0$, and the chosen boundary conditions define the shape of the total wave function and a process proceeding in the field of the potential $V_{1}(r)$. There is a similar situation for the potential $V_{2}(r)$, which remains reflectionless with the variation of the S-matrix phase.
Now let’s analyze the form of the wave function (\[eq.3.2.1.2\]) in asymptotic region. According to (\[eq.3.1.1\]), $W(r) \to 0$ at $r \to +\infty$ and we obtain: $$\chi_{l=0}^{(2)}(k,r) =
\bar{N}_{2} \biggl(e^{-ikr} - S_{l=0}^{(2)} e^{ikr}\biggr).
\label{eq.3.2.1.8}$$ One can see, that two components $\chi_{l=0}^{(\pm)}(k,r)$ in (\[eq.3.2.1.2\]) represent convergent and divergent waves, that can be useful for analysis of propagation of the particle in the field $V_{2}(r)$. Thus, we have found an *exact analytical division of the total radial wave function into its convergent and divergent components* (as for regular and singular Coulomb functions for the known Coulomb potential) in the description of scattering of the particle in the inverse power potential (\[eq.3.1.5\]).
If for the convergent and divergent waves to define radial flows as: $$j^{\pm} (k,r) =
\displaystyle\frac{i\hbar}{2m}
\biggl( \chi_{l=0}^{(\pm)}(k,r)
\displaystyle\frac{d \chi_{l=0}^{(\pm), *}(k,r)}{dr}-
\chi_{l=0}^{(\pm), *}(k,r)
\displaystyle\frac{d \chi_{l=0}^{(\pm)}(k,r)}{dr} \biggr),
\label{eq.3.2.1.9}$$ then for both waves we obtain coincided absolute values of their flows: $$j^{\pm} (k,r) = \pm\displaystyle\frac{\hbar k}{m} |\bar{N}_{2}|^{2}.
\label{eq.3.2.1.10}$$ We see, that the flows do not vary in dependence on $r$, and this gives a fulfillment of a conservation law for the flows from each wave and the total flow. Therefore, the convergent wave $\chi_{l=0}^{(-)}(k,r)$ propagates into the center without the smallest reflection by the field, because it is defined and is continuous on the whole region of the definition of the potential (\[eq.3.1.5\]) and it forms the constant radial flow $j^{-}(r)$. Now we can tell with confidence, that the *inverse power radial potential (\[eq.3.1.5\]), for which we have found the radial wave function (\[eq.3.2.1.2\])–(\[eq.3.2.1.4\]) for scattering, is reflectionless at $l=0$*.
Further, one can find the radial wave functions at $l \ne 0$ on the basis of the same analysis, if for the radial wave function (\[eq.3.2.1.1\]) for the potential with zero value to use spherical Hankel functions instead of factors $\exp{(\pm ikr)}$.
### Wave functions for the reflectionless potential with the barrier \[sec.3.2.2\]
One can use Exp. (\[eq.3.1.4\]) for calculation of a new reflectionless potential $V_{2}(r)$ with a barrier on the basis of the known reflectionless inverse power potential $V_{1}(r)$. Let’s assume, that these potentials are connected with one superpotential $W_{2}(r)$. Let’s consider the wave function for the reflectionless inverse power potential $V_{1}(r)$ at $l=0$ in the form: $$\chi_{l=0}^{(1)}(k,r) =
\bar{N}_{1} \Bigl(f^{-}(r) e^{-ikr} -
S_{l=0}^{(1)} f^{+}(r) e^{ikr} \Bigr).
\label{eq.3.2.2.1}$$ Then the radial wave function at $l=0 $ for the reflectionless potential $V_{2}(r)$ with the barrier can be found on the basis of the second expression of (\[eq.2.3.6\]). Taking into account (\[eq.2.1.4\]) and (\[eq.2.4.9\]), we obtain: $$\begin{array}{lcl}
\chi_{l=0}^{(2)}(k,r) & = &
% \displaystyle\frac{1}{N_{2}} A_{2} \chi_{l=0}^{(1)}(k,r) =
\displaystyle\frac{\bar{N}_{1}}{N_{2}}
\biggl( \alpha\displaystyle\frac{d}{dr} + W_{2}(r) \biggr)
\Bigl(f^{-}(r) e^{-ikr} - S_{l=0}^{(1)} f^{+}(r) e^{ikr} \Bigr) = \\
& = &
\displaystyle\frac{\bar{N}_{1}}{N_{2}}
\biggl[
\biggl(\alpha \displaystyle\frac{d f^{-}(r)}{dr} -
ik\alpha f^{-}(r) + W_{2}(r) f^{-}(r) \biggr) e^{-ikr} - \\
& - &
S_{l=0}^{(2)}
\biggl(\alpha \displaystyle\frac{d f^{+}(r)}{dr} +
ik\alpha f^{+}(r) + W_{2}(r) f^{+}(r) \biggr) e^{ikr}
\biggr].
\end{array}
\label{eq.3.2.2.2}$$ In this expression one can see the division of the total radial wave function into the convergent and divergent components, that can be interesting in analysis of scattering (with possible tunneling) of the particle in the field of the reflectionless potential $V_{2}(r)$ with the barrier.
So, if to use the potential (\[eq.3.1.5\]) as the first reflectionless inverse power potential, then we find: $$\beta_{2} = 2 \alpha
\label{eq.3.2.2.3}$$ and $$\begin{array}{lcl}
f^{\pm}(r) =
1 \pm \displaystyle\frac{i}
{k \biggl(\bar{r} + \displaystyle\frac{1}{C\alpha} \biggr)}, &
\displaystyle\frac{d f^{\pm}(r)}{dr} =
\mp \displaystyle\frac{i}
{k \biggl(\bar{r} +
\displaystyle\frac{1}{C\alpha}\biggr)^{2}}, &
W_{2}(r) =
\displaystyle\frac{\alpha}{\bar{r}}
\displaystyle\frac{1 - 6C\alpha\bar{r}^{3}}
{1 + 3C\alpha\bar{r}^{3}}.
\end{array}
\label{eq.3.2.2.4}$$ Substituting these expressions into (\[eq.3.2.2.2\]), one can find the total radial wave function for the reflectionless potential with the barrier. The value of the partial component of the S-matrix $S_{l=0}^{(2)}$ can be found from a boundary condition of this wave function at point $r=0$, as it was made in the previous paragraph for the inverse power reflectionless potential (\[eq.3.1.5\]).
Conclusions \[sec.conclusions\]
===============================
In finishing we note main conclusion and new results.
- The new exactly solvable radial reflectionless potential with barrier, which in the spatial semiaxis of radial coordinate $r$ has one hole and one barrier, after which it falls down monotonously to zero with increasing of $r$, is proposed. It has shown, that at its shape such potential looks qualitatively like radial scattering potentials in two-partial description of collision between particles and nuclei or radial decay potentials in the two-partial description of decay of compound spherical nuclear systems.
- The found reflectionless potential with the barrier depends on parameters $\gamma_{n}$ and $C$. One can deform the shape of this potential: by discrete values of $\gamma_{n}$ (from the sequence (\[eq.3.1.7\])) and by continuous values of $C$. The parameter $\gamma_{n}$ at its variation does not displace visibly a maximum of the barrier and a minimum of the hole along the semiaxis $r$, but it changes their absolute values. The parameter $C$ allows to displace continuously both the barrier maximum and the hole minimum.
- A new approach for construction of a hierarchy of the radial reflectionless potentials with barriers is proposed.
- An exact analytical form for the total radial wave function, its convergent and divergent components (as for regular and singular Coulomb functions for the known Coulomb potential) has found in the description of scattering of a particle in the field of the inverse power reflectionless potential and in the field of the reflectionless potential with the barrier (at $\beta=2\alpha$).
- It has shown for the inverse power potential, that the radial flows for the convergent and divergent components of the radial wave function are constant on the whole semiaxis of $r$, have opposite directions and coincide by absolute values. This proves the reflectionless property of the inverse power potential (with a possible tunneling near the point $r=0$) on the whole semiaxis $r$. Such analysis is applicable for the found potential with the barrier also.
The analysis has shown, that any selected region of the reflectionless potential with the barrier (with take into account both the barrier region, and the small vicinity near $r=0$) does not influence on the propagation of the particle. During scattering in the spherically symmetric field with such radial potential, the particle propagates through it without the smallest reflection and without any change of angle of direction of its motion (or tunneling). One can conclude, that the found radial potential with the barrier is reflectionless for the propagation of the particle with any kinetic energy. If to use it for the two-partial description of the scattering of the particle on the nucleus with the spherical shape, then one can conclude, that such nucleus shows itself as *invisible* for the incident particle.
[^1]: E-mail: [email protected]
| ArXiv |
---
abstract: 'We study the waves at the interface between two thin horizontal layers of immiscible fluids subject to high-frequency horizontal vibrations. Previously, the variational principle for energy functional, which can be adopted for treatment of quasi-stationary states of free interface in fluid dynamical systems subject to vibrations, revealed existence of standing periodic waves and solitons in this system. However, this approach does not provide regular means for dealing with evolutionary problems: neither stability problems nor ones associated with propagating waves. In this work, we rigorously derive the evolution equations for long waves in the system, which turn out to be identical to the ‘plus’ (or ‘good’) Boussinesq equation. With these equations one can find all time-independent-profile solitary waves (standing solitons are a specific case of these propagating waves), which exist below the linear instability threshold; the standing and slow solitons are always unstable while fast solitons are stable. Depending on initial perturbations, unstable solitons either grow in an explosive manner, which means layer rupture in a finite time, or falls apart into stable solitons. The results are derived within the long-wave approximation as the linear stability analysis for the flat-interface state \[D.V. Lyubimov, A.A. Cherepanov, Fluid Dynamics [**21**]{}, 849–854 (1987)\] reveals the instabilities of thin layers to be long-wavelength.'
author:
- 'D. S. Goldobin'
- 'A. V. Pimenova'
- 'K. V. Kovalevskaya'
- 'D. V. Lyubimov'
- 'T. P. Lyubimova'
title: 'Running interfacial waves in two-layer fluid system subject to longitudinal vibrations'
---
Introduction {#sec_intro}
============
In [@Wolf-1961; @Wolf-1970] Wolf reported experimental observations of the occurrence of steady wave patterns on the interface between immiscible fluids subject to horizontal vibrations. The build-up of the theoretical basis for these experimental findings was initiated with the linear instability analysis of the flat state of the interface [@Lyubimov-Cherepanov-1987; @Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999] (see Fig.\[fig1\] for the sketch of the system considered in these works). Specifically, it was found that in thin layers the instability is a long-wavelength one [@Lyubimov-Cherepanov-1987]. In [@Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999], the linear stability was determined for the case of arbitrary frequency of vibrations.
In spite of the substantial advance in theoretical studies, the problem proved to require subtle approaches; a comprehensive straightforward weakly-nonlinear analysis of the system subject to high-frequency vibrations still remains lacking in the literature (as well as the long-wavelength one). The approach employed in [@Lyubimov-Cherepanov-1987] can be (and was) used for analysis of time-independent quasi-steady patterns (including non-linear ones) only, but not the evolution of these patterns over time. This “restricted” analysis of the system revealed that quasi-steady patterns can occur both via sub- and supercritical pitchfork bifurcations, depending on the system parameters. Later on, specifically for thin layers, which will be the focus of our work, the excitation of patterns was shown to be always subcritical [@Zamaraev-Lyubimov-Cherepanov-1989] (paper [@Zamaraev-Lyubimov-Cherepanov-1989] is published only in Russian, although the result can be derived from [@Lyubimov-Cherepanov-1987] as well). Within the approach of [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989] neither time-dependent patterns nor the stability of time-independent patterns can be analyzed. Specifically for the case of subcritical excitation, time-independent patterns may belong to the stability boundary between the attraction basins of the flat-interface state and the finite-amplitude pattern state in the phase space. [^1]
In this work we accomplish the task of derivation of the governing equations for dynamics of patterns on the interface of two-layer fluid system within the approximation of inviscid fluids. In Wolf’s experiments [@Wolf-1961; @Wolf-1970], the viscous boundary layer in the most viscous liquid was an order of magnitude thinner than the liquid layer, meaning the approximation of inviscid liquid is relevant. The layer is assumed to be thin enough for the evolving patterns to be long-wavelength [@Lyubimov-Cherepanov-1987]. With the governing equations we analyze the dynamics of the system below the linear instability threshold, where the system turns out to be identical to the ‘plus’ Boussinesq equation. The system admits soliton solutions, these solutions are parameterized with single parameter, soliton speed. The maximal speed of solitons equals the minimal group velocity of linear waves in the system; the soliton waves move always slower than the packages of linear waves. Stability analysis reveals that the standing and slow solitons are unstable while fast solitons are stable. The system, as the ‘plus’ Boussinesq equation, is known to be fully integrable.
Recently, the problem of stability of a liquid film on a horizontal substrate subject to tangential vibrations was addressed in the literature [@Shklyaev-Alabuzhev-Khenner-2009]. The stability analysis for space-periodic patterns and solitary waves for the latter system was reported in [@Benilov-Chugunova-2010]. The similarity of this problem with the problem we consider and expected similarity of results are illusive. Firstly, for the problem of [@Shklyaev-Alabuzhev-Khenner-2009] the liquid film is involved into oscillating motion only due to viscosity, an inviscid liquid will be motionless over the tangentially vibrating substrate, while in the system we consider the inviscid fluid layers will oscillate due to motion of the lateral boundaries of the container and fluid incompressibility [@Lyubimov-Cherepanov-1987; @Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999]. Secondly, the single-film case corresponds to the case of zero density of the upper layer in a two-layer system; in the system we consider this is a very specific case. These dissimilarities have their reflection in the resulting mathematical models; the governing equations for long-wavelength patterns derived in [@Shklyaev-Alabuzhev-Khenner-2009] are of the 1st order with respect to time and the 4th order with respect to the space coordinate and describes purely dissipative patterns in the viscous fluid, while the equation we will report is of 2nd order in time, 4th order in space and describes non-dissipative dynamics.
The paper is organized as follows. In Sec.\[sec\_statement\] we provide a physical description and mathematical model for the system under consideration. In Sec.\[sec\_deriv\] the governing equations for long-wavelength patterns are derived and discussed. In Sec.\[sec\_solitons\] soliton solutions are presented and their stability properties are analyzed. Conclusions are drawn in Sec.\[sec\_concl\].
Problem statement and governing equations {#sec_statement}
=========================================
We consider a system of two horizontal layers of immiscible inviscid fluid, confined between two impermeable horizontal boundaries (see Fig.\[fig1\]). The system is subject to high-frequency longitudinal vibrations of linear polarization; the velocity of vibrational motion of the system is $be^{i\omega
t}+c.c.$ (here “$c.c.$” stands for complex conjugate). For simplicity, we consider the case of equal thickness, say $h$, of two layers, which is not expected to change the qualitative picture of the system behavior [^2] but makes calculations simpler. The density of upper liquid $\rho_1$ is smaller than the density of the lower one $\rho_2$. We choose the horizontal coordinate $x$ along the direction of vibrations, the $z$-axis is vertical with origin at the unperturbed interface between layers.
In this system, at the limit of infinitely extensive layers, the state with flat interface $z=\zeta(x,y)=0$ is always possible. In real layers of finite extent, the oscillating lateral boundaries enforce liquid waves perturbing the interface; however, at a distance from these boundaries the interface will be nearly flat as well. For inviscid fluids, this state (the ground state) features spatially homogeneous pulsating velocity fields $\vec{v}_{j0}$ in both layers; $$\begin{array}{c}
\displaystyle
\vec{v}_{j0}=a_j(t)\vec{e}_x,\qquad
a_j(t)=A_je^{i\omega t}+c.c.,\\[10pt]
\displaystyle
A_1=\frac{\rho_2 b}{\rho_1+\rho_2},\qquad
A_2=\frac{\rho_1 b}{\rho_1+\rho_2},
\end{array}
\label{eq01}$$ where $j=1,2$ and $\vec{e}_x$ is the unit vector of the $x$-axis. All equations and parameters in this subsection are dimensional. The time instant $t=0$ is chosen so that $b$ and $A_j$ are real. The result (\[eq01\]) follows from the condition of zero pressure jump across the uninflected interface and the condition of the total fluid flux through the vertical cross-section being equal $\int_{-h}^{+h}v^{(x)}dz=2h(be^{i\omega t}+c.c.)$ (which is due to the system motion with velocity $be^{i\omega t}+c.c.$).
Considering the flow of inviscid fluid, it is convenient to introduce the potential $\phi_j$ of the velocity field; $$\vec{v}_j=-\nabla\phi_j\,.
\label{eq02}$$ The mass conservation law for incompressible fluid, $\nabla\cdot\vec{v}_j=0$, yields the Laplace equation for the potential, $\Delta\phi_j=0$. The kinematic conditions on the top and bottom boundaries $$\phi_{1z}(z=h)=\phi_{2z}(z=-h)=0
\label{eq03}$$ and on the interface $z=\zeta(x,y)$ $$\begin{aligned}
\dot{\zeta}&=&-\phi_{1z}+\nabla\phi_1\cdot\nabla\zeta\,,
\label{eq04}
\\[5pt]
\dot{\zeta}&=&-\phi_{2z}+\nabla\phi_2\cdot\nabla\zeta
\label{eq05}\end{aligned}$$ are also to be taken into account. (In what follows, the upper dot stands for the time-derivative and letter in subscript denotes partial derivative with respect to the corresponding coordinate.) Equations (\[eq04\]) and (\[eq05\]) can be derived from the condition that the points of zero value of the distance function $F=z-\zeta(x,y)$, which correspond to the position of the interface, move with fluid, i.e., the Lagrangian derivative (material derivative) $dF/dt=\partial F/\partial t+\vec{v}\cdot\nabla{F}$ is zero on the interface: $-\dot\zeta+v^{(z)}-\vec{v}\cdot\nabla\zeta=0$, and this holds for both fluids.
After substitution of the potential flow, the Euler equation takes the following form: $$\nabla\left(-\dot{\phi}_j+\frac{1}{2}\left(\nabla\phi_j\right)^2\right)
=\nabla\left(-\frac{1}{\rho_j}p_j-gz\right),$$ where $g$ is the gravity. The latter equation provides the expression for the pressure field in the volume of two fluids for a given flow field; $$p_j=p_{j0}+\rho_j\left(\dot{\phi}_j
-\frac{1}{2}\left(\nabla\phi_j\right)^2-gz\right).
\label{eq06}$$
Now the stress on the interface needs to be included to make the equation system self-contained, by providing the required boundary conditions for $\phi_j$ on the interface between the two fluids. The pressure jump across the interface is due to the surface tension; $$z\!=\!\zeta(x,y):
\quad
p_1-p_2 =-\alpha\nabla\cdot\vec{n}\quad
\mbox{with }\vec{n}=\frac{\nabla F}{\left|\nabla F\right|},
\label{eq07}$$ where $\alpha$ is the surface tension coefficient and $\vec{n}$ is the unit vector normal to the interface.
The system we consider does not possess any internal instability mechanisms in the absence of vibrations (unlike, e.g., [@Thiele-Vega-Knobloch-2006; @Nepomnyashchy-Simanovskii-2013]). Vibrations discriminate one of horizontal directions and there are no reasons to expect that close to the threshold of vibration-induced instabilities the excited patterns will experience spatial modulation along the $y$-direction, which is perpendicular to the vibration polarization direction. Furthermore, the linear stability analysis revealed the marginal vibration-induced instability of the flat-interface state to be long-wavelength [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989]. Hence, we restrict our consideration to the case of $(x,z)$-geometry and the long-wavelength approximation, $\left|{\partial_x\vec{v}}\right|\ll\left|{\partial_z\vec{v}}\right|$.
Governing equations for large-scale patterns {#sec_deriv}
============================================
Derivation of equations
-----------------------
In this section we derive the governing equation for long-wavelength ([*or*]{} large-scale) patterns. We employ the standard multiscale method with small parameters $(T^{-1}/\omega)$ and $(h/l)$, where $T$ is the characteristic time scale of the evolution of interface patterns (to be specified below), and $l$ is the reference horizontal length of patterns, $\partial_x\sim
l^{-1}$. The hierarchy of small parameters and the orders of magnitude of fields will be determined in the course of derivation.
Within the long-wavelength approximation, the solutions to the Laplace equation for $\phi_j(x,t)$ satisfying boundary conditions (\[eq03\]) in the most general form read $$\begin{aligned}
\phi_1=-a_1(t)x+\Phi_1(x,t)-\frac{1}{2}(h-z)^2\Phi_{1xx}(x,t)
\nonumber\\[5pt]
{}+\frac{1}{4!}(h-z)^4\Phi_{1xxxx}(x,t)-\dots\,,
\label{eqa01}\end{aligned}$$ $$\begin{aligned}
\phi_2=-a_2(t)x+\Phi_2(x,t)-\frac{1}{2}(h+z)^2\Phi_{2xx}(x,t)
\nonumber\\[5pt]
{}+\frac{1}{4!}(h+z)^4\Phi_{2xxxx}(x,t)-\dots\,.
\label{eqa02}\end{aligned}$$ Here the ground state (the flat-interface state) is represented by the terms $-a_j(t)x$; $\Phi_j(x,t)$ describe perturbation flow, they are as yet arbitrary functions of $x$ and $t$. After substitution of $p_j$ from expression (\[eq06\]) and $\phi_j$ from expressions (\[eqa01\])–(\[eqa02\]), the condition of stress balance on the interface (\[eq07\]) reads
$$\begin{aligned}
\displaystyle
\hspace{-40pt}
p_{1\infty}-p_{2\infty}
+\rho_1\Bigg[-\dot{a}_1x+\dot{\Phi}_1-\frac{(h-\zeta)^2}{2}\dot{\Phi}_{1xx}
-\frac{1}{2}\left(-a_1+\Phi_{1x}-\frac{(h-\zeta)^2}{2}\Phi_{1xxx}\right)^2\quad
\nonumber\\[10pt]
\displaystyle
\qquad
{}-\frac{((h-\zeta)\Phi_{1xx})^2}{2}+\dots\Bigg]
\nonumber\\[10pt]
\displaystyle
{}-\rho_2\Bigg[-\dot{a}_2x+\dot{\Phi}_2-\frac{(h+\zeta)^2}{2}\dot{\Phi}_{2xx}
-\frac{1}{2}\left(-a_2+\Phi_{2x}-\frac{(h+\zeta)^2}{2}\Phi_{2xxx}\right)^2\quad
\nonumber\\[10pt]
\displaystyle
{}-\frac{((h+\zeta)\Phi_{2xx})^2}{2}+\dots\Bigg]
\nonumber\\[10pt]
\displaystyle
{}+(\rho_2-\rho_1)g\zeta=\alpha\frac{\zeta_{xx}}{(1+\zeta_x^2)^{3/2}}\,.
\nonumber\end{aligned}$$
Here “…” stand for terms $\mathcal{O}_1(\dot\Phi_jh^4/l^4)+\mathcal{O}_2(\Phi_j^2h^4/l^6)+\mathcal{O}_3(a_j\Phi_jh^4/l^5)$; here and in what follows, $\mathcal{O}_j(Z)$ stand for unspecified contributions of the same order of smallness as their argument $Z$, and index $j$ is used to distinguish several nonidentical contributions to one and the same equation. We specify the order of the neglected terms so as to facilitate tracking the correctness of the derivations. The difference of constants $p_{1\infty}-p_{2\infty}$ is to be determined from the condition that in the area of vanishing perturbations of the pulsation flow, i.e. $\Phi_j(x,t)=const$, the interface remains flat, i.e. $\zeta(x,t)=0$. This condition yields $p_{1\infty}-p_{2\infty}-(\rho_1a_1^2(t)-\rho_2a_2^2(t))/2=0$. One can choose the following units of measurements for length: $L=\sqrt{\alpha/[(\rho_2-\rho_1)g]}$, for time: $T=L/b$, and for the fluid densities: $\rho_\ast$—which mean replacement $$\begin{array}{c}
(x,z)\to(Lx,Lz),\quad t\to Tt,\quad \zeta\to L\zeta,\\[5pt]
\Phi_j\to(L^2/T)\Phi_j,\quad \rho_i\to\rho_\ast\rho_i
\end{array}
\label{rescaling1}$$ in equations—and rewrite the last equation in the dimensionless form
$$\begin{array}{l}
\displaystyle
B\Bigg[\frac{\rho_1a_1^2-\rho_2a_2^2}{2}+\rho_1\dot{\Phi}_1-\rho_2\dot{\Phi}_2
-\frac{\rho_1(h-\zeta)^2}{2}\dot{\Phi}_{1xx}+\frac{\rho_2(h+\zeta)^2}{2}\dot{\Phi}_{2xx}
-\frac{\rho_1}{2}\left(a_1-\Phi_{1x}+\frac{1}{2}(h-\zeta)^2\Phi_{1xxx}\right)^2 \\[15pt]
\displaystyle\quad
{}+\frac{\rho_2}{2}\left(a_2-\Phi_{2x}+\frac{1}{2}(h+\zeta)^2\Phi_{2xxx}\right)^2
-\frac{\rho_1}{2}\left((h-\zeta)\Phi_{1xx}\right)^2
+\frac{\rho_2}{2}\left((h+\zeta)\Phi_{2xx}\right)^2
+\dots\Bigg]+\zeta
=\frac{\zeta_{xx}}{(1+\zeta_x^2)^{3/2}}\,.
\end{array}
\label{eqa03}$$
Here the dimensionless vibration parameter $$B\equiv\frac{\rho_\ast b^2}{\sqrt{\alpha(\rho_2-\rho_1)g}}=B_0+B_1
\label{eqa04}$$ ($\rho_j$ is dimensional here), where $B_0 $ is the critical value of the vibration parameter above which the flat-interface state becomes linearly unstable, $B_1$ is a small deviation of the vibration parameter from the critical value. Further, kinematic conditions (\[eq04\]) and (\[eq05\]) turn into $$\dot\zeta=\left(-(h-\zeta)\Phi_{1x}+\frac{1}{3!}h^3\Phi_{1xxx}
-a_1\zeta+\dots\right)_x ,
\label{eqa05}$$ $$\dot\zeta=\left((h+\zeta)\Phi_{2x}-\frac{1}{3!}h^3\Phi_{2xxx}
-a_2\zeta+\dots\right)_x .
\label{eqa06}$$ Here “…” stand for $\mathcal{O}_1(\Phi_jh^2\zeta/l^3)+\mathcal{O}_2(\Phi_jh^4/l^5)$. Equations (\[eqa03\]), (\[eqa05\]) and (\[eqa06\]) form a self-contained equation system.
It is convenient to distinguish two main time-modes in various fields: the average over vibration period part and the pulsation part; $$\begin{array}{l}
\zeta=\eta(\tau,x)+\xi(\tau,x)e^{i\omega t}+c.c.+\dots\,, \\[10pt]
\Phi_j=\varphi_j(\tau,x)+\psi_j(\tau,x)e^{i\omega t}+c.c.+\dots\,,
\end{array}$$ where $\tau$ is a “slow” time related to the average over vibration period evolution and “…” stand for higher powers of $e^{i\omega t}$.
In order to develop an expansion in small parameter $\omega^{-1}$, we have to adopt a certain hierarchy of smallness of parameters, fields, etc. We adopt small deviation from the instability threshold $B_1\sim\omega^{-1}$. Then $\eta\sim\omega^{-1}$ and $\partial_x\sim\omega^{-1/2}$ (cf. [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989]). It is as well established (e.g., [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989]) that for finite wavelength perturbations (finite $k\ne0$) $B_0(k)=B_0(0)+Ck^2+\mathcal{O}(k^4)$. Generally, the expansion of the exponential growth rate of perturbations in the series for $B_1$ near the instability threshold possesses a non-zero linear part, and $B_0(k)-B_0(0)\sim k^2$; therefore, $\partial_\tau\sim\mathcal{O}_1(B_1)+\mathcal{O}_2(k^2)\sim\omega^{-1}$. It is more convenient to determine the order of magnitude of $\xi$, $\varphi_j$ and $\psi_j$ in the course of development of the expansion.
Collecting terms with $e^{i\omega t}$ in equations (\[eqa05\]) and (\[eqa06\]), one finds $$\begin{aligned}
i\omega\xi+\xi_\tau=\Big(-(h-\eta)\psi_{1x}+\frac{1}{3!}h^3\psi_{1xxx}\qquad
\nonumber\\[5pt]
{}+\xi\varphi_{1x}-A_1\eta+\dots\Big)_x\,,
\label{eqa07}\end{aligned}$$ $$\begin{aligned}
i\omega\xi+\xi_\tau=\Big((h+\eta)\psi_{2x}-\frac{1}{3!}h^3\psi_{2xxx}\qquad
\nonumber\\[5pt]
{}+\xi\varphi_{2x}-A_2\eta+\dots\Big)_x\,,
\label{eqa08}\end{aligned}$$ where “…” stand for $\mathcal{O}_1((\xi\varphi+\eta\psi)h^2/l^4)+\mathcal{O}_2(\psi\,h^4/l^6)$. Terms constant with respect to $t$ sum up to $$\eta_\tau=\Big(-(h-\eta)\varphi_{1x}+\xi\psi_{1x}^\ast+c.c.-A_1\xi^\ast+c.c.+\dots\Big)_x\,,
\label{eqa09}$$ $$\eta_\tau=\Big((h+\eta)\varphi_{2x}+\xi\psi_{2x}^\ast+c.c.-A_2\xi^\ast+c.c.+\dots\Big)_x\,,
\label{eqa10}$$ where the superscript “$*$” stands for complex conjugate and “…” stand for $\mathcal{O}_1((\eta\varphi+\xi\psi)h^2/l^4)+\mathcal{O}_2(\varphi
h^4/l^6)$. The difference of equations (\[eqa07\]) and (\[eqa08\]) yields $\psi_j\sim\omega^{-1/2}$, and the difference of (\[eqa09\]) and (\[eqa10\]) yields $\varphi_j\sim\omega^{-1}$. For dealing with non-linear terms in what follows, it is convenient to extract the first correction to $\psi_j$ explicitly, i.e. write $\psi_j=\psi_j^{(0)}+\psi_j^{(1)}+\dots$, where $\psi_j^{(1)}\sim\omega^{-1}\psi _j^{(0)}\sim\omega^{-3/2}$. Equation (\[eqa07\]) (or (\[eqa08\])) yields in the leading order ($\sim\omega^{-3/2}$) $$\xi=\frac{i}{\omega}(h\psi_{1x}+A_1\eta)_x\sim\omega^{-\frac{5}{2}}\,.
\label{eqa11}$$
Considering the difference of (\[eqa08\]) and (\[eqa07\]), one has to keep in mind that we are interested in localized patterns for which $\Phi_{jx}(x=\pm\infty)=0$, $\zeta(x=\pm\infty)=0$. Hence, this difference can be integrated with respect to $x$, taking the form $$\begin{aligned}
h(\psi_1+\psi_2)_x-\eta(\psi_1-\psi_2)_x
-\frac{1}{6}h^3(\psi_1+\psi_2)_{xxx}\qquad
\nonumber\\[5pt]
{}-\xi(\varphi_1-\varphi_2)_x+(A_1-A_2)\eta+\dots=0\,,
\nonumber\end{aligned}$$ which yields in the first two orders of smallness $$h(\psi_1^{(0)}+\psi_2^{(0)})_x=-(A_1-A_2)\eta=-\frac{\rho_2-\rho_1}{\rho_2+\rho_1}\eta\,,
\label{eqa12}$$ $$h(\psi_1^{(1)}+\psi_2^{(1)})_x=(\psi_1^{(0)}-\psi_2^{(0)})_x\eta
+\frac{1}{6}h^3(\psi_1^{(0)}+\psi_2^{(0)})_{xxx}\,.
\label{eqa13}$$ The difference and the sum of equations (\[eqa09\]) and (\[eqa10\]) yield in the leading order, respectively, $$\varphi_1=-\varphi_2\equiv\varphi\,,
\label{eqa14}$$ $$\eta_\tau=-h\varphi_{xx}\,.
\label{eqa15}$$
Let us now consider equation (\[eqa03\]). We will collect groups of terms with respect to power of $e^{i\omega t}$ and the order of smallness in $\omega^{-1}$.\
$\underline{\sim\omega^{+\frac{1}{2}}e^{i\omega t}}$: $$i\omega B_0(\rho_1\psi_1^{(0)}-\rho_2\psi_2^{(0)})=0\,.$$ We introduce $$\psi^{(0)}\equiv\rho_j\psi_j^{(0)}\,.
\label{eqa16}$$ The last equation and equation (\[eqa12\]) yield $$\psi_x^{(0)}=-\frac{1}{h}\frac{\rho_1\rho_2(\rho_2-\rho_1 )}{(\rho_2+\rho_1)^2}\eta\,.
\label{eqa17}$$ $\underline{\sim\omega^0e^{i\omega t}}$: $$\mbox{No contributions.}$$ $\underline{\sim\omega^{-\frac{1}{2}}e^{i\omega t}}$: $$\begin{array}{l}
\displaystyle
i\omega B_1\underbrace{(\rho_1\psi_1^{(0)}-\rho_2\psi_2^{(0)})}_{\quad=0}
+i\omega B_0(\rho_1\psi_1^{(1)}-\rho_2\psi_2^{(1)})
\\[20pt]
\displaystyle\qquad
{}+B_0\underbrace{(\rho_1\psi_1^{(0)}-\rho_2\psi_2^{(0)})_\tau}_{\quad=0}
\\[20pt]
\displaystyle\qquad\qquad
{}+i\omega B_0\frac{h^2}{2}\underbrace{(\rho_2\psi_{2xx}^{(0)}-\rho_1\psi_{1xx}^{(0)})}_{\quad=0}=0\,.
\end{array}$$ (We marked the combinations which are known to be zero from the leading order of expansion.) Similarly to (\[eqa16\]), we introduce $$\psi^{(1)}\equiv\rho_j\psi_j^{(1)}\,.
\label{eqa18}$$ The last equation and equation (\[eqa13\]) yield $$\begin{aligned}
&&\hspace{-20pt}\displaystyle
\psi_x^{(1)}=\frac{1}{h}\frac{\rho_2-\rho_1}{\rho_2+\rho_1}\psi_x^{(0)}\eta
+\frac{h^2}{6}\psi_{xxx}^{(0)}
\nonumber\\[10pt]
&&\displaystyle
=-\frac{\rho_1\rho_2(\rho_2-\rho_1)^2}{h^2(\rho_2+\rho_1)^3}\eta
-\frac{h\rho_1\rho_2(\rho_2-\rho_1)}{6(\rho_2+\rho_1)^2}\eta_{xx}\,.
\label{eqa19}\end{aligned}$$ $\underline{\sim\omega^{-1}(e^{i\omega t})^0}$: $$B_0[-\rho_2(A_2\psi_{2x}^{(0)\ast}+c.c.)+\rho_1(A_1\psi_{1x}^{(0)\ast}+c.c.)]+\eta=0\,.$$ Substituting (\[eqa16\]) and (\[eqa17\]) into the last equation gives $$\left[-\frac{2B_0\rho_1\rho_2(\rho_2-\rho_1)^2}{h(\rho_2+\rho_1)^3}+1\right]\eta=0\,.$$ Thus we obtain the solvability condition, which poses a restriction on $B_0$; this restriction determines the linear instability threshold $$B_0=\frac{(\rho_2+\rho_1)^3h}{2\rho_1\rho_2(\rho_2-\rho_1)^2}\,.
\label{eqa20}$$ $\underline{\sim\omega^{-2}(e^{i\omega t})^0}$: (using (\[eqa14\]) for $\varphi_j$) $$\begin{array}{l}
\displaystyle
B_1\underbrace{[-\rho_2(A_2\psi_{2x}^{(0)\ast}+c.c.)
+\rho_1(A_1\psi_{1x}^{(0)\ast}+c.c.)]}_{\qquad=-\eta/B_0}\\[5pt]
\displaystyle
\qquad
{}+B_0\bigg[(\rho_2+\rho_1)\varphi_\tau+\rho_2|\psi_{2x}^{(0)}|^2
\\[15pt]
\displaystyle
{}-\rho_2\Big(A_2\psi_{2x}^{(1)\ast}+c.c.-A_2\frac{h^2}{2}\psi_{2xxx}^{(0)\ast}+c.c.\Big)
-\rho_1|\psi_{1x}^{(0)}|^2
\\[15pt]
\displaystyle\quad
{}+\rho_1\Big(A_1\psi_{1x}^{(1)\ast}+c.c.
-A_1\frac{h^2}{2}\psi_{1xxx}^{(0)\ast}+c.c.\Big)\bigg]=\eta_{xx}\,.
\end{array}$$ Substituting $\psi_j^{(n)}$ from (\[eqa16\])–(\[eqa19\]), one can rewrite the latter equation as $$\begin{array}{r}
\displaystyle
-\frac{B_1}{B_0}\eta+B_0\Bigg[(\rho_2\!+\!\rho_1)\varphi_\tau
-\frac{\rho_2\!-\!\rho_1}{\rho_2\rho_1}\left(\!\frac{\rho_1\rho_2(\rho_2\!-\!\rho_1)\eta}{h(\rho_2\!+\!\rho_1)^2}\!\right)^2
\\[15pt]
\displaystyle
{}-\frac{2\rho_1\rho_2(\rho_2-\rho_1)^3}{h^2(\rho_2+\rho_1)^4}\eta^2
-\frac{h\rho_1\rho_2(\rho_2-\rho_1)^2}{3(\rho_2+\rho_1)^3}\eta_{xx}
\qquad\\[15pt]
\displaystyle
{}+h^2\frac{\rho_1\rho_2(\rho_2-\rho_1)^2}{h(\rho_2+\rho_1)^3}\eta_{xx}\Bigg]=\eta_{xx}\,.
\end{array}$$ Together with equation (\[eqa15\]) the latter equation form the final [*system of governing equations for long-wavelength perturbations*]{} of the flat-interface state: $$\left\{
\begin{array}{rcl}
\displaystyle
B_0(\tilde\rho_2\!+\!\tilde\rho_1)\tilde\varphi_{\tilde\tau}&
\displaystyle \!\!=\!\!&
\displaystyle \left[1\!-\!\frac{\tilde{h}^2}{3}\right]\tilde\eta_{\tilde x\tilde x}
+\frac{3}{2\tilde h}\frac{\tilde\rho_2\!-\!\tilde\rho_1}{\tilde\rho_2\!+\!\tilde\rho_1}\tilde\eta^2
+\frac{B_1}{B_0}\tilde\eta\,,\\[15pt]
\displaystyle
\tilde\eta_{\tilde\tau}&
\displaystyle \!\!=\!\!&
\displaystyle -\tilde h\tilde\varphi_{\tilde x\tilde x}\,.
\end{array}
\right.
\label{eq08}$$ Here we explicitly mark the dimensionless variables and parameters with the tilde sign to distinguish them from original dimensional variables and parameters. Above in this paragraph, the tilde sign was omitted to make calculations possibly less laborious. For convenience we explicitly specify how to read rescaling (\[rescaling1\]) with the tilde-notation: $x=L\tilde
x$, $t=(L/b)\tilde t$, $\rho_i=\rho_\ast\tilde\rho_i$, etc. The expression for $B_0$ (\[eqa20\]) in the original dimensional terms reads $$B_0=\frac{\rho_\ast(\rho_2+\rho_1)^3h}{2\rho_1\rho_2(\rho_2-\rho_1)^2}
\sqrt{\frac{(\rho_2-\rho_1)g}{\alpha}}\,.
\label{eq10}$$
We remark that equation system (\[eq08\]) is valid for $B_1$ small compared to $B_0$, otherwise one cannot stay within the long-wavelength approximation. On rare occasions it is possible to use long-wavelength for finite deviations from the linear instability threshold and derive certain information on the system dynamics (e.g., in [@Goldobin-Lyubimov-2007] for Soret-driven convection from localized sources of heat or solute in a thin porous layer, an unavoidable appearance of patterns similar to hydraulic jumps [@Watson-1964] was predicted within the long-wavelength approximation though for a finite deviation from the linear instability threshold).
On the long-wavelength character of the linear instability
----------------------------------------------------------
In the text above, we relied on the fact that instability is long-wavelength for thin enough layers. Now we have appropriate quantifiers to specify quantitatively, what“thin enough” means. According to [@Lyubimov-Cherepanov-1987], we require that $\tilde
h<\sqrt{3}$. Remarkably, we can see a footprint of this fact from equation system (\[eq08\]) with multiplier $[1-\tilde{h}^2/3]$ ahead of $\tilde\eta_{\tilde x\tilde x}$. Indeed, the exponential growth rate $\tilde\lambda$ of linear normal perturbations $(\tilde\eta,\tilde\varphi)\propto\exp(\tilde\lambda\tilde
t+i\tilde k\tilde x)$ of the trivial state obeys $$\tilde\lambda^2=\frac{\tilde h\,\tilde{k}^2}{B_0(\tilde\rho_2+\tilde\rho_1)}
\left(-\Bigg[1-\frac{\tilde{h}^2}{3}\Bigg]\tilde{k}^2+\frac{B_1}{B_0}\right).
\label{eq11}$$ Below the linear instability threshold of infinitely long wavelength perturbations, i.e. for $B_1<0$, there are no growing perturbations for $\tilde{h}<\sqrt{3}$, while the perturbations with large enough $\tilde k$ grow for $\tilde{h}>\sqrt{3}$. Of course, this analysis of equation system (\[eq08\]) only highlights the long-wavelength character of the linear instability, since it deals with the limit of small $\tilde{k}$ and does not provide information on the linear stability for finite $\tilde{k}$. A comprehensive proof of the long-wavelength character of the instability for $\tilde{h}<\sqrt{3}$ comes from [@Lyubimov-Cherepanov-1987].
In the following we will consider system behavior below the linear instability threshold, i.e. for negative $B_1$. It is convenient to make further rescaling of coordinates and variables: $$\begin{array}{l}
\displaystyle
\tilde x\to x\sqrt{\frac{B_0}{(-B_1)}\Bigg[1-\frac{\tilde{h}^2}{3}\Bigg]}\,,
\\[20pt]
\displaystyle
\tilde t\to t\sqrt{\frac{\tilde\rho_2-\tilde\rho_1}{\tilde{h}}\frac{B_0^3}{B_1^2}
\Bigg[1-\frac{\tilde{h}^2}{3}\Bigg]}\,,\\[20pt]
\displaystyle
\tilde\eta\to\eta\,\tilde{h}\frac{\tilde\rho_2+\tilde\rho_1}{\tilde\rho_2-\tilde\rho_1}\frac{(-B_1)}{B_0}\,,
\\[20pt]
\displaystyle
\tilde\varphi\to\varphi\sqrt{\frac{(\tilde\rho_2+\tilde\rho_1)^2}{(\tilde\rho_2-\tilde\rho_1)^3}
\frac{B_1^2}{\tilde{h}B_0^3}\Bigg[1-\frac{\tilde{h}^2}{3}\Bigg]}\,.
\end{array}
\label{eq12}$$ We note that this implies the following rescaling of [*initial dimensional*]{} coordinates and variables: $$\begin{array}{l}
\displaystyle
x\to x\,L\sqrt{\frac{B_0}{(-B_1)}\Bigg[1-\frac{h^2}{3L^2}\Bigg]}\,,\\[20pt]
\displaystyle
t\to t\sqrt{\frac{\rho_2-\rho_1}{\rho_\ast}\frac{L^3B_0^3}{h\,b^2B_1^2}
\Bigg[1-\frac{h^2}{3L^2}\Bigg]}\,,\\[20pt]
\displaystyle
\eta\to\eta\,h\frac{\rho_2+\rho_1}{\rho_2-\rho_1}\frac{(-B_1)}{B_0}\,,\\[20pt]
\displaystyle
\varphi\to\varphi\sqrt{\frac{\rho_\ast(\rho_2+\rho_1)^2}{(\rho_2-\rho_1)^3}
\frac{L^3B_1^2}{h\,b^2B_0^3}\Bigg[1-\frac{h^2}{3L^2}\Bigg]}\,.
\end{array}
\label{eq13}$$ After this rescaling, equation system (\[eq08\]) takes the zero-parametric form; $$\begin{aligned}
\dot\varphi&=&\eta_{xx}+\frac{3}{2}\eta^2-\eta\,,
\label{eq14}
\\
\dot\eta&=&-\varphi_{xx}\,.
\label{eq15}\end{aligned}$$
The derivation of the latter equation system itself is one of the main results we report with this paper, as it allows consideration of the evolution of quasi-steady patterns in the two-layer fluid system under the action of the vibration field.
The ‘plus’ Boussinesq equation and the original Boussinesq equation for gravity waves in shallow water
------------------------------------------------------------------------------------------------------
The equation system (\[eq14\])–(\[eq15\]) can be rewritten in the form of a ‘plus’ Boussinesq equation (plus BE); $$\ddot{\eta}-\eta_{xx}+\left(\frac{3}{2}\eta^2+\eta_{xx}\right)_{xx}=0\,.
\label{eq_plBE}$$ Meanwhile, the original Boussinesq equation B (BE B) for gravity waves in a shallow water layer [@Boussinesq-1872] or in a two-layer system without vibrations [@Choi-Camassa-1999] reads $$\ddot{\eta}-\eta_{xx}-\left(\frac{3}{2}\eta^2+\eta_{xx}\right)_{xx}=0\,.
\label{eq_BEB}$$ Both systems are fully integrable and multi-soliton solutions are known for them from the literature (e.g., [@Manoranjan-etal-1984; @Manoranjan-etal-1988; @Bogdanov-Zakharov-2002]). However, their dynamics is essentially different; the original BE B suffers from the short-wave instability, while the plus BE is free from this instability. Solitons in the plus BE can be unstable, decaying into pairs of stable solitons or experiencing explosive formation of sharp peaks in finite time [@Bogdanov-Zakharov-2002; @Bona-Sachs-1988; @Liu-1993]. In the sections below we will provide overview of the soliton dynamics for equation (\[eq\_plBE\]) in relation to the fluid dynamical system we deal with. Prior to doing so, in this subsection, we would like to focus more on discussion of different kinds of the generalized Boussinesq equation and their relationships with dynamics of systems of fluid layers.
Small-amplitude gravity waves in shallow water are governed by the set A Boussinesq equations (equation system (25) in [@Boussinesq-1872]) which read in our terms after proper rescaling as $$\left\{
\begin{array}{rcl}
\displaystyle\dot{\eta}+\varphi_{xx}&\!\!=\!\!&
\displaystyle-(\eta\,\varphi_x)_x+\frac{1}{6}\varphi_{xxxx}\,,
\\[10pt]
\displaystyle\dot{\varphi}+\eta&\!\!=\!\!&
\displaystyle-\frac{1}{2}(\varphi_x)^2+\frac{1}{2}\dot\varphi_{xx}\,,
\end{array}
\right.$$ where the terms in the right hand side of equations are small, i.e., both nonlinearity and dispersion are small. To the leading corrections pertaining to nonlinearity and dispersion, the latter equation system can be recast as $$\ddot{\eta}-\eta_{xx}=\left((\varphi_x)^2+\frac{1}{2}\eta^2+\eta_{xx}\right)_{xx}\,,$$ where small terms are collected in the r.h.s. part of the equation. For waves propagating in one direction $\partial_x\approx\pm\partial_t$ and, to the leading corrections, one can make substitution $(\varphi_x)^2\approx(\dot\varphi)^2\approx\eta$, which yields equation (\[eq\_BEB\]). Thus, the Boussinesq equation for the classical problem of waves in shallow water is not only inaccurate far from the edge of the spectrum of soliton speed (near $c=1$) but is also inappropriate for consideration of collisions of counterpropagating waves (as $|\varphi_x|\ne|\dot\varphi|$ for them). In contrast, the equations we derived for our physical system are accurate close to the vibration-induced instability threshold for the entire range of soliton speeds and all kinds of soliton interactions as long as the profile remains smooth.
It is also noteworthy, that character of the original Boussinesq equation B is inherent to the dynamics of inviscid fluid layers in force fields and does not change without special external fields, the action of which cannot be formally represented by any correction to the gravity. The case of a vibration field turns out to be one such and yields the dynamics governed by the plus BE.
Long waves below the linear instability threshold {#sec_solitons}
=================================================
In this section we consider waves in the dynamic system (\[eq14\])–(\[eq15\]). In equations (\[eq13\]), one can see how the rescaling of each coordinate and variable depends on $(-B_1)=B_0-B$. From these dependencies it can be seen, that for patterns in the dynamic system (\[eq14\])–(\[eq15\]) the corresponding patterns in real time–space will obey the following scaling behavior near the linear instability threshold: spatial extent $x_\ast\propto1/\sqrt{B_0-B}$, reference time $t_\ast\propto1/(B_0-B)$, reference profile deviation $\eta_\ast\propto(B_0-B)$.
Linear waves: dispersion equation, group velocity
-------------------------------------------------
Let us first describe propagation of small perturbation, linear waves, in the dynamic system (\[eq14\])–(\[eq15\]). For normal perturbations $(\eta,\varphi)\propto\exp(-i\Omega t+ikx)$ the oscillation frequency reads $$\Omega(k)=k\sqrt{1+k^2}\,.
\label{eq16}$$ The corresponding phase velocity is $$v_\mathrm{ph}=\Omega/k=\sqrt{1+k^2}\,,
\label{eq17}$$ and the group velocity, which describes propagation of envelopes of wave packages, is $$v_\mathrm{gr}=\frac{\mathrm{d}\Omega}{\mathrm{d}k}=\frac{1+2k^2}{\sqrt{1+k^2}}\,.
\label{eq18}$$ One can see that the minimal group velocity is 1 and the group velocity $v_\mathrm{gr}$ monotonously increases as wavelength decreases (see Fig.\[fig2\]).
Solitons
--------
The dynamic system (\[eq14\])–(\[eq15\]) admits time-independent-profile solutions, solitons $\eta(x,t)=\eta(x-ct)$, where $c$ is the soliton velocity. With identical equality $\partial_t\eta(x-ct)=-c\partial_x\eta(x-ct)$, for localized patterns, which vanish at $x\to\pm\infty$, equation (\[eq15\]) can be once integrated and yields $\varphi^\prime=c\eta$ (here the prime denotes the differentiation with respect to argument). Eq.(\[eq14\]) takes the form $$0=\eta^{\prime\prime}+\frac32\eta^2-(1-c^2)\eta\,.
\label{eq19}$$ The latter equation admits the soliton solution $$\eta_0(x,t)=\frac{1-c^2}{\displaystyle\cosh^2\frac{\sqrt{1-c^2}(x\pm ct)}{2}}\,,
\label{eq20}$$ the propagation direction ($+c$ or $-c$) is determined by the flow, $\varphi^\prime=\pm c\eta$. The family of soliton solutions turns out to be one-parametric, parameterized by the speed $c$ only. The speed $c$ varies within the range $[0,1]$; standing soliton ($c=0$) is the sharpest and the tallest one and for the fastest solitons, $c\to1$, the spatial extent tends to infinity, while the height tends to $0$.
Considering in the same way a non-rescaled equation system (\[eq08\]), one can see, that for a given physical system with vibration parameter $B$ as a control parameter, the shape of a soliton solution is controlled by combination $$[(-B_1)/B_0-\tilde{c}^2\tilde{h}^{-1}B_0(\tilde\rho_2+\tilde\rho_1)]\,.
\label{eq21}$$ This means that one and the same interface inflection soliton can exist for different values of $B$, though, since the shape-controlling parameter (\[eq21\]) should be the same, the non-rescaled soliton run speed $\tilde{c}$ grows as the departure from the threshold $(-B_1)$ increases.
Since $v_\mathrm{gr}\ge1$ (see Fig.\[fig2\]) and $c^2\le1$, solitons of arbitrary height travel slower than any small perturbations of the flat-interface state. The maximal speed of solitons, $c_{\max}=1$, coincides with the minimal group velocity of linear waves. This yields notable information on the system dynamics. Fast solitons with $c$ tending to $1$ from below are extended and have a small height (see equation (\[eq20\])), while envelopes of long linear waves propagate with velocity $v_\mathrm{gr}$ tending to $1$ from above. This means that envelopes of small-height soliton packages travel faster than solitons in these packages. The issue of the generality of situations where the ranges of the possible soliton velocities and the group velocities of linear waves do not overlap but only touch each other is interestingly addressed in [@Akylas-1993; @Longuet-Higgins-1993] from the view point of emission of wave packages by the soliton (or the impossibility of such an emission).
Stability of solitons
---------------------
The stability properties of solitons in the ‘plus’ Boussinesq equation were addressed in literature [@Manoranjan-etal-1988; @Bona-Sachs-1988; @Liu-1993]; in [@Bona-Sachs-1988] the solitons with $1/2\le c\le1$ were proved to be stable and in [@Liu-1993] the solitons with $c<1/2$ were proved to be unstable. One can add more subtle details to this information: the spectrum of Lyapunov exponents (exponential growth rate) and the dependence of the scenario of nonlinear growth of perturbations on the initial perturbation.
The problem of linear stability of the soliton $\eta_0(x-ct)$ to perturbations $\big(e^{\lambda t}\eta_1(x_1),e^{\lambda
t}\varphi_1(x_1)\big)$ in the copropagating reference frame $x_1=x-ct$ reads $$\begin{aligned}
\lambda\varphi_1+c\varphi_1^{\prime}&=&
\eta_1^{\prime\prime}+3\eta_0(x_1)\,\eta_1-\eta_1\,,
\label{eq22}
\\
\lambda\eta_1+c\eta_1^{\prime}&=&-\varphi_1^{\prime\prime}
\label{eq23}\end{aligned}$$ with boundary conditions $$\eta_1(\pm\infty)=\varphi_1(\pm\infty)=0\,.
\label{eq24}$$
The eigenvalue problem (\[eq22\])–(\[eq24\]) was solved numerically with employment of the shooting method. The spectra of eigenvalues $\lambda$ for different $c$ are plotted in Fig.\[fig3\] and the first two eigenmodes of perturbations of the standing soliton ($c=0$) are plotted in Fig.\[fig4\](b). In Fig.\[fig4\](a), one can see the exponential growth rate ${\mathrm{Re}}(\lambda)$ of perturbations; the standing and slow solitons with $c<0.5$ are unstable, while the fast solitons with $c\ge 0.5$ are stable.
The scenarios of evolution of unstable solitons were observed numerically by means of direct numerical simulation of the dynamic system (\[eq14\])–(\[eq15\]) with the finite difference method in an $x$-domain of length $200$ with periodic boundary conditions and the space step size $h_x=0.05$. [^3] As in [@Manoranjan-etal-1984; @Manoranjan-etal-1988; @Bogdanov-Zakharov-2002], two possible scenarios were observed: (i) soliton explosion with formation of a finite amplitude relief or, possibly, layer rupture; (ii) falling-apart of the soliton into two stable solitons (Fig.\[fig5\]). Since the phase space of the system is infinite-dimensional, the problem of discrimination of the initial perturbations leading to explosion and those leading to falling-apart may be generally nontrivial. However, in Fig.\[fig3\] one can see that there is only one instability mode for $c<1/2$ and the nonlinear evolution of perturbations turns out to depend only on projection of the small initial perturbation on this unstable direction. If the instability mode is normalized in such a way that $\eta_1(x_1=0)>0$ (cf. Fig.\[fig4\](b)), the initial perturbations with a positive scalar product with the instability mode lead to explosion, while the perturbations with a negative scalar product lead to falling-apart into two stable solitons.
Soliton gas
-----------
In Fig. \[fig6\], a sample of the system dynamics from arbitrary initial conditions is presented in domain $x\in[0;250]$ with periodic boundary conditions. One can see that, beyond the locations of formation of singularities [@Goldobin-etal-EPL-2014-solitons], this dynamics can be well treated as the kinetics of a gas of stable solitons. For wave dynamics in soliton-bearing systems, statistical physics approaches which describe the dynamics of dense soliton gases can be developed [@El-Kamchatnov-2005].
Knowing that the system dynamics can be viewed as a kinetics of a soliton gas, we can readdress the question of relationships between the group velocity of linear waves and the speed of solitons. The fast solitons with $c\to1$ have height $\eta_\mathrm{max}=[1-c^2]\to0$ and width $\delta\propto1/\sqrt{1-c^2}\to\infty$, i.e. must obey the laws established for the linear waves with wavenumber $k\to0$. Meanwhile for the latter waves we know the group velocity $v_\mathrm{gr}=1+(3/2)k^2+\mathcal{O}(k^4)$ \[see Eq.(\[eq18\])\]. Thus, the envelopes of traveling solitons always travel faster than these solitons. For an envelope of a nearly monochromatic wave the possibility of such a behavior is obvious, while for a gas of quasi-particles additional explanations are needed. For waves of density of quasi-particles (correspond to waves of envelope) it is actually possible to travel faster than the fastest particles if these particles have a finite “collision diameter”. (For a better intuition on this, one can imagine the elastic collision of two hard spheres moving along a line. These spheres exchange their momentums from the distance equal to the sum of their radii. If they are identified only by the momentum, they effectively jump for the distance of their interaction and proceed to move with the initial momentums.) Indeed, colliding copropagating solitons exchange their momentums, which means they efficiently exchange their locations, not crossing one another, but approaching for a certain finite distance, the collision diameter. Thus the momentum efficiently jumps in the direction of soliton motion for this distance and, during one and the same time interval, the wave in a gas can cover a longer distance than the gas quasi-particles.
As soon as one can speak of the soliton density waves in a soliton gas, the question of the criterion for this gas to be considered as a continuous medium or a vacuum arises. For instance, it is obvious that one cannot speak of density waves or envelope waves for a system state with a single soliton; this is a vacuum. Whereas, for a continuous medium the concept of the group velocity should work well. Let us consider a gas of solitons with characteristic width $\delta\gg1$, i.e., quasi-particle speed $c^2=1-\delta^{-2}$ \[see Eq.(\[eq20\])\]. The signal transfer speed due to collisions (as described in the paragraph above) is larger that the particle speed, according to $v_\mathrm{gr}\approx
c/(1-n\delta)$, where $n$ is the soliton number density. Mathematically, the criterion for $n$ is of interest. For us to be able to consider the soliton gas as a continuous medium, with the group velocity featured by Eq.(\[eq18\]), $v_\mathrm{gr}$ should at least reach $1$. Hence, $c/(1-n_\mathrm{min}\delta)=1$, which can be rewritten as $1-\delta^{-2}/2=1-n_\mathrm{min}\delta$, and one finally finds $$n_\mathrm{min}\sim\delta^{-3}\,.
\label{eq:g01}$$ Interestingly, the last equation means that the maximal characteristic inter-soliton distance $\delta_\ast=1/n_\mathrm{min}$ for the gas to be a continuous medium but not a vacuum scales with the soliton width $\delta$ as $$\delta_\ast\sim\delta^3\,.
\label{eq:g02}$$
Conclusion {#sec_concl}
==========
We have considered the dynamics of patterns on the internal surface of the horizontal two-layer system of inviscid fluids subject to tangential vibrations. For thin layers ($h<\sqrt{3\alpha/[(\rho_2-\rho_1)g]}$) the instability is known to be long-wavelength and subcritical [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989]. The governing equations for long-wavelength patterns below the linear instability threshold have been derived—equation system (\[eq14\])–(\[eq15\])—allowing for the first time theoretical analysis for time-dependent patterns in the system and for stability of time-independent (quasi-steady) patterns. We note that the stability analysis for the only time-independent localized patterns in the system, standing solitons, has revealed them to be unstable.
The system dynamics is found to be governed by dynamic system (\[eq14\])–(\[eq15\]) which is equivalent to the ‘plus’ Boussinesq equation. For dynamic system (\[eq14\])–(\[eq15\]), one-parametric family of localized solutions of time-independent profile, solitons, exists (equation (\[eq20\])). These solitons are up-standing embossments of the interface (cf. black curve in Fig.\[fig4\](b)) and are parameterized by the soliton speed $c$ only, which varies from $c=0$ (the tallest and sharpest solitons) to $c=1$ (solitons with width tending to infinity and height tending to zero). The standing and slow solitons ($c<1/2$) are unstable [@Liu-1993], while the fast solitons ($c\ge 1/2$) are stable [@Bona-Sachs-1988]. The group velocity of linear waves in the system is $v_\mathrm{gr}\ge1$, meaning that all the solitons travel more slowly than any wave packages of small perturbations of the flat-interface state.
Two scenarios of development of the instability of slow solitons are possible, depending on the initial perturbation: explosion (probably leading to further layer rupture) and splitting into a pair of fast stable solitons. No other localized waves have been detected with direct numerical simulation, meaning that this one-parameter family of solitons is the only localized waves in the system. The system dynamics can be fully represented as the kinetics of gas of solitons before an explosion (and after it).
It is not possible to compare our results to the results presented by Wolf [@Wolf-1961; @Wolf-1970] in detail. Wolf presented the wave patterns of the interface for the inverted state (the heavy liquid above the light one) above the linear instability threshold of the flat-interface non-inverted state. Meanwhile, we consider waves on the interface for the non-inverted state below the threshold, and our non-trivial findings pertain specifically to this case but not to the case of the inverted stratification.
We are thankful to Dr. Maxim V. Pavlov and Dr. Takayuki Tsuchida for their useful comments on the work and drawing our attention to the fact that system (\[eq14\])–(\[eq15\]) is identical to the ‘plus’ Boussinesq equation. We thank Prof.Jeremy Levesley for his help with manuscript preparation. The work has been supported by the Russian Science Foundation (grant no. 14-21-00090).
To the memory of our teachers and friends A. A. Cherepanov, D. V. Lyubimov, and S. V. Shklyaev.
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[^1]: In the idealistic dissipation-free case, when there are no attracting states, the basins of attraction are replaced with the basins of dynamics around corresponding states; i.e., of running perturbations around the flat-interface state and of evolving finite-amplitude patterns.
[^2]: For problems considered in [@Lyubimov-Cherepanov-1987; @Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999; @Zamaraev-Lyubimov-Cherepanov-1989], moderate discrepancies between two thicknesses resulted only in quantitative corrections.
[^3]: The fact, that unperturbed unstable analytical solution used as initial conditions could persist for up to $100$ time units, suggests the discretisation and numerical integration schemes introduce quite a small inaccuracy into the system simulation.
| ArXiv |
---
abstract: 'ALOHA-type protocols became a popular solution for distributed and uncoordinated multiple random access in wireless networks. However, such distributed operation of the Medium Access Control (MAC) layer leads to sub-optimal utilization of the shared channel. One of the reasons is the occurrence of collisions when more than one packet is transmitted at the same time. These packets cannot be decoded and retransmissions are necessary. However, it has been recently shown that it is possible to apply signal processing techniques with these collided packets so that useful information can be decoded. This was recently proposed in the Irregular Repetition Slotted ALOHA (IRSA), achieving a throughput $T \simeq 0.97$ for very large MAC frame lengths as long as the number of active users is smaller than the number of slots per frame. In this paper, we extend the operation of IRSA with *i)* an iterative physical layer decoding processing that exploits the capture effect and *ii)* a Successive Interference Cancellation (SIC) processing at the slot-level, named intra-slot SIC, to decode more than one colliding packet per slot. We evaluate the performance of the proposed scheme, referred to as Extended IRSA (E-IRSA), in terms of throughput and channel capacity. Computer-based simulation results show that E-IRSA protocol allows to reach the maximum theoretical achievable throughput even in scenarios where the number of active users is higher than the number of slots per frame. Results also show that E-IRSA protocol significantly improves the performance even for small MAC frame lengths used in practical scenarios.'
author:
-
title: 'Intra-Slot Interference Cancellation for Collision Resolution in Irregular Repetition Slotted ALOHA'
---
random access protocols, slotted ALOHA, irregular repetition slotted ALOHA, bipartite graphs, capture effect, intra-slot interference cancellation, successive interference cancellation, collision resolution, iterative decoding.
Introduction
============
Uncoordinated Medium Access Control (MAC) protocols, such as ALOHA or Carrier Sensing Multiple Access (CSMA), are used in today’s communication networks due to their capability for managing the access to a shared communication channel in a distributed manner. A clear example is the operation of the Random Access Channel (RACH) of LTE which consists in a framed slotted ALOHA scheme where slots represent orthogonal preambles, which users use to contend for the access to the resources [@LAYA2014].
Despite the congestion problems that these protocols suffer from in highly dense networks, they are still the best solutions available for completely distributed access in wireless networks. There are many scenarios where centralized-based access is not possible due to the long propagation delays (e.g. satellite communications) or due to scalability issues when the number of contending devices is extremely high and unpredictable, e.g. Machine-to-Machine (M2M) networks.
Therefore, when it comes to highly dense dynamic networks, random-based distributed protocols are the only viable solution known to date. It has been proven in the literature that, among the existing alternatives, frame-based ALOHA-type protocols can perform best when optimally configured. However, the high probability of collision will still yield low performance. To overcome this limitation, the use of Successive Interference Cancellation (SIC) techniques is becoming a hot topic in the area of MAC design. The combination of the MAC layer with SIC techniques, traditionally employed at the PHY layer for coding purposes, is deemed to lead a major breakthrough in the performance of MAC protocols by turning collisions into useful information.
Recently, approaches based on multiple packet transmission [@DA] and iterative interference cancellation (IC) [@CRDSA], [@IRSA] have shown to yield dramatic performance improvements in terms of throughput with respect to previous existing solutions.
The Contention Resolution Diversity Slotted ALOHA (CRDSA) protocol proposed in [@CRDSA] was the first ALOHA-based protocol providing the adoption of SIC techniques for resolving collisions. More specifically, each packet is transmitted in two different randomly selected slots within a MAC frame. Even though this approach apparently increases the network load, it provides time-domain diversity through the transmission of a redundant copy of each packet. The replicas of each packet possess a pointer to the slot where the other replica was sent. Whenever a packet is successfully decoded, the pointer is extracted and the interference contribution caused by the twin replica on the corresponding slot is removed. The procedure is iterated, eventually permitting the recovery of the whole set of packets transmitted within the same frame. CRDSA achieves a maximum throughput, defined as probability of successful packet transmission per slot, of $T \simeq 0.55$, while the peak throughput for Framed Slotted ALOHA is just $T \simeq 0.37$.
The CRDSA protocol was later generalized in [@IRSA], allowing users to transmit more than 2 copies of the same packet per frame. In particular, the actual number of packet replicas is drawn from a probability mass function, referred to as *degree distribution* [@IRSA], that is optimized to achieve the maximum supportable load on the shared medium. Since the number of transmitted replicas is different from user to user, this scheme is dubbed Irregular Repetition Slotted Aloha (IRSA). In [@IRSA], the operation of IRSA is described by borrowing concepts from graph codes such as belief propagation on a packet level for resolving collisions. It provides a bipartite graph representation allowing a fast analytical characterization of the IRSA performance. The convergence analysis of the SIC process shows that IRSA provides a throughput equal to $T \simeq 0.97$ if a suitable degree distribution is selected and as long as the number of available slots is greater than the number of contending devices. Despite these promising performance figures, IRSA cannot perform optimally when the number of devices is greater than the number of slots. This behavior can represent a boundary in scenarios suffering of channel overload problems such as M2M networks where a massive number of devices limits its application in realistic scenarios.
This is the main motivation for the work presented in this paper, where we propose an extension of IRSA, referred to as Extended IRSA (E-IRSA), which can operate excellently even when the number of devices is above the number of available slots per frame. In the proposed scheme, the receiver attempts to recover as many data packets as possible for each single slot exploiting the capture effect, which enables to decode those packets received with the strongest signal in a given slot. Whenever a packet is decoded, its interference contribution is subtracted first from the overall signal received in that slot, i.e., *intra-slot SIC*, and then, as well as in IRSA, from signals received in the slots where the related packet replicas have been also transmitted, i.e., *inter-slot SIC*.
In summary, E-IRSA extends IRSA in two ways: $(i)$ it applies iterative physical-layer decoding that exploits the capture effect in order to decode more than one data packet per slot, and $(ii)$ it applies intra-slot SIC, in order to increase the decoding probability of the next colliding packets.
This extension has been motivated by the promising results published in [@MUDIRSA] and [@Stephan]. The work in [@MUDIRSA] presents a theoretical study on a generalized IRSA scheme assuming that the receiver is capable of decoding multiple colliding packets jointly using *multiuser detection* (MUD) techniques in systems adopting code-division multiple access (CDMA).
In its turn, the work in [@Stephan] describes a practical implementation of a further generalization of IRSA, the so-called Coded Slotted Aloha (CSA) [@CSA], where several options for decoding more than one packet per slot in case of collision are considered. This work relies on concepts from physical layer network coding (PNC) [@PNC1], [@PNC2], and MUD and it also shows how it is possible to perform intra-slot SIC removing one or more packets from the overall signal received in a slot.
The rest of the paper is organized as follows. Section II introduces the system model and notations of E-IRSA. The description of the proposed collision resolution scheme is then provided in Section III. Simulation results are provided in Section IV. Finally, Section V concludes the paper.
System Model and Notation
=========================
We consider a network composed by one receiver (also referred to as coordinator) and $m$ devices (also referred to as users) located at one-hop distance from the coordinator, forming a star topology. Every user is frame- and slot-synchronous, and has only one data packet (also referred to as burst or message) ready to transmit to the coordinator, per MAC frame. The latter is divided into $n$ slots of equal length. The transmission of a packet takes at most one slot. According to [@IRSA], each of the $m$ users performs a random number of replicas of the same packet, referred to as *repetition rate*, selected by a probability mass function we dubbed *degree distribution*. Furthermore, the users transmit in randomly selected slots and without performing carrier sensing. Hence, each slot can be in one of three states: $(i)$ empty, i.e., no user has transmitted in the slot; $(ii)$ clean, i.e., one user has transmitted in the slot; or $(iii)$ collision, i.e., more users have transmitted in the same slot. As introduced by [@IRSA], the IRSA operation can be described by a bipartite graph $\mathcal{G}=(U,S,E)$ consisting of a set $U$ of $m$ *user nodes*, i.e., one for each packet that is transmitted, a set $S$ of $n$ *sum nodes*, i.e., one for each slot in the MAC frame, and a set $E$ of *edges*. An edge connects a user node (UN) $u_i \in U$ to a sum node (SN) $s_j \in S$ if and only if a replica of the $i$-th packet is transmitted in the $j$-th slot. Loosely speaking, UNs correspond to packets, SNs correspond to slots, and each edge corresponds to a packet replica. Hence, a packet with $l$ replicas is represented by a UN with $l$ neighbors. A slot where $l$ replicas collide corresponds to a SN with $l$ connections. The number of edges connected to a node is referred to as the *node degree*. In Fig. \[frameSIC\] an example of IRSA MAC frame is displayed while Fig. \[graphSIC\] shows the corresponding bipartite graph model.
![An example of MAC frame composed by $n$ slots, with $m$ active users adopting IRSA protocol to transmit their packets.[]{data-label="frameSIC"}](framexample.pdf){width="2.8in"}
![Bipartite graph model corresponding to the above MAC frame status.[]{data-label="graphSIC"}](graphexample.pdf){width="2.3in"}
We define the *logical system traffic load* $G$ as the average number of packet transmissions per slot and it is given by $G=m/n$. It provides a direct measure of the traffic handled by the scheme in contrast with the physical system load, which also takes in account the replicas. We also define the *throughput* $T$ as the average number of successful packet transmission per slot and it is given by $T=G(1-P_L)$ where $P_L$ is the probability that a transmission attempt does not succeed, referred to as *burst loss probability*. Hence, throughput performance depends on $G$, $P_L$ and accordingly on the variable repetition rate selected by the user. It measures the number of received packets per slot and can assume values greater than one due to the capability of receiver to decode more than one packet per slot. It is worth noticing that, this new definition of throughput is a generalization of that one provided by [@IRSA], i.e., the probability of successful packet transmission per slot. The channel is modelled as block fading, where the fading coefficient $h$ follows a Rayleigh distribution with $h \sim \mathcal{CN}(0,\sqrt{\textrm{SNR}})$. It is constant during one slot, but it can vary user by user and slot by slot. The packet of the generic user $k \in {1, 2, ..., m}$ is denoted as $\textbf{u}_k$. Every packet is coded using a linear block code of length $C_L$ and rate $R$. The corresponding codeword symbol is $\textbf{c}_k=\textbf{u}_k\textbf{Z}$ where $\textbf{Z}$ is a common generator code matrix, which is the same for all the users. The codeword symbols are then mapped, for example, to BPSK symbols and the mapping function $\mu$ is defined element-wise as $\mu(0)=-1$, $\mu(1)=1$. Hence, the codeword symbol of a generic bit position $p$ transmitted by a generic user $k$ is given by $x_{k,p}=\mu(c_{k,p}) \in \{-1,1\}$. The choice of modulation does not change the principles of the following analysis. Assuming that in a single slot $K$ users collide, with $K \leq m$, the received signal can be written as:
$$\label{eq:receivedSignal}
\textbf{y} = \sum_{k=1}^K{h_k\textbf{x}_k}+\textbf{w}$$
where $\textbf{w}$ is the channel noise (AWGN) with $\textbf{w} \sim \mathcal{N}(0,1)$. Fig. \[channelModel\] shows the $K$-user multiple-access channel as described above.
![K-user multiple-access channel with block fading. All users apply the same channel code.[]{data-label="channelModel"}](channelmodel.pdf){width="3.3in"}
Extended IRSA
=============
The IRSA protocol efficiently works when a large frame size is considered and the number of active users is lower than the number of slots in the MAC frame, i.e., the logical system traffic load $G<1$. In these conditions, the probability to have a loop in the corresponding graph representation is reduced. Loop is a specific combination of collisions that interrupts the iterative IC process. For instance, there is a loop when a couple of packets are transmitted in the same couple of slots (see Fig.\[graphloop\]). Furthermore, the assumption on very large frame size requires a more complex receiver, introduces delay and, as a consequence, it could not be reasonable in a practical context.
![An example of loop in the IRSA MAC frame graph representation between two user nodes and two slot nodes.[]{data-label="graphloop"}](graphloopexample.pdf){width="2.5in"}
Extended IRSA (E-IRSA) aims to provide an efficient collision resolution scheme by decoding multiple colliding packets per slot, by supporting logical system traffic load $G \geq 1$, and by offering optimal performance in terms of throughput, even for small MAC frame sizes. Furthermore, our work provides a complete and concrete analysis in a realistic context, such as applications for M2M scenarios. We extend the IRSA protocol presented in [@IRSA] by considering the physical layer decoding combined with intra-slot and inter-slot IC processes. More specifically, E-IRSA works as follows. Each user transmits, in randomly selected slots, a number of replica packets depending on a common probability mass function. The receiver stores the received signal of the entire frame and attempts to resolve collisions by performing a physical-layer decoding. In order to maintain low both delays and receiver complexity we considered the simplest and most ordinary decoding approach for extracting information from colliding packets. A detailed description of the adopted decoding method can be found in [@Stephan Section III.C]. This approach separately decodes each packet belonging to $\textbf{y}$ (i.e., the overall signal formed by colliding packets within a single slot), considering all other packets as interference. It exploits the channel state information (CSI) of all the users and the known transmit alphabet, i.e., the BPSK constellation. The decoding process is carried out putting in a soft-input decoder, (such as a Viterbi, a turbo, or an LDPC decoder) the log-likelihood value (L-value) of the generic user $k$ and the generic bit position $p$,
$$\label{eq:Lvalue}
L_{k,p} \triangleq \ln \frac{P[c_{k,p}=1~|~y_p]}{P[c_{k,p}=0~|~y_p]} = \ln \frac{P[x_{k,p}=1~|~y_p]}{P[x_{k,p}=-1~|~y_p]}.$$
For each slot by starting sequentially from the first to the last one, the receiver tries to recover as many user packets as possible, based on the received signal $\textbf{y} = [y_1, y_2, ..., y_{C_L}]$ exploiting capture effect, i.e., starting to decode from the colliding packet having the strongest signal in the slot. Whenever a packet $\textbf{u}_{\widetilde{k}}$ is successfully decoded, the receiver can obtain the corresponding codeword $\textbf{c}_{\widetilde{k}}$ and the symbol sequence $\textbf{x}_{\widetilde{k}}$. Hence, it performs the *intra-slot SIC*, subtracting the corresponding interference contribution $h_{\widetilde{k}}\textbf{x}_{\widetilde{k}}$ from the received signal $\textbf{y}$, according to
$$\label{eq:intraSIC}
\textbf{y} = \sum_{k=1}^K{h_k\textbf{x}_k}+\textbf{w}-h_{\widetilde{k}}\textbf{x}_{\widetilde{k}}.$$
Then, as well as IRSA, the receiver performs the *inter-slot SIC* by extracting the packet header, which contains the pointers to the slots where the replicas of the decoded packet have been transmitted, and by subtracting the interference contributions caused by the replicas, from the received signals of the corresponding slots. The IC process can be represented through a message-passing along the edges of the graph. All the details on SIC operation are described in [@CRDSA], [@IRSA]. This procedure is iterated until the decoding process of all remaining packets of the current slot fails. When a failure occurs, E-IRSA procedure focuses to the next slot. At the end of the frame, if there are still undecoded packets, the whole E-IRSA procedure is repeated one more time. E-IRSA procedure can be executed only under the following conditions:
- packets have a pointer to their $l-1$ respective replicas. The overhead due to the inclusion of pointers in the packet header may be reduced by adopting the efficient techniques described in [@IRSA];
- the receiver is able to get a good estimation of the channel state information. Under this assumption the interference cancellation and the estimation of the channel parameters necessary to perform it are ideal, i.e., the receiver will estimate, as for a conventional one, the fading coefficient of every user in every slot. These assumptions simplify the analysis without substantially affecting the performance, as shown in [@CRDSA],[@IRSA].
It is worth noticing that, differently to IRSA, E-IRSA is scalable since the presence of loops in the MAC frame graph representation does not interrupt the iterative IC process. Every loop can be resolved by performing intra-slot SIC after a successful packet decoding. In the future, we will analyze the convergence properties of the proposed IC process.
Differently to [@CSA], we did not introduce an additional physical decoding step in order to minimize the user device operations with a consequence reduction of the user-side power consumption, delay and hardware complexity.
Differently to [@MUDIRSA], which considers multiuser detection techniques to jointly decode colliding packets with a fixed *joint capability*, we adopt the simplest decoding approach and the number of decoded packets per slot may vary slot by slot according to the instantaneous channel quality conditions and the expected SNR value.
Simulation Results
==================
A numerical evaluation is conducted by using MATLAB^^, to assess the effectiveness of the proposed E-IRSA. In order to make the simulative scenario as realistic as possible, our simulator includes: $(i)$ all the user-side steps from the setting of the replica repetition rate to the packet generation, $(ii)$ the generation of the pointers to the replicas, $(iii)$ a complete symbol-level implementation of the signal received by the coordinator, $(iv)$ the soft-input decoder/encoder, $(v)$ the channel model and $(vi)$ the successive interference cancellation by exploiting the pointers of the decoded packet.
Scenario
--------
We implemented the channel model as block fading with additive noise. The fading coefficients, which follow a Rayleigh distribution, are constant during one slot but independent for each user and each slot. For all simulations, we use an LDPC code from the standard [@WiMAX] with code rate $R=1/2$, word length $C_L=576$ bits, and message length $RC_L=288$ bits, i.e., the packet payload size including the pointers to the other replicas. The fixed packet size for all users has been chosen just for ease of implementation. However, a variable packet size does not affect the algorithm operation. Altough the E-IRSA application is not restricted to M2M networks, it appears to be the most natural application of this scheme. For these reasons, we have taken in account a realistic packet payload size for a possible M2M scenario, e.g. smart metering.
The probability mass function, which sets the replica repetition rate of each user, is fixed to $0.5x^2+0.28x^3+0.22x^8$. According to [@IRSA], this is the optimal degree distribution, for a maximum repetition rate equal to 8, that maximizes the *system load threshold* $G^*$, allowing transmission with vanishing error probability ($P_L\rightarrow0$) for any offered traffic up to $G^*$, which represents the channel capacity. Finally, each simulation run has been repeated several times to get 95% confidence intervals.
Table \[tab:sim\] lists the main simulation parameters related to the considered scenario.
----------------------------------------------------------- --
**Parameter & **Value\
MAC frame length $n$ & 10, 20, 100, 1000 slots, variable\
SNR \[dB\] & 10, 20, 30, 40, variable\
Number of users $m$ & 80, 100, 120, 150, variable\
Codeword length $C_L$ \[bit\] & 576\
Code Rate $R$ & $1/2$\
Message length $RC_L$ \[bit\] & 288\
Modulation & BPSK\
Symbol level decoder & LDPC\
Fading & Rayleigh Block fading\
Probability mass function & $0.5x^2+0.28x^3+0.22x^8$\
Maximum Repetition Rate & 8\
****
----------------------------------------------------------- --
: Simulation Parameters[]{data-label="tab:sim"}
Performance Evaluation
----------------------
For the sake of completeness, in all the proposed numerical evaluation the performance IRSA approach are taken into account.
The first set of simulations we assume a fixed MAC frame size $n=20$, and a variable number of users $m$ transmitting within the MAC frame in order to have values of system traffic load $G \in [0,2]$. In Fig. \[n20\_TvsG\], the throughput is shown by varying the system traffic load and for different SNR values: 10, 20, 30 and 40 dB, respectively.
The relation throughput versus system load provided by E-IRSA, for higher SNRs is linear, i.e., every packet is decoded, hence, the whole traffic turns into throughput. In these cases, the throughput is the maximum theoretical achievable throughput, according to the expression $T=G(1-P_L)$.
In E-IRSA, the capability of the receiver to decode more than one packet per slot allows to manage systems with traffic loads $G \geq 1$, i.e., systems where the number of users is higher than the number of slots. This result is proved by the threshold effect occurring for $G \simeq 1.5$, even for lower SNR value such as 10 dB. Indeed, the throughput increases linearly up to the threshold then it degrades due to the threshold effect related to the iterative interference cancellation process. Similar behavior is obtained increasing the MAC frame size to $n=100$, as shown in Fig. \[n100\_TvsG\].
By increasing the MAC frame length, the iterative IC process provides the best performance for $n \rightarrow \infty$. Indeed, in Fig. \[n1000\_TvsG\], for a MAC frame size $n=1000$, IRSA scheme provides an improved throughput $T \simeq 0.8$ when $G \simeq 0.8$ and for an SNR value at least equal to 20 dB. E-IRSA scheme still provides the maximum achievable throughput as in the previous scenario. This concept is emphasized in Fig. \[G08\_TvsSlot\].
The second set of simulations assumes a fixed MAC frame size $n=100$ and evaluates the throughput by varying the number of active users $m$ (80, 100, 120, 150 users) and for different values of $SNR \in [10,40]$. The results shown in Fig. \[n100\_TvsSNR\], highlight the opposite throughput behavior between E-IRSA and IRSA when $G$ increases. Indeed, when the number of active users per frame increases, the achievable throughput increases for E-IRSA approach, but decreases for IRSA scheme. For instance, when $G=1$, i.e., $m=100$, E-IRSA scheme provides a throughput $T=1$ and IRSA scheme provides a throughput value no greater than 0.2, while for $G=1.5$, E-IRSA and IRSA scheme provide a throughput value equal to $T \simeq 1.5$ and $T \simeq 0$, respectively. This behavior occurs when the maximum system traffic load supportable by IRSA scheme is exceeded.
The last set of simulations assumes a fixed $G$ value equal to 0.8 and evaluates the throughput by varying the MAC frame size (from 10 to 1000 slots) and the SNR values (10, 20, 30, 40 dB). Fig. \[G08\_TvsSlot\] further demonstrates that E-IRSA performance does not depend on the MAC frame size, while IRSA scheme provides the maximum throughput when $n \rightarrow \infty$.
Conclusions
===========
The E-IRSA protocols has been presented in this paper as an extension of IRSA. By considering physical-layer decoding in combination with intra-slot and inter-slot successive interference cancellation, it is possible to decode multiple colliding packets in a given slot, thus boosting the performance of the wireless network. The results presented in this paper show that the traditional definition of throughput can reach values higher than one due to the capability of the receiver to decode more than one data packet per slot. For this reason, we have adopted the average number of successful packet transmission per slot as key performance metric, showing how E-IRSA can reach the maximum theoretical achievable throughput, i.e., the whole traffic load turns into throughput, even if the system traffic load $G \geq 1$, and independently of the MAC frame lengths and the number of contending devices. In addition, the gain in throughput leads to fewer retransmissions therefore the proposed scheme also promises improvements on device energy consumption and average delay of communications at the network level. Future work will aim at further exploring the performance of E-IRSA in terms of delay and average energy consumed by the users, separately, following the same methodology used in [@SIC-FSA] with respect to Framed Slotted ALOHA with SIC.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work has been partially funded by European Research Projects ADVANTAGE (FP7-607774) and P2P-SMARTEST (H2020-646469), and by the Catalan Government under grant (2014-SGR-1551).
[30]{} A. Laya, L. Alonso, and J. Alonso-Zarate, “Is the Random Access Channel of LTE and LTE-A Suitable for M2M Communications? A Survey of Alternatives,” *IEEE Tutorials and Survey Communications Magazine*, vol. 16, no.1, January 2014. L. Choudhury Gagan, and S. Rappaport Stephen, “Diversity ALOHA - A Random Access Scheme for Satellite Communications,” *Communications, IEEE Transactions on*, vol.31, no.3, pp.450, 457, March 1983. E. Casini, R. De Gaudenzi, and Od.R. Herrero, “Contention Resolution Diversity Slotted ALOHA (CRDSA): An Enhanced Random Access Scheme for Satellite Access Packet Networks,” *Wireless Communications, IEEE Transactions on*, vol.6, no.4, pp.1408, 1419, April 2007. G. Liva, “Graph-Based Analysis and Optimization of Contention Resolution Diversity Slotted ALOHA,” *Communications, IEEE Transactions on*, vol.59, no.2, pp.477, 487, February 2011. M. Ghanbarinejad, and C. Schlegel, “Irregular repetition slotted ALOHA with multiuser detection,” *Wireless On-demand Network Systems and Services (WONS), 2013 10th Annual Conference on*, pp.201, 205, 18-20 March 2013. S. Pfletschinger, M. Navarro, and G. Cocco, “Interference cancellation and joint decoding for collision resolution in slotted ALOHA,” *Network Coding (NetCod), 2014 International Symposium on*, pp.1, 6, 27-28 June 2014. E. Paolini, G. Liva, and M. Chiani, “Coded slotted ALOHA: A graph-based method for uncoordinated multiple access,” *Information Theory, IEEE Transactions on*, January 2014. S. Zhang, S. C. Liew, and P. P. Lam, “Hot topic: Physical-layer network coding,” in *ACM MobiCom*, Los Angeles, California, USA, Sept. 2006. P. Popovski, and H. Yomo, “The anti-packets can increase the achievable throughput of a wireless multi-hop network,” in *IEEE International Conference on Communication (ICC)*, Istanbul, Turkey, June 2006. *IEEE Standard for Local and metropolitan area networks. Part 16: Air Interface for Broadband Wireless Access Systems*, IEEE Standard for Local and metropolitan area networks Std., May 2009. F. Vazquez-Gallego, M. Rietti, J. Bas, J. Alonso-Zarate, and L. Alonso, “Performance Evaluation of Frame Slotted-ALOHA with Successive Interference Cancellation in Machine-to-Machine Networks,” *European Wireless 2014; 20th European Wireless Conference; Proceedings of*, pp.1, 6, 14-16 May 2014.
| ArXiv |
---
abstract: 'The paper deals with the program of determining the complexity of various homeomorphism relations. The homeomorphism relation on compact Polish spaces is known to be reducible to an orbit equivalence relation of a continuous Polish group action (Kechris-Solecki). It is shown that this result extends to locally compact Polish spaces, but does not hold for spaces in which local compactness fails at only one point. In fact the result fails for those subsets of ${\mathbb{R}}^3$ which are unions of an open set and a point. In the end a list of open problems is given in this area of research.'
author:
- 'Vadim Kulikov[^1]'
bibliography:
- 'ref1.bib'
title: 'Classification and Non-classification of Homeomorphism Relations'
---
**MSC2010:** 03E15, 57N10, 57M25, 57N65.
#### Acknowledgments. {#acknowledgments. .unnumbered}
I am grateful to Professors Alexander Kechris and Su Gao for providing useful answers to my e-mails which helped and encouraged me to proceed with this work. During the time of the preparation I also had several valuable discussions on this topic with my Ph.D. supervisor Tapani Hyttinen and my friends and collegues Rami Luisto, Pekka Pankka and Marcin Sabok. Last but not least I would like to thank a dear to me woman (whose identity is concealed) for the infinite inspiration that she has given me during the time of this work.
This research was supported by the Austrian Science Fund (FWF) under project number P24654.
Introduction {#sec:Intro}
============
It is known that the homeomorphism relation on Polish spaces is $\Sigma^1_2$ [@Gao] and ${{\Sigma_1^1}}$-hard [@FerLouRos Thm 22]. On the other hand, it is known that restricted to compact spaces this homeomorphism relation is reducible to an orbit equivalence relation induced by continuous action of a Polish group which is known to be strictly below ${{\Sigma_1^1}}$-complete. This result is extended to locally compact spaces in Theorem \[thm:locCom\].
The main result of this paper is that this “nice” property of locally compact spaces breaks when just one point is added to them: The homeomorphism relation on the ${\sigma}$-compact spaces of the form $V\cup \{x\}$ where $x\in {\mathbb{R}}^3$ is fixed and $V\subset {\mathbb{R}}^3$ is open falls somewhere in between: it is ${{\Sigma_1^1}}$ and the equivalence relation known as $E_1$ is continuously reducible to it (Theorem \[thm:NonClass\]). This implies that this homeomorphism relation is not classifiable in a Borel way by any orbit equivalence relation arising from a Borel action of a Polish group.
The proof relies on known results in knot theory and low dimensional topology. We hope that these methods can be helpful in approaching Question \[open:Main\] and other questions listed in Section \[sec:Further\].
Sections \[sec:KnotTheory\] and \[sec:BkgrndDST\] are devoted to the required preliminaries. In Section \[sec:NonClass\] we prove the main non-classification result. In the final sections the research topic of classifying homeomorphism relations is looked at in more detail: In Section \[sec:Other\] it is reviewed what positive results there are in classification of homeomorphism relations and in Section \[sec:Further\] a list of open questions is given in the area.
Preliminaries in Topology and Knot Theory {#sec:KnotTheory}
=========================================
In this section we go through those definitions and lemmas in knot theory and topology that we need in the proofs later. We assume that the reader is familiar with the notion of the first homology group $H_1(X)$ of a topological space $X$. The standard definitions can be found for example in [@Hatcher].
We denote by ${\mathbb{R}}^n$ the $n$-dimensional Euclidean space and by ${\mathbb{S}}^n$ the one-point compactification of it, i.e. ${\mathbb{S}}^n={\mathbb{R}}^n\cup\{\infty\}$ and the neighborhoods of $\infty$ are the sets of the form $\{\infty\}\cup({\mathbb{R}}^n\setminus C)$ where $C$ is compact. By ${{\operatorname{int}}}A$ we denote the topological interior of $A$ and by $\bar A$ the closure.
Hausdorff Metric and Path Connected Subspaces
---------------------------------------------
\[def:HausdorffMetric\] Let $X$ be a compact metric space. The space of all non-empty compact subsets of $X$ is denoted by $K(X)$. We equip $K(X)$ with the Hausdorff-metric: An *${\varepsilon}$-collar* of a set $C\subset X$ is the set $$C_{\varepsilon}=\{x\mid d(x,C)< {\varepsilon}\}$$ and the Hausdorff-distance between two sets in $K(X)$ is determined by $$d_{K(X)}(C,C')=\max\{\inf\{{\varepsilon}\mid C\subset C'_{\varepsilon}\},\inf\{{\varepsilon}\mid C'\subset C_{\varepsilon}\}\}.$$
The following facts are standard to verify.
\[fact:Hausdorffmetric\] Let $X$ be a compact metric space. Then $K(X)$ is compact and if $(C_i)_{i\in{\mathbb{N}}}$ is a converging sequence in $K(X)$ and $C_*$ is its limit, then
1. for every $x_*$ we have $x_*\in C_*$ if and only if there is a sequence $x_i$ converging to $x_*$ with $x_i\in C_i$ for all $i\in{\mathbb{N}}$. \[fact:Haus1\]
2. if every $C_i$ is connected, then $C_*$ is connected. \[fact:Haus2\] \[fact:Haus3\]
\[def:DenselyPathConnected\] A subset $A\subset {\mathbb{R}}^n$ is *path metric* if the distance between two points is given by $$d_E(x,y)=\inf\{L({\gamma})\mid {\gamma}\subset A\text{ is a path joining }x\text{ and }y\}$$ where $d_E$ is the Euclidean distance and $L({\gamma})$ is the length of the path. Equivalently $A$ is path metric if and only if for every two points $x,y\in A$ and ${\varepsilon}>0$ there is a path ${\gamma}\subset A$ connecting $x$ to $y$ and $L({\gamma})<(1+{\varepsilon})d_E(x,y)$.
\[lem:HDPathMetric\] If the Hausdorff dimension of a closed $A\subset{\mathbb{R}}^n$ is less than $n-1$, then ${\mathbb{R}}^n\setminus A$ is path metric.
Let $D_0$ be the $(n-1)$-dimensional unit disc $$D_0=\{(x_1,\dots,x_{n-1})\mid x_1^2+\dots+x_{n-1}^2<1\}.$$ and let $C_0$ be the cylinder $$D_0\times [0,1]\subset {\mathbb{R}}^n.$$ For $x,y\in{\mathbb{R}}^n$ denote by $[x,y]$ the straight line segment connecting $x$ and $y$. Suppose $A_0\subset C_0$ and assume that for every $(x_1,\dots,x_{n-1})\in D_0$ the set $$A_0\cap [(x_1,\dots,x_{n-1},0),(x_1,\dots,x_{n-1},1)]$$ is non-empty. Then $A_0$ must have Hausdorff dimension at least $n-1$: $A_0$ can be projected onto $D_0$ with the Lipschitz map $$(x_1,\dots,x_{n-1},x_n)\mapsto (x_1,\dots,x_{n-1},0),$$ the latter has Hausdorff dimension $n-1$ and the Hausdorff dimension cannot increase in a Lipschitz map. Therefore we have the following claim:
If $A_0\subset C_0$ has Hausdorff dimension less than $n-1$, then there is $(x_1,\dots,x_{n-1})\in D_0$ such that $[(x_1,\dots,x_{n-1},0),(x_1,\dots,x_{n-1},1)]\cap A_0={\varnothing}.$
Let $x,y\in {\mathbb{R}}^n\setminus A$ and let ${\varepsilon}>0$. Since $A$ is closed there is ${\delta}<{\varepsilon}/2$ such that $\bar B(x,{\delta})\cap A=\bar B(y,{\delta})\cap A={\varnothing}$. Let $P_x$ and $P_y$ be $(n-1)$-dimensional affine hyperplanes passing through $x$ and $y$ respectively and which are orthogonal to $x-y$. Then there is an affine map $f\colon {\mathbb{R}}^n\to{\mathbb{R}}^n$ such that $f[P_x]={\mathbb{R}}^{n-1}\times\{0\}$, $f[P_y]={\mathbb{R}}^{n-1}\times \{1\}$ and $f[P_x\cap \bar B(x,{\delta})]=D_0\times\{0\}$. Since $\dim_H(A)<n-1$, also $\dim_H(f[A])<n-1$ (because $f$ is Lipschitz) and so by the claim above there is a line segment $s$ passing from $f[\bar B(x,{\delta})\cap P_x]$ to $f[\bar B(y,{\delta})\cap P_y]$ outside $f[A]$ which is orthogonal to ${\mathbb{R}}^{n-1}\times\{0\}$. By applying $f^{-1}$ to $s$, we obtain a straight line segment passing from $\bar B(x,\delta)\cap P_x$ to $\bar B(y,\delta)\cap P_y$ orthogonal to $P_x$. Now by connecting the endpoints of $f^{-1}[s]$ to $x$ and $y$ we obtain a path outside $A$ of length at most $d(x,y)+2{\delta}=d(x,y)+{\varepsilon}$ connecting these two points.
\[lemma:CauchyComponent\] Suppose $X,X'\subset {\mathbb{R}}^n$ are such that $X$ is a path metric space and there is a homeomorphism $h\colon X\to X'$. If $(x_i)_{i\in{\mathbb{N}}}$ is a Cauchy sequence in $X$ converging in ${\mathbb{R}}^n$ to some point $x\in {\mathbb{R}}^n\setminus X$, then all the accumulation points of $(h(x_i))_{i\in{\mathbb{N}}}$ lie in the same component of ${\mathbb{R}}^n\setminus X'$. In particular, if this component is a singleton, then $(h(x_i))_{i\in{\mathbb{N}}}$ is also a Cauchy sequence.
Let $(x_i)_{i\in{\mathbb{N}}}$ be as in the statement. Suppose for a contradiction that $y^1$ and $y^2$ are two points in two different components of ${\mathbb{R}}^n\setminus X'$ that are accumulation points of $(h(x_i))_{i\in{\mathbb{N}}}$ and let $(x^1_{i})_{i\in{\mathbb{N}}}$ and $(x^2_{i})_{i\in{\mathbb{N}}}$ be subsequences of $(x_i)_{i\in{\mathbb{N}}}$ such that $(h(x^1_i))_{i\in{\mathbb{N}}}$ and $(h(x^2_i))_{i\in{\mathbb{N}}}$ converge to $y^1$ and $y^2$ respectively. For $k\in\{1,2\}$ and $i\in{\mathbb{N}}$ denote $y^k_i=h(x^k_i)$.
For each $i\in {\mathbb{N}}$ let ${\gamma}_i$ be a path in $X$ connecting $x^1_i$ to $x^2_{i}$ such that $L({\gamma})<(1+2^{-i})d(x^1_i,x^2_{i})$. We think of the paths as compact subsets of ${\mathbb{S}}^n$. The sequence $\{{\gamma}_i\mid
i\in{\mathbb{N}}\}$ converges in $K({\mathbb{S}}^n)$ to $\{x\}$. Consider the sequence $(h[\gamma_i])_{i\in{\mathbb{N}}}$. It is a sequence of compact subsets of ${\mathbb{S}}^n$, so it is a sequence of elements of $K({\mathbb{S}}^n)$. The latter is compact, so there is a converging subsequence: $(h[\gamma_{i(k)}])_{k\in{\mathbb{N}}}$. Denote by $\gamma$ the limit of that sequence. By Fact \[fact:Hausdorffmetric\].\[fact:Haus1\], we have $y^1,y^2\in {\gamma}$ since $y^1_i,y^2_i\in h[\gamma_i]$ for all $i\in{\mathbb{N}}$ and additionally, since every element in the sequence is connected, ${\gamma}$ is also connected by Fact \[fact:Hausdorffmetric\].\[fact:Haus2\].
Since $y^1$ and $y^2$ lie in different components of ${\mathbb{R}}^n\setminus X'$, there must be a point $z$ in ${\gamma}$ which is in $X'$. Now, by Fact \[fact:Hausdorffmetric\].\[fact:Haus1\] we can find a sequence $z_k\in h[{\gamma}_{i(k)}]$, $k\in{\mathbb{N}}$, such that $(z_k)_{k\in{\mathbb{N}}}$ converges to $z$. But $h^{-1}(z_k)$ lies in ${\gamma}_{i(k)}$ and so $(h^{-1}(z_k))_{k\in{\mathbb{N}}}$ converges to $x$. This is a contradiction, because $x\notin {\operatorname{dom}}h=X$, but $z\in {\operatorname{ran}}h= X'$.
Separation Theorems {#sec:Sep}
-------------------
Here we state, for the sake of completeness, two known results from finite dimensional topology that we will need.
\[thm:JordanBrouwer\] Let $h\colon {\mathbb{S}}^{n-1}\to {\mathbb{S}}^n$ be an embedding. Then ${\mathbb{S}}^n\setminus h[{\mathbb{S}}^{n-1}]$ consists of two open connected components.
(A Generalization of the Schönflies Theorem by M. Brown [@Brown])\[thm:Brown\] Let $h\colon {\mathbb{S}}^{n-1}\times [0,1]\to {\mathbb{S}}^n$ be an embedding. Then the closures of the complementary domains of $h[{\mathbb{S}}^{2}\times \{\frac{1}{2}\}]$ are topological $n$-cells, i.e. homeomorphic to closed balls. In particular there is a self-homeomorphism of ${\mathbb{S}}^n$ which takes $h[{\mathbb{S}}^{n-1}\times \{\frac{1}{2}\}]$ to the standard ${\mathbb{S}}^{n-1}$.
Knot Theory
-----------
We present the basics of knot theory here as neatly as possible and account only for the facts necessary for the present paper. Unless a specific reference is given below, the reader is referred to the classical textbooks on knot theory [@BurZie; @Kauf; @Mur] for the details and omitted proofs.
A *knot* is an embedding $K\colon {\mathbb{S}}^1\to {\mathbb{S}}^3$. We often identify a knot with its image, ${\operatorname{ran}}K$. This is in particular justified by the following equivalence relation on knots:
Two knots $K_0,K_1\colon {\mathbb{S}}^1\to {\mathbb{S}}^3$ are equivalent, if there is a homeomorphism $h\colon {\mathbb{S}}^3\to {\mathbb{S}}^3$ with $$K_0=h\circ K_1.$$ In literature this homeomorphism is often required to be orientation preserving in which case this equivalence relation coincides with the so-called *ambient isotopy*, but we do not require $h$ to be orientation preserving.
A knot is *trivial* if it is equivalent to the standard embedding ${\mathbb{S}}^1{\hookrightarrow}{\mathbb{S}}^3$. A knot is *tame* if it is equivalent to a smooth or a piecewise linear knot. As usual in knot theory, we consider only tame knots.
The following is a basic fact of knot theory:
There are infinitely many non-equivalent knots.
A not so basic fact is the following theorem:
([@GorLue])\[thm:GordonLuecke\] If two knots have homeomorphic complements, then they are equivalent.
Let $K$ be a knot in ${\mathbb{R}}^3$. A *Seifert surface* $S$ of $K$ is a compact orientable connected $2$-manifold with boundary $M\subset {\mathbb{R}}^3$ whose interior lies in ${\mathbb{R}}^3\setminus K$ and the boundary is exactly $K$.
\[fact:ExistsSeifertS\] For every open ball $B$ containing $K$ there exists a Seifert surface $S\subset B$ of $K$.
([@Rolfsen 5.D])\[fact:LinkingSeifert\] Let $K$ be a knot and $w$ a closed curve in ${\mathbb{R}}^3\setminus K$. The following are equivalent:
- $w\cap S\ne{\varnothing}$ for every Seifert surface $S$ of $K$,
- $w$ represents a non-trivial element in $H_1({\mathbb{R}}^3\setminus K)$.
\[fact:NonTrHomology\] For every knot $K$ we have $H_1({\mathbb{S}}^3\setminus K)\cong {\mathbb{Z}}$.
Preserving Knot Types
---------------------
The goal of this section is to prove Lemma \[lemma:FindPoints\] which says that if we carve out infinitely many knots from ${\mathbb{R}}^3$ in a certain way, then a self-homeomorphism of the left-over space will, in an approximate way, respect the knot-types of the carved knots.
\[def:Properties\] Let $(B_n)_{n\in{\mathbb{N}}}$ be a sequence of closed balls in ${\mathbb{R}}^3$, $(K_n)_{n\in{\mathbb{N}}}$ a sequence of knots, $Q\subset {\mathbb{R}}^3$ and $P\subset {\mathbb{R}}^3$. Here we list some properties for these sets which we will later refer to.
- All the balls are disjoint from each other and are contained in a bounded region, i.e. there is $r$ such that ${\bigcup}_{n\in{\mathbb{N}}}B_n\subset B(0,r)$.
- If $x$ is a limit of a sequence $(x_i)_{i\in{\mathbb{N}}}$ such that for all $i\in{\mathbb{N}}$ the point $x_i$ is in the ball $B_{n_i}$ and for all $i<j$, $n_i\ne n_j$, then $x$ is not in any of the balls. $Q$ is the set of such points $x$.
- $P\supset Q$, every connected component of $P$ contains a point in $Q$ and for all $n$ there is ${\varepsilon}>0$ such that $P\cap (B_n)_{{\varepsilon}}={\varnothing}$. (Recall the definition of ${\varepsilon}$-collar, Definition \[def:HausdorffMetric\]).
- $K_n\subset {{\operatorname{int}}}B_n$.
- $X={\mathbb{R}}^3\setminus (P\cup{\bigcup}_{n\in{\mathbb{N}}}K_n)$ is path metric (Definition \[def:DenselyPathConnected\]).
\[lemma:FindPoints\] Suppose $(B_n)_{n\in{\mathbb{N}}}$, $(K_n)_{n\in{\mathbb{N}}}$, $Q$ and $P$ as well as $(B_n')_{n\in{\mathbb{N}}}$, $(K_n')_{n\in{\mathbb{N}}}$, $Q'$ and $P'$ satisfy the properties B1 – B5. Let $$X={\mathbb{S}}^3\setminus (P\cup{\bigcup}_{n\in{\mathbb{N}}}K_n)$$ and $$X'={\mathbb{S}}^3\setminus (P'\cup{\bigcup}_{n\in{\mathbb{N}}}K'_n).$$ Suppose further that $X$ and $X'$ are homeomorphic and $h$ is the homeomorphism. Then there is a bijection $\rho\colon{\mathbb{N}}\to{\mathbb{N}}$ such that for all $n\in{\mathbb{N}}$ we have that $K_n$ and $K_{\rho(n)}'$ have the same knot-type and for some $z\in B_n\setminus K_n$ we have $h(z)\in B_{\rho(n)}'\setminus K_{\rho(n)}'$.
Fix $n\in {\mathbb{N}}$. By the Jordan-Brouwer separation theorem (Theorem \[thm:JordanBrouwer\]) the complement of $h[\partial B_n]$ in ${\mathbb{S}}^3$ consists of two open connected components, say $Y_1$ and $Y_2$. In this case, however, we can prove even more, namely that $Y_1$ and $Y_2$ are homeomorphic to open balls and that there is a self-homeomorphism of ${\mathbb{S}}^3$ wich takes $h[\partial B_n]$ to ${\mathbb{S}}^2$. Let ${\varepsilon}$ be small enough so that $(\partial B_n)_{\varepsilon}\cap
B_k={\varnothing}$ for all $k\ne n$, $(\partial B_n)_{\varepsilon}\cap P={\varnothing}$ and $(\partial B_n)_{{\varepsilon}}\cap K_n={\varnothing}$. This is possible by B2, B3 and B4. Let $$f\colon {\mathbb{S}}^2\times [0,1]\to (\partial B_n)_{{\varepsilon}}$$ be a homeomorphism such that $f[{\mathbb{S}}^2\times \{\frac{1}{2}\}]=\partial B_n$. We can think of $h\circ f$ as an embedding of ${\mathbb{S}}^2\times [0,1]$ into ${\mathbb{S}}^3$. Now apply the the generalized Schönflies theorem (Theorem \[thm:Brown\]) to $h\circ f$. Since $\partial B_n$ divides $X$ into two disjoint components as well as $h[\partial B_n]$ divides $X'$, $h$ takes them to one another. Assume without loss of generality that $h[{{\operatorname{int}}}B_n\setminus K_n]=Y_1\cap X'$.
\[claim:ConnectedThing\] The space $Y_1\setminus X'$ is connected.
For this we need a slight modification of the argument used to prove Lemma \[lemma:CauchyComponent\]. (Note that $B_n\setminus K_n$ is path metric.) Suppose there was two components $A$ and $B$ of $Y_1\setminus X'$ and let $(x_1,y_1,x_2,y_2,\dots)$ be a sequence such that $(x_i)_{i\in{\mathbb{N}}}$ converges (in ${\mathbb{S}}^3$) to a point in $A$ and $(y_i)_{i\in{\mathbb{N}}}$ converges to a point in $B$. Now $(h^{-1}(x_1),h^{-1}(y_1)\cdots)$ can only have accumulation points in $K_n$ (because the accumulation points cannot be in $X$). Pick Cauchy subsequences from both $(h^{-1}(x_i))_{i\in{\mathbb{N}}}$ and $(h^{-1}(y_i))_{i\in{\mathbb{N}}}$ and denote $(z_i)_{i\in{\mathbb{N}}}$ and $(w_i)_{i\in{\mathbb{N}}}$. Since $z=\lim_{i\to\infty}z_i$ and $w=\lim_{i\to\infty}w_i$ lie both in the knot, using the fact that $B_n\setminus K_n$ is path metric, it is possible to connect $z_i$ to $w_i$ by a curve ${\gamma}_i$ lying in $B_n\setminus K_n$ such that the sequence $({\gamma}_i)_{i\in{\mathbb{N}}}$ converges in $K({\mathbb{S}}^3)$ to a subset of $K_n$. Now pick (in $K({\mathbb{S}}^3)$) a converging subsequence $(\xi_{j})_{j\in{\mathbb{N}}}$ of $(h[{\gamma}_i])_{i\in{\mathbb{N}}}$. These are connected sets containing $h(z_i)$ and $h(w_i)$. Therefore the limit in $K({\mathbb{S}}^3)$ must intersect both $A$ and $B$ and since it is connected, it must contain a point $p$ in $Y_1\cap X'= h[{{\operatorname{int}}}B_n]$. By Fact \[fact:Hausdorffmetric\] there is a Cauchy sequence $(p_j)_{j\in{\mathbb{N}}}$ with $p_j\in \xi_{j}$ converging to $p$ but $((h^{-1}(p_j))_{j\in{\mathbb{N}}}$ does not have accumulation points in $B_n\setminus K_n$. This is a contradiction.
Thus, $Y_1\setminus X'$ is a connected component of ${\mathbb{S}}^3\setminus X'$ Note that this component must be in the interior of $\bar Y_1$, so it cannot be a subset of $P$, by B2, B3 and B4. Thus, it is $K_m'$ for some $m$. Since $h$ is a homeomorphism we have that $${{\operatorname{int}}}B_n\setminus K_n\approx Y_1\setminus K_m'.$$ Since $Y_1\approx {{\operatorname{int}}}B_n\approx {\mathbb{R}}^n$, we can conclude from Theorem \[thm:GordonLuecke\] that $K_n$ and $K_{m'}$ have the same knot-type.
By symmetry arguments using the fact that $h$ is a homeomorphism, this establishes a map $n\mapsto m$ which is actually bijective, so denote this bijection by $\rho$.
Let ${\gamma}\subset B_n\setminus K_n$ be a closed curve representing a non-trivial cycle in $H_1(B_n\setminus K_n)$ (such exists by Fact \[fact:NonTrHomology\]). Then $h[{\gamma}]$ will be a non-trivial cycle in $h[B_n]$. We would like to show that $h[{\gamma}]$ is also non-trivial in $({\mathbb{S}}^3\setminus Y_1)\cup h[B_n]$. But since we established a homeomorphism of ${\mathbb{S}}^3$ to itself taking $h[\partial B_n]$ to ${\mathbb{S}}^2$, we know that if ${\gamma}$ bounds a disk $D\subset ({\mathbb{S}}^3\setminus Y_1)\cup h[B_n]$, this disk can be isotoped to a disk $D'\subset Y_1$ keeping $Y_1\cap D$ fixed.
Let $S$ be a Seifert surface of $K_m'$ contained in $B_m'$ (see Fact \[fact:ExistsSeifertS\]). Then by Fact \[fact:LinkingSeifert\] there is a point $z'\in h[{\gamma}]\cap S$. Let $z=h^{-1}(z')$. This completes the proof, since $z'\in B_{m'}$.
Preliminaries in Descriptive Set Theory {#sec:BkgrndDST}
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A *Polish space* is a separable topological space which is homeomorphic to a complete metric space.
The most common examples of Polish spaces are ${\mathbb{R}}$, ${\mathbb{C}}$ and ${\mathbb{N}}^{\mathbb{N}}$ in the Tychonov product topology. Less common examples include the space of all homeomorphisms ${{\operatorname{Hom}}}(X)$ of a compact Polish space $X$ in the $\sup$-metric (see Fact \[fact:HomPolish\]) and the space of compact subsets of a compact space $X$ in the Hausdorff metric denoted $K(X)$ (see Fact \[fact:Hausdorffmetric\]).
([@Kechris])\[fact:Gdelta\] A subset of a Polish space is Polish in the subspace topology if and only if it is a $G_\delta$ subset.
([@Kechris Theorem 3.11 and Example 9B(8)]) \[fact:HomPolish\] For a compact Polish space $X$ equipped with the metric $d_X$, the space ${{\operatorname{Hom}}}(X)$ of homeomorphisms of $X$ in the $\sup$-metric, $\delta(h,g)=\sup\{d_X(h(x),g(x))\mid x\in X\}$ is a Polish space.
\[def:Reduction\] Suppose $E$ and $E'$ are equivalence relations on Borel subsets $A$ and $A'$ of Polish spaces $X$ and $X'$ respectively. The equivalence relation $E$ is *Borel reducible to* $E'$, denoted $E{\leqslant}_B E'$, if there is a Borel map $f\colon A\to A'$ such that $$\forall x,y\in A\big((x,y)\in E\iff (f(x),f(y))\in E'\big).$$ We say that an equivalence relation $E$ is *universal* among a set $X$ of equivalence relations, if $E\in X$ and for all $E'\in X$ we have $E'{\leqslant}_B E$.
A lot is known about the partial order ${\leqslant}_B$ on analytic equivalence relations which are defined on standard Borel spaces. A thorough treatment can be found in [@Gao]. Preface in [@Hjorth] gives a good glimpse of available applications. Here is an example of an equivalence relation which we will need:
\[def:E1\] Let $(2^{\mathbb{N}})^{\mathbb{N}}$ be the space of sequences of elements of $2^{\mathbb{N}}$ (the Cantor space). The topology on both $2^{\mathbb{N}}$ and $(2^{\mathbb{N}})^{{\mathbb{N}}}$ is given by the Tychonov product topology. Let $E_1$ be the equivalence relation given by: $$((r_n)_{n\in {\mathbb{N}}},(s_n)_{n\in {\mathbb{N}}})\in E_1\iff
\exists m\forall k>m(r_k=s_k).$$
Another wide class of equivalence relations is given by Polish group actions:
Let $G$ be a Polish group acting in a Borel way on a Polish space $X$. Let $E^X_G$ be the equivalence relation where $x,y\in X$ are equivalent if and only if there exists $g\in G$ such that $y=gx$. This is called the *orbit equivalence relation* induced by this (Borel) action of a Polish group.
Many natural equivalence relations, in particular the isomorphism on countable structures (see the end of this section), can be viewed as orbit equivalence relations induced by Polish group actions. A proof of the following can be found in [@Gao Theorem 10.6.1].
\[thm:KecLou\](Kechris-Louveau [@KecLou]) Let $E$ be any orbit equivalence relation induced by a Borel action of a Polish group. Then $E_1\not{\leqslant}_B E$.
\[def:StrangeSpaces\] Let $X$ be a compact Polish space. For a fixed closed (and hence compact) subset $F\subset X$, let $$K^{F}(X)=\{A\in K(X)\mid F\subset A\}.$$ (See Definition \[def:HausdorffMetric\] for the definition of $K(X)$.) Then $K^{F}(X)$ is a closed subspace of $K(X)$ and so Polish itself by Fact \[fact:Gdelta\]. Let $$K^{F}_*(X)=\{(X\setminus A)\cup F\mid A\in K^F(X)\}.$$ The Polish topology on $K^{F}_*(X)$ is induced by the bijection $K^F(X)\to K^F_*(X)$ given by $A\mapsto (X\setminus A)\cup F$.
Let $F\subset X$ be closed. Then elements of $K^F_*(X)$ are of the form $U\cup F$ where $U$ is an open set disjoint from $F$. Therefore elements of this space are $\sigma$-compact $G_\delta$-subsets. Using Fact \[fact:Gdelta\] we obtain:
\[fact:TheyreAllPolish\] For a fixed closed $F\subset X$ and $X$ compact $K^F_*(X)$ consists of $\sigma$-compact Polish spaces.
\[def:Relations\] For a fixed closed $F\subset {\mathbb{S}}^3$, let $\approx^F$ be the homeomorphism relation on the space $K^F_*({\mathbb{S}}^3)$.
The main result of this paper (Theorem \[thm:NonClass\]) can be now stated: for a fixed $x\in {\mathbb{S}}^3$, $E_1{\leqslant}_B\ \approx^{\{x\}}$.
A countable model in a fixed vocabulary with universe ${\mathbb{N}}$ can be coded as an element of $2^{{\mathbb{N}}}$ in such a way that each $\eta\in 2^{\mathbb{N}}$ in fact represents some model. There are many nice ways to do this, see for example [@Gao]. Let $\cong$ be the equivalence relation of isomorphism. It is well known that given a vocabulary and any collection of countable models in this vocabulary whose set of codes is Borel, $\cong$ is reducible to $\cong_G$ where $\cong_G$ is the isomorphism of graphs, i.e. vocabulary consists of one binary symbol and the models are infinite graphs with domain ${\mathbb{N}}$. This equivalence relation is induced by the action of the infinite symmetric group $S_\infty$ (which is Polish in the standard product topology). A corollary to Theorem \[thm:NonClass\] which follows from Theorem \[thm:KecLou\] is that $\approx^{\{x\}}$ is not reducible to $\cong_G$, although this has been proved for the homeomorphism relation on compact spaces already by Hjorth [@Hjorth].
The original motivation of this research was the following, stronger, question:
\[open:Main\] Is $\approx^{{\varnothing}}$ reducible to $\cong_G$?
Note that $\approx^{{\varnothing}}$ is just the homeomorphism relation on open subsets of ${\mathbb{S}}^3$. See Section \[sec:Further\] for a discussion on this and other open questions.
Parametrization {#ssec:Par}
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As was pointed out, the space $K^F_*(X)$ consists of ${\sigma}$-compact Polish spaces (Fact \[fact:TheyreAllPolish\]). However, not *all* ${\sigma}$-compact Polish spaces are found in $K^F_*(X)$. There are different ways to parametrize different classes of Polish spaces such as compact, locally compact, ${\sigma}$-compact, $n$-manifolds and so on. In this section we will present these different ways and show that essentially it does not matter which one we choose, all of them being essentially equivalent in some sense. Additionally in this section we introduce many new notations for various homeomorphism relations. A helpful list of notations can be found in Section \[sec:Conclusion\].
In [@HjoKec1] Hjorth and Kechris give a simple parametrization of all Polish spaces. Their parametrization, let us call it the *Hjorth-Kechris parametrization*, consists of two-fold sequences $\eta\in {\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$ which satisfy the requirements for a metric on ${\mathbb{N}}$. The set of such $\eta$ is easily seen to be Borel. Then the space $X(\eta)$ is obtained as a completion of this countable metric space. Another way to parametrize all Polish spaces is to view them as closed subsets of the Urysohn universal space $U$. Denote the space of all closed subsets of $U$ by $F(U)$. It can be equipped with a standard Borel structure which is inherited from $K(\bar U)$ where $\bar U$ is a compactification of $U$ (see [@Kechris Thm 12.6]). The Borel sets of $F(U)$ are generated by the sets of the form $$\{F\in F(U)\mid F\cap O\ne {\varnothing}\}$$ for some open $O\subset U$. This Borel structure is also generated by the *Fell topology* generated by the sets of the form
$$\label{eq:StandBor}
\{F\in F(U)\mid F\cap K={\varnothing}\land F\cap O_1\ne{\varnothing}\land\dots\land F\cap O_n\ne{\varnothing}\},$$
where $K$ varies over $K(U)$ and $O_i$ are open sets in $U$ [@Kechris Exercise 12.7].
Let us show that these parametrizations are essentially equivalent. The universality property of $U$ is that given any finite metric space $H$ and $x\in H$, every isometric embedding of $H\setminus \{x\}$ into $U$ extends to an isometric embedding of $H$ into $U$. Thus, given a countable metric space as defined by $\eta$ as above, it can be isometrically embedded into $U$. The closure of the image will then be homeomorphic (and even isometric) to $X(\eta)$. We want to show that there are Borel reductions reducing the homeomorphism of Polish spaces in one parametrization to the other. To show this, let us define an “intermediate” parametrization. Let $U^{\mathbb{N}}$ be the set of all countable sequences in $U$. Each such sequence $\xi$ corresponds to the Polish space $Y(\xi)$ obtained as its closure taken in $U$. Let $f_1\colon U^{\mathbb{N}}\to {\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$ be defined by $f_1(\xi)=\eta$ where $\eta(n,m)=d_U(\xi(n),\xi(m))$. Obviously $X(\eta)$ and $Y(\xi)$ are isometric and $f_1$ is continuous. Let $f_2\colon U^{\mathbb{N}}\to F(U)$ be the map which takes $\xi$ to the closure of $\{\xi(n)\mid n\in{\mathbb{N}}\}$ in $U$.
There are Borel functions $g_1\colon {\operatorname{ran}}(f_1)\to U^{\mathbb{N}}$ and $g_2\colon F(U)\to U^{\mathbb{N}}$ such that $f_1\circ g_1={\operatorname{id}}$ and $f_2\circ g_2={\operatorname{id}}$.
For $g_2$ we will use [@Sri Cor 5.4] which says that if $f\colon X\to Y$ is a Borel function between Polish spaces $X$ and $Y$ such that $f[V]$ is open for all open $V\subset X$, $f^{-1}[V]$ is $F_{\sigma}$ for all open $V\subset Y$ and $f^{-1}\{y\}$ is $G_\delta$ for all $y\in Y$, then there is $g\colon Y\to X$ such that $f\circ g={\operatorname{id}}_Y$.
It is easy to see that the inverse image under $f_2$ of a set of the form is $F_{\sigma}$ in $U^{\mathbb{N}}$, so in particular $f_2$ is Borel. Additionally, given a closed set $C\in F(U)$, the inverse image of the singleton $f_2^{-1}\{C\}$ is $G_\delta$. To see this, let $Q=\{q_n\mid n\in{\mathbb{N}}\}$ be a dense countable subset of $C$. Then $f_2^{-1}\{C\}$ is the intersection of $\{\xi\mid {\operatorname{ran}}(\xi)\subset C\}$ and the sets $$O(k,m)=\{\xi\mid \exists n\in{\mathbb{N}}(\xi(n)\in B(q_k,1/m))\}.$$ The former is closed and the latter are open, so the intersection is $G_{\delta}$. Let $V\subset U^{\mathbb{N}}$ be a basic open set. It is of the form $$O_0\times\cdots\times O_n\times U^{{\mathbb{N}}\setminus \{0,\dots,n\}}.\label{eq:openset}$$
To see that $f_2[V]$ is open in $U^{\mathbb{N}}$ note that it can be represented in the form of with $K={\varnothing}$ and $O_i$ as in ; thus $f_2$ is an open map. By [@Sri Cor 5.4] there is a Borel $g_2\colon F(U)\to U^{\mathbb{N}}$ such that $f_2\circ g_2={\operatorname{id}}$.
Now consider $f_1$. It is continuous and the inverse image of a singleton is closed. By [@Kechris Thm 35.46] (or again by [@Sri Cor 5.4]) there is $g_1\colon {\operatorname{ran}}(f_1)\to U^{\mathbb{N}}$ such that $f_1\circ g_1={\operatorname{id}}$. Note that ${\operatorname{ran}}(f_1)$ is merely the Borel subset of ${\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$ on which the operation $\eta\mapsto X(\eta)$ is well defined and produces a Polish space.
Let $\approx_P$ be the equivalence relation on ${\operatorname{ran}}(f_1)\subset{\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$ where $\eta$ and $\eta'$ are equivalent if and only if $X(\eta)$ and $X(\eta')$ are homeomorphic, let $\approx_P'$ be the equivalence relation on $U^{\mathbb{N}}$ where two sequences $\xi$ and $\xi'$ are equivalent if and only if $Y(\xi)$ and $Y(\xi')$ are homeomorphic, and let $\approx''_P$ be the equivalence relation on $F(U)$ where two closed $C$ and $C'$ are equivalent if and only if they are homeomorphic.
Then $f_1$, $g_1$, $f_2$ and $g_2$ witness that these three equivalence relations, $\approx_P$, $\approx'_P$ and $\approx''_P$ are all Borel reducible to each other.
Hjorth and Kechris showed that the set of those $\eta$ for which $X(\eta)$ is compact and the set of those for which it is locally compact are both Borel subsets of ${\mathbb{R}}^{{\mathbb{N}}\times{\mathbb{N}}}$. Taking Borel inverse images under $f_1$ and $g_2$ we obtain the same conclusion for the other parametrizations.
In [@HjoKec1] it is shown that the set of complex $n$-manifolds is Borel. One has to replace “biholomorphic” by “homeomorphic” in order to relax from complex manifolds to (conventional) manifolds. But as also proved in [@HjoKec1], in the case of locally compact spaces, a function is defined to be a homeomorphism in a Borel way. Thus, the set of those $\eta$ for which $X(\eta)$ is an $n$-manifold is Borel. Using the functions $f_1$, $g_1$, $f_2$ and $g_2$ we finally obtain that the sets of $n$-manifolds in all the other parametrizations are also Borel.
Denote by $\approx_P$ the homeomorhism relation on all Polish spaces and by $\approx_{loc}$, $\approx_c$ and $\approx_n$ the same relation restricted to the sets of locally compact, compact Polish spaces and $n$-manifolds respectively. From what is shown above it follows that the chosen parametrization is irrelevant. Recall also the notation $\approx^{\{x\}}$ from Definition \[def:Relations\]. All of these equivalence relations are defined on Borel subsets of Polish spaces.
The following easily follows from [@Kechris Exercise (27.9)]:
The set of those $C\in F(U)$ which are ${\sigma}$-compact is ${{\Pi_1^1}}$-complete.
Because, as custom is, we require in Definition \[def:Reduction\] that the domains of equivalence relations are Borel subsets of Polish spaces, we do not talk directly about the homeomorphism relation restricted to the ${\sigma}$-compact spaces. However, by removing that requirement and relaxing from Borel sets to *relatively Borel* one could also talk about the Borel reducibility of $\approx_{\sigma}$, the homeomorphism relation on ${\sigma}$-compact spaces, to other equivalence relations. From our results it would follow in particular that $E_1{\leqslant}_B\ \approx_{\sigma}$ and that $\approx_{\sigma}$ is not classifiably by any equivalence relation induced by a Borel group action.
Yet another way to parametrize compact and locally compact spaces is to view them as subsets of the Hilbert cube $I^{\mathbb{N}}$ where $I$ is the unit interval. It is known that for every compact Polish space there is a homeomorphic copy as a subset of $I^{\mathbb{N}}$. For locally compact spaces we also obtain a parametrization: By [@Kechris Theorem 5.3], the one-point compactification of every locally compact Polish space is a compact Polish space. Now fix a point $x\in I^{\mathbb{N}}$ and for each $\xi\in (I^{\mathbb{N}}\setminus\{x\})^{\mathbb{N}}$ let $Z(\xi)$ be the space $\overline{\{\xi(n)\mid n\in{\mathbb{N}}\}}\setminus \{x\}$. If $P$ is any locally compact space, then let $\bar P=P\cup \{\infty\}$ be its one-point compactification. There is an embedding of $\bar P$ into $I^{\mathbb{N}}$ and since $I^{\mathbb{N}}$ is homogenous (see e.g. [@Fort]), there is an embedding such that $\infty$ is mapped to $x$. Thus, in the notation of \[def:StrangeSpaces\], $K^{\{x\}}(I^{\mathbb{N}})$ is a space parametrizing all locally compact spaces. By using the fact that $I^{\mathbb{N}}$ can be also isometrically embedded into the Urysohn space $U$, one can use the methods from above to conclude that this parametrization is in our sense equivalent to all the other parametrizations (i.e. the homeomorphism relation is Borel bireducible with the corresponding relation in other parametrizations and the relevant subsets such as $n$-manifolds are Borel subsets).
#### Summary.
When proving a classification or a non-classification result for any of $\approx_P$, $\approx^{\{x\}}$, $\approx_{loc}$, $\approx_{c}$, $\approx_n$ it is irrelevant which of the parametrizations is used. Additionally the sets of locally compact and compact spaces, of $n$-manifolds and of the spaces in $K^{\{x\}}_*({\mathbb{S}}^3)$ are Borel no matter which parametrization is used.
Non-classification of $\approx^{\{x\}}$ {#sec:NonClass}
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This section is devoted to proving the main result:
\[thm:NonClass\] The equivalence relation $E_1$ (Definition \[def:E1\]) is continuously reducible to the homeomorphism relation on $K^{\{x\}}_*({\mathbb{S}}^3)$ for any fixed $x\in{\mathbb{S}}^3$.
As before, we parametrize ${\mathbb{S}}^3$ as ${\mathbb{R}}^3\cup \{\infty\}$. Obviously the choice of $x$ does not matter. In our case $x=(1,1,\frac{1}{2})\in {\mathbb{R}}^3$ as will be seen below.
For every $n\in{\mathbb{N}}$, $k\in{\mathbb{N}}$ and $l\in \{0,1\}$, let $B_{n,k,l}\subset {\mathbb{R}}^3$ be a closed ball with the center at $(1-2^{-n},1-2^{-k},l)$ and radius $2^{-4(n+1)(k+1)}$. Define $Q$, $P'$ and $P$ as follows: $$\begin{aligned}
Q&=&\{(1-2^{-n},1,l)\mid n\in{\mathbb{N}},l\in\{0,1\}\}
\cup\{(1,1-2^{-k},l)\mid k\in{\mathbb{N}},l\in\{0,1\}\},\\
P'&=&Q\cup {\bigcup}_{n\in{\mathbb{N}}}\{(1-2^{-n},1,t)\mid t\in [0,1]\}\\
P&=&P'\cup \{(1,1,t)\mid t\in [0,1]\}\setminus \{(1,1,\frac{1}{2})\}.
\end{aligned}$$
Thus, $(B_{n,k,l})$, $Q$ and $P$ satisfy the assumptions B1, B2 and B3 from Definition \[def:Properties\].
Let $\{P_{n,k,l}\mid n\in{\mathbb{N}},k\in{\mathbb{N}},l\in\{0,1\}\}$ be the set of all (mutually different) knot types indexed by the set ${\mathbb{N}}\times{\mathbb{N}}\times \{0,1\}$. Let $\bar r=(r_n)_{n\in{\mathbb{N}}}\in (2^{{\mathbb{N}}})^{{\mathbb{N}}}$ be a sequence of elements of $2^{\mathbb{N}}$. For each $(n,k,l)\in (2^{{\mathbb{N}}})^{{\mathbb{N}}}$, let $K^{\bar r}_{n,k,l}$ be a (piecewise linear) knot inside the interior of $B_{n,k,l}$. The knot-type of $K^{\bar r}_{n,k,l}$ is determined as follows:
- If $n$ is odd, then it is $P_{n,k,l}$,
- If $n$ is even and $r_{n/2}(k)=0$, then it is $P_{n,k,l}$,
- If $n$ is even and $r_{n/2}(k)=1$, then it is $P_{n,k,1-l}$.
Let $R(\bar r)$ be ${\mathbb{S}}^3\setminus (P\cup{\bigcup}_{n,k,l}K^{\bar r}_{n,k,l}).$ Note that $R(\bar r)$ corresponds to $X$ in Definition \[def:Properties\] and properties B4 and B5 are now also satisfied (B5 follows easily from Lemma \[lem:HDPathMetric\] and the fact that ${\mathbb{S}}^3\setminus X$ is a countable union of piecewise linear curves and points). Notice also that $R(\bar r)\setminus \{(1,1,\frac{1}{2})\}$ is an open set, so $R(\bar r)\in K^{\{(1,1,\frac{1}{2})\}}_*({\mathbb{S}}^3)$. In the following three claims we will show that $F$ is a continuous reduction: $\bar r$ and $\bar r'$ are $E_1$-equivalent if and only if $R(\bar r)$ and $R(\bar r')$ are homeomorphic.
Suppose $\bar r$ and $\bar r'$ are $E_1$-equivalent. Then $R(\bar r)$ and $R(\bar r')$ are homeomorphic.
For every $(n,k)\in {\mathbb{N}}\times{\mathbb{N}}$ let $C_{n,k}$ be the convex hull of $B_{n,k,0}\cup B_{n,k,1}$, a “capsule” containing $B_{n,k,0}$ and $B_{n,k,1}$ disjoint from all other balls and from $P$. Denote for simplicity $X=R(\bar r)$ and $X'=R(\bar r')$. Now $C_{n,k}\cap X$ and $C_{n,k}\cap X'$ are homeomorphic because both are complements of two knots of types $P_{n,k,0}$ and $P_{n,k,1}$. If $n$ is odd or $n$ is even and $r_{n/2}(k)=r'_{n/2}(k)$ then identity on $C_{n,k}$ witnesses this. Otherwise there is a homeomorphism $g_{n,k}$ of ${\mathbb{S}}^3$ fixing ${\mathbb{S}}^3\setminus C_{n,k}$ and taking $C_{n,k}\cap X$ to $C_{n,k}\cap X'$. For each $(n,k)$, if $n$ is even and $r_{n/2}(k)\ne r'_{n/2}(k)$, let $h_{n,k}=g_{n,k}$. Otherwise let $h_{n,k}$ be the identity on ${\mathbb{S}}^3$. Let $\pi\colon {\mathbb{N}}\to {\mathbb{N}}\times{\mathbb{N}}$ be a bijection and define a sequence of functions $(t_m)_{m\in{\mathbb{N}}}$ by induction as follows: $$\begin{aligned}
t_0&=&h_{\pi(0)}\\
t_{m+1}&=&h_{\pi(m+1)}\circ t_m.
\end{aligned}$$ We claim that for every $x\in R(\bar r)$ the limit $t(x)=\lim_{m\to\infty}t_m(x)$ exists and defines a homeomorphism $t$ from $R(\bar r)$ to $R(\bar r')$. Let us define a *support* of a homeomorphism $h$ to be the set ${\operatorname{sprt}}h=\{x\in{\operatorname{dom}}h\mid h(x)\ne x\}$. Now obviously for $m\ne
m'$, the supports of $h_{\pi(m)}$ and $h_{\pi(m')}$ are disjoint, so the existence of the limit follows easily. In fact if $x\in
C_{n,k}$ for some $n,k\in{\mathbb{N}}$, then $t(x)=h_{(n,k)}(x)$ and $t(x)=x$ otherwise. Same argument leads that $t$ is bijective. Let $(x,y,z)\in X$ and let us show that $t$ is continuous at $(x,y,z)$. If $y\ne 1$ and $x\ne 1$, then $(x,y,z)$ has a neighborhood intersecting only finitely many $C_{n,k}$, so $t$ is determined by a finite composition of continuous functions in this neighborhood. If $y=1$ and $x\notin \{1\}\cup\{1-2^{-n}\mid
n\in{\mathbb{N}}\}$, then the same holds again and also if vice versa: If $x=1$ and $y\notin \{1\}\cup\{1-2^{-n}\mid n\in{\mathbb{N}}\}$. If $y=1$ and $x\in \{1-2^{-n}\mid n\in{\mathbb{N}}\}$, then $(x,y,z)\in X$ only if $z\notin [0,1]$ (by the definition of $P$) and in this case $(x,y,z)$ has again an open neighborhood intersecting only finitely many $C_{n,k}$. If $x=1$ and $y\in
\{1\}\cup\{1-2^{-n}\mid n\in{\mathbb{N}}\}$, then every neighborhood intersects infinitely many $C_{n,k}$. Let $n_*$ be such that for all $n>n_*$ we have $r_{n}(k)=r'_{n}(k)$ which exists because $\bar r$ and $\bar r'$ are $E_1$-equivalent and let $U$ be a neighborhood of $(x,y,z)$ of radius $2^{-2n_*}$. Then $U$ intersects only those $C_{n,k}$ for which $n/2>n_*$ and so by the definition of $h_{n,k}$ it is identity on $C_{n,k}$ for all such $n$. Thus, $t_{m}$ is identity in $U$ for all $m$ and so $t$ is continuous. Now we should check that the inverse is also continuous. But with just a little care in the definition of $g_{n,k}$ we can assume that $g_{n,k}=g_{n,k}^{-1}$ and so $t=t^{-1}$. Thus by symmetry, $t^{-1}$ is also continuous.
Suppose $\bar r$ and $\bar r'$ are not $E_1$-equivalent. Then $R(\bar r)$ and $R(\bar r')$ are not homeomorphic.
Denote again $X=R(\bar r)$ and $X'=R(\bar r')$ and assume on contrary that there is a homeomorphism $h\colon X\to X'$. Since $\bar r$ and $\bar r'$ are not $E_1$-equivalent, there is a sequence $(n_i,k_i)_{i\in{\mathbb{N}}}$ such that $(n_i)_{i\in{\mathbb{N}}}$ is increasing and unbounded in ${\mathbb{N}}$ and for all $i$, $r_{n_i}(k_i)\ne r'_{n_i}(k_i)$.
Suppose first that $(k_i)_{i\in{\mathbb{N}}}$ is bounded in ${\mathbb{N}}$. Then there exists a subsequence $(n_{i(j)},k_{i(j)})_{j\in{\mathbb{N}}}$ such that $k_{i(j)}=k_*$ for all $j$ for some fixed $k_*$. By the construction each knot-type appears exactly once in either of the sets $$\{K^{\bar r}_{n,k,l}\mid (n,k,l)\in{\mathbb{N}}\times{\mathbb{N}}\times\{0,1\}\}$$ and $$\{K^{\bar r'}_{n,k,l}\mid (n,k,l)\in{\mathbb{N}}\times{\mathbb{N}}\times\{0,1\}\}.$$ For each $m\in{\mathbb{N}}$ define the point $x_{m}$ as follows: Let $x_{m}$ be the point in $B_{m,k_*,0}\setminus K^{\bar r}_{m,k_*,0}$ given by Lemma \[lemma:FindPoints\]. We know that if $m$ is odd, then $K^{\bar r}_{m,k_*,0}$ has the same knot-type as $K^{\bar r'}_{m,k_*,0}$ and if $m/2=n_{i(j)}$ for some $j$, then $K^{\bar r}_{m,k_*,0}$ has the same knot-type as $K^{\bar r'}_{m,k_*,1}$. Thus there are infinitely many $m$ such that $h(x_m)\in B_{m,k_*,0}$ and infinitely many $m$ such that $h(x_m)\in B_{m,k_*,1}$. Thus, both points $(1,1-2^{-k_*},0)$ and $(1,1-2^{-k_*},1)$ are accumulation points of $(h(x_m))_{m\in{\mathbb{N}}}$. But only $(1,1-2^{-k_*},0)$ is an accumulation point of $x_m$ which is a contradiction with Lemma \[lemma:CauchyComponent\], because both $\{(1,1-2^{-k_*},0)\}$ and $\{(1,1-2^{-k_*},1)\}$ are connected components of both ${\mathbb{S}}^3\setminus X$ and ${\mathbb{S}}^3\setminus X'$.
Suppose now that $(k_i)_{i\in {\mathbb{N}}}$ is unbounded in ${\mathbb{N}}$. Now pick a subsequence $(n_{i(j)},k_{i(j)})_{j\in{\mathbb{N}}}$ such that not only $n_{i(j)}$ is strictly increasing, but also $k_{i(j)}$ is. For all $j$, let $x_{2j}$ be the point in $B_{2n_{i(j)},k_{i(j)},0}$ given by Lemma \[lemma:FindPoints\]. By similar argumentation as above we know that $h(x_{2j})\in B_{2n_{i(j)},k_{i(j)},1}$. Now again for all $j$, define the point $x_{2j+1}$ to be a point in $B_{2j+1,k_{i(j)},0}$ given again by Lemma \[lemma:FindPoints\]. By the construction we know that $h(x_{2j+1})$ is in $B_{2j+1,k_{i(j)},0}$ too. Thus $(x_m)_{m\in{\mathbb{N}}}$ is now a Cauchy sequence converging to $(1,1,0)$ and $(h(x_m))_{m\in{\mathbb{N}}}$ is sequence with two accumulation points $(1,1,0)$ and $(1,1,1)$. The first of these points belongs to the connected component $\{(1,1,t)\mid 0{\leqslant}t<\frac{1}{2}\}$ of both ${\mathbb{S}}^3\setminus X$ and ${\mathbb{S}}^3\setminus X'$ (by the definition of $P$) and the second belongs to the other connected component $\{(1,1,t)\mid \frac{1}{2}<t{\leqslant}1\}$. Thus, we obtain a contradiction with Lemma \[lemma:CauchyComponent\] again.
$F$ is continuous.
The inverse image of an ${\varepsilon}$-neighborhood of $R(\bar r)$ consists of all $\bar r$ which are mapped inside the ${\varepsilon}$-collar of $R(\bar r)$ and in whose ${\varepsilon}$-collar $R(\bar r)$ is contained. It is evident that only finitely many of the knot-types are determined by the ${\varepsilon}$-collar, since the ${\varepsilon}$-collar of $Q$ (or $P$) “swallows” all but finitely many knots.
By Theorem \[thm:KecLou\] we have:
The homeomorphism relation $\approx^{\{x\}}$, is not Borel reducible to any orbit equivalence relation induced by a Borel action of a Polish group.
There is a collection $D$ of Polish spaces homeomorphic to subsets of ${\mathbb{S}}^3$ of the form $V\cup \{x\}$ where $V$ is open and $x\in {\mathbb{R}}^3$ such that all elements of $D$ have the same fundamental group, but are not classifiable up to homeomorphism by any equivalence relation arising from a Borel action of a Polish group.
The fundamental group of $R(\bar r)$ is the same for all $\bar r$ – the free product of the knot groups – as can be witnessed by the Seifert-van Kampen theorem by considering $R(\bar r)$ as the union of its open subsets $A_{n}={\mathbb{S}}^3\setminus (P\cup K_n\cup {\bigcup}_{k\ne n}B_k)$ (here we fall back to the easier enumeration of the balls by just one index used in Definition \[def:Properties\]).
Positive Classification Results {#sec:Other}
===============================
The results in this section are either known or follow easily from what is known. We give some of the proofs for the sake of completeness. Since the main result of the paper deals with $\sigma$-compact spaces, we begin this section with the following relatively simple observation:
The homeomorphism relation on any Borel collection of ${\sigma}$-compact spaces, such as $K^{\{x\}}_{*}$ is ${{\Sigma_1^1}}$.
Two ${\sigma}$-compact spaces $X$ and $X'$ are homeomorphic if and only if there exist sequences of compact sets $(C_n)_{n\in{\mathbb{N}}}$ and $(C_n')_{n\in{\mathbb{N}}}$ and homeomorphisms $h_n\colon C_n\to C_n'$ for each $n$ such that
1. $C_n\subset C_{n+1}$ and $C'_n\subset C'_{n+1}$ for all $n$,
2. $h_n\subset h_{n+1}$ for all $n$,
3. $X={\bigcup}_{n\in{\mathbb{N}}}C_n$ and $X'={\bigcup}_{n\in{\mathbb{N}}}C'_n$.
To see this suppose $h\colon X\to X'$ is a homeomorphism. Since $X$ is ${\sigma}$-compact, there is a sequence of compact sets $(C_n)_{n\in{\mathbb{N}}}$ which satisfies the first parts of (1) and (3). Let $C'_n=h[C_n]$. Then $(C_n')_{n\in{\mathbb{N}}}$ and $(h_n)_{n\in{\mathbb{N}}}$ where $h_n=h{\!\restriction\!}C_n$ satisfy all the rest. On the other hand suppose that such sequences $(C_n)_{n\in{\mathbb{N}}}$, $(C_n')_{n\in{\mathbb{N}}}$ and $(h_n)_{n\in{\mathbb{N}}}$ exist. Then obviously ${\bigcup}_{n\in{\mathbb{N}}}h_n$ is a homeomorphism $X\to X'$.
Consider the space $F(U)$ defined in Section \[ssec:Par\] parametrizing all Polish spaces. Then, according to the above, the homeomorphism relation restricted to ${\sigma}$-compact spaces can be defined by saying that $C$ and $C'$ are equivalent if there exist sequences satisfying (1), (2) and (3). But these properties are all Borel properties. Also for $h_n$ to be a homeomorphism is Borel, because the domains $C_n$ are compact. This shows that the equivalence relation is ${{\Sigma_1^1}}$.
Now we turn to compact and locally compact spaces.
Let $I^{{\mathbb{N}}}$ be the Hilbert cube. A set of *infinite deficiency* $A\subset I^{{\mathbb{N}}}$ is a closed set whose projection onto infinitely many interval-coordinates is a singleton. A closed set $A$ is a *$Z$-set*, if for every open $U\subset I^{\mathbb{N}}$ with all homotopy groups vanishing, all the homotopy groups of $U\setminus A$ vanish too.
Anderson proved in [@Anderson] the following:
([@Anderson])\[Anderson\]
1. If a set is of an infinite deficiency, then it is a $Z$-set.
2. Each homeomorphism between two closed $Z$-subsets of $I^{\mathbb{N}}$ can be extended to a homeomorphism of $I^{\mathbb{N}}$ onto itself.
From this it is not difficult to obtain the following theorem:
\[thm:KechSol\] The homeomorphism relation on compact Polish spaces is continuously reducible to an orbit equivalence relation induced by a Polish group action.
Let $h_1\colon I^{\mathbb{N}}\to I^{\mathbb{N}}$ be an embedding defined as follows: $$h_1((x_i)_{i\in{\mathbb{N}}})=(y_i)_{i\in{\mathbb{N}}}$$ where for all $n$, $y_{2n}=x_n$ and $y_{2n+1}=0$. Let $X={\operatorname{ran}}(h_1)$. Then $X$ is homeomorphic to $I^{\mathbb{N}}$. Let $\approx^*_{c}$ be the equivalence relation on $K(I^{\mathbb{N}})$ where two compact sets $C$ and $C'$ are equivalent if there exists a homeomorphism $h\colon I^{\mathbb{N}}\to I^{\mathbb{N}}$ taking $C$ onto $C'$. By Fact \[fact:HomPolish\] this equivalence relation is induced by a Polish group action, and it is standard to verify that this action is continuous.
Let $\approx_{c}$ be the homeomorphism relation on $K(I^{\mathbb{N}})$ where $C$ and $C'$ are equivalent if they are homeomorphic. Thus, it is sufficient to find a reduction of this into $\approx^*_{c}$. For each $C\in K(I^{\mathbb{N}})$ let $F(C)=h_1[C]$. Now, if $C\approx_{c} C'$, then there is a homeomorphism between $F(C)$ and $F(C')$. But these are of infinite deficiency, so by Theorem \[Anderson\] there is a homeomorphism $h\colon I^{\mathbb{N}}\to I^{\mathbb{N}}$ taking $F(C)$ onto $F(C')$ and so $F(C)\approx^*_{c} F(C')$. If $C$ and $C'$ are not homeomorphic, then so are not $F(C)$ and $F(C')$, so no such homeomorphism can exist.
Arguments along the same lines give us a stronger result:
\[thm:locCom\] The homeomorphism relation on all locally compact Polish spaces is Borel reducible to an orbit equivalence relation induced by a continuous Polish group action.
Let ${{\operatorname{Hom}}}^{\{x\}}(I^{\mathbb{N}})$ be the subgroup of ${{\operatorname{Hom}}}(I^{\mathbb{N}})$ which consists of those homeomorphisms $h$ such that $h(x)=x$. As a closed subgroup of ${{\operatorname{Hom}}}(I^{\mathbb{N}})$ it is also Polish and acts continuously on $K^{\{x\}}(I^{\mathbb{N}})$ (see Definition \[def:StrangeSpaces\]).
As shown in Section \[ssec:Par\], the space of locally compact spaces can be parametrized as $K^{\{x\}}(I^{\mathbb{N}})$ each locally compact space being homeomorphic to $C\setminus \{x\}$ for some $C\in K^{\{x\}}(I^{\mathbb{N}})$. Applying the homeomorphism $h_1$ from the proof of Theorem \[thm:KechSol\], we may as well assume that this $C$ is of infinite deficiency. Now, if the two spaces $C\setminus \{x\}$ and $C'\setminus \{x\}$ are homeomorphic, the homeomorphism extends to their one-point compactifications and thus to $x$. Further, since $C$ and $C'$ are $Z$-sets, the homeomorphism extends to an element of ${{\operatorname{Hom}}}^{\{x\}}(I^{\mathbb{N}})$. On the other hand, it is obvious that if $C\setminus \{x\}$ and $C'\setminus \{x\}$ are not homeomorphic, then no such element of ${{\operatorname{Hom}}}^{\{x\}}(I^{\mathbb{N}})$ can exist.
By combining these results with Theorem \[thm:NonClass\] we can conclude that “not locally compact at one point” is in a sense the strongest requirement for Polish spaces to be non-classifiable by such an orbit equivalence relation. This is also reflected in the following Corollary:
\[cor:locsig\] Then $\approx^{\{x\}}\ \not{\leqslant}_B\ \approx_{loc}$.
We would like to apply Theorem \[thm:locCom\] to the homeomorphism on $n$-manifolds. It was discussed in Section \[ssec:Par\] that the set $M_n$ of $n$-manifolds is Borel as a subset of the space of all Polish spaces (in any of the parametrizations). As before, denote the homeomorphism relation on $M_n$ by $\approx_n$. Since manifolds are locally compact the inclusion into the locally compact spaces is a reduction $\approx_n\ {\leqslant}_B\ \approx_{loc}$. By applying Theorem \[thm:locCom\] we get the following:
$\approx_n$ is reducible to an orbit equivalence relation induced by a Polish group action.
More is known in the case $n=2$. There is a classification of $\approx_2$ by algebraic structures using cohomology groups by Goldman [@Goldman]. It is probably routine to verify that this gives a Borel reduction into the isomorphism on countable structures, but for now I leave it open in the form of a conjecture:
\[con:firstCon\] The classification in [@Goldman] is a Borel reduction into $\cong_G$, thus $\approx_2\ {\leqslant}_B\ \cong_G$.
If the conjecture holds, we obtain a consequence which follows from Theorem \[thm:graphstomanifolds\] below:
\[conj:2to3\] $\approx_2\ {\leqslant}_B\ \approx_3$.
The converse to Conjecture \[con:firstCon\] is known to hold: $\cong_G\ {\leqslant}_B\ \approx_2$. In fact $\cong_G\ {\leqslant}_B \ \approx_n$ for all $n{\geqslant}2$. We sketch two proofs of this fact – one is based on results by Camerlo and Gao and extension theorems from topology – the other one, for $n=3$, is based on the methods used in this paper, just to illustrate how these methods can be used.
\[thm:graphstomanifolds\] For all $n{\geqslant}2$ we have $\cong_G\ {\leqslant}_B\ \approx_n$.
I would like to thank Clinton Conley who came up with this proof at `mathoverflow.net`. It is sufficient to find a reduction into the homeomorphism relation on open subsets of ${\mathbb{S}}^n$ or ${\mathbb{R}}^n$. As shown in [@CamGao], $\cong_G$ is Borel reducible to the homeomorphism relation on $K(2^{\mathbb{N}})$. On one hand it is known that every homeomorphism of a totally disconnected compact subset of the plane extends to the whole plane ([@Moise Ch. 13, Thm 7]). On the other hand, by an application of Lemma \[lemma:CauchyComponent\], every homeomorphism of ${\mathbb{R}}^2\setminus C$ where $C$ is compact and totally disconnected, induces a homeomorphism of $C$. Thus, we can define a reduction from the homeomorphism relation on $K(2^{\mathbb{N}})$ to $\approx_2$: let $f\colon 2^{\mathbb{N}}\to {\mathbb{R}}^2$ be the standard embedding (the Cantor set) and with $C\subset 2^{\mathbb{N}}$ associate the open set ${\mathbb{R}}^2\setminus f[C]$. Of course these homeomorphisms extend to ${\mathbb{R}}^n$ for every $n>2$ as well, so in fact we have $\cong_G\ {\leqslant}_B\ \approx_n$ for all $n$.
Again, we consider only open subsets of ${\mathbb{R}}^3$. It was proved by H. Friedman and L. Stanley in [@FrSt] that $\cong_G$ is reducible to the isomorphism relation on countable linear orders.
Given a countable linear order $L$ with domain $\{x_n\mid n\in{\mathbb{N}}\}$, construct first a set of disjoint open intervals $U_n\subset [0,1]$ such that ${\bigcup}_{n\in{\mathbb{N}}}U_n$ is open and dense in $[0,1]$, $\sup U_n{\leqslant}\inf U_m$ if and only if $x_n<_Lx_m$ and if $x_m$ is an immediate successor of $x_n$ then $\sup U_n=\inf U_m$. Then, by considering $[0,1]$ as a subset of ${\mathbb{R}}^3$ in a canonical way, replace each open interval with a copy of the chain depicted on Figure \[fig:VadimsSuperLink\]. Let $C(L)$ be the closure of the union of all these chains in ${\mathbb{R}}^3$ By using methods similar to those above, one can show that two linear orders $L$ and $L'$ are isomorphic if and only if the complements of $C(L)$ and $C(L')$ are homeomorphic.
![The singular link.[]{data-label="fig:VadimsSuperLink"}](Singular_link_trefoil.mps){width="70.00000%"}
The idea is that the knot-types fix the orientation within the chain, and the set $Q$ – in this case, the set $[0,1]\setminus {\bigcup}_{n\in{\mathbb{N}}}U_n$ – is totally disconnected and the homeomorphism of the complement extends to it. Moreover it extends to it in an order preserving way and also preserves end-points of the chains. On the other hand all these chains are similar to one another, so any isomorphism of $L$ can be realized as a homeomorphism of the complement of $C(L)$.
Further Research {#sec:Further}
================
Let $O_n({\mathbb{S}}^n)$ be the space of all open subsets of ${\mathbb{S}}^n$ and let $\approx^o_n$ be the homeomorphism relation on this space. As before, $\approx_n$ is the homeomorphism relation on general non-compact $n$-manifolds without boundary. As before, let $\approx_{P}$ be the homeomorphism relation on all Polish, spaces, $\approx_{loc}$ the one on locally compact, $\approx_{c}$ the one on compact Polish spaces and $\approx^{\{x\}}$ as in Definition \[def:Relations\]. An open-ended research direction is to establish the places of these and other topological equivalence relations in the hierarchy of analytic equivalence relations. Positive and negative, new and old results have been reviewed in this paper.
We already stated the main open question:
Is $\approx_3\ {\leqslant}_B\ \cong_G$? If not, is it universal among orbit equivalence relations induced by a Polish group action?
And further one can ask:
\[q:whichnm\] For which $n$ and $m$ do we have $\approx_n\ {\leqslant}_B\ \approx_m$? The same for $\approx_n^o$ and $\approx_{m}^o$.
For which $n\in{\mathbb{N}}$ and known equivalence relations $E$ do we have or $E{\leqslant}_B\ \approx_n$ and same for $\approx^o_n$?
What about open subsets of the separable Hilbert space $\ell_2$? By the results of Henderson [@Hend] this covers all reasonable concepts of infinite-dimensional manifolds.
\[q:S12\] What is the exact complexity of $\approx_P$? It is known that it is $\Sigma^1_2$ [@Gao] and ${{\Sigma_1^1}}$-hard [@FerLouRos].
What are the precise locations of $\approx_{loc}$ and $\approx_{c}$ and in particular are they bireducible? Hjorth has shown using turbulence theory [@Hjorth] that $\cong_G\ <_B\ \approx_c$ (notice strict inequality) and by the results above, $\approx_c$ as well as $\approx_{loc}$ are below the universal equivalence relation induced by the Polish group action.
The following question has been asked already in [@FaToTo2]:
Is $\approx_c$ universal among all equivalence relations that are reducible to an orbit equivalence relation induced by a Polish group action?
And of course the conjectures from the end of the previous section:
\[conj:2to3dubl\] The classification in [@Goldman] is a Borel reduction into $\cong_G$, thus $\approx_2\ {\leqslant}_B\ \cong_G$. In particular $\approx_2\ {\leqslant}_B\ \approx_3$.
Concerning Question \[q:whichnm\] and Conjectures \[conj:2to3\] and \[conj:2to3dubl\]: at first it might seem that it holds that $\approx_n\ {\leqslant}_B\ \approx_{n+1}$. However, the obvious candidate for a reduction $M\mapsto M\times{\mathbb{R}}$ does not work: as shown in [@McMillan] there are open subsets $O$ of ${\mathbb{R}}^3$ which are not homeomorphic with ${\mathbb{R}}^3$, yet $O\times {\mathbb{R}}\approx {\mathbb{R}}^4$. There are no such manifolds in dimension $2$ [@Daverman], but it is still unclear to the author whether the general map $M\mapsto M\times{\mathbb{R}}$ from $2$- to $3$-manifolds provides a reduction between the homeomorphism relations.
Conclusion {#sec:Conclusion}
==========
As a conclusion we provide a diagram of all the relevant equivalence relations and which relations are knownbetween them. We omit some obvious arrows that follow e.g. from transitivity. In the diagram we use the following notation:
$$\begin{aligned}
\xymatrix{E\ar[r] & \ E'} & & E {\leqslant}_B E',\\
\xymatrix{E\ar@{.>}[r]|? & E'} & & \text{Not known whether or not } E {\leqslant}_B E',\\
\xymatrix{E\ar@{.>}[r]|C & E'} & & \text{Conjectured in this paper that } E {\leqslant}_B E',\\
\xymatrix{E\ar@{-|}[r] &\ E'} & & E \not{\leqslant}_B E',\\
E_1 && \text{Definition \ref{def:E1},}\\
E_0 && \text{Two sequences $\eta,\xi\in {\mathbb{N}}^{\mathbb{N}}$ are equivalent if $\exists n\forall (m>n)(\eta(m)=\xi(m))$,}\\
E_{Gr} && \text{The universal equivalence relation induced by a Borel Polish group action,}\\
E_{{{\Sigma_1^1}}} && \text{The universal ${{\Sigma_1^1}}$ equivalence relation,}\\
\approx^{\{x\}}&& \text{See Definition~\ref{def:Relations},}\\
\approx_{loc}&& \text{Homeomorphism on locally compact Polish spaces,}\\
\approx_c&& \text{Homeomorphism on compact Polish spaces,}\\
\approx_P&& \text{Homeomorphism on all Polish spaces,}\\
\approx_n && \text{Homeomorphism on $n$-manifolds.}\\
\cong_G && \text{Isomorphism on countable graphs.}\end{aligned}$$
$$\xymatrix{
& \approx_P \ar@<1ex>@{.>}[d]|?& \\
& E_{{{\Sigma_1^1}}}\ar[u] \ar@{.>}[dl]|? &\\
\approx^{\{x\}}\ar@<1ex>[ur] &\approx_{loc}\ar@{|->}[u]\ar@<1ex>@{.>}[d]|? \ar@{|->}[l]\ar@<1ex>[r]& E_{Gr}\ar@{|->}[ul]\ar@{.>}[l]|? \\
& \approx_c \ar[u] & \approx_n \ar@<1ex>@{.>}[dl]|?\ar[u]\ar[ul]\ar@<1ex>@{.>}[d]|?&\approx_m\ar@{<.>}[l]|?\\
E_1 \ar[uu]& \cong_G\ar@{|->}[u]\ar[ur]\ar@<-1ex>[r]& \approx_2\ar@{.>}[l]|C\ar@<.5ex>@{.>}[u]|C \ar@/^/[uul]\\
& E_0 \ar@{|->}[ul]\ar@{|->}[u]\ar@{|->}[ur]&
}$$
[^1]: Affiliation: Kurt Gödel Research Center, University of Vienna, [email protected], phone: 00436804461218
| ArXiv |
---
abstract: 'We present the Photon-Plasma code, a modern high order charge conserving particle-in-cell code for simulating relativistic plasmas. The code is using a high order implicit field solver and a novel high order charge conserving interpolation scheme for particle-to-cell interpolation and charge deposition. It includes powerful diagnostics tools with on-the-fly particle tracking, synthetic spectra integration, 2D volume slicing, and a new method to correctly account for radiative cooling in the simulations. A robust technique for imposing (time-dependent) particle and field fluxes on the boundaries is also presented. Using a hybrid OpenMP and MPI approach the code scales efficiently from 8 to more than 250.000 cores with almost linear weak scaling on a range of architectures. The code is tested with the classical benchmarks particle heating, cold beam instability, and two-stream instability. We also present particle-in-cell simulations of the Kelvin-Helmholtz instability, and new results on radiative collisionless shocks.'
author:
- 'Troels Haugb[ø]{}lle'
- Jacob Trier Frederiksen
- '[Å]{}ke Nordlund'
bibliography:
- 'references.bib'
title: 'Photon-Plasma: a modern high-order particle-in-cell code'
---
Introduction
============
Particle-In-Cell models have gained widespread use in astrophysics as a means to understand plasma dynamics, particularly in collisionless plasmas, where non-linear instabilities can play a crucial role for launching plasma waves and accelerating particles. The advent of of tera- and now peta-flop computers has made it possible to study the macroscopic properties of both relativistic and non-relativistic plasmas from first principles in large scale 2D and 3D models, and sophisticated methods, such as the extraction of synthetic spectra is bridging the gap between models and observations.
While Particle-In-Cell codes were some of the first codes to be developed for computers[@Harlow:1957; @Harlow:1964], and several classic books have been written on the subject[@birdsall:1985; @hockney:1988], modern numerical methods are in use today in the community, which did not exist then, and the temporal and spatial scales of the problems have grown enormously. Furthermore, in the context of astrophysics the modeling of relativistic plasmas has become of prime importance.
In this paper we present the relativistic particle-in-cell [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} in use at the University of Copenhagen, and the numerical and technical methods implemented in the code. The code was initially created during the ’Thinkshop’ workshop at Stockholm University in 2005, and has since then been under continuous development. It has been used on many architectures from SGI, IBM, and SUN shared memory machines to modern hybrid Linux GPU clusters. Currently our main platforms of choice are Blue-Gene and Linux clusters with 8-16 cores per node and infiniband. We have also developed a special GPU version that achieves a 20x speedup compared to a single 3GHz Nehalem core (to be presented in a forthcoming paper). The code has excellent scaling, with more than 80% efficiency on Blue-Gene/P scaling weakly from 8 to 262.144 cores, and on Blue-Gene/Q from 64 to 524.288 threads. The I/O and diagnostics is fully parallelized and on some installations we reach more than 45 GB/s read and 8 GB/s write I/O performance.
In Section II and III we introduce the underlying equations of motion, and the numerical techniques for solving the equations. We present our formulation of radiative cooling in Section IV, and in Section V the various initial and boundary conditions supported by the code. Section VI presents on-the-fly diagnostics, including the extraction of synthetic spectra. Section VII describes the binary collision modules, while Section VIII contains test problems. In Section IX we discuss aspects of parallelization and scalability, and finally in section X we finish with concluding remarks.
Equations of motion
===================
The [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} is used to find an approximate solution to the relativistic Maxwell-Vlasov system $$\label{eq:vlasov}
{\frac{\partial f^s}{\partial t}} + {\bm{u}}\cdot\frac{\partial f^s}{\partial {\bm{x}}} +
\frac{q^s}{m^s}({\bm{E}}+ {\bm{u}}\times{\bm{B}})\cdot\frac{\partial f^s}{\partial ({\bm{u}\gamma})} = \mathcal{C}$$ $$\begin{aligned}
\label{eq:gauss}
{\nabla\cdot}{\bm{E}}& = \frac{\rho_c}{\epsilon_0} \\
\label{eq:divb} {\nabla\cdot}{\bm{B}}& = 0 \\
\label{eq:faraday} {\frac{\partial {\bm{B}}}{\partial t}} & = - {\nabla\times}{\bm{E}}\\
\label{eq:ampere} \frac{1}{c^2}{\frac{\partial {\bm{E}}}{\partial t}} & = {\nabla\times}{\bm{B}}- \mu_0{\bm{J}}\,,\end{aligned}$$ where $s$ denotes particle species in the plasma (electrons, protons, …), $\gamma = {(1-(u/c)^2)^{-1/2}}$ is the Lorentz factor, and $\mathcal{C} \equiv \partial{f^s}/\partial{t}\big|_{coll}$ denotes some collision operator.
In a completely collisionless plasma $\mathcal{C}$ is zero, but in the code we also allow for binary collisions between particles. As shown below in the tests, discretization effects in the interpolation of fields and sources between the mesh and the particles and the integration of the equations of motion lead to a non-zero, non-physical, collision term, which should be minimized, especially in the case of collisionless plasmas, but also with respect to any collisional modeling term introduced explicitly. The charge and current densities are given by taking moments of the distribution function over momentum space $$\begin{aligned}
\label{eq:rho}
\rho_c({\bm{x}}) = \int \textrm{d}{\bm{u}}\sum_s q^s f^s({\bm{x}},{\bm{u}}) \\
{\bm{J}}({\bm{x}}) = \int \textrm{d}{\bm{u}}\sum_s {\bm{u}}\, q^s f^s({\bm{x}},{\bm{u}}) \,.\end{aligned}$$ To find an approximate representation for this six-dimensional system in the particle-in-cell method so-called macro particles are introduced to sample phase space. Macro particles can either be thought of as Lagrangian markers that measure the distribution function in momentum space at certain positions, or as ensembles of particles that maintain the same momentum while moving through the volume. If the trajectory of a macro particle is a solution to the Vlasov equation, given the linearity, a set of macro particles will also be a solution to the system. Other continuum fields, which only depend on position, are sampled on a regular mesh. Macro particles are characterized by their positions ${\bm{x}}_p$ and proper velocities ${\bm{p}}_p={\bm{u}\gamma}$. They have a weight factor $w_p$, giving the number density of physical particles inside a macro particle, and a physical shape $S$. The shape is chosen to be a symmetric, positive and separable function, with a volume that is normalized to 1. For example in three dimensions it can be written $S({\bm{x}}-{\bm{x}}_p) = S_\textrm{1D}(x-x_p)S_\textrm{1D}(y-y_p)S_\textrm{1D}(z-z_p)$, and $\int S({\bm{x}}-{\bm{x}}_p) \textrm{d}{\bm{x}}= 1$. The full distribution function of a single macro particle is then $$\label{eq:pseudop}
f_p({\bm{x}},{\bm{p}}) = w_p\, \delta({\bm{p}}-{\bm{p}}_p)\,S({\bm{x}}-{\bm{x}}_p)\,.$$ Inserting the above in [Eq. \[eq:vlasov\]]{} and taking moments [@Lapenta:2006; @hockney:1988; @birdsall:1985] we find the equations of motion for a single macro particle, $$\begin{aligned}
\label{eq:pmotion}
\frac{\textrm{d}{\bm{x}}_p}{\textrm{d}t} &= {\bm{u}}_p &
\frac{\textrm{d}{\bm{u}}_p\gamma_p}{\textrm{d}t} &= \frac{q}{m}\left({\bm{E}}_p + {\bm{u}}_p \times {\bm{B}}_p \right)\,,\end{aligned}$$ where the electromagnetic fields are sampled at the macro particle position through the shape functions $$\begin{aligned}
{\bm{E}}_p &= {\bm{E}}({\bm{x}}_p) = \int \!\! d{\bm{x}}\, {\bm{E}}({\bm{x}}) \, S({\bm{x}}-{\bm{x}}_p) \\
{\bm{B}}_p &= {\bm{B}}({\bm{x}}_p) = \int \!\! d{\bm{x}}\, {\bm{B}}({\bm{x}}) \, S({\bm{x}}-{\bm{x}}_p)\,.\end{aligned}$$ The Vlasov equation ([Eq. \[eq:vlasov\]]{}) is linear, and if a single macro particles obeys [Eqs. \[eq:pmotion\]]{} a collection of macro particles, describing the plasma, will also obey it. Using that the shape functions are symmetric, and assuming that the electromagnetic fields are constant inside each cell volume we find $$\begin{aligned}
\label{eq:pfields}
{\bm{E}}_p &= \!\!\!\!\!\! \sum_{{\bm{x}}_c = \textrm{cell vertices}} \!\!\!\!\!\! {\bm{E}}({\bm{x}}_c) \, W({\bm{x}}_c-{\bm{x}}_p) \\
{\bm{B}}_p &= \!\!\!\!\!\! \sum_{{\bm{x}}_c = \textrm{cell vertices}} \!\!\!\!\!\! {\bm{B}}({\bm{x}}_c) \, W({\bm{x}}_c-{\bm{x}}_p)\,,\end{aligned}$$ where the weight function $W$ is given by integrating the shape function over the cell volume $$\label{eq:weightfunction}
W({\bm{x}}_c-{\bm{x}}_p) = \int_{{\bm{x}}_c-\frac{\Delta{\bm{x}}}{2}}^{{\bm{x}}_c+\frac{\Delta{\bm{x}}}{2}} \!\!\!\!\!\! \textrm{d}{\bm{x}}\, S({\bm{x}}- {\bm{x}}_p)\,.$$
In principle any shape function would be valid. However, in practice most PIC codes employ shape functions belonging to a family of basis functions known as $B$-splines. They have a number of useful properties[@Chaniotis2004]:
- The particle interpolation function, a $B$-spline of order $\mathcal{O}$, is ${\mathcal{O}}-1$ times differentiable continuous, for ${\mathcal{O}}>1$
- Particle weight functions also become $B$-splines of order ${\mathcal{O}}+1$
- Their support is bounded; their width in number of mesh points is equal to their order + 1.
- The sum on mesh points over the support is $\Sigma^\textrm{support}_{i} B_i^\mathcal{O} \equiv 1$, for $\mathcal{O}>0$
In the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} one can select particle shape functions from one of the four lowest order $B$-splines (see also [Fig. \[fig:bsplines\]]{}) giving the weight functions:
NGP
: ’Nearest grid point’, $$W^0(x) =
\begin{cases}
1 & \text{if } \left|\delta\right| \leq \frac{1}{2} \\
0 & \text{otherwise }
\end{cases}$$
CIC
: ’Cloud-In-Cell’, $$W^1(x) =
\begin{cases}
1 - \left|\delta\right| & \text{if } \left|\delta\right| < 1 \\
0 & \text{otherwise }
\end{cases}$$
TSC
: ’Triangular Shaped Cloud’, $$W^2(x) =
\begin{cases}
\frac{3}{4} - \delta^2 & \text{if } \left|\delta\right| < \frac{1}{2} \\
\frac{1}{2} \left(\frac{3}{2} - \left|\delta\right|\right)^2 & \text{if } \frac{1}{2} \leq \left|\delta\right| < \frac{3}{2} \\
0 & \text{otherwise }
\end{cases}$$
PCS
: ’Piecewise Cubic Spline’, $$W^3(x) =
\begin{cases}
\frac{1}{6}\left(4-6\delta^2+3\left|\delta\right|^3\right) & \text{if } 0 \leq \left|\delta\right| < 1 \\
\frac{1}{6}\left(2-\left|\delta\right|\right)^3 & \text{if } 1 \leq \left|\delta\right| < 2 \\
0 & \text{otherwise }
\end{cases}\,,$$
where $W^i(x)$ is the one dimensional weight function, and $\delta \equiv (x_c-x_p) / \Delta x$.
{width="0.4\linewidth"} {width="0.4\linewidth"}
Normally, there are many more particles than mesh points in a particle-in-cell model. One important benefit of introducing a higher order particle shape function is a reduction in the aliasing effects associated with the under-sampling when interpolating particle data on the mesh. When employing a higher order field solver, also, a higher order particle shape function such that the effective width of the particle shape function response and the field differencing operator response are ’not too far apart’. If the particle shape function has a very low order — say $\mathcal{O}=0$ — the strong aliasing at high frequency may be visible to a high order — say $\mathcal{O}=6$ — differential operator, which will introduce spurious contributions from the Maxwell source term(s). One should thus take care to keep the order of the particle shape function ’high’, if the field solver difference operators have order ’high’. On balance, using the highest order cubic spline interpolation with 64 mesh points in three dimensions to couple the fields and the particles has turned out to be effectively the cheapest method in most applications. The large support leads to better noise properties, and hence a lower amount of particles can be used to reach the same quality compared to using more particles and a lower order $B$-spline. The cost is also partially offset by the increased number of FLOPS per memory transfer, when compared to a lower order scheme integration cycle resource consumption.
Discretization and time integration
===================================
To solve the equations of motion for the coupled particle-field system ([Eq. \[eq:faraday\]]{}, [Eq. \[eq:ampere\]]{}, and [Eq. \[eq:pmotion\]]{}) we need to choose both a spatial and temporal discretization, and use either explicit or implicit spatial derivatives and time integration techniques. To optimally exploit the resolution, and taking into account the symmetries of the Maxwell equations, we use a Yee lattice[@yee] to stagger the fields. The charge density is located at cell centers. To comply with Gauss law ([Eq. \[eq:gauss\]]{}) the electric fields and the current density, both entering with the same spatial distribution in Ampère’s law ([Eq. \[eq:ampere\]]{}), are staggered upwards to cell faces, while magnetic fields, to be consistent with the curl operator in [Eq. \[eq:ampere\]]{}, are placed at cell edges. With this distribution (see [Fig. \[fig:yee\]]{}) the derivatives in [Eqs. \[eq:gauss\]]{}-\[eq:ampere\] are automatically calculated at the right spatial positions, and no interpolations are needed. Because of the spatial staggering the numerical derivatives commute, and the magnetic field evolution conserves divergence to round-off precision. The electric potential $\phi_E$ is located at cell centers, and the magnetic potential $\phi_B$ is at cell corners. For the time integration we stagger the proper velocity of the macro particles ${\bm{u}\gamma}_p$ and the associated current density ${\bm{J}}$ backwards time. In the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} the natural order to do the different updates in a time step is (see [Fig. \[fig:time\]]{})
![Spatial staggering in the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}.[]{data-label="fig:yee"}](staggering2){width="0.8\linewidth"}
![Time staggering and integration order in the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}. ${\bm{u}\gamma}_p$ and ${\bm{J}}$ are staggered $\Delta t/2$ backwards in time, while everything else is time centered.[]{data-label="fig:time"}](timestep){width="\linewidth"}
1. [Update the proper velocity ${\bm{u}\gamma}_p$ of the macro particles using either the Boris or Vay particle pusher]{}
2. [Calculate the charge density on the mesh $\rho^i({\bm{x}}_p^i)$, update the macro particle positions to ${\bm{x}}_p^{i+1}$, and recalculate the charge density $\rho^{i+1}({\bm{x}}_p^{i+1})$]{}
3. [If using a charge conserving method, find the current density from the time derivative of the charge density $\hat\partial_t \rho = [\rho^{i+1}({\bm{x}}_p^{i+1}) - \rho^i({\bm{x}}_p^i)]/\Delta t$, else calculate it directly from the particle flux ${\bm{u}}_p$ interpolated on the mesh.]{}
4. [Given the time staggered current density, update the magnetic field using an implicit method]{}
5. [Combine the old and new magnetic field values to make a time centered update of the electric field]{}
In between step 3 and 4 first the particle-, charge- and current boundary conditions are applied, together with an exchange of particles between MPI threads. Then the particle array, local to each thread, is sorted sequentially according to the cell position. The sorting assures optimal cache locality when computing the charge density, and step 1 to 3 can be executed in one sweep, and using small cache friendly buffer arrays for accumulating $\rho_c$ and $\hat\partial_t\rho_c$ or ${\bm{J}}$. The boundary conditions for the magnetic and electric fields are applied while solving their time evolution in step 4 and 5. The different steps are described in more detail below.
The time update shown above corresponds to a second order Leap Frog method. Compared to for example Runge-Kutta methods, or other methods where all variables are time centered, the advantage is that it requires zero extra storage for intermediate steps, and the particle update is symplectic, giving stable particle orbits. In general, symplectic integrators use a pair of conjugate variables. In this case they are the position, electromagnetic fields and charge density and the particle proper velocities and current densities. As detailed below, the leap frog update is only possible because we evaluate the ${\bm{u}}\times{\bm{B}}$ term in the Lorentz force using an implicit evaluation of the velocity. There exists higher order symplectic integrators with several substeps of the general positions and momenta composing a full time update. But only for a subset of these integrators the position is always at the mid-distance in time between the current and new momenta; a necessary condition to be able to update the particle momenta.
We have implemented a symmetric fourth order symplectic integrator[@Yoshida:1990; @Forest:1990; @Candy:1991] in the code. Due to its symmetric nature, a single, full timestep is performed by simply taking four leap frog substeps, making it an almost trivial change to implement, with updates needed only in the main driver. The method significantly increases the long term stability of the evolution for e.g. streaming plasma simulations, at the price of increasing the cost of a single time step by a factor of 4. For three dimensional simulations, where increasing the resolution by a factor of 2 costs a factor of 16 in cpu time this can be very worth while, compared to using the second order leap frog method with a $\sqrt{2}$ higher resolution.
Particle motion
---------------
To move the macro particles forward in time we have to solve [Eq. \[eq:pmotion\]]{}. Because of the time staggering it is straight forward to make an $\mathcal{O}(\Delta t^2)$ precise position update $${\bm{x}}_p^{i+1} = {\bm{x}}_p^{i} + \Delta t\, {\bm{u}}_p^{i+1}\,,$$ where the upper index $i$ indicates the iteration number. The position ${\bm{x}}_p^i$ is evaluated at $t^i$, while the proper velocity ${\bm{u}\gamma}_p^i$ is staggered backwards in time and evaluated at $t^{i-1/2}=t^{i}-\Delta t/2$.
The proper velocity update is a bit more delicate. To find a time centered Lorentz force we need a time centered value for the velocity ${\bm{u}}_p$. This gives an implicit equation for ${\bm{u}\gamma}_p^{i+1}$, and traditionally the Boris particle pusher[@boris] has been used. It is formulated so that first the time centered proper velocity is computed as the average ${\bm{u}\gamma}_p(t^i)=({\bm{u}\gamma}_p^i + {\bm{u}\gamma}_p^{i+1})/2$, and from that the normal velocity ${\bm{u}}_p(t^i)$ is derived. It is still an option in the code, but Vay has showed[@vay:2008] that the Boris pusher is not Lorentz invariant, and gives incorrect solutions in simple relativistic test cases. Instead, the Vay particle pusher calculates ${\bm{u}}_p(t^i)=({\bm{u}}_p^i + {\bm{u}}_p^{i+1})/2$ directly, as the average of the two three velocities. Even though the resulting equations for ${\bm{u}\gamma}_p^{i+1}$ involve the root of a fourth order polynomial, there is an analytic solution, and the end result is a Lorentz invariant proper velocity update. Therefore, the Vay particle pusher is now the standard option for updating the proper velocity. It is $\mathcal{O}(\Delta t^2)$ precise.
Solving Maxwell’s Equations
---------------------------
In the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} Maxwell’s equations are solved by means of an implicit scheme for evolving the magnetic field, ${\bm{B}}^{i} \rightarrow {\bm{B}}^{i+1}$, followed by an explicit update of the electric field, ${\bm{E}}^{i} \rightarrow {\bm{E}}^{i+1}$. De-centering of the integrator may be employed, such that the implicit magnetic field term, ${\bm{B}}^{i+1}$, is weighted higher than the explicit term, ${\bm{B}}^i$, by setting a parameter $\alpha$, at run initialization.
With this de-centering of the implicit averaging of ${\bm{B}}$ taken at two consecutive time steps, the discretized forms of Faraday’s and Ampère’s Laws read $$\label{eq:bnplus1}
\frac{{\bf B}^{i+1}-{\bf B}^i}{\Delta
t} = - \hat\nabla^+ \times \left( \alpha {\bf E}^{i+1} + \beta {\bf E}^i
\right)\,,$$ and $$\label{eq:enplus1}
\frac{{\bf E}^{i+1}-{\bf E}^i}{\Delta t} = c^2 \left( \hat\nabla^- \times \left( \alpha {\bf B}^{i+1} + \beta {\bf B}^i \right)
- \mu_0 {\bf J}^{i+1} \right)\,,$$ where, $\alpha$ and $\beta=1-\alpha$ with $\alpha\geq1/2$, quantifies the forward de-centering of the implicit magnetic field term and the high frequency wave damping strength and spectral width that accompanies the choice. The hat and sign denotes that it is the discrete version of the differential operator and that it is applied downwards or upwards. $i$ indicates the iteration. The current is downstaggered in time, and the $i+1$ iteration is already calculated from the macro particle distribution, when starting to solve the Maxwell equations.
Isolating ${\bm{E}}^{i+1}$ in [Eq. \[eq:enplus1\]]{} and taking the curl, we can insert it in to [Eq. \[eq:bnplus1\]]{}. Using the vector identity $\nabla\times\nabla\times{\bm{B}}=\nabla\left({\nabla\cdot}{\bm{B}}\right)-{\nabla^2}{\bm{B}}$, which also holds for the staggered discretized operators, and using ${\nabla\cdot}{\bm{B}}=0$, produces an elliptic equation for ${\bm{B}}^{i+1}$ $$\begin{aligned}
(1-c^2\alpha^2\Delta t^2\hat\nabla^2){\bm{B}}^{i+1} =& {\bm{B}}^i + c^2 \alpha \beta {\Delta t}^2 \hat\nabla^2 {\bm{B}}^i \nonumber \\
& - \Delta t \hat\nabla^+ \times {\bm{E}}^i \nonumber \\ \label{eq:EllipticB}
& - c^2 \alpha \mu_0 {\Delta t}^2 \hat\nabla^+ \times {\bm{J}}^{i+1}\,.\end{aligned}$$ The right hand side of [Eq. \[eq:EllipticB\]]{} contains only known terms (at time $t_i$ for ${\bm{E}}^i$ and ${\bm{B}}^i$, and $t_{i} + \Delta t/2$ for ${\bm{J}}^{i+1}$), but the operator on the left hand side is elliptic, complicating a direct solve. We have implemented a simple iterative solver taking ${\bm{B}}^{i+1}={\bm{B}}^{i}$ as a first guess, with a solution found by successive relaxation. Elliptic equations are non-local and our solver requires repeated updates of boundaries and ghost-zones. However, the limitation to the parallel scalability is not serious, in that convergence is normally reached in 1-10 iterations for most simulation setups, with tolerances on the residual error of about $10^{-6}$. Once the relaxed solution has been found, and provided that the initial simulation setup had ${\nabla\cdot}{\bm{B}}=0$, going forward ${\nabla\cdot}{\bm{B}}=0$ is guaranteed, due the constraint (${\nabla\cdot}{\bm{B}}=0$) being implicitely built into the derivation of [Eq. \[eq:EllipticB\]]{}. Having found ${\bm{B}}^{i+1}$, the electric field is simply updated explicitly by [Eq. \[eq:enplus1\]]{}.
The field integrator is unconditionally stable for $1/2^+ < \alpha < 1^-$. However, the scheme tends to damp out high-frequency waves due to the de-centered implicit nature of the scheme, and the solver is only second order accurate for values $\alpha\approx1/2^+$. Besides providing a tunable stabilization of the field integration scheme, this parameter also determines how large a time step can be chosen, and the damping of high frequency waves. Empirically, with our highest order scheme (6$^\textrm{th}$ order fields, PCS particle assignment), for values $\alpha\geq0.525$ the scheme is numerically stable.
Forming spatial derivatives of the field quantities is done by finite differencing on a uniform (but not necessarily isotropic), mesh. A set of operators identical to those used in the [<span style="font-variant:small-caps;">Stagger</span> code]{} – also developed and maintained in Copenhagen[@1997LNP...489..179N] – are implemented, with a choice between 2$^\textrm{nd}$, 4$^\textrm{th}$ and 6$^\textrm{th}$ order accuracy in space. Staggering of the variables on a Yee lattice [@yee] leads to highly simplified computations for the difference equations. For example, for Faraday’s Law ([Eq. \[eq:bnplus1\]]{}) the $x$-component, namely $\left[{\partial}_t {\bm{B}}\right]_x \equiv {\partial}_y E_z - {\partial}_z E_y$, along the $y-axis$ reduces to ${\partial}_t B_x = {\partial}_y E_z$. From [Fig. \[fig:yee\]]{} we see that this computation yields the desired value exactly where needed, provided that we compute the central differences at the half-staggered mesh point; a single component of the ${\nabla\times}$ operator is illustrated in [Fig. \[fig:diffoper\]]{}.
![Example of the sixth order difference operation ${\nabla\times}$ in 1D (along the y-axis). Due to the Yee mesh staggered layout of variables, the central difference is computed exactly where needed. The differential operator called `’ddyup’` (in the code) produces the derivative w.r.t. the Y-axis, up-shifted one half mesh point on the axis. For the example in the figure this produces the correct derivative of $B_y$ at the desired mesh point location, $y_{j+1/2}$. In the nomenclature adopted here, this operator is denoted $^6\hat{{\partial}}_{y}^+$. Cell centers are marked in blue, and cell edges in red. Further, compare this figure with the respective components in [Fig. \[fig:yee\]]{}.[]{data-label="fig:diffoper"}](6th_order_diff_example){width="\linewidth"}
In the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}, for the 6$^\textrm{th}$ order accurate finite difference first derivative with respect to the $y$-axis the expression reads $$\begin{aligned}
^6\hat{{\partial}}_y^+ f_{j+1/2} = & ~ ~ ~ ~ a \left(\frac{f_{j+1}-f_{j}}{\Delta y}\right) \nonumber \\
& + b \left(\frac{f_{j+2}-f_{j-1}}{\Delta y}\right) \nonumber \\
& + c \left(\frac{f_{j+3}-f_{j-2}}{\Delta y}\right)~,\end{aligned}$$ with coefficients $a = {25}/{21}$, $b = {-25}/{384}$ and $c = {3}/{640}$, matching those of the [<span style="font-variant:small-caps;">Stagger</span> code]{}. For our example of Faraday’s Law above, the corresponding expression to compute becomes $$\begin{aligned}
^6\hat{{\partial}}_y^+ E_z(y_{j+1/2}) = & ~~~~ \frac{25}{21} \left(\frac{E_z(y_{j+1})-E_z(y_{j}) }{\Delta y}\right) \nonumber \\
& + \frac{-25}{384}\left(\frac{E_z(y_{j+2})-E_z(y_{j-1})}{\Delta y}\right) \nonumber \\
& + \frac{3}{640} \left(\frac{E_z(y_{j+3})-E_z(y_{j-2})}{\Delta y}\right)\end{aligned}$$ for that component along the $y$-direction.
This choice of coefficients (based on a Taylor expansion) for the higher order differential operators has enabled us to consistently port simulation data from the [<span style="font-variant:small-caps;">Stagger</span> code]{} to the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}, for coupling of MHD simulations and PIC simulations.
Charge conserving current density
---------------------------------
Assuming the charge and current density on the mesh to be volume averaged, just like the fields, and inserting the phase space density of a set of macro particle ([Eq. \[eq:pseudop\]]{}) into [Eq. \[eq:rho\]]{} for the charge density, we find the charge density in a cell as $$\label{eq:rho2}
\rho_c({\bm{x}}_c) = \!\!\!\! \sum_{\textrm{particles}} \!\!\!\! q_p \, w_p \, W({\bm{x}}_p-{\bm{x}}_c)\,,$$ where $q_p$ and $w_p$ are the charge and number density of the macro particle. Another reason for choosing volume averaging, just like with the electromagnetic fields, is that the computation of the fields at the particle positions and the charge density on the mesh has to use the same interpolation technique, or particles can induce self-forces[@birdsall:1985; @hockney:1988]. In principle, one could use a similar definition for the current density $$\label{eq:jj2}
{\bm{J}}_c({\bm{x}}_c) = \!\!\!\! \sum_{\textrm{particles}} \!\!\!\! q_p \, w_p \, {\bm{u}}_p W({\bm{x}}_p-{\bm{x}}_c)\,,$$ but for the discretized version of Gauss law ([Eq. \[eq:gauss\]]{}) to hold true the correspondingly discretized charge conservation law, $$\label{eq:cc1}
\hat\partial_t \rho_c + {\hat{\nabla}\cdot}{{\bm{J}}} = 0\,,$$ has to be satisfied. In general, with the above definitions for the charge and current densities, this will not be the case. While it holds for particles moving inside a cell, particles that move through a cell boundary in a single time step can violate charge conservation. Methods have been developed[@eastwood:1991; @villasenor:1992; @esirkepov:2001; @umeda:2003] for second order field solvers that instead of using [Eq. \[eq:jj2\]]{} to compute ${\bm{J}}$ use different schemes to directly find ${\bm{J}}$ via [Eq. \[eq:cc1\]]{}. In particular Esirkepov showed[@esirkepov:2001] how to make a unique linear decomposition of the change in charge density $\hat\partial_t \rho_c$ for arbitrary shape functions into different spatial directions, and decouple [Eq. \[eq:cc1\]]{} into a set of differential equations, each involving only one component of the current density $$\label{eq:jj3}
\hat\partial_i^- J_i = \left.\hat\partial_t \rho_c\right|_i \equiv \mathcal{D}\rho_i \,.$$ The directional time derivatives of the charge density $\mathcal{D}\rho_i$ are computed on a macro particle basis, as described by Esirkepov[@esirkepov:2001], and for a second order differential operator, $$^2\hat\partial_i^- J_i = \frac{J_i^0 - J_i^-}{\Delta x_i}\,,$$ it is straight forward to compute $J_i$ from a single particle with a simple prefix sum, assuming that the contribution of a particle to the current density has compact support on the mesh and far away is zero. In the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} the current density is found using this method when a second order field solver is in use. Note that *all* PIC codes based directly on the published charge conserving methods[@eastwood:1991; @villasenor:1992; @esirkepov:2001; @umeda:2003] work with second order field solvers *only*, since the order of the discretized differential operators for Gauss law and charge conservation has to be the same. The Esirkepov method has a very concise formulation and is well suited to generalize to higher order field solvers, but it is not clear how to couple the methods by Eastwood[@eastwood:1991], Villasenor and Buneman[@villasenor:1992], or Umeda[@umeda:2003] to higher order field solvers.
For higher order field solvers it is more complicated to solve [Eq. \[eq:jj3\]]{}. For example a sixth order differential operator involves sixth mesh points $$\label{eq:jj6}\footnotesize{
^6\hat\partial_i^- J_i\! = \frac{1}{\Delta x_i}\!\left[c(J_i^{++}\!\!\!\!\!- J_i^{---}) +
b(J_i^{+}\!\! - J_i^{--}) + a(J_i^0\!\! - J_i^-)\right]}$$ and the simple prefix sum that can be used at second order turns into a linear set of equations. It is also not clear what the boundary conditions, even for a single particle with compact support, should be. In the simplest case, for a single particle, the matrix will look something like $$\footnotesize{
\left( \begin{array}{cccccccc}
a & b & c & 0 & &\ldots & 0 \\
-a & a & b & c & 0 &\ldots & 0 \\
-b & -a & a & b & c &\ldots & 0 \\
0 & \vdots & & \ddots & &\vdots & 0 \\
0 & \ldots & -b & -a & a & b & c \\
0 & \ldots & -c & -b & -a & a & b \\
0 & \ldots & 0 & -c & -b & -a & a
\end{array}\right) \left(\begin{array}{c}
J_i^{5-} \\
J_i^{4-} \\
J_i^{3-} \\
\vdots \\
J_i^{3+} \\
J_i^{4+} \\
J_i^{5+}
\end{array}\right) = \left( \begin{array}{c}
0 \\
0 \\
\mathcal{D}\rho_i^{3-} \\
\vdots \\
\mathcal{D}\rho_i^{3+} \\
0 \\
0 \\
\end{array}\right)} \,,$$ where we for simplicity have absorbed $\Delta x_i$ in the definition of $a$, $b$, and $c$. There are several problems with using the above set of equations: i) it is not clear how the “no current far away” condition is implemented, and because of the few points involved the cutoff will have some impact on the result ii) the equation for a single macro particle with cubic interpolation easily involves a 10$\times$10 matrix, and iii) as formulated above, the problem is highly unstable, because the coefficients are alternating and of very different size ($a=25/21$, $b=-25/384$, $c=3/640$). Furthermore to solve the above system of equations for three directions and every single macro particle is both very costly, and will introduce noticeable round-off error on the mesh, when all contributions from all particles are summed up. In the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} we have developed a fast and stable alternative, similar in structure to the one published recently[@londrillo:2010] for the Aladyn code. Instead of solving the current density for each macro particle, taking advantage of the linearity of the problem, we sum up $\mathcal D\rho_i$ directly on the mesh. Then [Eq. \[eq:jj6\]]{} can be solved on the whole domain. There is still the problem of the alternating coefficients, but that can be dealt with, by first solving for the difference $\Delta J_i \equiv J_i^0 - J_i^-$, and then using a prefix sum, just like in the second order method, to find $J_i(x_i)$. In terms of the differences the differential operator becomes $$\label{eq:jj6d}
^6\hat\partial^-_i J_i\! = \!\tilde c (\Delta J_i^{++}\!+ \Delta J_i^{--}) +
\tilde b (\Delta J_i^+ + \Delta J_i^-) + \tilde a\Delta J_i^0\,,$$ where the coefficients are $\tilde a = (a+b+c)/\Delta x_i$, $\tilde b = (b+c)/\Delta x_i$, $\tilde c = c/\Delta x_i$, and the corresponding linear system, which now has to be solved on all of the mesh, is a symmetric penta-diagonal system $$\label{eq:penta}
\footnotesize{
\left( \begin{array}{cccccccc}
\tilde a & \tilde b & \tilde c & 0 & & \ldots & 0 \\
\tilde b & \tilde a & \tilde b & \tilde c & 0 & \ldots & 0 \\
\tilde c & \tilde b & \tilde a & \tilde b & \tilde c & \ldots & 0 \\
0 & \vdots & & \ddots & & \vdots & 0 \\
0 & \ldots & \tilde c & \tilde b & \tilde a & \tilde b & \tilde c \\
0 & \ldots & 0 & \tilde c & \tilde b & \tilde a & \tilde b \\
0 & \ldots & & 0 & \tilde c & \tilde b & \tilde a \\
\end{array}\right) \left(\begin{array}{c}
\Delta J_i^0 \\
\Delta J_i^1 \\
\Delta J_i^2 \\
\vdots \\
\Delta J_i^{n-2} \\
\Delta J_i^{n-1} \\
\Delta J_i^n
\end{array}\right) = \left( \begin{array}{c}
\mathcal{D}\rho_i^{0} \\
\mathcal{D}\rho_i^{1} \\
\mathcal{D}\rho_i^{2} \\
\vdots \\
\mathcal{D}\rho_i^{n-2} \\
\mathcal{D}\rho_i^{n-1} \\
\mathcal{D}\rho_i^{n}
\end{array}\right) \,.
}$$ Pentadiagonal systems have stable and efficient solvers[@penta:1996], and it costs a negligible amount of cpu time compared to the rest of the code to find the current density on the mesh, ${\bm{J}}$, from the decomposed time derivative in the charge density, $\bm{\mathcal{D}\rho}$. Furthermore, because the evaluation is done directly on the mesh, the accumulated errors in the charge conservation are smaller than if the current density is computed on a macro particle basis. To avoid having to specify boundary conditions for $\bm{\mathcal{D}\rho}$ and introduce new terms into the matrix in [Eq. \[eq:penta\]]{} we use an extra virtual cell layer. Before a timestep, by definition, the charge density will always be zero in the virtual cell layer, and the current density can then be computed correctly with a “zero-current” ansatz for the current density on the lower boundary. Afterwards the normal boundary conditions for the current density can be applied, just like when using [Eq. \[eq:jj2\]]{} for computing the current density. This method also completely decouples the solution, when running the code with multiple MPI threads. If instead of a sixth order field solver a fourth order field solver is used the system [Eq. \[eq:penta\]]{} becomes tridiagonal.
In the original version of the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} [Eq. \[eq:jj2\]]{} is used to compute the current density. The error introduced into Gauss law ([Eq. \[eq:gauss\]]{}) is mostly on the Nyquist scale, and a simple iterative Gauss-Seidel filtering technique is used to correct ${\bm{E}}$. The module also computes the error, and has been used to validate the charge conserving methods. In most applications, the relative error with the old method can be kept on a $10^{-4}$-$10^{-5}$ level by running the filter 5 to 10 times per iteration, but there is no unique way to correct the electric field, and the divergence cleaning introduces tiny electric fluctuations, which couple back to the particles through the Lorentz force. Apart from the higher cost of the elliptic filter when running on many cores, the method is worse at conserving energy and has a larger numerical heating rate for cold plasma beams than a charge conserving method, and without a current filter it can be unstable. When running the code with charge conservation and in single precision at high resolution (e.g. $\sim\!\!1000^3$ cells and 10-100 billion particles), over time the numerical round-off noise will eventually build up errors both in Gauss’ law and in the solenoidal nature of the magnetic field ([Eq. \[eq:divb\]]{}). A single filtering step on the electric and magnetic fields every $\sim$5$^\textrm{th}$ iteration is enough to keep the relative errors at the $10^{-5}$ level. When running the code in double precision we have not seen any need to use divergence cleaning, and if the initial and boundary conditions obey [Eqs. \[eq:gauss\]]{} and \[eq:divb\], then the relative error typically stays below $\sim\!\!10^{-7}$.
Radiative cooling
=================
Very high energy electrons loose momentum by emitting radiation. The emission in itself is a very valuable diagnostic and the extraction of the spectrum is treated below, but the energy loss for the high energy particles is normally not computed in particle-in-cell codes. Building on the work on radiative losses done by Hededal[@Hededal:2005b] in our old PIC code, we have developed a numerically stable method for correctly calculating radiative losses in the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}. The radiated power from a single particle $P_\textrm{rad}$ can be written as[@Hededal:2005b] $$\label{eq:pradv}
P_\textrm{rad} = \frac{\mu_0 q^2}{6\pi c} \left[ \gamma^4 \dot{\bm{u}}^2 +
\frac{\gamma^6}{c^2} (\dot{\bm{u}}\cdot{\bm{u}})^2\right]\,,$$ where dot denotes the time derivative.
Denoting the proper velocity ${\bm{p}}={\bm{u}\gamma}$, and using the identities ${\bm{p}}\cdot\dot{\bm{p}}= \gamma^4 {\bm{u}}\cdot\dot{\bm{u}}$, and $\dot{\bm{p}}= \gamma \dot{\bm{u}}+ c^{-2}\gamma^3 {\bm{u}}({\bm{u}}\cdot\dot{\bm{u}})$, we can rewrite [Eq. \[eq:pradv\]]{} to $$\begin{aligned}
\nonumber
P_\textrm{rad} & = \frac{\mu_0 q^2}{6\pi c} \left[ \gamma^2 \dot{\bm{p}}^2 -
\frac{1}{c^2} (\dot{\bm{p}}\cdot{\bm{p}})^2\right] \\ \label{eq:prad}
& = \frac{\mu_0 q^2}{6\pi c} \left[ \dot{\bm{p}}^2 + \frac{1}{c^2}
(\dot{\bm{p}}\times{\bm{p}})^2\right] \,.\end{aligned}$$ Notice how [Eq. \[eq:prad\]]{} only involves proper velocities, and is numerically stable at both high and low Lorentz factors, while the first version relies on a cancellation between $\gamma^2$ and ${\bm{p}}^2$, and [Eq. \[eq:pradv\]]{} uses three velocities, prone to numerical errors.
The radiative cooling always acts in the opposite direction to the proper velocity vector, and therefore the change in the length of the proper velocity vector is directly related to the change in the kinetic energy and $P_{rad}$: $$\begin{aligned}
\nonumber
\dot{\bm{p}}_\textrm{rad} &= \frac{{\bm{p}}\cdot\dot{\bm{p}}_\textrm{rad}}{{\bm{p}}^2} {\bm{p}}\\ \label{eq:udot}
&= - \frac{\mu_0 q^2}{6 \pi m c} \frac{\gamma}{{\bm{p}}^2}
\left[ \dot{\bm{p}}^2 + \frac{1}{c^2} (\dot{\bm{p}}\times{\bm{p}})^2\right] {\bm{p}}\,,\end{aligned}$$ where we have used that ${\bm{p}}\cdot\dot{\bm{p}}= c^2 \gamma\,\dot\gamma$, and $m c^2 (\gamma - 1\dot{)\,} = -P_\textrm{rad}$.
In the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} the Boris or the Vay pusher is used to advance the particles. To integrate the effect of radiative cooling, for simplicity and to keep the scheme explicit, we assume that the cooling in a single timestep only changes the energy with a minor amount. The particle pushers advances the four velocities from time step $t - \Delta t/2$ to $t + \Delta t/2$, while accelerations are computed time centered at $t$. Below we denote the three times with $-$, $+$, and $0$ respectively.
To get a time centered cooling rate ${\bm{p}}_0$, $\dot{\bm{p}}_0$, and $\gamma_0$ are needed. The proper acceleration $\dot{\bm{p}}_0$ is already naturally time centered, but with the Vay Pusher, for consistency one should use the time centered three velocity to compute $ww_0$ and $\gamma_0$, which however is numerically imprecise when using single precision. To get a more numerically precise, albeit ever so slightly inconsistent, measure for $\dot{\bm{p}}_0$ and $\gamma_0$ we use the time averaged proper velocities $$\begin{aligned}
{\bm{p}}_0 & = \frac{{\bm{p}}_+ + {\bm{p}}_-}{2} &
\gamma_0 & = \sqrt{1 + {\bm{p}}_0^2 c^{-2}} \,.\end{aligned}$$ These values can then be plugged into [Eqs. \[eq:prad\]]{} and \[eq:udot\] to find the change in momentum due to radiative cooling as $$\begin{aligned}
\nonumber
{\bm{p}}_+ & = {\bm{p}}_- + \dot{\bm{p}}_0 \Delta t \\ \nonumber
& = {\bm{p}}_- + (\dot{\bm{p}}_\textrm{EM} + \dot{\bm{p}}_\textrm{rad})\Delta t \\
& = {\bm{p}}_+^\textrm{EM} + \dot{\bm{p}}_\textrm{rad} \Delta t\,.\end{aligned}$$ In principle one could find the converged solution to the above non-linear equation system, under the assumption that cooling is a small correction, by iterating the radiative cooling computation a couple of times, while updating ${\bm{p}}_+$ and $\dot{\bm{p}}_0$. But in practice we have found that the direct explicit calculation of the cooling rate done by making no iterations is acceptable.
Initial and boundary conditions
===============================
Setting up consistent boundary and initial conditions for a particle-in-cell code can be non-trivial, due to the mixed particle and cell nature and the staggering in space and time. In the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} the initial and boundary conditions are supported through the loading of different boundary and initial condition modules, and over time a number of modules have been developed.
Particle Injection
------------------
Macro particles in a particle-in-cell code sample phase space and are by construction placed using a random generator. Any call to a random generator is related to the position in the mesh, where the pseudo-random number is needed. To make the random number generation scalable, but independent of the parallelization technique, we have implemented two types of random generators: i) We have developed a multi stream variant of the Mersenne Twister random generator[@mersenne] to generate high quality random numbers, and setup one stream per $xy$-slice of cells. This generator is used in e.g. shock simulations where the particle average density and velocity is a function of a single coordinate. ii) In the case of more complicated setups, e.g. when using snapshots from MHD simulations as described below, a simpler random generator is used, where the state is contained in a single 32-bit integer, but each mesh point has a its own state. The Mersenne Twister generator is used to generate the initial seed in each cell for the simple random generator. To sample the velocity phase-space we have implemented cumulative 2D and 3D relativistic Maxwell distributions[@Dunkel:2009] using an inverted lookup table[@birdsall:1985]. It is important to use the correct dimensionality when initializing the velocities, or the corresponding 2D or 3D temperature will be incorrect. The particle positions are correspondingly injected either uniformly in an $xy$-slice or in a single cell. Notice that using an injection method in a full $xy$-slice allow for larger density fluctuations inside the cells, while if using a cell-by-cell injection method there will only be fluctuations on the sub-cell level. Depending on the physical model on hand one or the other method may be more desirable. We typically use a standard technique to inject particles in pairs of different charge type to avoid having free charges initially. But the code is flexible, and has a built in module to correct Gauss law. Therefore it is also possible to inject particles completely at random and correct the electric field to include the non trivial effect of electrostatic fluctuations from the initial non-neutral charge distribution.
Boundary conditions
-------------------
We handle periodic boundaries trivially by padding the domain with ghostzones, a technique also used at the edge of MPI domains, and copying fields from the top of the domain to the bottom and vice versa while updating boundaries. The particles, on the other hand, are simply allowed to stream freely and the position is in all but a few cases calculated modulo the domain size. To maintain uniform numerical precision even for very large domains the position is decomposed internally as an integer cell number, and a floating point number giving the fractional position inside the cell.
Reflective boundaries are implemented using “virtual particles” on the other side of the reflecting boundary, taking in to account the symmetries of the Maxwell equations and the staggering of the mesh. By convention the boundary is placed at the center of the cell (i.e. the charge density is *on* the boundary), below for simplicity taken to be an upper boundary. When the charge and current densities on the mesh are calculated for each particle inside the boundary a corresponding virtual “ghost particle” outside the boundary has to be accounted for. Particles close to the reflecting boundary will contribute to the charge and current densities outside the boundary, while the virtual particles will contribute a corresponding charge and current density inside the boundary. This can most efficiently be calculated by disregarding the boundary at first after a particle update, and calculate the charge and current density as if the particles were streaming freely. The contribution to the charge and current densities inside the boundary from the virtual particles can then be calculated by taking into account the symmetry. It is easily seen that this corresponds, up to a sign, to the contribution of the normal particles outside the boundary $$\begin{aligned}
\rho_c(x_\textrm{b} - \delta x) &= \rho_c(x_\textrm{b} - \delta x) + \rho_c(x_\textrm{b} + \delta x) \\
{\bm{J}}_\parallel(x_\textrm{b} - \delta x) &= {\bm{J}}_\parallel(x_\textrm{b} - \delta x) + {\bm{J}}_\parallel(x_\textrm{b} + \delta x) \\
J_\perp(x_\textrm{b} - \delta x) &= J_\perp(x_\textrm{b} - \delta x) - J_\perp(x_\textrm{b} + \delta x)\,,\end{aligned}$$ where $x_\textrm{b}$ is the position of the boundary, $\delta x$ is the distance; i.e. integer $\Delta x$ for centered quantities, and half integer for staggered, $\parallel$ indicates components parallel and $\perp$ perpendicular to the boundary. The perpendicular component of the current density is staggered and anti-symmetric across the boundary. Only after calculating the current density on the mesh are all particles outside the boundary reflected according to $$\begin{aligned}
x &\to 2\, x_\textrm{b} - x & v\gamma_\parallel &\to v\gamma_\parallel & v\gamma_\perp &\to - v\gamma_\perp
\,.\end{aligned}$$ When the charge and current densities are correctly calculated inside and on the boundary we can start considering the values outside. The staggered fields, i.e. the perpendicular current density and electric field, the parallel magnetic field, and the magnetic potential, may be shown to be antisymmetric across the boundary $$\begin{aligned}
J_\perp(x_\textrm{b} + \delta x) &= - J_\perp(x_\textrm{b} - \delta x) \\
E_\perp(x_\textrm{b} + \delta x) &= - E_\perp(x_\textrm{b} - \delta x) \\
{\bm{B}}_\parallel(x_\textrm{b} + \delta x) &= - {\bm{B}}_\parallel(x_\textrm{b} - \delta x) \\
\phi_B(x_\textrm{b} + \delta x) &= - \phi_B(x_\textrm{b} - \delta x) \,.\end{aligned}$$ Conversely, the centered fields (in the direction of the boundary), charge density, parallel current densities, electric fields, the electric potential, and the perpendicular component of the magnetic field are symmetric across the boundary $$\begin{aligned}
\rho_c(x_\textrm{b} + \delta x) &= \rho_c(x_\textrm{b} - \delta x) \\
{\bm{J}}_\parallel(x_\textrm{b} + \delta x) &= {\bm{J}}_\parallel(x_\textrm{b} - \delta x) \\
\phi_E(x_\textrm{b} + \delta x) &= \phi_E(x_\textrm{b} - \delta x) \\
{\bm{E}}_\parallel(x_\textrm{b} + \delta x) &= {\bm{E}}_\parallel(x_\textrm{b} - \delta x) \\
B_\perp(x_\textrm{b} + \delta x) &= B_\perp(x_\textrm{b} - \delta x) \,,\end{aligned}$$ and we only need to determine the values of the centered fields at the boundary. The charge and current densities are derived from the particle distribution and are therefore already given at all points, including the boundary. $B_\perp(x_\textrm{b})$ is the only unknown component of the magnetic field, and it can be computed from the solenoidal constraint $\hat\nabla^+\cdot {\bm{B}}=0$ calculated at the boundary. Given that the magnetic field is the first to be evolved forward in time it is then possible to self consistently calculate the parallel electric field on the boundary ${\bm{E}}_\parallel(x_\textrm{b})$, directly from the evolution equation.
Outflow boundaries are less constrained than reflecting boundaries, given the extrapolating nature, and various types of damping layers and extrapolations have been considered[@PML2D; @PML3D; @umeda:2001]. We use a damping layer of mesh points — typically 20 — to damp all perpendicular components of the electromagnetic fields, effectively absorbing reflected waves, and combine it with an extrapolating boundary condition (i.e. symmetric first derivative). We allow particles to still generate current and charge densities on the mesh inside the box until they are well outside the boundary. As long as the disturbances in the outgoing flow are small this works well and is stable for extended run times.
Sliding window and injection of particles
-----------------------------------------
In highly relativistic flows it can be advantageous to simulate the plasma in a frame where the region of interest is relativistic. This constrains the time such a region can be followed, because the computational domain has to be continuously expanding or has to have an enormous aspect ratio in the flow direction. This is the case for example for relativistic collisionless shocks. An alternative to this is to use a sliding window as the computational domain centered on the region of interest and moving with the same velocity maintaining it at the center of the box[@movingframe]. We have implemented this technique in the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}, together with a moveable open boundary and particle injector. If the window is moving with a velocity $v$ relative to the lab frame then in each iteration we check if $v (t_\textrm{i} - t_{\textrm{old}}) > \Delta x$ and in that case we move the box a full mesh point, if necessary. I.e. if $v = 0.5 c$ and $\Delta t = 0.5 \Delta x / c$ the code will roughly move the window one point in every 4 iterations. The move is implemented by removing one cell at one end of the box, translating everything one point, and injecting an extra layer of inflow particles at the other end. This technique was used to successfully capture the long term evolution of an 3D ion-electron collisionless shock[@haugboelle:2011].
Embedded particle-in-cell models {#sec:embed}
--------------------------------
MHD models have successfully been applied for many years to study the large scale structure of plasmas from laboratory length scales to the largest scales in the universe. MHD can reach over such enormous scales, because a statistical description of the plasma is employed, where the microscopic state is captured using statistical quantities like the temperature, viscosity and resistivity. On the other hand, the kinetic description of a plasma used in a particle-in-cell code is ideally suited to investigate non-thermal processes, such as particle acceleration and particle-wave interactions, and can be used to model and understand a much broader range of plasma instabilities than MHD. The drawback is that exactly because the plasma is described in kinetic terms, explicit particle-in-cell codes, like the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}, have to respect microscopic constraints and resolve the Debye length, the plasma skin depth, and the light crossing time of a single cell. Recently, we have developed a technique to couple the two approaches using the results from MHD simulations to supply initial and boundary conditions for our PIC code. This has enabled us to for the first time make a realistic particle-in-cell description of active coronal regions[@baumann:2012a; @baumann:2012b], and investigate the mechanism that accelerates particles in the solar corona. The MHD code is used to evolve the global plasma over several solar hours, and a snapshot just before e.g. a major reconnection event is used to study a small time sequence, of the order of tens of solar seconds, using the PIC code.
Given an MHD snapshot, typically a smaller cutout from a larger simulation of the region of interest, we first interpolate to the resolution that is to be used in the PIC code. The interpolated magnetic fields are corrected with a divergence cleaner that uses the same numerical derivative operator that are used in the PIC code, to assure that the initial magnetic fields are solenoidal to roundoff precision. In a given cell the density in the MHD snapshot is used to set the weight of individual particles. The particles are placed randomly inside the cell, but in pairs, so that initially there are no free charges. The velocity of the particles have three contributions, $$v\gamma = v\gamma_{\textrm{bulk}} + v\gamma_{\textrm{thermal}} + v\gamma_{\textrm{current}}\,.$$ The bulk momentum is taken from the MHD snapshot. The thermal velocity is sampled from a Maxwell distribution using the MHD temperature, and finally the current speed is found from the ideal MHD current $\mu_0{\bm{J}}= \mu_0 \sum_i q^i n^i v^i_\textrm{current} = \nabla \times {\bm{B}}$. The average momentum has to correspond to the bulk momentum in the MHD snapshot. Taking into account the mass ratio, then for example in a neutral two component proton-electron plasma, with $n = \rho_{\textrm{MHD}} / (m_e + m_p)$, the weighting is $$\begin{aligned}
v^e_\textrm{current} &= - \frac{m_p}{\mu_0 |q| \rho } \nabla \times {\bm{B}}&
v^p_\textrm{current} &= \frac{m_e}{\mu_0 |q| \rho } \nabla \times {\bm{B}}\,,\end{aligned}$$ and in general the correct way is to use harmonic weighting. Finally, an initial condition has to be specified for the electric field ${\bm{E}}$. One possibility is ${\bm{E}}=0$. It satisfies Gauss law – the plasma is neutral initially – but is inconsistent with the EMF from the MHD equations. If using this initial condition, in the beginning of the run a powerful small scale electromagnetic wave is launched throughout the box, when the $\partial_t {\bm{E}}$ term in Ampère’s law adjusts the electric field on a plasma oscillation time scale. Another possibility is to set ${\bm{E}}= - {\bm{u}}\times {\bm{B}}$, in accordance with the ideal MHD equations. Then there is no guarantee that Gauss law is satisfied. In the code the second choice is used, but small scale features in the electric field are corrected by running the build-in Gauss law divergence cleaner for a few iterations. The remaining difference is adjusted by changing slightly the ion- and electron-density, making the plasma charged. Typically this only leads to small scale changes. The resulting initial electric field then both satisfies Gauss law, and is almost in accordance with the MHD EMF. Apart from using the MHD EMF, we have also options to add the Hall and Battery effect terms.
To evolve the model, boundary conditions on all six boundaries are needed. For the plasma they are constructed exactly like the initial conditions. They can easily be made time dependent, by loading several MHD snapshots and interpolating in time. In every time step, when applying the boundary conditions, first all particles in the boundary zones are removed – also particles that have crossed the boundary from the interior of the box – and are then replaced with a fresh plasma, according to the MHD snapshot. The boundary is typically 3 zones broad; enough to allow for the sixth order differential operators on the interior of the mesh, and enough to make a well defined charge and current density with the cubic spline interpolation. This plasma is retained when evolving, and particles from the boundary zones are allowed to cross into the interior of the computational domain. By maintaining a correct thermal distribution in the boundary zones, and simply letting the dynamics decide which particles stream into the box, the inflow maintains a perfect Maxwell distribution, and the resulting plasma is practically identical to what would have been obtained with the open boundary method of Birdsall et al[@birdsall:1985], but is much simpler to implement, and correctly accounts for bulk velocities and currents in and out of the box. If there is a differential between the charge or electric current in the boundary. Inside the domain the $\partial_t {\bm{E}}$ term adjusts the plasma almost instantaneously, and the balance is maintained. This boundary condition is very similar to a perfect thermal bath, but with in- and out-going bulk velocities and currents.
We do not keep the fields fixed at the boundary condition, but instead let them evolve freely, only subject to reasonable symmetry conditions at the boundary, which keep them consistent with the Maxwell equations. To respect the symmetry of the equations we let
- [$E_\perp$ symmetric, ${\bm{E}}_\parallel$ antisymmetric]{}
- [$\partial_{\perp} B_\perp$ symmetric, ${\bm{B}}_\parallel$ symmetric]{}
- [$\rho_c$ and ${\bm{J}}$ specified according to MHD snapshot]{}
- [$\phi_E$ symmetric, $\phi_B$ antisymmetric]{},
where $\parallel$ are the components parallel with the boundary and $\perp$ the component perpendicular to the boundary. For the relatively short times that we have evolved imbedded PIC simulations[@baumann:2012a; @baumann:2012b] these boundaries are stable.
A severe limitation for coupling PIC and MHD codes is that in many situations the Debye length, plasma frequency and other microphysical length and time scales are many orders of magnitude smaller than the scales of interest. For example, in the solar corona the Debye length is measured in millimeters, while interesting macroscopic scales are measured in megameters. If we were to simulate the true system using an explicit PIC code we would need roughly $(10^8)^3$ cells, which is computationally unfeasible in the foreseeable future. To circumvent this problem we have developed a novel method, in which we rescale the physical units while maintaining the hierarchy of time, length and velocity scales. The [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} is flexible and can be employed with a range of different unit systems. Furthermore all natural constants are maintained in the code. The rescaling technique is discussed in detail in Baumann et al[@baumann:2012a].
Diagnostics
===========
Particle tracking and field slicing
-----------------------------------
Particle-in-Cell simulations are routinely run with billions of particles, with each particle taking up $\sim$50 bytes, and for $10^3$-$10^6$ iterations. To store the full data set from every single iteration would take up petabytes of storage, and is impracticable. Instead, a standard practice when running PIC simulations is to only store every n$^\textrm{th}$ particle in a snapshot, and dump snapshots with a reduced frequency, decreasing the data volume dramatically. But to understand the underlying physics of for example particle acceleration in detail a more fine-grained approach is warranted. To that end we have implemented dumping of field slices and tracking of individual particles. Any particle in the code can be tagged for particle tracking, according to a number of criteria. For example based on its energy, at random, or according to the specific ID of the particle, which is reproducible between runs. The tagged particles are harvested by each MPI thread individually, and together with the position and momentum the local values of the current, density, electric and magnetic field are recorded, by interpolation from the mesh to the particle position. Everything is arranged in a single array that is sent to the master thread. The master thread then dumps the particle records to a single file, appended to in each iteration. On x86 clusters we can sustain tracing a million particles without significant performance degradation, while on clusters with weaker CPUs, such as BlueGene/P, we are limited to $\sim10^5$ particles. To put the particle tracks into context, data on of the field evolution is also needed. To save time-resolved field data we have made a field slicing module, where a large selection of fields (e.g. the electric field ${\bm{E}}$, ${\bm{E}}\cdot{\bm{B}}$, ${\bm{J}}\times{\bm{B}}$ etc) may be stored as 2D slices. The extent of a slice, and the number of field layers in the perpendicular direction to the slice, used for averaging, is user selectable. These two techniques have been used in concert, to understand the mechanism behind particle acceleration in reconnection events in the solar corona [@baumann:2012a; @baumann:2012b], and in collisionless shocks [@haugboelle:2011]. Both diagnostics a interactively steerable: parameters can be changed, and diagnostics can be turned on and off while a simulation is running.
Synthetic Spectra
-----------------
The radiation signature from an large number of of accelerated charges is not easily computed analytically from first principle, for plasmas with complex fields topologies, rich phase space structure, and temporal evolution. Application examples include relativistic outflows and collisionless shocks; more specifically, for example, gamma-ray bursts, where magneto-bremsstrahlung is very likely to constitute a major part of the observational signal.
However, since particle-in-cell codes automatically provide all variables needed for producing a radiation spectrum, namely ${\bm{r}}$ (position), ${\bm{\beta}}$ (${\bm{v}}/c$, velocity), and $\dot{{\bm{\beta}}}$ ($\dot{{\bm{v}}}/c$, acceleration), a radiation spectral synthesizer has been integrated into the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}. We need only designate observer position(s) and match the frequency range to the plasma conditions to complete the setup for the computing the radiation integral ([Eq. \[eq:radpower\]]{} below). During run-time the synthesizer computes the radiation signature for an ensemble of charged particles (in most cases electrons) in the simulation volume,; the formula for the spectrum is given by $$\begin{aligned}
\nonumber
\frac{d^2W}{d\Omega d\omega} & = \frac{\mu_0cq_e^2}{16\pi^3} \times \\ \label{eq:radpower}
& \!\!\!\! \left| \int^{\infty}_{-\infty} \frac{\textbf{n} \times \big(
(\textbf{n}-{\bm{\beta}}) \times
\dot{{\bm{\beta}}}\big)}{(1-\textbf{n}\cdot{\bm{\beta}})^2}e^{i\omega(t'-\textbf{n}\cdot{\bm{r}}_0(t')/c)}dt'\right|^2\end{aligned}$$ with $t'$ the retarded time and $\textbf{n}$ the direction of the observer. A first comprehensive and thorough study of the spectral synthesis method is given by Hededal[@Hededal:2005b], which also covers a range of test examples. While in that study, the spectral synthesis was done as post-production, in the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} all parts of the integration are done at run-time, with very little overhead, even for large numbers of particle traces[@Trier:2010; @Medvedev:2011].
The discretization of [Eq. \[eq:radpower\]]{} is done in four parts:
1. *Frequency range* is specified as an interval and is discretized into $N_{\omega}$ bins, typically of order $10^3$, either with linear or (more often in practice) with logarithmic binning.
2. *Observer positions*, often more than one ($N_{obs}>1$), are specified at run initiation time (input), typically with directionality perpendicular to a sphere centered on the simulation volume, or any important direction.
3. *Time subsampling* may be chosen; this partitions the integration for every simulation time step into a number of subcycled integration intervals: Subcycling is employed on particles selected for synthetic tracing since, for highly relativistic situations, the retarded electric field can be extremely compressed in spikes (for example in the case of synchrotron motion with $\gamma(v)\ll1$). In such situations the subcycling provides a much cheaper alternative than to restrict the Courant condition for the entire simulation.
4. *Radiative regions* are defined in a uniform meshing (independently of MPI and simulation mesh geometries), which gives the advantage of offering the possibility to sample very local volumes of the plasma. This may be of interest in simulations with — at the same time — subvolumes of very high and low anisotropy, such as is the case in fully resolved shock simulations, and relativistic streams.
A seamless integration into the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} has made the synthesis module computationally efficient, and due to the embarrassingly parallel nature of the spectra collection procedure plasma simulations have been run with millions of particles used for sampling the synthetic spectra. Particles are chosen for spectral integration before or during a run by tagging for synthetic sampling. The detailed sampling is important in highly relativistic cases with bulk flows – for example when investigating the radiation signature from relativistic collisionless shocks, and streaming instabilities, more generally [@Hededal:2005b; @Trier:2010; @Medvedev:2011].
Binary collision operators
==========================
The classic particle-in-cell framework does not take into account physical collision processes, and all particle interactions are mediated through the electromagnetic fields on the mesh. Low energy electromagnetic waves, and large impact parameter electrostatic scatterings between charged particles can be resolved directly on the grid through particle-wave interaction, but the photon energy is limited by the grid resolution and binary electro static scattering is not correctly represented. To allow for binary interactions, high energy photons, and in general interactions of neutral and charged particles and gas drag forces, we have to model them explicitly in cases where they are of importance, such as in high energy density plasmas, and partially ionized mediums. The [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} supports the inclusion of particle-particle interactions, decay of particles, and allow for neutral particles, in particular photons, in the model. The first implementation of binary interactions—in particular Compton scattering—together with tests of the method was given by Haugb[ø]{}lle[@Haugboelle:2005] and Hededal[@Hededal:2005b]. This first implementation motivated the name for the code, the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}. The Compton scattering module was used to model the interaction of a gamma-ray burst with a circumstellar medium[@Trier:2008.2; @Trier:2008.3; @Trier:2008.1]. Coulomb collisions have later been incorporated into the framework, to study particle acceleration in solar active regions [@Baumann:2012c].
Compton Scattering and splitting of particles
---------------------------------------------
The classic Monte Carlo approach to scattering is based on a cut-off probability: first a probability for the process is computed and then it is compared with a random number. If the random number is lower than the threshold the scattering for the full macro particle pair is carried through, and otherwise nothing happens. This probabilistic approach is straight forward both numerically and conceptually, but it can be noisy, in particular when interaction effects are strong but have low probability.
In the code the natural domain to consider is a single cell, partly because that is by definition the volume of a single macro particle, partly because some interactions (e.g. electro static interactions) are mediated by the grid at larger scales. In a PIC simulation typical numbers are $10-10^3$ particles per species per cell, and a probabilistic approach would result in an unacceptable level of noise. Consider a beam incident on a thermal population: The first generation of scattered particles may be computed relatively precise, but the spectra of later generations will require an excessive amount of macro particles, if they all represent an equal amount of physical particles, given the exponentially lower number density of later generations. Another well known consequence is that the precision scales inversely proportional to *the square root* of the number of particles. This is a problematic limitation, when the higher order generations are important ingredients of the physics, and the scattering process is not just a means to thermalizing or equilibrating the phase space distribution.
![Sketch of the scattering mechanism in the code of an incident macro particle (shown as blue/dark gray) on a target macro particle (shown as red/light gray) resulting in the creation of a scattered macro particle pair.[]{data-label="fig:detailedcollisions"}](scattering.eps){width="48.00000%"}
To circumvent these problems, the Compton scattering module is instead based on an explicit splitting approach for particles using the calculation of cross sections and allowing for individual weights for each macro particle. To implement the scattering of two macro particles we transform to the rest frame of the target particle and compute the probability $P(n)$ that a single incident particle during a single timestep is scattered on the $n$ target particles. If the incident macro particle has weight $m$, then $k=P(n) m$ particles will interact and two new macro particles representing the fraction of scattered particles are created (see [Fig. \[fig:detailedcollisions\]]{}). In the case of Compton scattering, the target will always be the charged particle, while the incident is a photon, and the scattering amplitude is calculated with the Klein-Nishina formula, which covers the full energy range from Thompson to high energy Compton scattering. To make the process computationally more efficient prior probabilities can be applied. If instead of selecting all pairs in a given computational cell, one only selects pairs with a prior probability $Q$, then the weight of the scattered macro particles has to be changed to $k/Q$. By cleverly selecting the prior probability such that for example the Thompson regime is avoided, the computational load can be greatly decreased.
Each scattering leads to the creation of a new macro particle pair, and if untamed, the number of macro particles will grow exponentially. To keep the number of particles under control, we use particle merging in cells where the number of particles is above a certain threshold. The algorithm is described in detail by Haugb[ø]{}lle[@Haugboelle:2005] and Hededal[@Hededal:2005b].
The Compton scattering module of the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} has allowed us to approach exciting new topics in high energy plasma astrophysics, where plasmas are excited and populations modified by photons, and the back-reaction of the plasma on the (kinetic) photons produces interesting and detailed descriptions of for example the production of inverse Compton components[@Trier:2008.2], and photon beam induced plasma filamentation[@Trier:2008.1].
Coulomb scattering
------------------
Coulomb scattering of charged particles in an astrophysical context is for example important in order to understand the solar chromosphere and the lower parts of the corona, where the mean free path is similar to the dynamical length scales in the system. In the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} we have integrated a collision model, where all macro particle pairs in a cell are considered for scattering. We calculate the scattering process in the elastic Rutherford regime. The collision process is implemented using a physical description for each macro particle pair, starting by calculating the time of closest approach. Only if that time is less than $\Delta t/2$ from the current time, ie. inside the current time interval, is the scattering carried through. In the limit of very small time steps this makes the algorithm independent of the size of the time step. When calculating the impact parameter between two macro particles we have to take in to account that each particle represents a large number of physical particles, therefore the impact parameter is rescaled with the typical distance between each physical particle $n^{-1/3}$, where $n$ is the number density. Because macro particles carry a variable weight, we use the geometric mean of the number density of the particle pairs to calculate the effective number density $n = (n_1 n_2)^{1/2}$.
We consider three different regimes, based on the impact parameter: i) If the impact parameter is larger than the local Debye length we assume that Debye screening between the two particles is so effective that no scattering happens. ii) If the impact parameter is so large that the effective scattering angle is less than $\theta_c$ radians (normally taken to be 0.2 in the code) we use a statistical approach: At small angles the scattering angle is inversely proportional to the impact parameter, and we can replace the many small angle scattering by fewer large angle scatterings, comparing the ratio of the cut-off to the impact parameter $b_c / b$ with a random number. If it is lower the scattering is carried through, but using the cut-off impact parameter $b_c$ for greater computational efficiency. iii) If the impact parameter is smaller than the cut-off parameter $b_c$ then we make a detailed computation of the scattering. In both of the two last cases the scattering angle is calculated in the center-of-mass frame as an inelastic Rutherford scattering, which conservers the energy of each macro particle.
The explicitly physical implementation at the macro particle level of Coulomb scattering is conceptually completely different from the Compton scattering module, in particular because the main consequence of Coulomb scattering is the thermalization and isotropization of the plasma.
Test Problems
=============
Testing the accuracy and precision of a particle-in-cell code is particularly difficult, because of the non linear nature of plasma dynamics, Monte-Carlo particle sampling, and the few examples of realistic test problems with analytic counterparts. To facilitate cross comparison with other codes, below we apply the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} to a set of classic test problems, and in some cases compare different order splines and differential operators to highlight the impact of using high order methods. We also give an example of of the more non-standard feature of radiative cooling. Further tests have been published in the case of Compton scattering[@Haugboelle:2005; @Hededal:2005b] and synthetic spectra[@Hededal:2005b].
Numerical heating and collision artifacts
-----------------------------------------
Numerical heating is a well studied feature of all PIC codes[@birdsall:1985]. It happens due to the interaction of particles with the mesh; the so-called grid-collisions. If particle populations with different energy distributions exist in the plasma, the interaction through the mesh will tend to equilibrate the kinetic energy of each particle species. The equilibration of temperatures happens in a laboratory plasma too, albeit normally at a much slower rate, and the difference between a laboratory and the computational plasma is the much smaller number of macro-particles used to represent the plasma in the latter case. It is also worth pointing out that the numerical mesh heating is an equilibration of kinetic energies, and as such much more severe for ion-electron plasmas. It is important to keep this energy equilibration in mind when simulating plasmas with greatly varying temperatures, or when analyzing heating rates due to real physics. If the numerical heating rate is close to the physical rate in question, the results cannot be trusted. It is interesting to note that the heating rate is practically invariant with respect to the numerical technique used, and instead mainly depends on the number of time steps taken.
The test is done in two dimensions, with three velocity components. There is no bulk velocity, and the kinetic energy corresponds to a thermal velocity of both ions and electrons of $v_{th,e} = v_{th,i} = 0.1\,c$ per component, or $E_\textrm{kin}^\textrm{s} = 0.015\,m_\textrm{s}\, c^2$. The size of the box is $12.8^2$ electron skin depths with 10 cells per skin depth for a $128^2$ resolution. We perform two tests with 5 and 50 particles per cell. The mass ratio is $m_i / m_e = 16$. We used TSC or cubic spline interpolation, 2$^\textrm{nd}$, 4$^\textrm{th}$, or 6$^\textrm{th}$ order field solver, and in the case of the 6$^\textrm{th}$ order field solver with cubic interpolation we also use the charge conserving (CC) method for the current density with second or fourth order time stepping.
{width="0.23\linewidth"} {width="0.23\linewidth"} {width="0.23\linewidth"} {width="0.23\linewidth"}
A relativistic cold beam {#sec:coldbeam}
------------------------
A particle-in-cell code is not Galilean invariant. When a cold plasma beam travels through the box at constant velocity the electrostatic fluctuations inside the Debye sphere or, if it is less than the mesh spacing, inside a single cell, will have resonant modes with the mesh spacing. This leads to an effective drag and redistribution of kinetic energy from the stream direction to the parallel direction, and general warm up of the beam. When the temperature reaches a critical level the instability is quenched. For relativistic beams there is the additional complication that electromagnetic waves are represented on the mesh. The solver has an effective dispersion relation, and short wavelength waves travel below the speed of light. On the other hand, particles are Lagrangian, and if relativistic they can effectively travel above the speed of short wavelength waves, giving rise to numerical Cherenkov radiation.
A classic method to limit the impact the of the cold beam instability is applying filters to either the current density or the electric field. This may to some extent filter out the effects, but will also filter out some of the physics. Alternatively, higher order field solvers, interpolation techniques, and time stepping can mitigate the effects.
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To test the numerical methods used in the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}, we have made a cold beam test with 9 different versions of the code. We do not apply any filters to the current density, to show the actual performance of the different code versions. Apart from the 8 methods used for the numerical heating test there is also a version where instead of the implicit field solver a simple (but 6$^\textrm{th}$ order) FDTD explicit solver is used. Notice how the explicit solver has numerical heating, while implicit solvers have numerical cooling[@birdsall:1985], and how the stability of the beam is greatly enhanced by the implicit solver, compared to using an explicit solver. But only the combination of the implicit solver with a charge conserving current deposition gives stable beams, with the smallest heating (see figure \[fig:coldbeam\]). It is also interesting to notice that the heating is non-isotropic. It is therefore not the same to initialize a plasma with a low but stable temperature, as to use a very low temperature and let the cold beam stability warm up the beam.
The test is done in 2D2V with a $\Gamma=10$ streaming pair plasma through a 256 $\times$ 256 cell domain with 10 particles per species per cell and a relativistic skin depth $\delta = [(m c^2 \Gamma)/(4\pi n q^2)]^{1/2}$ of ten cells. The initial temperature is measured in the rest frame of the plasma and has a root-mean-square per velocity component of $0.025c$, so that $\sum \textrm{rms}(v_\textrm{th}) = 0.05c$.
By rerunning at higher resolutions we have investigated at what resolution other methods give comparable results to the charge conserving method with fourth order time stepping, by comparing stream velocities, and parallel and perpendicular temperatures at $\omega_p t=1000$ (see table \[tab:coldbeam\]). It is only at higher resolutions that the non-charge conserving methods do not suffer from the catastrophic instability seen in figure \[fig:coldbeam\], and cost-wise the sixth order charge conserving method is marginally the cheapest. Had the test been in 3D, where the cost goes like resolution to the fourth power, the fourth order time integration method would have been the cheapest for this particular beam test. At high enough resolution the cold beam instability is quenched. For the fourth order charge conserving method this happens at a resolution of approximately 3072$^2$, where instead of a large heating rate, and then a new stable temperature, we observe a gradual heating over time. A resolution of 3072$^2$ corresponds to resolving the Debye length, $\lambda_D = v_\textrm{th} / (\Gamma c) \delta_e$, with 1.9 cells. Taking into account that we do not apply any damping, this is in good agreement with the common wisdom that the Debye length has to be resolved by roughly one cell.
[l c c c]{} Method & Res & Cost & $\mu$s/part\
CCo 6$^\textrm{th}\!$ order field; 4$^\textrm{th}\!$ order time & 256$^2$ & 1 & 6.60\
CCo 6$^\textrm{th}\!$ order field & 380$^2$ & 0.9 & 1.72\
Cubic interpolation; 6$^\textrm{th}\!$ order field & 512$^2$ & 1.3 & 1.05\
TSC interpolation; 2$^\textrm{nd}\!$ order field & 950$^2$ & 4.9 & 0.63\
Relativistic two-stream instability
-----------------------------------
Relativistically counter-streaming plasmas have previously been established to be subject to a general instability class; the oblique (or mixed-mode) two-stream-filamentation instability (MMI), which mixes the two-stream (TSI) and filamentation (FI) instabilities. A thorough and exhaustive analysis of the MMI was given by Bret et al[@Bret:2004; @Bret:2007], and has been investigated numerically by several groups (see e.g. Tzoufras et al[@Tzoufras:2006] and Dieckmann et al[@Dieckmann:2006]). Due to its mixed nature, the MMI contains both an electrostatic and an electromagnetic wave component[@Bret:2004].
Potentially, both electrostatic and electromagnetic turbulence (wavemode coupling leading to cascades/inverse cascades in k-space) is possible in such systems. This potential for producing very broadband plasma turbulence (in both E and B fields) is highly relevant to inertial confinement fusion experiments. Other examples where electromagnetic wave turbulence[@Frederiksen:2008] and the general MMI are important are astrophysical jets and shocks from gamma-ray bursts and active galactic nuclei, where ambient plasma streams through a shock interface moving at relativistic speeds.
To test physical scenarios responsible for observational signatures from these astrophysical sources, and from plasma experiments, we must construct plausible shocked outflow conditions[@bib:Medvedev1999; @Frederiksen:2004; @haugboelle:2011] and then subsequently synthesize radiation signatures[@Hededal:2005b; @Trier:2010; @Medvedev:2011] to test the assumed physical conditions against observational evidence. Studying the MMI, in both its linear and non-linear evolution is therefore well motivated.
Growth rates of the (general) MMI, the special case of FI and the TSI, respectively, are calculated as in[@Bret:2004]: $$\begin{aligned}
\gamma_{MMI} & = & 2^{-4/3}~ \sqrt{3}~ \alpha^{1/3}~ \Gamma(v_b)^{-1/3} \label{eq:mmigrowth}\\
\gamma_{FI} & = & \beta~ \alpha^{1/2}~ \Gamma(v_b)^{-1/2}~ \label{eq:figrowth}\\
\gamma_{TSI} & = & 2^{-4/3}~ \sqrt{3}~ \alpha^{1/3}~ \Gamma(v_b)^{-1} \label{eq:tsigrowth}\end{aligned}$$ where $\beta \equiv v_b/c$ is the beam velocity, $\alpha \equiv n_b/n_p$ is the beam-to-background density ratio, and $\Gamma(v_b)$ is the beam bulk flow Lorentz factor. For thin, high-Lorentz factor beams, the MMI is the fastest growing mode, dominating over both the FI and the TSI. In the test we have chosen the MMI is dominant, with subdominant FI and TSI components.
We perform six runs with identical initial conditions and physical scaling, using combinations of finite difference operators and particle shape functions as given in Table\[tab:orders\] below. This way, we test the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} for differences/similarities between interpolation schemes.
To capture the MMI as the fastest growing mode, we initialize a simulation volume with a cold thin neutral beam (electrons + ions) through a warm thick neutral background (electrons + ions), with no fields initially. The beam and background densities are $n_b=0.1$ and $n_p=0.9$, respectively. The beam velocity is chosen to have $\Gamma(v_b)=4$. Temperatures of the beam and background are $T_b=0.01$ and $T_p=0.1$, respectively. With these choices of physical properties, the growth rates of the fastest growing MMI, FI, and TSI modes become $\gamma_{MMI}=0.201$, $\gamma_{FI}=0.153$, $\gamma_{TSI}=0.080$.
The computational domain is $\{L_x,L_y,L_z\}=\{12.8\delta_e,12.8\delta_e,12.8\delta_e\}$, with $\{N_x,N_y,N_z\}=\{128,128,128\}$ cells. We use 20 particles/cell/species, or a total of 80 particles/cell (beam+background). The physical constants are scaled as $c=1$, $q_e=1$, $m_e=1$ and $m_i/m_e\equiv1836$.
[cccccc]{} Run & Fields & Particles & charge-conservation & Time order\
1 & 2$^\textrm{nd}$ & tsc & no & 2\
2 & 2$^\textrm{nd}$ & cubic & no & 2\
3 & 6$^\textrm{th}$ & tsc & no & 2\
4 & 6$^\textrm{th}$ & cubic & no & 2\
5 & 6$^\textrm{th}$ & cubic & yes & 2\
6 & 6$^\textrm{th}$ & cubic & yes & 4\
For our choice of run parameters a MMI mode develops with a propagtion wave vector, ${\bf{k}}_{MMI}$, that is oblique with respect to the streaming direction, at an angle given by $\theta_{MMI}=\angle({\bf{k}}_{MMI},{\bf k}_\textrm{beam})=
\textrm{arctan}\left(\sqrt{v_b/v_p}\right)\approx74^\circ$. The corresponding electric field component is almost parallel to the direction of propagation[@Bret:2004].
To find growth rates of the MMI we calculate the volume integrated electrostatic energy as a function of time $$E_{E,tot}(\theta_{MMI}) = \int_{V} |{\bm{E}}_\perp| sin(\theta_{MMI}) + E_\parallel cos(\theta_{MMI})~dV$$ of the electric field projected on the propagation direction, ${\bm{E}}({\bm{r}})\cdot{\bf{k}}_{MMI}$. Similarly, we also measure the TSI mode growth rate ([Eq. \[eq:tsigrowth\]]{}) by the same calculation, but for the TSI we have $\theta_{TSI} = \angle({\bf{k}}_{TSI},\hat{\bf{z}}=0)$. Our results are summarized in Table \[tab:growth\].
[ccccccccc]{} Run & 1 & 2 & 3 & 4 & 5 & 6 & theory\
$\gamma_{MMI}$ & 0.185 & 0.190 & 0.203 & 0.206 & 0.205 & 0.205 & **0.201**\
$\gamma_{TSI}$ & 0.060 & 0.079 & 0.065 & 0.082 & 0.088 & 0.088 & **0.080**\
From figure \[fig:mmi\_growth\] we see that the initial noise build-up prior to instability dominance is strongest (as expected) for Run1. This results in lower growth rates for lower order runs, since the noise tends to ’flatten« the total energy history, with a lower $\gamma_{MMI}$ as a result. This error in the measurement decreases with increasing run number (Run1$\rightarrow$Run2$\rightarrow$…), with higher order runs’ growth rates less susceptible to noise distortion. This is similar to what was seen in the cold beam tests: in a lower order integration scheme more energy is lost to artificial heating and Cerenkov radiation, making less energy available for the physical instabilities, and changing the parameters (for example temperatures) of the plasma components, modifying the overall setup.
Because of the lower level of noise, the onset of instability is delayed for runs of increasing order, while the peak energies and peak times coincide roughly for all runs; this effect correlates with the increased growth rates of the higher order runs, relatively. Both of the effects mentioned above are caused by the difference in dynamic response of the various schemes. A higher order scheme is desirable since the noise levels, heating, numerical stopping power and dynamical friction are all reduced considerably. The differences between using either TSC or cubic interpolation, second or sixth order field integration, and (no) charge conservation are all clearly reflected in the growth rates of the MMI and TSI.
Nonetheless, growth rates are seen to converge with different methods to a value which deviates from the theoretical prediction by about 2% for the MMI component, and $\sim$10% for the TSI component, for the highest order schemes, and the overall development is in qualitative agreement for all test cases.
![Growth of the volume-totaled electrostatic energy, for the electric field projected onto the direction of propagation of the fastest growing mode, ${\bm{E}}\cdot {\bf{k}}_{MMI}$. Runs 1-6 are compared in the plot. Thickened line segment Run1 (black) gives the fitting interval. Runs have decreasing initial energy for increasing run number designation (Table \[tab:orders\]). Runs 5 (orange) and 6 (yellow) are completely coinciding.[]{data-label="fig:mmi_growth"}](mmi_growth){width=".48\textwidth"}
Concluding this test, we have verified the growth rate of the relativistic two-stream (or oblique or mixed-mode) instability, and found very good agreement with growth rates also when selecting the TSI branch. The slight excesses in values of $\gamma_{TSI}([Run1,...Run6])$ is likely caused by the fact that the relativistic beam is perfectly grid aligned, thus subjected to the finite grid instability, which introduces additional electric field energy in the beam direction, while for lower orders, this is more than compensated by the overall dissipation to all electromagnetic and particle components.
Radiatively cooled collisionless shocks
---------------------------------------
The radiative cooling currently implemented is inspired by the early work of Hededal; he validated it for a simple test case of a radiatively cooling charged particle in a homogeneous magnetic field[@Hededal:2005b]. For a more non trivial application we here for the first time present a series of simulations of radiatively cooled initially non-magnetized collisionless shocks. If we assume that the acceleration of a particle in a collisionless shock is mostly due to a homogeneous magnetic field $B$, we can estimate the synchrotron cooling time for an initial Lorentz factor $\gamma_0$ as $$t^\textrm{syn}_\textrm{cool} = \frac{6 \pi \epsilon_0 m^3 c^3}{q^4 B^2 \gamma_0}$$ Consider a relativistic collisionless shock in the contact discontinuity frame, with the upstream moving at a Lorentz factor $\Gamma$ and having a number density $n$. The kinetic energy density in the upstream is $$E_{kin} = (\Gamma - 1) n \sum m c^2 = (\Gamma - 1) M c^2 n\,.$$ We assume that the magnetic energy density at the shock interface $E_B = B^2 / 2 \mu_0$ is related through some efficiency factor $\alpha\simeq0.1$ to the upstream kinetic energy density. The relativistic plasma frequency in the upstream medium is $\omega_{pe}^2 = q^2 n / (\epsilon_0 m_e \Gamma)$. Putting it all together we can find the synchrotron cooling time at the shock interface, for a particle with a gamma factor $\gamma_0$, to be $$T_\textrm{cool} = \omega_{pe} t^\textrm{syn}_\textrm{cool} =
\frac{3 \pi^{1/2} \epsilon_0^{3/2} m_e^{5/2} c^3}{\alpha q^3 n^{1/2} \Gamma^{1/2} (\Gamma - 1) M \gamma_0} \,,$$ For a pair plasma $M=2m_e$, and if we consider upstream particles $\gamma_0=\Gamma$ it reduces to $$T_\textrm{cool} =
\frac{3 \pi^{1/2} \epsilon_0^{3/2} m_e^{3/2} c^3}{2 \alpha q^3 (\Gamma - 1) \Gamma^{3/2} n^{1/2}}\,,$$ We use this definition to label the different runs. Our definition of the cooling time is similar to the one given in Medvedev and Spitkovsky[@Medvedev:2009], but differs because they considered the downstream skin depth: $T^\textrm{syn}_\textrm{cool,our} = 9/4 \Gamma^{1/2} T^\textrm{syn}_\textrm{cool,M-S}$.
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We have made long term 2D2V simulations of the shock, using reflecting boundaries and different cooling times, including a run without cooling for reference, and with electron-ion and pair plasmas. The runs were done with cubic interpolation, sixth order field pusher and a 17-point current density filter. We used the sliding window technique to be able to follow the development of the shock up to ${\omega_{pe}\,t}=5000$, and used more than 5 billion particles to model the shocks. In all cases the upstream Lorentz factor is $\Gamma=10$. While below we present results from runs with 20 cells per skin depth, we have used both 10, 20, and 40 cells per skin depth and between 12 and 24 particles per species per cell and find converged results. Examples of the magnetic field and phase space density for different cooling times can be seen in figure \[fig:cooling\]. The shocks can phenomenologically be categorized into three types: shocks in the strong cooling regime, radiative shocks, and weakly radiative shocks. In the strong cooling regime, there are no traces of a power law tail of accelerated particles, and both the evolution, shock jump conditions, and the micro structure itself are qualitatively different from a normal non-radiating shock. In particular, the upstream is completely unperturbed by the existence of the shock. This is found for cooling times $T_\textrm{cool} \lesssim 200$. In a strongly radiating shock the micro structure is disturbed, and the downstream magnetic islands only survive very close to the shock interface, but the upstream does have resemblance to a normal collisionless shock. The power law tail of accelerated particles is completely gone, and the shock jump conditions are altered. This is found for cooling times $200 \lesssim T_\textrm{cool} \lesssim 1000$. The weakly radiative shocks are similar in structure to a non-radiating shock, but with mildly perturbed shock jump conditions and propagation velocity (see figure \[fig:jump\]). The power law index for the high energy population of accelerated electrons is steeper than for a non-radiating shock, with a dependence on the cooling time. The magnetic energy density decreases significantly approximately 150 $\delta_e$ away from the shock interface, but the time a high energy particle spends that close to the shock transition, where the bulk of the cooling happens, is a stochastic function of its angle to the shock normal, and the number of scatterings on fluctuations in the electromagnetic field. In principle, if we simulate for a long enough time, with a long cooling time and with high statistics, the powerlaw tail of accelerated particles will grow over more than a decade in energy, and a well defined cooling break should emerge with two different slopes clearly visible. In practice, given the tangled nature and stochastic propagation, the break will most probably be smooth, significantly smeared out around the characteristic energy, where electrons start to be efficiently cooled. In these exploratory simulations the emergence of a cooling break does not occur. Instead, in the case of weakly radiative shocks, the high-energy part of the particle distribution (PDF) is a powerlaw with an exponential cut-off, but with a steeper powerlaw index than in the non-radiating case (see fig. \[fig:pdf\]). It is well known that in PIC simulations of non-radiating shocks the upstream region affected by high energy particles produced at the shock interface only grows with time, and it has been an open question what the long term structure looks like[@Keshet:2009]. This is different for radiatively cooled shocks, where at large times the shock settles down to a quasi-steady state, making it possible to draw conclusions about the long term behavior. The extent of the upstream is limited, and the powerlaw part of the PDF does not grow in time. Any collisionless shock is radiatively cooling, given long enough time, and from our simulations it is clear that the impact of cooling for weakly radiating shocks is greater than speculated in e.g. Medvedev & Spitkovsky[@Medvedev:2009], where analytic estimates were used. Given the possibility of collisionless shocks to mediate secondary instabilities such as the Bell instability far upstream, by the generation of streaming cosmic rays[@Niemiec:2008], it would be interesting to understand the impact on collisionless shock for the very large cooling times expected in e.g. GRB afterglows. We speculate that using progressively larger cooling times in sufficiently large simulations could give insight in how to scale the solution to arbitrarily long cooling times.
![Top panel: Density ratio between up and down stream at ${\omega_{pe}\,t}= 1600$, as a function of cooling time. Bottom panel: Effective adiabatic index, derived from the shock velocity. The right most point is for a run without radiative cooling.[]{data-label="fig:jump"}](jump_conditions){width="\linewidth"}
![The particle distribution function sampled in the downstream region for the case of no cooling and $T_\textrm{cool}=25600$, firmly in the weakly radiative regime. The power law index is indicated with the dashed line, and given in the legend.[]{data-label="fig:pdf"}](pdf_nocool5_c25600){width="\linewidth"}
The linear magnetized Kelvin-Helmholtz instability
--------------------------------------------------
Earth’s magnetopause constitutes an important region in space. It is the boundary layer, separating the Earth’s magnetosphere from the solar wind. In this region, the Kelvin-Helmholtz instability (KHI) is driven by velocity shears between the magnetosheath and the magnetospheric plasma at low latitude (close to the magnetic equator). We have selected the problem of the linear magnetized KHI to compare with results using our local MHD [<span style="font-variant:small-caps;">Stagger</span> code]{}. Essentially, we use the setup for the SWIFF Magnetopause Challenge [@SWIFF_D2.1_KH] code comparison, except that the amplitude is slightly different and there are half the number of skin depths in the kinetic case. The MHD and PIC code setups are nearly identical, the only difference being the addition of microphysical parameters, and that in the PIC case we must ensure that the initial condition is a kinetic equilibrium respecting constraints like Gauss law.
We compare our PIC results against those obtained with the [<span style="font-variant:small-caps;">Stagger</span> code]{}, also developed and maintained at the Niels Bohr [email protected]. This finite difference mesh based MHD code is a fully 3D resistive and compressible MHD code. The MHD variables are located on staggered meshes, and the discretization is very similar to the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}, with sixth order spatial derivatives, and fifth order interpolation of variables. The time integration of the MHD equations is performed using an explicit 3rd order low storage Runge-Kutta method[@1980JCoPh..35...48W].
The experiment is a periodic 2D3V setup, and the box size is $L_x=90\pi$, $L_y=30\pi$. The initial velocity field $\textbf{V} = V_y(x) \textbf{e}_y$ contains a periodic double shear layer, to avoid boundary effects, with a velocity amplitude $A_{eq}=1$. The sheared velocity jumps are located at $L_x / 4$ and $3/4 \ L_x$, and the transition width is $a=3$. With these parameters, the initial velocity profiles is defined as: $$V_y(x) = \tanh \left( \frac{x - L_x / 4}{3} \right) - \tanh \left( \frac{x - 3/4 L_x}{3} \right)-1~.$$
Initially, $J_{eq} = {\bm{E}}= 0$, ${\bm{B}}= B_0 sin(\theta) \textbf{e}_y + B_0 cos(\theta) \textbf{e}_z$, with $B_0 = 1$, $\theta = 0.05$. The density, pressure and Alfvén velocity are all unity: $\rho=P=V_A=1$. For the MHD we use an adiabatic equation of state with $\gamma_\textrm{gas} = 5/3$, and impose a small perturbation in the velocity field to seed the KHI $\delta\textbf{V}=\textbf{e}_z\times\nabla\psi$, where $$\psi = \epsilon f(x) \sum_{m=1}^{N_y/4} cos(2 \pi m y / L_y + \phi_m )/m ~,$$ and $$f(x) = \exp \left[- \left(\frac{x - L_x/4} {3}\right)^2 \right] + \exp \left[ - \left(\frac{x - 3/4L_x} {3}\right)^2 \right]$$ and $\epsilon$ is such that $\max(| \delta V |) \simeq 10^{-3}$. $\phi_m$ are random phases.
In the MHD case we resolve the box with $N_x=1536$, $N_y=512$ cells. The PIC setup was prepared with physical initial conditions essentially identical to the MHD setup, using the technique of section \[sec:embed\] to reach an approximate kinetic equilibrium. The objective was to produce the ion-scale KHI, while resolving the electron skin-depth $\delta_e$ and having well separated inertial scales with $m_i/m_e =64$. We use a setup similar to what was used for a hybrid code in the SWIFF comparison[@SWIFF_D2.1_KH], but limit the number of ion skin depths to $45\pi \times 15\pi$, and the resolution to $N_x=6144$, $N_y=2048$. To have a reasonable plasma frequency, we rescale the speed of light to $c=10$, and use $\delta_e = 6 \Delta x$, and 20 particles per cell per species with a total of roughly 500 million particles in the box. Our choice of the shearing jump amplitude ($V_0=\pm1.0$) and width ($a=3.0$) selects a fastest growing mode (FGM) leading to production of two vortices, which pair up and merge during the early and late non-linear stages of the KHI. Figure \[fig:small\_PIC\_KH\_fig\] shows the vortices just prior to, and well after the vortex merging in the PIC model.
{width="\textwidth"}
The growth rates for the two extremum cases of transverse (${\bm{B}}_0 \perp {\bm{v}}_0$) and parallel (${\bm{B}}_0 \parallel
{\bm{v}}_0$) configurations were investigated theoretically and further calculated by Miura[@Miura:1982.1; @Miura:1982.2]. We may assume that for a weak parallel component, i.e. $B_{0z} \ll B_{0y}$ ($B_{0z} = 0.05 B_{0y}$), the instability evolves almost as the ideal transverse case to a good approximation. It was also determined that the stability criterion for $V_0=2$ is $M_f\equiv{V_0}/{\sqrt{v_A^2+c_s^2}}<2$. Here, $M_f$ is the fast mode magneto-acoustic Mach number.
For our setup $M_f=\sqrt{2}$, with both $v_A^2=1$,$c_s^2=1$, using $V_0=2$ and $a=3$. From Miura 1982[@Miura:1982.1], we find a growth rate for our specific setup to be ${2a}\gamma_{KH}/{V_0}\approx0.162$, which leads to the final result of $$\gamma_{KH,FGM}\approx0.055\pm0.002.$$
To measure the KHI rate growth — in both the MHD and PIC cases — we calculate the quantity $$\left|\tilde{V_x}(x_i,k_y,t_n)\right|^2 = \left| \frac{1}{N_y}
\sum^{N_y-1}_{j=0} V_x(x_i,y_j,t_n) e^{-i2\pi\frac{k_yj}{N_y}} \right|^2~,$$ i.e. the power spectra along the $y$-axis of the $x$-component of velocity, $V_x$, at constant $x_i$. This is then averaged in the $x$-direction $$\mathcal{Q}(t_n) = \frac{1}{x_{hi}-x_{lo}} \sum^{x_{hi}}_{x_{lo}} \left|\tilde{V_x}(x_i,k_y,t_n)\right|^2~,$$ for two different sets of $\{x_{lo},x_{hi}\}$, namely i) centered on one shearing layer, 5$\Delta_x$ wide, and ii) across the entire half-volume in the x-direction. We then make a fit to exponential growth to obtain $\gamma_{KI}(mode)$ for the FGM.
[ l c c c ]{} & PIC & & MHD\
Right & 0.058 & & 0.058\
Total & 0.050 & & 0.056\
We find that growth rates from both the MHD and PIC simulations agree well with theory, to within about 10% (see table \[tab:kh\_growth\_rates\_compare\]). Averaging over all layers in the simulation half-plane yields a lower growth rate with the MHD in closest agreement with theory. This is expected, since averaging over only 5 layers does not capture the entire width of the shearing layer which is $W_{shear}\sim50\Delta_x$. For the PIC growth rate results discrepancies in physics and uncertainties in data fitting, due to mode coupling are higher than for the MHD case; and growth rates therefore differ by as much as 10% in the in the volume-averaged case. The explanation for a slightly lower growth rate in the total volume averaged PIC case may be due to enhanced dissipation and intrinsic noise properties of the PIC code, or the development of secondary instabilities.
Concluding this test, we emphasize that a PIC code has been used to run a fully MHD problem in PIC explicit mode, with almost identical growth rates and vortex structures in the linear phase. For the vortex-merging epoch of the PIC run, results deviate qualitatively and quantitatively from MHD, due to kinetic effects and secondary tearing-like instabilities developing during the merging stage.
Parallelization, scaling and performance
========================================
Modern 3D Particle-in-Cell experiments in astrophysics use billions of particles to simulate the macroscopic structure of plasmas. To run these simulations the code has to be massively parallel. The [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} started with a simple domain decomposition along one axis, using MPI. Later it was changed to support a 3D MPI decomposition, and in the current version we use a hybrid parallelization with OpenMP and MPI to scale the code effectively up to hundreds of thousands of cores (with 262.144 cores being the largest case tried so far). The OpenMP hybrid approach is also necessary when running the GPU version of the code, because normally the ratio of CPU cores to GPUs in a GPU system is larger than one.
MPI
---
The MPI parallelization has been designed to be as transparent as possible, and consists of different modules, all collected in a single file. This makes it easy to compile the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} both with and without the MPI library.
- [Particles: After moving the particles on a node, and applying the physical boundary conditions, on each thread we check sequentially in the $x$-, $y$-, and $z$-direction which particles have moved to other threads, and interchange particles accordingly. A reverse transversion of the linked list structure containing the send particles makes it efficient to store the received particles in the same slots.]{}
- [Fields: When the physical boundary conditions are applied for the fields, ghostzones are also exchanged between threads.]{}
- [I/O: The snapshot and restart file format is binary, and the code uses MPI-IO. For the particles each attribute (ie. the $x$-position) is stored in a single block, and the IO-types are all 64-bit, making it possible to store billions of particles in a single snapshot with GB/s of performance. The code stores everything in a few files, and restarts can be done on an arbitrary number of threads.]{}
- [Synthetic Spectra: The spectra are sampled on sub-volumes of the data, and each thread only allocates data for the sub-volumes that intersect with the local domain. This makes the mechanism scalable to $\mathcal{O}(10^5)$ sub-volumes, without wasting memory. The lowest ranked thread for a given sub-volume writes the data, making the I/O scalable too.]{}
- [Particle Tracing: Particle tracing is done locally on each thread, while only the master thread writes the I/O. This is a potential bottleneck, and on systems with relatively weak CPUs, e.g. Blue-Gene/P we are limited to tracing $\sim$1 million particles, while for x86-based clusters with global Lustre filesystems we can effectively trace up to $\sim$10 million particles, without significant slowdowns.]{}
- [2D-slices: The code can extract (averaged) 2D slices on-the-fly while running. While this is very convenient for movie making, for large runs thread contention becomes a problem when reducing the data on a single thread: When thousands of threads try to communicate with a single thread, the slow down can be significant. We use single-sided MPI to effectively circumvent the thread contention, making slice writing scalable to at least 100.000 MPI threads on network transports that support RDMA.]{}
A major problem with simple domain decomposed particle-in-cell codes is that systematic local over-densities of particles easily occur, for example in collisionless shock or laser wake-field simulations. This leads to severe load imbalance. To work around this the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} has a simple dynamic load-balance feature. If enabled, mesh-slices along the $z$-direction can be exchanged in-between neighboring threads, according to a user defined cost formula. In the current version particles and cells are assigned a certain cost, and this makes it possible to dynamically sample local spikes in the particle density (see Figure \[fig:loadbalance\]). When a slice of cells is moved from a thread to the neighbor thread the corresponding field values, particle data, synthetic spectra sub-volumes, and local random number generators are updated. The same mechanism is used to make an optimal distribution of cells when restarting a run. We have tested the MPI performance in a simple, but realistic setup, with two counter streaming plasma beams on the Blue-Gene/P machine JUGENE running in pure MPI mode. Figure \[fig:mpi\_scaling\] shows weak scaling behavior of the code from 8 to 262.144 cores. This was done a few years ago, with a version of the code without charge conservation, and the scaling of the current code is significantly better.
![Density profile of a 2D collisionless shock. The dashed lines indicate the $z$-boundaries of MPI domains. They are updated dynamically to equalize the load.[]{data-label="fig:loadbalance"}](density){width="\linewidth"}
![Weak scaling in pure MPI mode when running on the BG/P machine JUGENE. The setup is a relativistic two-stream experiment with periodic boundaries and a $\Gamma=10$ streaming motion. There are roughly 80 particles per cell with $16^3$ cells per MPI domain. This experiment is not using the charge conserving current deposition, and the scaling of the current code is significantly better.[]{data-label="fig:mpi_scaling"}](mpi_scaling){width="\linewidth"}
{width="0.32\linewidth"} {width="0.32\linewidth"} {width="0.32\linewidth"}
OpenMP
------
The 3D MPI domain decomposition has served well, scaling the code, for most applications, to 10.000 cores on x86 clusters and more than 100.000 cores on Blue-Gene/P. To scale the code to a million threads or more on for example Blue-Gene/Q, one must introduce a new layer of parallelization. We have used OpenMP for a number of reasons: It is relatively easy to get almost perfect scaling for a low number of cores, it can be done incrementally, and it is a natural thing to do for the GPU version of the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{}. Furthermore, using a hybrid parallelization, the size of each MPI domain becomes bigger, and the particle load-balance requirements, due to fluctuations and variations in number density, become smaller. This is important for experiments with some level of density fluctuations, even if the scaling per se is good in pure MPI mode. In the current version the most CPU consuming parts of the code have been OpenMP parallelized:
- [The Mover / Charge deposition is trivially parallelized by allocating one set of charge density fields per thread, and parallelizing the update of particles on a per cell basis.]{}
- [The field solver consist of simple differential operators, and we use loop-based parallelization for perfect speedup.]{}
- [The sorting of particles is more challenging: We first parallelize on the number of species, this is embarrassing parallel. Then on a nested level we first partition the particle data with nested parallel sweeps in 128 sets, and afterwards use quick-sort to sort each set of particles in parallel.]{}
- [Sending particles between threads: We first parallelize on the number of species, this is embarrassing parallel. Then on a nested level we use loop parallelization to select particles to be send, and multiplex MPI communication with OMP sections.]{}
- [Several other auxiliary routines have been parallelized and made thread safe: The random generators, synthetic spectra sampling, initial- and boundary conditions.]{}
The rest of the code, mostly diagnostics and I/O, can be incrementally OpenMP parallelized as needed. We find excellent scaling inside x86 nodes, with optimal performance at 4 MPI threads per socket / 8 per node, only loosing about 5% when using 1 MPI thread per socket. The 5% is easily regained in higher MPI efficiency and better load-balance when running on a large number of nodes (top panel in [Fig. \[fig:openmp\]]{}). We also find that we gain roughly 15% in performance by running with hyperthreading (32 compared to 16 threads in bottom panel in [Fig. \[fig:openmp\]]{}). Finally, on Blue-Gene/Q it is absolutely crucial to use all 64 hardware threads on a 16 core node. It improves overall performance by a factor of 2. But to effectively use the massive amount of threads exposed in the system, we need OpenMP, since otherwise the domain size is too small. Currently we use 4 or 16 OpenMP threads per node, depending on the scale of the problem. Even though we loose 16% performance by going from 4 to 16 OpenMP threads, for runs with e.g. particle tracing enabled or other IO done only through the master thread, and for very large runs, where load imbalance can be more than 20%, it is advantageous to use 16 OpenMP threads per MPI thread. In simpler and smaller scale runs we use 4 OpenMP threads per MPI thread.
Concluding Remarks
==================
In this paper we present the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} particle-in-cell code, with new numerical methods, physical extensions, and parallelization techniques. Originally the main motivation for developing a new particle-in-cell code from scratch was to implement a modern, modularized and extendible code, with modern numerical techniques. In the paper we have demonstrated how the extension of the classical PIC framework to higher order spatial and temporal derivatives, together with a novel charge conservation scheme, gives a much higher accuracy and better stability than traditional, second order PIC codes, when using the same number of grid points. Conversely, fewer mesh points are needed to reach a given level of fidelity in realistic three dimensional kinetic astrophysical setups, which leads to large savings in computational costs. The excellent scalability of the code, and the ability to do diagnostics on-the-fly, makes it possible to achieve a high scientific turnaround, where numerical experiments with more than 100 billion particles can be performed in a day, with dedicated use of petascale computational resources.
Among the novel physical extensions of the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} code are the description of binary interactions by particle splitting, the ability to include radiative cooling in a self-consistent manner, the possibility of embedding the kinetic model in an MHD snapshot, and the application of a sliding simulation window. Other extensions are in development or are planned for the future, among them nested grids, proper inclusion of neutral particles and dust, self gravity, smooth transition from the kinetic to the MHD regime, and simpler binary collision drag descriptions, relevant for studying fractionation processes.
The [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} has already been in full production mode for some years, the user base is growing steadily, and we expect that it will be used extensively at the Niels Bohr Institute and elsewhere for many years to come. Hopefully the detailed description of the algorithms documented in this paper will also be of use to others, when developing particle-in-cell techniques and applying them astrophysical plasma physics problems in the future.
We thank Juri Poutanen for funding the workshop ’Thinkshop 2005’ in Stockholm 2005; <http://comp.astro.ku.dk/Twiki/view/NumPlasma/ThinkShop2005>, which initialized the development of the [<span style="font-variant:small-caps;">PhotonPlasma</span> code]{} code. Further, we wish to acknowledge Boris Stern for ample help with Compton scattering Large Particle Monte-Carlo techniques. TH and [Å]{}N are supported by the Centre for Star and Planet Formation, which is financed by the Danish National Science Foundation. [Å]{}N and JTF are supported by SWIFF; the research leading to these results has received funding from the European Commission’s Seventh Framework Programme (FP7/2007-2013) under the grant agreement SWIFF (project n$^\texttt{o}$ 263340, www.swiff.eu). Computer time was provided by the Danish Center for Scientific Computing (DCSC), and through the PRACE project ’Ab Initio Modeling of Solar Active Regions’ running on JUGENE and JUQUEEN at the Jülich Supercomputing Centre; we thank the centre staff for their support, in particular in connection with I/O scaling issues, and for help with execution of full machine jobs.
| ArXiv |
---
abstract: |
Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index quasiconvex subgroup $H$ of $W$ and any finite subset $\{ \gamma_1, \ldots , \gamma_n \} \subset W \setminus H$ there is a surjection $f$ from $W$ to a finite alternating group such that $f (\gamma_i) \notin f (H)$. A corollary is that a right-angled Artin group splits as a direct product of cyclic groups and groups with many alternating quotients in the above sense.
Similarly, finitely generated subgroups of closed, orientable, hyperbolic surface groups can be separated from finitely many elements in an alternating quotient, answering positively the conjecture of Wilton [@wilton2012alternating].
author:
- Michal Buran
bibliography:
- '../mybib.bib'
title: 'Alternating quotients of right-angled Coxeter groups'
---
Introduction
============
It is often fruitful to study an infinite discrete group via its finite quotients. For this reason, conditions that guarantee many finite quotients can be useful.
One such notion is residual finiteness. A group $G$ is said to be *residually finite* if for every $g \in G \setminus \{e \}$, there exists a homomorphism $f: G \rightarrow F$, where $F$ is a finite group and $f(g) \neq e$.
We could try to strengthen this notion by requiring that any finite set of non-trivial elements is not killed by some map to a finite group. But these two notions are equivalent as we could simply take product of maps, which don’t kill the individual elements.
Another way to modify this is to require that the image of $\gamma$ avoids the image of a specified subgroup $H < G$, which does not contain $\gamma$. If this is true for all finitely generated subgroups $H$, we say that $G$ is *subgroup separable*.
Finitely generated free groups are subgroup separable [@hall1949coset Theorem 5.1]. The finite quotient $F$ of a free group could be a priori anything. Wilton proved that (for a free group with at least two generators) we can require $f$ to be a surjection onto a finite alternating group, thus giving us some control over the maps which ‘witness’ subgroup separability [@wilton2012alternating].
Scott showed that closed, orientable, hyperbolic surface groups are subgroup separable [@scott1978subgroups].
Extending and combining methods from both papers, our main theorem shows that even in the case of hyperbolic surface groups, we can require the image to be a finite alternating group.
Let $H$ be a subgroup of a finitely generated group $G$, let $\mathcal{C}$ be a class of groups. We say that $H$ is *$\mathcal{C}$-separable* if for any choice of $\{ \gamma_1, \ldots , \gamma_m \} \subset G \setminus H$ there is a surjection $f$ from $G$ to a group in $\mathcal{C}$ such that $f (\gamma_i) \notin f (H)$ for all $i$.
Note the difference between this terminology and the one above. We talk about subgroups as $\mathcal{C}$-separable in contrast with subgroup separability, which is a property of the entire group.
We will usually take $\mathcal{C}$ to be the class of alternating groups or symmetric groups. We will denote these classes by $\mathcal{A}$ and $\mathcal{S}$, respectively.
In this case, there is a difference between taking a single $\gamma_1$ and multiple group elements as a product of maps surjecting alternating groups is not a map onto an alternating group. In particular, if $G = A_n \times A_m$ then any $\gamma \in G \setminus \{e\}$ does not map to $e$ under at least one of the projections onto factors. However, if we take $\gamma_1, \ldots, \gamma_k$ to be an enumeration of $G \setminus \{e\}$, then the image of any map injective on these elements is isomorphic to $G$ and hence not an alternating group.
The following is our main result.
Let $\mathcal{G}$ be a non-discrete finite simplicial graph of size at least $3$. Every infinite index quasiconvex subgroup of a right-angled Coxeter group $W$ associated to $\mathcal{G}$ is $\mathcal{A}$-separable (and $\mathcal{S}$-separable) if and only if $\mathcal{G}^c$ is connected.
If $\mathcal{G}$ was a discrete graph, there would be difficulties in controlling a permutation parity of the images of generators. It is possible that this can be resolved.
We require infinite index as otherwise the finite quotient by the normal subgroup contained in $H$ could potentially have no alternating quotients.
Quasiconvexity is required as not all finitely generated subgroups of RACG are $\mathcal{C}$-separable, where $\mathcal{C}$ is the set of finite groups [@haglund2008special Example 10.3].
Every finitely generated right-angled Artin group is a direct product of cyclic group and groups whose infinite index quasiconvex subgroups are $\mathcal{A}$-separable.
Infinite index quasiconvex subgroups of closed, orientable, hyperbolic surface groups are $\mathcal{A}$-separable.
Preliminaries
=============
$\mathcal{A}$-separability
--------------------------
We will establish some properties of $\mathcal{A}$-separability.
\[product\] Let $A$ and $B$ be non-trivial finitely generated groups. Then $\{e \} < A \times B$ is not $\mathcal{A}$-separable.
There are only finitely many surjections from $A \times B$ onto $A_2, A_3$ and $A_4$. If $A \times B$ is infinite, then there is a non-identity element $g$ in the kernel of all these maps. Consider elements $g, (e,b), (a,e)$, where $a \neq e$, $b \neq e$. Suppose $f:A \times B \rightarrow A_n$ is a surjection, which does not map these elements to $e$.
By the choice of $g$, we have $n>4$. The group $f(A \times e)$ is a normal subgroup of $A_n$, so it is $e$ or $A_n$. Similarly for $e \times B$. However $A_n$ is not commutative, so one of $A \times e, e \times B$ is mapped to $e$.
If both $A$ and $B$ are finite and $\{e \} < A \times B$ is $\mathcal{A}$-separable, enumerate $A \times B$ as $\gamma_1, \ldots \gamma_m$. Applying the $\mathcal{A}$-separability condition with respect to this set, we get an isomorphism $f : A \times B \rightarrow A_n$. However, $A_n$ is not a direct product, so one of $A, B$ is $A_n$ and the other is trivial.
This implies that passing to a finite degree extension does not in general preserve $\mathcal{A}$-separability of quasiconvex subgroups. However passing to finite-index subgroup does:
\[fi\] Let $G$ be a finitely generated group, let $H$ be a finite-index subgroup of $G$, and let $K$ be an infinite index subgroup of $H$. If $K$ is $\mathcal{A}$-separable in $G$, then it is $\mathcal{A}$-separable in $H$.
We need $K$ to be infinite index in $H$, as otherwise it’s possible that $K = N(H)$ in the notation of the proof below. E.g. take $G = A_n$, $H$ a proper subgroup, $K = \{e\}$.
Suppose $\gamma_1, \ldots, \gamma_n \in H \setminus K$.
Let $N(H) = \bigcap_{g \in G} H^g$ be a normal subgroup contained in $H$. Then $N(H)$ is still finite index and let $M = [G : N(H)]$ be this index. Since $G$ is finitely generated, there are only finitely many surjections $f : G \twoheadrightarrow A_m$ with $m \leq M$. The intersection of preimages of $f(K)$ over such surjections is a finite intersection of finite index subroups, hence a finite index subgroup. So there exists some $\gamma_0 \in G \setminus K$ such that $f(\gamma_0) \in f(K)$ for all $f : G \twoheadrightarrow A_m$ with $m \leq M$.
As $K$ is $\mathcal{A}$-separable in $G$, there exists a surjection $f : G \twoheadrightarrow A_m$, such that $f(\gamma_i) \notin f(K)$ for all $i \in \{0, \ldots n\}$. By the choice of $\gamma_0$ we have $m > M$. But $[A_m:f(N(H))] \leq M$, so $f(N(H)) = A_m$. In particular, $f(H) = A_m$ and $f|_H$ is the desired surjection.
Cube complexes
--------------
For further details of the definitions from this section, the reader is referred to [@haglund2008special].
*An $n$-dimensional cube $C$* is $I^n$, where $I = [-1,1]$. *A face* of a cube is a subset $F = \{\underline{x}: x_i = (-1)^\epsilon \}$, where $1 \leq i \leq n$, $\epsilon =0,1$.
Suppose $\mathcal{C}$ is a set of cubes and $\mathcal{F}$ is a set of maps between these cubes, each of which is an inclusion of a face. Suppose that every face of a cube in $\mathcal{C}$ is an image of at most one inclusion of a face $f \in \mathcal{F}$. Then *the cube complex $X$ associated to $(\mathcal{C},\mathcal{F})$* is $$X = (\bigsqcup_{C \in \mathcal{C}} C) / \sim$$ where $\sim$ is the smallest equivalence relation containing $x \sim f(x)$ for every $f \in \mathcal{F}$, $x \in Dom(f)$.
*A midcube $M$* of a cube $I^n$ is a set of the form $\{\underline{x}: x_i = 0 \}$ for some $ 1 \leq i \leq n$.
If $f: C \rightarrow C'$ is an inclusion of a face and $M$ is a midcube of $C$, then $f(M)$ is contained in unique midcube $M'$ of $C'$. Moreover $f|_M :M \rightarrow M'$ is an inclusion of a face.
Let $X$ be a cube complex associated to $(\mathcal{C},\mathcal{F})$. Let $\mathcal{M}$ be the set of midcubes of cubes of $\mathcal{C}$. Let $\mathcal{F'}$ be the set of restrictions of maps in $\mathcal{F}$ to midcubes.
The pair $(\mathcal{M}, \mathcal{F}')$ satisfies that every face is an image of at most one inclusion of a face, so there is an associated cube complex $X'$. Moreover, inclusions of midcubes descend to a map $\varphi : X' \rightarrow X$. *A hyperplane $H$* is a connected component of $X'$ together with a map $\varphi|_H$.
Hyperplanes are analogous to codimension-1 submanifolds.
Suppose $X$ is a cube complex.
Define a relation of *elementary parallelism* on oriented edges of $X$ by $\overrightarrow{e_1} \sim \overrightarrow{e_2}$ if they form opposite edges of a square. Extend this to the smallest equivalence relation. *The wall $W(\overrightarrow{e})$* is the equivalence class containing $\overrightarrow{e}$. Similarly, we can define an elementary parallelism on unoriented edges and *an unoriented wall $W(e)$*.
We denote by $\overleftarrow{e}$ the edge $\overrightarrow{e}$ with the opposite orientation.
There is a bijective correspondence between unoriented walls and hyperplanes, where $W(e)$ corresponds to $H(e)$, a hyperplane which contains the unique midcube of $e$. We say $H(e)$ is dual to $e$. By abuse of notation, we sometimes identify $H(e)$ with its image.
The following notion was invented by Haglund and Wise and was originally called *A-special* [@haglund2008special Definition 3.2].
A cube complex is *special* if the following holds.
1. For all edges $\overrightarrow{e} \notin W(\overleftarrow{e})$. We say the hyperplanes are $2$-sided.
2. Whenever $\overrightarrow{e_2} \in W(\overrightarrow{e_1})$, then $e_1$ and $e_2$ are not consecutive edges in a square. Equivalently, each hyperplane embeds.
3. Whenever $\overrightarrow{e_2} \in W(\overrightarrow{e_1})$, $\overrightarrow{e_2} \neq \overrightarrow{e_1}$, then the initial point of $\overrightarrow{e_2}$ is not the initial point of $\overrightarrow{e_1}$. We say that no hyperplane directly self-osculates.
4. Whenever $\overrightarrow{e_2} \in W(\overrightarrow{e_1})$ and $\overrightarrow{f_2} \in W(\overrightarrow{f_1})$ and $e_1$ and $f_1$ form two consecutive edges of a square and $\overrightarrow{e_2}$ and $\overrightarrow{f_2}$ start at the same vertex, then $\overleftarrow{e_2}$ and $\overrightarrow{f_2}$ are two consecutive edges in some square. We say that no two hyperplanes inter-osculate.
Haglund and Wise have shown that $CAT(0)$ cube complexes are special [@haglund2008special Example 3.3.(3)]. In this paper, we will only ever use specialness of these complexes.
Every special cube complex is contained in a nonpositively curved complex with the same $2$-skeleton [@haglund2008special Lemma 3.13]. The hyperplane $H(e)$ separates a $CAT(0)$ cube complex $X$ into two connected components.
\[Half-space, [@haglund2008finite]\] Let $X \backslash \backslash H$ be the union of cubes disjoint from $H$. If $X$ is $CAT(0)$, $X \backslash \backslash H$ has two connected components. Call them *half-spaces $H^-$ and $H^+$*
Define *$N(H)$* to be the union of all cubes intersecting $H$. Let *$\partial N(H)$* consist of cubes of $N(H)$ that don’t intersect $H$. In the case of a simply connected special cube complex $\partial N(H)$ has two components; call them *$\partial N(H)^ +$* and *$\partial N(H)^ -$*.
\[Convex subcomplex\] A subcomplex $Y$ of a cube complex $X$ is *(combinatorially geodesically) convex* if any geodesic in $X^{(1)}$ with endpoints in $Y$ is contained in $Y$.
The components of the boundary of a hyperplane $\partial N(H)^ +$, $\partial N(H)^ -$ and half-spaces are combinatorially geodesically convex [@haglund2008finite Lemma 2.10]. Any intersection of half-spaces is convex [@haglund2008finite Corollary 2.16] and a convex subcomplex of a $CAT(0)$ cube complex coincides with the intersection of all half-spaces containing it [@haglund2008finite Proposition 2.17].
\[Bounding hyperplane\] A hyperplane *bounds* a convex cubical subcomplex $Y \subset X$ if it is dual to an edge with endpoints $v \in Y$ and $v' \notin Y$.
Right-angled Coxeter and Artin groups
-------------------------------------
\[Right-angled Coxeter group\] Given a graph $\mathcal{G}$ with vertex set $I$, let $S = \{ s_i : i \in I \}$. *The right-angled Coxeter group* associated to $\mathcal{G}$ is the group $C(\mathcal{G})$ given by the presentation $\langle S \mid s_i^2 = 1 \text{ for } i \in I, [s_i, s_j]=1 \text{ for } (i,j) \in E(\mathcal{G}) \rangle$.
The right-angled Coxeter group $C(\mathcal{G})$ acts on *the Davis–Moussong Complex $DM(\mathcal{G})$* [@haglund2008special]. Througout the paper if we talk about the action of $C(\mathcal{G})$ on a cube complex, we mean this action. The Davis–Moussong complex is similar to Cayley complex, but it doesn’t contain ‘duplicate squares’ and it contains higher dimensional cubes.
Fix a vertex $v_0 \in DM(\mathcal{G})$. There is a bijection between the vertices of $DM(\mathcal{G})$ and the elements of $C(\mathcal{G})$ given by $gv_0 \longleftrightarrow g$. Vertices $g v_0$ and $g s v_0$ are connected by an edge $ge_s$ labelled $s$. If the generators $s_{i_1}, s_{i_2}, \ldots s_{i_n}$ pairwise commute, there is an $n$-cube with the vertex set $\{ g (\Pi_{j \in P} s_{i_j}) v : P \subset \{1, \ldots, n \} \}$.
Note that $g s_i g^{-1}$ acts on the left on $DM(\mathcal{G})$ as a reflection in $H(ge_{s_i})$. There is also a right action of $C(\mathcal{G})$ on $DM(\mathcal{G})^0$, where $s_i$ sends $g v_0$ to $g s_i v_0$ – the vertex to which $g$ is connected by an edge labeled $s_i$. This action does not extend to $DM(\mathcal{G})$.
More generally, if $\Gamma$ is a subgroup of $C(\mathcal{G})$, the action of $C(\mathcal{G})$ on the right cosets of $\Gamma$ can be realized geometrically as an action of $C(\mathcal{G})$ on $\Gamma \backslash DM(\mathcal{G})^0$. This action is given by $ (\Gamma hv_0 ).g= \Gamma hg v_0$. If $\Gamma$ acts on $DM(\mathcal{G})$ co-compactly, this gives a finite permutation action. We will use this to construct maps from $C(\mathcal{G})$ to $S_n$.
If $G$ acts on a cube complex $X$, we say $H < G$ is *quasiconvex* if there is a non-empty convex subcomplex $Y \subset X$, which is invariant under $H$ and moreover $H$ acts on $Y$ cocompactly. We say, that $H$ acts on $X$ with *core* $Y$.
If $G$ is hyperbolic, this coincides with the usual notion of quasiconvexity [@haglund2008finite].
The right-angled Artin group associated to a simplicial graph $\mathcal{G}$ is $A(\mathcal{G}) = \langle g_v: g \in V(\mathcal{G}) \mid g_u g_v = g_v g_u \text{ for } \{u,v\} \in E(\mathcal{G}) \rangle$.
The next lemma relates RAAGs and RACGs.
[@davis2000right] \[RAAGtoRACG\] Given a graph $\mathcal{G}$, define a graph $\mathcal{H}$ as follows:
- $V(\mathcal{H})=V(\mathcal{G}) \times \{0,1\}$
- $(u,1)$ and $(v,1)$ are connected by an edge if $\{u, v\}$ is an edge of $\mathcal{G}$. The $(u,0)$ and $(v,1)$ are connected by an edge if $u$ and $v$ are distinct. Similarly, $(u,0)$ and $(v,0)$ are connected by an edge if $u$ and $v$ are distinct.
The right-angled Artin groups $A(\mathcal{G})$ is a finite-index subgroups of the right-angled Coxeter group $C(\mathcal{H})$ via the inclusion $\iota$ extending $g_u \longrightarrow s_{(u,0)} s_{(u,1)}$.
A right-angled Artin group $A(\mathcal{G})$ acts on Salvetti complex $X=X(\mathcal{G})$, which consists of the following:
- $X^0 = A(\mathcal{G})$
- If generators $g_{u_1}, g_{u_2}, \ldots g_{u_n}$ pairwise commute and $g \in A(\mathcal{G})$, there is a unique $n$-cube with the vertex set $\{ g (\Pi_{j \in P} g_{u_{j}}) : P \subset \{1, \ldots, n \} \}$.
The action of the right-angled group on the vertex set is by the left multiplication and it extends uniquely to the entire cube complex.
For the rest of the paper whenever we talk about the action of a RACG or RAAG on a cube complex, we mean the canonical action on the associated Davis-Moussong Complex or Salvetti complex, respectively.
Jordan’s Theorem
----------------
A subgroup $G < S_n$ is called *primitive* if it acts transitively on $\{1, \ldots ,n \}$ and it does not preserve any nontrivial partition.
If $n$ is a prime and $G$ is transitive, then the action is primitive.
Our main tool is the following.
[@dixon1996permutation Theorem 3.3D] For each $k$ there exists $n$ such that if $G < S_n$ is a primitive subgroup and there exists $\gamma \in G \setminus \{ e \}$, which moves less than $k$ elements, then $G = S_n$ or $A_n$.
The main theorem and its proof
==============================
Our main theorem relates combinatorics of $\mathcal{G}$ and $\mathcal{A}$-separability of $C(\mathcal{G})$.
\[main\] Let $\mathcal{G}$ be a non-discrete finite simplicial graph of size at least $3$. Then all infinite-index quasiconvex subgroups of the right-angled Coxeter group associated to $\mathcal{G}$ are $\mathcal{A}$-separable and $\mathcal{S}$-separable if and only if $\mathcal{G}^c$ is connected.
Recall that here quasiconvex means that it acts cocompactly on a convex subcomplex of Davis-Moussong complex. Similar result holds for RAAGs.
\[AsepRAAG\] Let $\mathcal{G}$ be a finite simplicial graph of size at least $2$. Then all infinite index quasiconvex subgroups of the right-angled Artin group associated to $\mathcal{G}$ are $\mathcal{A}$-separable if and only if $\mathcal{G}^c$ is connected.
Here quasiconvex means that the subgroup acts cocompactly on a convex subcomplex of Salvetti complex. There is another action of the Artin group on a cube complex given by embedding the group in right-angled Coxeter group as described in the Lemma \[RAAGtoRACG\]. We will first show that quasiconvexity with respect to Salvetti complex implies quasiconvexity with respect to Davis-Moussong complex.
\[qcinRACGtoqcinRAAG\] Suppose $\mathcal{G}$ is a simplicial complex, and $K$ a quasiconvex subgroup of $A(\mathcal{G})$ with respect to the action on $X(\mathcal{G})$. Let $\mathcal{H}$ be as in lemma $\ref{RAAGtoRACG}$ and identify $A(\mathcal{G})$ with a subgroup of $C(\mathcal{H})$ in the same lemma. Then $K$ is quasiconvex in $C(\mathcal{H})$ with respect to the action on $DM(\mathcal{H})$.
Recall that $N(H)$ is the union of all cubes intersecting a hyperplane $H$. For a hyperplane $H$ in a $CAT(0)$ cube complex $X$, $N(H) \simeq H \times [0,1]$. We can collapse $N(H)$ onto $H$. Formally, say $(x,t) \sim (x, t')$ for all $x \in H$ and $t,t' \in [0,1]$. *Collapse of neighbourhood of $H$* is the quotient map $X \longrightarrow X / \sim$. We can collapse multiple neighbourhoods simultaneously by quotienting by the smallest equivalence relation, which contains the equivalence relation for each hyperplane.
Let $f: (DM(\mathcal{H}), v_0) \longrightarrow (Y,y)$ be the simultaneous collapse of all hyperplanes labelled by $s_{(v,0)}$ for all $v \in \mathcal{G}$. See Figure \[tree\]. The equivalence relation commutes with the action of $C(\mathcal{H})$, so there is an induced action of $C(\mathcal{H})$ on $Y$.
We collapsed all edges with labels from $\mathcal{G} \times \{0\}$ so for all $s_{(v,0)}$ and all $g \in C(\mathcal{H})$, we have $gs_{(v,0)}.y = g.y$. Let $f': X(\mathcal{G}) \longrightarrow Y$ be defined as follows
- Vertices: Send $g$ to $g.y$.
- Edges: Send the edge between $g$ and $gg_{v}$ to the edge between $g.y$ and $gg_{v}.y$. It is indeed an edge as $g.y = gs_{(v,0)}.y$ and $gs_{(v,0)}s_{(v,1)}.y$
- Squares: Send the square with vertices $g, gg_{v}, gg_u, gg_u g_v$ to the square with vertices $g.y, gg_{v}.y, gg_u.y, gg_u g_v.y$.
- Higher dimensions: Extend analogously.
The map $f'$ is $A(\mathcal{G})$-equivariant cube complex isomorphism.
For any $y \in Y$, we have that $(f')^{-1}(y)$ is contained in a cube of dimension at most $d$, where $d$ is the size of maximal clique in $\mathcal{G}$. Suppose $K$ acts cocompactly on a convex subcomplex $Z$ of $X(\mathcal{G})$. Then $K$ acts cocompactly on $W:= f^{-1} f'(Z) \subset DM(\mathcal{H})$.
It remains to show that $W$ is convex. Let $e$ be an edge in $DM(\mathcal{H})$ with exactly one endpoint in $W$. The edge $e$ is labelled by some $s_{(v,1)}$ as all edges labelled by $s_{(v,0)}$ either lie entirely in $W$ or have an empty intersection with it. Collapsing map sends parallel edges to parallel edges (unless it sends them both to a vertex) and any sequence of elementary parallelisms in codomain lifts to the domain, so $f(H(e)) = H(f(e))$. In particular, if $H(e)$ intersects $W$, then $H(f(e))$ intersects $Z$ and by the convexity of $Z$, $f(e)$ lies entirely in $Z$, which contradicts that $e$ doesn’t lie entirely in $W$.
So a quasiconvex with respect to the action on $X(\mathcal{G})$ implies quasiconvex with respect to the action on $DM(\mathcal{H})$.
(-29.91,-23.18) rectangle (27.36,21.97); (-8,-2)– (10,-2); (-2,4)– (-2,-2); (-8,-2)– (-8,16); (-2,4)– (-8,4); (-8,-2)– (-26,-2); (-14,4)– (-14,-2); (-14,4)– (-8,4); (-2,-8)– (-2,-2); (-8,-2)– (-8,-20); (-2,-8)– (-8,-8); (-8,-2)– (-26,-2); (-14,-8)– (-14,-2); (-14,-8)– (-8,-8); (4,-2)– (10,-2); (6,0)– (6,-2); (4,-2)– (4,4); (6,0)– (4,0); (4,-2)– (-2,-2); (2,0)– (2,-2); (2,0)– (4,0); (6,-4)– (6,-2); (4,-2)– (4,-8); (6,-4)– (4,-4); (4,-2)– (-2,-2); (2,-4)– (2,-2); (2,-4)– (4,-4); (-2,-2)– (10,-2); (4,2)– (6,2); (4.67,2.67)– (4.67,2); (4,2)– (4,4); (4.67,2.67)– (4,2.67); (3.33,2.67)– (3.33,2); (3.33,2.67)– (4,2.67); (4.67,1.33)– (4.67,2); (4,2)– (4,0); (4.67,1.33)– (4,1.33); (3.33,1.33)– (3.33,2); (3.33,1.33)– (4,1.33); (-10,-2)– (8,-2); (2,-2)– (8,-2); (2,2)– (4,2); (8.67,-1.33)– (8.67,-2); (8,-2)– (8,0); (8.67,-1.33)– (8,-1.33); (7.33,-1.33)– (7.33,-2); (7.33,-1.33)– (8,-1.33); (-2,4)– (2,0); (6,0)– (7.33,-1.33); (6,0)– (4.67,1.33); (2,0)– (3.33,1.33); (-8,-2)– (-8,16); (-2,4)– (-8,4); (-2,4)– (-2,-2); (-8,10)– (-8,16); (-6,12)– (-8,12); (-8,10)– (-2,10); (-6,12)– (-6,10); (-8,10)– (-8,4); (-6,8)– (-8,8); (-6,8)– (-6,10); (-10,12)– (-8,12); (-10,12)– (-10,10); (-8,10)– (-8,4); (-10,8)– (-8,8); (-10,8)– (-10,10); (-8,4)– (-8,16); (-4,10)– (-4,12); (-3.33,10.67)– (-4,10.67); (-4,10)– (-2,10); (-3.33,10.67)– (-3.33,10); (-3.33,9.33)– (-4,9.33); (-3.33,9.33)– (-3.33,10); (-4.67,10.67)– (-4,10.67); (-4,10)– (-6,10); (-4.67,10.67)– (-4.67,10); (-4.67,9.33)– (-4,9.33); (-4.67,9.33)– (-4.67,10); (-4,8)– (-4,10); (-7.33,14.67)– (-8,14.67); (-8,14)– (-6,14); (-7.33,14.67)– (-7.33,14); (-7.33,13.33)– (-8,13.33); (-7.33,13.33)– (-7.33,14); (-2,4)– (-6,8); (-6,12)– (-7.33,13.33); (-6,12)– (-4.67,10.67); (-6,8)– (-4.67,9.33); (-8,-2)– (-26,-2); (-14,4)– (-14,-2); (-14,4)– (-8,4); (-20,-2)– (-26,-2); (-22,0)– (-22,-2); (-20,-2)– (-20,4); (-22,0)– (-20,0); (-20,-2)– (-14,-2); (-18,0)– (-18,-2); (-18,0)– (-20,0); (-22,-4)– (-22,-2); (-22,-4)– (-20,-4); (-20,-2)– (-14,-2); (-18,-4)– (-18,-2); (-18,-4)– (-20,-4); (-14,-2)– (-26,-2); (-20,2)– (-22,2); (-20.67,2.67)– (-20.67,2); (-20,2)– (-20,4); (-20.67,2.67)– (-20,2.67); (-19.33,2.67)– (-19.33,2); (-19.33,2.67)– (-20,2.67); (-20.67,1.33)– (-20.67,2); (-20,2)– (-20,0); (-20.67,1.33)– (-20,1.33); (-19.33,1.33)– (-19.33,2); (-19.33,1.33)– (-20,1.33); (-18,2)– (-20,2); (-24.67,-1.33)– (-24.67,-2); (-24,-2)– (-24,0); (-24.67,-1.33)– (-24,-1.33); (-23.33,-1.33)– (-23.33,-2); (-23.33,-1.33)– (-24,-1.33); (-14,4)– (-18,0); (-22,0)– (-23.33,-1.33); (-22,0)– (-20.67,1.33); (-18,0)– (-19.33,1.33); (-8,-2)– (-8,16); (-14,4)– (-8,4); (-14,4)– (-14,-2); (-8,10)– (-8,16); (-10,12)– (-8,12); (-8,10)– (-14,10); (-10,12)– (-10,10); (-8,10)– (-8,4); (-10,8)– (-8,8); (-10,8)– (-10,10); (-6,12)– (-8,12); (-6,12)– (-6,10); (-8,10)– (-8,4); (-6,8)– (-8,8); (-6,8)– (-6,10); (-8,4)– (-8,16); (-12,10)– (-12,12); (-12.67,10.67)– (-12,10.67); (-12,10)– (-14,10); (-12.67,10.67)– (-12.67,10); (-12.67,9.33)– (-12,9.33); (-12.67,9.33)– (-12.67,10); (-11.33,10.67)– (-12,10.67); (-12,10)– (-10,10); (-11.33,10.67)– (-11.33,10); (-11.33,9.33)– (-12,9.33); (-11.33,9.33)– (-11.33,10); (-12,8)– (-12,10); (-8.67,14.67)– (-8,14.67); (-8,14)– (-10,14); (-8.67,14.67)– (-8.67,14); (-8.67,13.33)– (-8,13.33); (-8.67,13.33)– (-8.67,14); (-14,4)– (-10,8); (-10,12)– (-8.67,13.33); (-10,12)– (-11.33,10.67); (-10,8)– (-11.33,9.33); (-2,-8)– (-2,-2); (-8,-2)– (-8,-20); (-2,-8)– (-8,-8); (-8,-2)– (-26,-2); (-14,-8)– (-14,-2); (-14,-8)– (-8,-8); (-2,4)– (-2,-2); (-2,4)– (-8,4); (-8,-2)– (-26,-2); (-14,4)– (-14,-2); (-14,4)– (-8,4); (4,-2)– (10,-2); (6,-4)– (6,-2); (4,-2)– (4,-8); (6,-4)– (4,-4); (4,-2)– (-2,-2); (2,-4)– (2,-2); (2,-4)– (4,-4); (6,0)– (6,-2); (4,-2)– (4,4); (6,0)– (4,0); (4,-2)– (-2,-2); (2,0)– (2,-2); (2,0)– (4,0); (-2,-2)– (10,-2); (4,-6)– (6,-6); (4.67,-6.67)– (4.67,-6); (4,-6)– (4,-8); (4.67,-6.67)– (4,-6.67); (3.33,-6.67)– (3.33,-6); (3.33,-6.67)– (4,-6.67); (4.67,-5.33)– (4.67,-6); (4,-6)– (4,-4); (4.67,-5.33)– (4,-5.33); (3.33,-5.33)– (3.33,-6); (3.33,-5.33)– (4,-5.33); (2,-6)– (4,-6); (8.67,-2.67)– (8.67,-2); (8,-2)– (8,-4); (8.67,-2.67)– (8,-2.67); (7.33,-2.67)– (7.33,-2); (7.33,-2.67)– (8,-2.67); (-2,-8)– (2,-4); (6,-4)– (7.33,-2.67); (6,-4)– (4.67,-5.33); (2,-4)– (3.33,-5.33); (-8,-2)– (-8,-20); (-2,-8)– (-8,-8); (-2,-8)– (-2,-2); (-8,-14)– (-8,-20); (-6,-16)– (-8,-16); (-8,-14)– (-2,-14); (-6,-16)– (-6,-14); (-8,-14)– (-8,-8); (-6,-12)– (-8,-12); (-6,-12)– (-6,-14); (-10,-16)– (-8,-16); (-10,-16)– (-10,-14); (-8,-14)– (-8,-8); (-10,-12)– (-8,-12); (-10,-12)– (-10,-14); (-8,-8)– (-8,-20); (-4,-14)– (-4,-16); (-3.33,-14.67)– (-4,-14.67); (-4,-14)– (-2,-14); (-3.33,-14.67)– (-3.33,-14); (-3.33,-13.33)– (-4,-13.33); (-3.33,-13.33)– (-3.33,-14); (-4.67,-14.67)– (-4,-14.67); (-4,-14)– (-6,-14); (-4.67,-14.67)– (-4.67,-14); (-4.67,-13.33)– (-4,-13.33); (-4.67,-13.33)– (-4.67,-14); (-4,-12)– (-4,-14); (-7.33,-18.67)– (-8,-18.67); (-8,-18)– (-6,-18); (-7.33,-18.67)– (-7.33,-18); (-7.33,-17.33)– (-8,-17.33); (-7.33,-17.33)– (-7.33,-18); (-2,-8)– (-6,-12); (-6,-16)– (-7.33,-17.33); (-6,-16)– (-4.67,-14.67); (-6,-12)– (-4.67,-13.33); (-8,-2)– (-26,-2); (-14,-8)– (-14,-2); (-14,-8)– (-8,-8); (-20,-2)– (-26,-2); (-22,-4)– (-22,-2); (-20,-2)– (-20,-8); (-22,-4)– (-20,-4); (-20,-2)– (-14,-2); (-18,-4)– (-18,-2); (-18,-4)– (-20,-4); (-22,0)– (-22,-2); (-22,0)– (-20,0); (-20,-2)– (-14,-2); (-18,0)– (-18,-2); (-18,0)– (-20,0); (-14,-2)– (-26,-2); (-20,-6)– (-22,-6); (-20.67,-6.67)– (-20.67,-6); (-20,-6)– (-20,-8); (-20.67,-6.67)– (-20,-6.67); (-19.33,-6.67)– (-19.33,-6); (-19.33,-6.67)– (-20,-6.67); (-20.67,-5.33)– (-20.67,-6); (-20,-6)– (-20,-4); (-20.67,-5.33)– (-20,-5.33); (-19.33,-5.33)– (-19.33,-6); (-19.33,-5.33)– (-20,-5.33); (-18,-6)– (-20,-6); (-24.67,-2.67)– (-24.67,-2); (-24,-2)– (-24,-4); (-24.67,-2.67)– (-24,-2.67); (-23.33,-2.67)– (-23.33,-2); (-23.33,-2.67)– (-24,-2.67); (-14,-8)– (-18,-4); (-22,-4)– (-23.33,-2.67); (-22,-4)– (-20.67,-5.33); (-18,-4)– (-19.33,-5.33); (-8,-2)– (-8,-20); (-14,-8)– (-8,-8); (-14,-8)– (-14,-2); (-8,-14)– (-8,-20); (-10,-16)– (-8,-16); (-8,-14)– (-14,-14); (-10,-16)– (-10,-14); (-8,-14)– (-8,-8); (-10,-12)– (-8,-12); (-10,-12)– (-10,-14); (-6,-16)– (-8,-16); (-6,-16)– (-6,-14); (-8,-14)– (-8,-8); (-6,-12)– (-8,-12); (-6,-12)– (-6,-14); (-8,-8)– (-8,-20); (-12,-14)– (-12,-16); (-12.67,-14.67)– (-12,-14.67); (-12,-14)– (-14,-14); (-12.67,-14.67)– (-12.67,-14); (-12.67,-13.33)– (-12,-13.33); (-12.67,-13.33)– (-12.67,-14); (-11.33,-14.67)– (-12,-14.67); (-12,-14)– (-10,-14); (-11.33,-14.67)– (-11.33,-14); (-11.33,-13.33)– (-12,-13.33); (-11.33,-13.33)– (-11.33,-14); (-12,-12)– (-12,-14); (-8.67,-18.67)– (-8,-18.67); (-8,-18)– (-10,-18); (-8.67,-18.67)– (-8.67,-18); (-8.67,-17.33)– (-8,-17.33); (-8.67,-17.33)– (-8.67,-18); (-14,-8)– (-10,-12); (-10,-16)– (-8.67,-17.33); (-10,-16)– (-11.33,-14.67); (-10,-12)– (-11.33,-13.33);
\[Proof of corollary \[AsepRAAG\]\] $\Rightarrow$: If $H,K$ are components of $\mathcal{G}^c$, then $A(\mathcal{G}) = A(H^c) \times A(K^c)$ so by Lemma \[product\] the trivial subgroup $\{e\}$ is not $\mathcal{A}$-separable in $A(\mathcal{G})$.
$\Leftarrow$: Let $\mathcal{H}$ be as in Lemma \[RAAGtoRACG\].
Suppose $U$ is a proper component of $\mathcal{H}^c$. The vertices $(v,0)$ and $(v,1)$ are not connected by an edge in $\mathcal{H}$, so $U^0$ is of the form $V \times \{0,1\}$ for some $V \subsetneq \mathcal{G}^0$. But then looking at $V \times \{1\} \subset \mathcal{G} \times \{1\}$ gives that $V^0$ is a vertex set of a proper component of $\mathcal{G}^c$.
So $\mathcal{G}^c$ connected implies $\mathcal{H}^c$ is connected.
By lemma \[qcinRACGtoqcinRAAG\] the quasiconvexity condition of the Main theorem \[main\] is also satisfied.
(-1.45,-1.32) rectangle (1.69,1.13); (0,0) – (52.63:0.08) arc (52.63:142.63:0.08) – cycle; (0,0) circle (1cm); plot\[domain=1.53:1.978,variable=\]([1\*4.39\*cos(r)+0\*4.39\*sin(r)]{},[0\*4.39\*cos(r)+1\*4.39\*sin(r)]{}); plot\[domain=1.694:2.944,variable=\]([1\*1.39\*cos(r)+0\*1.39\*sin(r)]{},[0\*1.39\*cos(r)+1\*1.39\*sin(r)]{}); plot\[domain=0.482:1.697,variable=\]([1\*1.43\*cos(r)+0\*1.43\*sin(r)]{},[0\*1.43\*cos(r)+1\*1.43\*sin(r)]{}); (0.09,-0.61)– (-0.15,-0.43); (-0.15,-0.43)– (0.02,-0.21); (0.05,-0.62) node\[anchor=north west\] [$y$]{}; (-0.17,-0.47) node\[anchor=north west\] [$e_1$]{}; (-0.15,-0.17) node\[anchor=north west\] [$e_2$]{}; (0.28,-0.18) node\[anchor=north west\] [$H(e_1)$]{}; (0.27,-0.48) node\[anchor=north west\] [$H(e_2)$]{}; (-0.2, 0.05) node\[anchor=north west\] [$L$]{};
(0.09,-0.61) circle (1pt); (-0.15,-0.43) circle (1pt); (0.02,-0.21) circle (1pt); (0.06,-0.41) circle (1pt); (0.06,-0.3) node [$90\textrm{{\ensuremath{^\circ}}}$]{};
[@scott1985correction Correction to the proof of Theorem 3.1] \[fgtoqc\] A closed, orientable, hyperbolic surface group $G$ is a finite index subgroup of $C(C_5)$, where $C_5$ is a cycle of length $5$. Moreover, for a suitable embedding $G \xhookrightarrow{} C(C_5)$, all finitely generated subgroups of $G$ are quasiconvex in $C(C_5)$ with respect to the action on $DM(C_5)$.
Scott uses a different terminology, so it makes sense to summarise the proof. The natural generators of $C(C_5)$ act on the hyperbolic plane by reflections in the sides of a right-angled pentagon. Translates of the pentagon give a tiling of the hyperbolic plane. Dual to this cell complex is a square complex $DM(C_5)$. Under this identification, the geodesic lines bounding the the pentagons of the tiling become hyperplanes of $DM(C_5)$.
Suppose $H$ is a finitely generated subgroup of the surface group $G = \pi_1 (\Sigma)$. Let $\Sigma_H$ be the covering space associated to $H$. By Lemma 1.5 in $\cite{scott1978subgroups}$, there exists a closed, compact, incompressible subsurface $\Sigma' \subset \Sigma_H$ such that the induced map $\pi_1 \Sigma' \longrightarrow \pi_1\Sigma_H$ is surjective. Moreover, by [@scott1985correction Correction to the proof of Theorem 3.1] we can require $\Sigma'$ to have a geodesic boundary.
Let $\widetilde{\Sigma'}$ be the lift of $\Sigma '$ to $\mathbb{H}^2 = DM(C_5)$. Let $Y$ be the intersection of all half-spaces containing $\widetilde{\Sigma'}$. Suppose $y$ lies in $Y$, but not in $N_3(\widetilde{\Sigma'})$ and that $e_1, e_2$ are the first two edges of the combinatorial geodesic from $y$ to $\widetilde{\Sigma'}$. Since $y \in Y$, both $H(e_1)$ and $H(e_2)$ intersect $\widetilde{\Sigma'}$. Consequently, $H(e_1)$ intersects $H(e_2)$ as $H(e_2)$ does not separate $H(e_1)$ from $\widetilde{\Sigma'}$. Call the intersection $y'$. The point $y$ is a centre of a pentagon and $y'$ is a vertex of the same pentagon, so the distance between them doesn’t depend on $y$ (for example by specialness of $DM(\mathcal{G})$). See Figure \[correction\].
The closest boundary component $L$ of $\widetilde{\Sigma'}$ to $y$ is seen from $y'$ at more than the right angle (remember that the hyperplanes are geodesics). But such a point is within distance $\int_{t=0 }^{\pi/4} \frac{1}{\cos(t)}dt$ of $L$. To see this, take $L$ to be the vertical ray through $(0,0)$ in the upper half-plane model to the. Then the set of points with obtuse subtended angle is contained between rays $y = x$ and $y =-x$. Geodesic between these rays and $L$ is an arc of length $\int_{t=0}^{\pi/4} \frac{r \sqrt{\cos^2(t) + \sin^2(t)}}{r \cos(t)} dt = \int_{t=0 }^{\pi/4} \frac{1}{\cos(t)}dt$.
(-0.66,-0.26) rectangle (1.05,1.09); (0.5,0.53) – (0.47,0.53) – (0.47,0.5) – (0.5,0.5) – cycle; (0,0) – (-25.22:0.04) arc (-25.22:90:0.04) – cycle; plot(,[(-0-0\*)/1]{}); (0,0) – (0,1.09); plot\[domain=0:3.14,variable=\]([1\*0.5\*cos(r)+0\*0.5\*sin(r)]{},[0\*0.5\*cos(r)+1\*0.5\*sin(r)]{}); (0.5,0.5) – (0.5,1.09); plot(,[(-0–0.5\*)/0.5]{}); plot(,[(-0–0.5\*)/-0.5]{}); (-0.08,0.65) node\[anchor=north west\] [$L$]{}; plot\[domain=0.79:1.57,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=0:1.13,variable=\]([1\*0.8\*cos(r)+0\*0.8\*sin(r)]{},[0\*0.8\*cos(r)+1\*0.8\*sin(r)]{}); (-0.46,0.73) – (-0.46,1.09); (-0.54,0.76) node\[anchor=north west\] [$$y'$$]{}; (0.21,0.72) node\[anchor=north west\] [a]{};
(0.44,0.53) node [$90\textrm{{\ensuremath{^\circ}}}$]{}; (-0.46,0.73) circle (0.5pt); (-0.35,0.75) node [$>90\textrm{{\ensuremath{^\circ}}}$]{};
Therefore $y'$ (and hence $y$) is at a uniformly bounded distance from $\widetilde{\Sigma'}$ and the action of $H$ on $Y$ is cocompact.
\[surfaces\] All finitely generated infinite index subgroups of closed, orientable, hyperbolic surface group $G$ are $\mathcal{A}$-separable in $G$.
By Lemma \[fgtoqc\], finitely generated subgroups of $G$ are quasiconvex in $C(C_5)$. By the main theorem \[main\] they are $\mathcal{A}$-separable in $C(C_5)$. By Lemma \[fi\], they are $\mathcal{A}$-separable in $G$.
\[Disjoint hyperplanes, bounding hyperplanes, positive half-space\] Let $X$ be a cube complex, $Y$ a convex subcomplex. Let $\mathcal{D}(Y)$ be the set of hyperplanes disjoint from $Y$. Let $\mathcal{B}(Y)$ the set of hyperplanes bounding $Y$.
If $H \in \mathcal{D}(Y)$, denote by *$H^+$* the half-space of $X\backslash \backslash H$ containing $Y$.
Recall that any intersection of half-spaces is convex and conversely any convex subcomplex is an intersection of the half-spaces containing it. Hence it is equivalent to specify a convex subcomplex or the half-spaces in which it is contained (or the set of disjoint hyperplanes if there can be no confusion about the choice of half-spaces, e.g. if only one choice gives a non-empty intersection).
Suppose $G$ acts on a cube complex $X$ with core $Y$. Define *deletion* as removing a bounding hyperplane $H_0$ and all its $G$-translates from $\mathcal{D}(Y)$. That is the result of deletion of $H_0$ is $Y' = \cap_{H \in \mathcal{D}(Y) \setminus G. \{H_0 \} } H^+$.
The cube complex $V = H_0^- \cap Y'$ is called a *vertebra*. See Figures \[fig:pentagons\] and \[vertebra\].
(-4.37,-3.17) rectangle (6.57,5.42); (0,0)– (0,1); (0,0)– (1,0); (1,1)– (1,0); (1,1)– (1,1.56); (0.71,1.71)– (0.85,1.71); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); (2,0)– (2,1); (2,0)– (1,0); (1,1)– (1,1.56); (1.29,1.71)– (1.15,1.71); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); (4,0)– (4,1); (4,0)– (3,0); (3,1)– (3,0); (3,1)– (3,1.56); (3.29,1.71)– (3.15,1.71); (2,0)– (3,0); (3,1)– (3,1.56); (2.71,1.71)– (2.85,1.71); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); (8,0)– (8,1); (8,0)– (7,0); (7,1)– (7,0); (7,1)– (7,1.56); (7.29,1.71)– (7.15,1.71); (6,0)– (6,1); (6,0)– (7,0); (7,1)– (7,1.56); (6.71,1.71)– (6.85,1.71); (4,0)– (5,0); (5,1)– (5,0); (5,1)– (5,1.56); (4.71,1.71)– (4.85,1.71); (6,0)– (5,0); 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plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); (0,0)– (-1,0); (-1,1)– (-1,0); (-1,1)– (-1,1.56); (-0.71,1.71)– (-0.85,1.71); (-2,0)– (-2,1); (-2,0)– (-1,0); (-1,1)– (-1,1.56); (-1.29,1.71)– (-1.15,1.71); (-4,0)– (-4,1); (-4,0)– (-3,0); (-3,1)– (-3,0); (-3,1)– (-3,1.56); (-3.29,1.71)– (-3.15,1.71); (-2,0)– (-3,0); (-3,1)– (-3,1.56); (-2.71,1.71)– (-2.85,1.71); (-8,0)– (-8,1); (-8,0)– (-7,0); (-7,1)– (-7,0); (-7,1)– (-7,1.56); (-7.29,1.71)– (-7.15,1.71); (-6,0)– (-6,1); (-6,0)– (-7,0); (-7,1)– (-7,1.56); (-6.71,1.71)– (-6.85,1.71); (-4,0)– (-5,0); (-5,1)– (-5,0); (-5,1)– (-5,1.56); (-4.71,1.71)– (-4.85,1.71); (-6,0)– (-5,0); (-5,1)– (-5,1.56); (-5.29,1.71)– (-5.15,1.71); (-1.1,0.65) node\[anchor=north west\] [$ s_1 $]{}; (-0.10,0.65) node\[anchor=north west\] [$ s_4 $]{}; (-0.90,1.13) node\[anchor=north west\] [$ s_2 $]{};
A vertebra is an intersection of two combinatorially geodesically convex sets, so it also is combinatorially geodesically convex. In particular, it is connected.
We say $G$ acts *without self-intersections* on a cube complex $X$, if $N(gH) \cap N(H) \neq \emptyset$ implies $gH = H$ for all hyperplanes $H$ of $X$ and $g \in G$.
\[scale=2\] (1,0) – (0,0); (1,0) node\[anchor=north\] [$V$]{}; (0,0) – ($ (-1/2,{sqrt(3)/2}) $); (0,0) – ($ (-1/2,{-sqrt(3)/2}) $); ($ (-1/2,{sqrt(3)/2}) $) – ($ (-1/2,{sqrt(3)/2}) +(1/2,{sqrt(3)/2})$); ($ (-1/2,-{sqrt(3)/2}) $) – ($ (-1/2,-{sqrt(3)/2}) +(1/2,{-sqrt(3)/2})$); ($ (-1/2,{sqrt(3)/2}) $) – ($ (-1/2,{sqrt(3)/2}) -(1,0)$); ($ (-1/2,{-sqrt(3)/2}) $) – ($ (-1/2,{-sqrt(3)/2}) -(1,0)$); (1/2,0.05) – (1/2,-0.05) node\[midway,anchor=north\] [$H_0$]{}; (0,-2) – (-3/2,-2) node\[midway,anchor=north\] [$Y'$]{}; (1,-2.3) – (-3/2,-2.3) node\[midway,anchor=north\] [$Y$]{};
\[lemmaA\] Suppose that $G$ acts without self-intersections on a locally compact $CAT(0)$ cube complex $X$ with core $Y$. Then the result $Y'$ of deletion of $H_0$ is also a core for $G$. Let $G_{H_0} (:= \{g \in G | g.H_0 = H_0 \}$ be the stabilizer of $H_0$ in $G$. If $C$ is a set of orbit representatives for the action of $G$ on the vertices of $Y$ and $D$ is a set of orbit representatives for the action of $G_{H_0}$ on the vertices of the vertebra $V = H_0^- \cap Y'$, then $C' = C \sqcup D$ is a set of orbit representatives for the action of $G$ on the vertices of $Y'$.
Recall that $CAT(0)$ implies special.
First note that $\mathcal{D}(Y') = \mathcal{D}(Y) \setminus G. \{H_0 \}$ by definition and $\mathcal{B}(Y) \setminus G. \{H_0 \} \subset \mathcal{B}(Y')$ as a bounding hyperplane $Y$ still bounds $Y'$ unless it is a translate of $H_0$.
The set of half-spaces containing $Y$ is invariant under $G$, hence $Y'$ is invariant. The subcomplex $Y'$ is an intersection of half-spaces, hence convex. Suppose $v \in Y' \setminus Y$. Let $v_0, v_1 \ldots v_k$ be a combinatorial geodesic from $v$ to $Y$ of shortest length with edges $e_1, \ldots e_k$ and suppose $k>1$. Let’s $H_i$ be the hyperplane dual to $e_i$. Then as $v_{k-1} \notin Y$, we have $H_k \in G. \{H_0\}$. Since $G$ acts on $X$ without self-intersections $H_{k-1} \notin G. \{H_0\}$. And $H_{k-1} \notin \mathcal{D}(Y')$, because $v_0, v_k \in Y'$ and $Y'$ is convex, so $e_{k-1} \in Y'$
Therefore $H_{k-1} \notin \mathcal{D}(Y)$. It must intersect $Y$, so it is not entirely contained in $H_k^-$ and it intersects $H_k$. Because the cube complex is special, $H_k$ and $H_{k-1}$ don’t interosculate. In particular, there is a square with two consecutive sides $e_{k-1}$ and $e_k$. Let $e'_j$ be the edge opposite $e_j$ in this square. We can now construct a shorter path from $v_0$ to $Y$ with edges $e_1, \ldots, e_{k-2}, e'_{k}$. Contradiction.
So $k \leq 1$ and $Y'$ lies in a $1$-neighbourhood of $Y$ and therefore the action is cocompact.
There is a unique edge connecting $v \in Y' \setminus Y$ to $Y$ as any path of length $2$ is a geodesic or is contained in some square. In the first case by convexity of $Y$, we have $v \in Y$. In the second, $H_0 \notin \mathcal{D}(Y)$.
By invariance of $Y$, the $G$-translates of $V$ don’t intersect $Y$. Suppose $v \in Y' \setminus Y$. There is a unique hyperplane in $G. \{H_0 \}$ dual to an edge $e_1$, which connects $v$ to $Y$. Say $g.H_0$. Then $v$ belongs to a unique translate of $V$, namely $g.V$.
Let $\mathcal{G}$ be a finite simplicial graph. If $K$ is a subgroup of a right-angled Coxeter group $C(\mathcal{G})$ and it acts on the Davis-Moussong complex with core $Y$, then deletion produces another core.
The Davis-Moussong complex $DM(\mathcal{G})$ is a $CAT(0)$ cube complex, hence simply connected special. The action of $C(\mathcal{G})$ on it preserves labels. In any square the consecutive edges have distinct labels, so the action is without self-intersections. The restriction to $K$ is also without self-intersections.
\[intersection\] Suppose $G$ acts on a $CAT(0)$ cube complex $X$ with core $Y$. If $Y' \subset X$ is constructed from $Y$ using a deletion of $H = H(e)$, then each edge in $V = H^- \cap Y'$ is dual to a hyperplane intersecting $H$.
Let $e'$ be an edge in $V$ and $H'$ a hyperplane dual to $e'$. If $H' \cap H = \emptyset$, $H'$ is contained entirely in $H^-$. But then $H'$ is disjoint from $Y$. In particular one of the endpoints of $e'$ is in the opposite half-space of $X \backslash \backslash H'$ than $Y$.
Since $Y'$ is the intersection of all half-spaces containing $Y$ with the exception of the $G$-translates of $H^+$, the hyperplane $H'$ is $gH$ for some $g \in G$.
The subcomplex $Y$ is $G$-invariant and $H$ bounds $Y$, hence $H'$ bounds $Y$. This contradicts $H' \subset H^-$.
\[commutation\] Suppose $G<C(\mathcal{G})$ acts on $DM(\mathcal{G})$ with core $Y$. If $Y' \subset X$ is constructed from $Y$ using a deletion of $H = H(e)$, then each edge in $V = H^- \cap Y'$ has a label which commutes with the label of $e$.
Suppose $Y$ is a subcomplex of $X$ and $p=e_1 e_2 \ldots e_n$ is a path in $X$ and then *deletion of hyperplanes along the path $p$* is the deletion of $H(e_1), H(e_2), \ldots H(e_n)$.
If $v \in X$, and $s_1, s_2, \ldots s_n$ is a sequence of edge labels, then *the deletion with labels $s_1, s_2, \ldots s_n$ at $v$* is the *deletion of hyperplanes along $p$*, where $p$ is a path $e_1, e_2, \ldots e_n$ starting at $v$ with $e_i$ labelled $s_i$.
Suppose $Y_n$ was built from $Y_0$ using a series of deletion of hyperplanes $H_1, \ldots H_n$. We call $T=Y_n \cap H_1^-$ *a tail*. Moreover, if $H_j$ corresponds to $s_{i_j}$ and there is no confusion about the initial vertex of the path, we say that $Y_n$ was built from $Y_0$ with respect to $i_1, \ldots i_n$.
\[reduce\] Suppose $\mathcal{G}$ is a finite simplicial graph. Suppose $\mathcal{G}^c$ is connected, $| \mathcal{G}| >1$ and $H$ acts on $DM(\mathcal{G})$ with a core $Y \subsetneq DM(\mathcal{G})$. Then there exists a core $Y'$ which can be obtained from $Y$ by deletion along a path $e_1, e_2 \ldots e_n$ with the vertebra $Y' \cap H(e_n)^-$ a single vertex.
The hypothesis that $\mathcal{G}^c$ is connected is necessary. Consider the situation when $\mathcal{G}$ is a square. Then $C(\mathcal{G}) = D_\infty \times D_\infty$ and $DM(\mathcal{G})$ is the standard tiling of $\mathbb{R}^2$. Let $H = D_\infty$ be the subgroup generated by two non-commuting generators of $C(\mathcal{G})$. The invariance of the core and cocompactness of the action imply that any core for $H$ is of the form $\mathbb{R} \times [k,l]$ for some $k,l \in \mathbb{Z}$.
Every hyperplane intersecting such a core divides it into two infinite half-spaces.
Since $Y$ is a proper subcomplex, there exists $e_1$ be such that $H(e_1) = H_1$ bounds $Y$. Let $v_0$ be the endpoint of $e_1$, which lies in $Y$. Let $v_1$ be the other endpoint. Say the label of $e_1$ is $s_1$. Let $Y_1$ be a cube complex obtained from $Y$ by deletion of $H_1$.
Let $S_1$ be the set of generators labelling the edges of vertebra $V_1$. Then by corollary $\ref{commutation}$, $s_1$ commutes with all generators in $S_1$.
If $e_2 \notin V_1$ is an edge with endpoint $v_1$, whose label $s_2$ does not commute with $s_1$, we can define $H_2, Y_2, V_2$ and $S_2$ similarly as before. Just as before the generators of $S_2$ commute with $s_2$
The hyperplanes $H_1$ and $H_2$ don’t intersect, so $N(H_2) \subset H_1^-$. There is an inclusion of $V_2$ into $V_1$ given by sending a vertex of $V_2$ to the unique vertex of $V_1$ to which it is connected by an edge labelled $s_2$. Extending this map to edges and cubes is a label preserving map between cube complexes $V_2$ and $V_1$. It follows that $S_2$ is a (not necessarily proper) subset of $S_1$.
We will now show that, by a series of such operations, we can reach a situation where $S_n = \emptyset$. I.e. the vertebra $V_n$ is a single vertex.
Suppose we have already applied deletion $i$ times and $S_i$ is non-empty. We will use a series of deletions to get $S_{k+1} \subsetneq S_{k} \subset S_{k-1} \subset \ldots \subset S_{i+1} \subset S_i$. By an abuse of notation, we’ll identify the vertices of $\mathcal{G}^c$ with the labels and with the generators of the right-angled Coxeter group. (Rather than having a generator $s_v$ for every vertex $v \in V(\mathcal{G})$ and using these as labels.)
Since the group does not split as a product, there exists some $a \in S_i$ and $b \notin S_i$ which don’t commute. Since $\mathcal{G}^c$ is connected, there exists a vertex path $s_{i-1}, \ldots s_k = b$ in $\mathcal{G}^c$ from the vertex $s_{i-1}$, which is the label of the hyperplane we removed last.
Succesive generators in this path don’t commute. Indeed assume that $s_j$ and $s_{j+1}$ commute. Take $v \in DM(\mathcal{G})$, let $e_1, e_2, e_3, e_4$ be edges of the path starting at $v$ with labels $s_j, s_{j+1}, s_j, s_{j+1}$. This is a closed loop, since $s_j$ and $s_{j+1}$ commute. The hyperplane $H(e_1)$ separates $v$ from $s_j s_{j+1} v$, so it has to be dual to one of $e_3$ and $e_4$. Parallel edges have the same labels so $H(e_1) = H(e_3)$. Similarly $H(e_2) = H(e_4)$. The hyperplane $H(e_1)$ separates $e_2$ from $e_4$, so $H(e_1)$ and $H(e_2)$ have to cross. Davis-Moussong complex is special, so there is a square where $e_1$ and $e_2$ are successive edges. By construction of the complex, $s_j$ and $s_{j+1}$ are connected by an edge is $\mathcal{G}$. This contradicts adjacency of $s_j$ and $s_{j+1}$ in $\mathcal{G}^c$.
Apply deletion of hyperplanes labeled $s_i, \ldots , s_k$ starting at some vertex of $v \in V_{i-1}$. Note that the $j$th hyperplane we remove belongs in a subset of $\mathcal{B}(Y_{j-1})$ as $s_i \ldots s_{j-1} v \in V_{j-1}$ and $s_j$ does not commute with $s_{j-1}$. Moreover, $S_j = \{s \in S_{j-1} : ss_j = s_j s\}$. In particular, $S_{k+1} \subset S_i$ and $a$ does not belong to $S_{k+1}$ as $as_k \neq s_ka$. Similarly, the hyperplane $H'$ dual to edge between $v$ and $s_{j+1}$ is dual
Therefore $S_{k+1}$ is a proper subset of $S_i$ and we can continue this process until we get an empty $S_n$.
We can even control the label of the hyperplane which was removed last. Indeed, if the last removed hyperplane had label $s_i$, and $b$ is some other generator, pick a vertex path between $s_i$ and $b$ in $\mathcal{G}^c$. Then remove hyperplanes labelled by vertices on this path, starting at the unique vertex of a vertebra.
By lemma \[lemmaA\] there is a set of orbit representatives $K$ for the action of $G$ on $Y_n$ with $T \subset K$.
Haglund shows the following [@haglund2008finite Proof of Theorem A].
\[Scott\] Suppose $G < C(\mathcal{G})$ acts on $DM(\mathcal{G})$ with a core $Y$ with a set of orbit representatives $K$. Let $\Gamma_0 <C(\mathcal{G})$ be generated by the reflections in the hyperplanes bounding $Y$. Let $\Gamma_1 = \Gamma_1 (Y)= \langle G, \Gamma_0 \rangle$. Then $Y$ is a fundamental domain for the action of $\Gamma_0$ on $X$ and $K$ is a a set of orbit representatives for the action of $\Gamma_1$ on $X$.
Let $C(\mathcal{G})$ act on the right cosets of $\Gamma_1 < C(\mathcal{G})$. We have that $s \in S$ sends $\Gamma_1 g$ to $\Gamma_1 g s = (\Gamma_1 g s g^{-1}) g$. But $gs g^{-1}$ is a reflection in the hyperplane $H(ge_s)$. By definition of $\Gamma_0$ if $H(ge_s)$ bounds $Y$, $gs g^{-1} \in \Gamma_0$ and $\Gamma_1 g$ is fixed by $s$.
Moreover, if $K = \{g_1 v_0, \ldots ,g_n v_0 \}$, then $\{g_0, \ldots g_n\}$ is a set of right coset representatives for $\Gamma_1$.
We will first prove that by a suitable sequence of deletions, we can satisfy the conditions of Jordan’s theorem. It follows that we can construct quotients that are either alternating or symmetric.
If $Y$ is a subset of $X$, then $N_1(Y)$ is union of closed cubes, which have non-empty intersection with $Y$. We define inductively $N_i(Y) = N_1(N_{r-1}(Y))$.
If $Y$ is convex, then so is $N_r(Y)$ (as a neighbourhood is obtained by removing bounding hyperplanes and therefore it is an intersection of convex subcomplexes). And if $H$ acts cocompactly on $Y$, it still acts cocompactly on $N_r(Y)$.
Let $C(\mathcal{G})$ be the right-angled Coxeter group associated to $\mathcal{G}$ a finite simplicial graph, $|\mathcal{G}| >2$ , and $H$ acts with a proper core $Y$. Let $\mathcal{C}$ be the class of symmetric and alternating groups. If $\mathcal{G}^c$ is connected $H$ is $\mathcal{C}$-separable.
As $H$ acts with a proper core, there exists a generator of $C(\mathcal{G})$ not contained in $H$. Say $s_0 \notin H$.
Suppose $\gamma_1, \ldots \gamma_n \notin H$.
Fix $v \in Y$. Without loss of generality, we may assume that $Y$ contains $N(v)$ and $\gamma_i v$ for all $i$ (otherwise replace $Y$ with $N_r(Y)$ for a sufficiently large $r$). Moreover, by lemma \[reduce\] we may assume that there exists a hyperplane $H_0 \notin \mathcal{D}(Y)$ with $|H_0^- \cap Y| = 1$ and by the remark after the proof the label of $H_0$ is $s_0$.
As $\mathcal{G}^c$ is connected, there exists a generator $s_1$ not commuting with $s_0$. Let $v_0$ be the unique vertex of $H_0^- \cap Y$. Delete $k$ hyperplanes labelled alternately by $s_1$ and $s_0$ starting at $v_0$ – delete hyperplanes $H(e_{s_1}), H(s_1 e_{s_2}), H(s_1 s_2 e_{s_3}), \ldots , H(s_1 \ldots s_{k-1} e_{s_k})$ etc. where $k$ is to be specified later and $e_{s_i}$ is the edge labelled $s_i$ starting at $v_0$. Call the resulting core $Y'$.
Let $\Gamma_0$ be the group generated by reflections in hyperplanes bounding $Y'$. Let $\Gamma_1 = \langle \Gamma_0, H \rangle$. Then $[C(\mathcal{G}): \Gamma_1] = |H \setminus Y'|$, where $|H \setminus Y'|$ denotes the number of vertices of $H \setminus Y'$. A suitable choice of $k$ makes this a prime. As $\Gamma_1 \setminus C(\mathcal{G}) \cong H \setminus Y'$ and $\gamma_i v \notin H.v$, we may choose $\gamma_i$ as one of the coset representatives. In particular, $\gamma_i$ does not fix $\Gamma_1$, so it does not act as an element of $H$.
Let $s_3$ be a generator distinct from $s_1$ and $s_2$.
By the remark after lemma \[Scott\], $s_3$ fixes the cosets corresponding to the vertices of the tail. So it moves at most $|H \setminus Y|$ elements. By taking $k$ large enough so that $|H \setminus Y'|$ is still a prime, we may ensure that the conditions of Jordan’s lemma are satisfied.
Changing parity
===============
We shall now prove that we may force the action to be alternating (similarly we can force it to be symmetric).
*The parity of $s_i$ with respect to the core $Y$* is the parity of $s_i$ acting on the right cosets of $\Gamma_1 (Y)$.
We will modify the construction of the tail in order to make each $s_i$ act as an even permutation (or we will make at least one of $s_i$ acts as an odd permutation).
Suppose $g.v_0$ is in the tail. If the edge between $g.v_0$ and $gs.v_0$ is in the tail, then $g.v_0$ and $gs.v_0$ map to distinct vertices in $\Gamma_1 \setminus X$, hence $\Gamma_1 g \neq \Gamma_1 g s$.
If $gs.v_0$ is not in the tail, then the hyperplane dual to this edge bounds $Y$ and the reflection in this hyperplane belongs to $\Gamma_1$. Therefore $\Gamma_1 = \Gamma_1 g s g^{-1}$ or equivalently $\Gamma_1 g = \Gamma_1 g s$.
More precisely, suppose $H$ acts with core $Y$ and $Y'$ is the core resulting from deletion of $H_0, \ldots, H_k$, and the label of $H_i$ is $s_i$. Moreover assume $H_0 \cap Y'$ is a single edge.
Then the parity of $s_1$ with respect to $Y'$ is the sum of the parity of $s_1$ with respect to $Y$ and the number of edges labelled $s_1$ in $H_0^- \cap Y'$. So we can control parity of $s_1$ by changing the number of edges with label $s_1$ in the tail. Suppose that the conditions of Jordan’s theorem are satisfied with a margin $M$ (i.e. the conditions are satisfied even if $s_3$ moves $|H \setminus Y| + M$ elements). Taking $M = (|\mathcal{G}|-2) (2d+1)+16$, where $d$ is the diameter of $\mathcal{G}^c$ will be sufficient.
First let us show that we can deal with parity of all generators other than $s_1$ and $s_2$.
\[parity\] For any $i \in I \setminus \{1,2\}$, if the tail of $Y$ is a path with labels $s_1, s_2, \ldots s_1, s_2, s_1$ of length at least $2d_{\mathcal{G}^c}(v_1,v_i)+1$ starting at vertex $V$, then there exists a core $Y'$ such that in the associated action the parity of $s_i$ changed and the parities of no $s_j$ changed for $j \in I \setminus \{1,2,i\}$. Moreover, $|H \setminus Y| = |H \setminus Y'|$ and $Y'$ contains a tail of the same length as $Y$ and the labels of these two paths are the same with the exception of a subpath labeled $s_1, s_2, \ldots s_1, s_2, s_1$ of length $2d_{\mathcal{G}^c}(v_1,v_i)+1$.
(0,0) – (10,0) ; (2,4) – (-6,4); in [0,4,8]{} (,0) – node\[anchor=east\] [$s_1$]{} (,1); in [2,6,10]{} (,0) – node\[anchor=east\] [$s_2$]{} (, 2); in [2,-2,-6]{} (,4) – node\[anchor=east\] [$s_2$]{} (,2); in [0,-4]{} (,4) – node\[anchor=east\] [$s_1$]{} (, 3);
in [-2,2,6]{} ([-1]{},2) – ([+1]{},2); in [1,5,9]{} (,0) node\[anchor=north\] [$s_3$]{}; in [3,7]{} (,0) node\[anchor=north\] [$s_4$]{}; in [-5.5,-2.5,-1.5]{} (,2) node\[anchor=north\] [$s_5$]{}; in [1.5,2.5,5.5,6.5,9.5]{} (,2) node\[anchor=south\] [$s_5$]{}; in [-3,1]{} (,4) node\[anchor=south\] [$s_3$]{};
(0,2) node\[anchor=south west\] [$s_4$]{}; (0,2) node\[anchor=south east\] [$s_4$]{}; (0,2) node\[anchor=north west\] [$s_4$]{}; (-4,2) node\[anchor=south west\] [$s_4$]{}; (-4,2) node\[anchor=south east\] [$s_4$]{}; (4,2) node\[anchor=north west\] [$s_4$]{}; (4,2) node\[anchor=north east\] [$s_4$]{}; (8,2) node\[anchor=north west\] [$s_4$]{}; (8,2) node\[anchor=north east\] [$s_4$]{};
(-6,2) – (-5,2); (10,2) – (9,2); (-3,2) arc (0:180:1); (0,1) arc (-90:180:1); (3,2) arc (-180:0:1); (7,2) arc (-180:0:1);
Say $v_1 = v_{i_0}, v_{i_1}, \ldots v_{i_d} = v_i$ is a path in $\mathcal{G}^c$ of the shortest length. Let $Y'$ be a subcomplex built using deletions of hyperplanes $s_{i_0}, s_{i_1}, \ldots s_{i_d}, s_{i_{d-1}}, \ldots, s_{i_0}, s_2, s_1, \ldots s_1$ starting at $v$.
Compared to $Y$, the tail of this complex contains two more edges labeled $s_{j_i}$ for $0 < j < d$. It also contains an extra edge labeled $s_{i_d} = s_i$, so the parity of $s_i$ changed and the parity of other generators $s_j $ remains the same for $j \neq 1,2,i$.
Now let’s change the parity of a generator that appears in the tail.
\[squares\] If the tail of $Y$ contains a path with labels $s_1, s_2, \ldots s_1, s_2, s_1$ of length at least $7$, then there exists a core $Y'$ such that in the associated action only the parity of $s_1$ changed. Moreover, $|H \setminus Y| = |H \setminus Y'|$ and $Y'$ is built from the same complex as $Y$ using a sequence of deletions, whose labels agree with that of $Y$ with the exception of $5$ deletions. (We allow a deletion to be replaced by no deletion.)
(0,0) node\[anchor=north\] [$v_1$]{} circle (0.05); (1,0) node\[anchor=north\] [$v_2$]{} circle (0.05); (1/2,[sqrt(3)/2]{}) node\[anchor=south\] [$v_3$]{} circle (0.05); (1,0) – (1/2,[sqrt(3)/2]{});
(2,0) – node\[anchor=south east\] [$s_2$]{} ([2+sqrt(2)]{},[sqrt(2)]{}) – node\[anchor= south west\] [$s_1$]{} ([2+2\*sqrt(2)]{},0) – node\[anchor=south east\] [$s_2$]{} ([2+3\*sqrt(2)]{},[sqrt(2)]{});
(7,0) – node\[anchor=south east\] [$s_2$]{} ([7+sqrt(2)]{},[sqrt(2)]{}) – node\[anchor=south west\] [$s_3$]{} ([7+2\*sqrt(2)]{},0) – node\[anchor=north west\] [$s_2$]{} ([7+sqrt(2)]{},[-sqrt(2)]{}) – node\[anchor=north east\] [$s_3$]{} (7,0);
1. Suppose there is some $s_3$ commuting with $s_2$, but not with $s_1$. Then instead of the deletion of the hyperplanes $s_2, s_1, s_2$, delete the hyperplanes labelled $s_2, s_3$. This creates a square. Continue building the tail starting from one of the vertices of the square using the deletions of the hyperplanes with the same labels as before. The new tail contains the same number of $s_2$ labels, two more of $s_3$ and one fewer $s_1$. Hence only the parity of $s_1$ changed.
To be precise, we need to take the path labelled $s_1, s_2, s_1$ which is a subpath of a path labelled $s_2,s_1,s_2,s_1,s_2$ in the tail, as otherwise deleting a hyperplane labelled $s_3$ could potentially introduce more than just a side of a square. Similarly for the other cases in this proof.
2. Suppose there is some $s_3$ commuting with $s_1$, but not $s_2$. Then instead of the deletion of the hyperplanes labelled $s_1, s_2, s_1, s_2, s_1$, delete the hyperplanes labelled $s_1, s_3$ and then delete the hyperplanes labelled $s_2$ at two of the vertices of the square. This creates a square with two spurs. Continue building the tail starting from the remaining vertex of the square. The new tail contains the same number of $s_2$ labels, two more of $s_3$ and one fewer $s_1$. Hence only the parity of $s_1$ changed.
3. Suppose there is no generator commuting with exactly one of $s_1, s_2$. As the graph $\mathcal{G}$ is non-discrete, there is a generator commuting with both $s_1$ and $s_2$. The component of $\mathcal{G}^c$ containing $v_1$ and $v_2$ is not a proper subgraph so there exist $s_3, s_4$ such that $s_3$ commutes with both $s_1$ and $s_2$, and $s_4$ does not commute with any of the $s_1,s_2,s_3$. Now instead of the deletion of the hyperplanes labelled $s_1, s_2, s_1, s_2, s_1$, delete the hyperplanes labelled $s_4, s_1, s_3$. This creates a square with labels $s_1, s_3, s_1, s_3$. Perform deletion with respect to $s_4$ at one of the vertices. Then continue building the tail.
Let $Y'$ be the new subcomplex. By construction $|H \setminus Y| = |H \setminus Y'|$ and the sequences of labels deleted hyperplanes for the two complexes differ at no more than $5$ places.
Using lemmas $\ref{parity}$ and $\ref{squares}$, we can now modify segments of the tail to make the parity of all elements even. This completes the proof of the main theorem.
| ArXiv |
---
abstract: |
The [[[Honey-Bee]{}]{}]{} game is a two-player board game that is played on a connected hexagonal colored grid or (in a generalized setting) on a connected graph with colored nodes. In a single move, a player calls a color and thereby conquers all the nodes of that color that are adjacent to his own current territory. Both players want to conquer the majority of the nodes. We show that winning the game is PSPACE-hard in general, NP-hard on series-parallel graphs, but easy on outerplanar graphs.
In the solitaire version, the goal of the single player is to conquer the entire graph with the minimum number of moves. The solitaire version is NP-hard on trees and split graphs, but can be solved in polynomial time on co-comparability graphs.
*Keywords:* combinatorial game; computational complexity; graph problem.
author:
- 'Rudolf Fleischer [^1]'
- 'Gerhard J. Woeginger [^2]'
date:
-
-
title: |
[**An Algorithmic Analysis of the\
Honey-Bee Game**]{}[^3]
---
Introduction {#s_intro}
============
The [[[Honey-Bee]{}]{}]{} game is a popular two-player board game that shows up in many different variants and at many different places on the web (the game is best be played on a computer). For a playable version we refer the reader for instance to Axel Born’s web-page [@Bor09]; see Fig. \[fig\_born\] for a screenshot. The playing field in [[[Honey-Bee]{}]{}]{} is a grid of hexagonal honey-comb cells that come in various colors; the coloring changes from game to game. The playing field may be arbitrarily shaped and may contain holes, but must always be connected. In the beginning of the game, each player controls a single cell in some corner of the playing field. Usually, the playing area is symmetric and the two players face each other from symmetrically opposing starting cells. In every move a player may call a color $c$, and thereby gains control over all connected regions of color $c$ that have a common border with the area already under his control. The only restriction on $c$ is that it cannot be one of the two colors used by the two players in their last move before the current move, respectively. A player wins when he controls the majority of all cells. On Born’s web-page [@Bor09] one can play against a computer, choosing from four different layouts for the playing field. The computer uses a simple greedy strategy: “Always call the color $c$ that maximizes the immediate gain.” This strategy is short-sighted and not very strong, and an alert human player usually beats the computer after a few practice matches.
(28000,18000)
In this paper we perform a complexity study of the [[[Honey-Bee]{}]{}]{} game when played by two players on some arbitrary connected graph instead of the hex-grid of the original game. We will show in Section \[s\_two\] that [[[Honey-Bee-2-Players]{}]{}]{} is NP-hard even on series-parallel graphs, and that it is PSPACE-complete in general. On outerplanar graphs, however, it is quite easy to compute a winning strategy.
In the *solitaire* (single-player) version of [[[Honey-Bee]{}]{}]{} the goal is to conquer the entire playing field as quickly as possible. Intuitively, a good strategy for the solitaire game will be close to a strong heuristic for the two-player game. For the solitaire version, our results draw a sharp separation line between easy and difficult cases. In particular, we show in Section \[s\_one\] that [[[Honey-Bee-Solitaire]{}]{}]{} is NP-hard for split graphs and for trees, but polynomial-time solvable on co-comparability graphs (which include interval graphs and permutation graphs). Thus, the complexity of the game is well-characterized for the class and subclasses of perfect graphs; see Fig. \[fig\_results\] for a summary of our results.
results.tex
Definitions {#s_def}
===========
We model [[[Honey-Bee]{}]{}]{} in the following graph-theoretic setting. The playing field is a connected, simple, loopless, undirected graph $G=(V,E)$. There is a set $C$ of $k$ colors, and every node $v\in V$ is colored by some color $col(v)\in C$; we stress that this coloring does not need to be proper, that is, there may be edges $[u,v]\in E$ with $col(u)=col(v)$. For a color $c\in C$, the subset $V_c\subseteq V$ contains the nodes of color $c$. For a node $v\in V$ and a color $c\in C$, we define the *color-$c$-neighborhood* $\Gamma(v,c)$ as the set of nodes in $V_c$ either adjacent to $v$ or connected to $v$ by a path of nodes of color $c$. Similarly, we denote by $\Gamma(W,c)=\bigcup_{w\in W}\Gamma(w,c)$ the color-$c$-neighborhood of a subset $W\subseteq V$. For a subset $W\subseteq V$ and a sequence $\gamma={\langle \gamma_1,\ldots,\gamma_b\rangle}$ of colors in $C$, we define a corresponding sequence of node sets $W_1=W$ and $W_{i+1}=W_i\cup \Gamma(W_i,\gamma_i)$, for $1\le i\le b$. We say that sequence $\gamma$ started on $W$ *conquers* the final node set $W_{b+1}$ in $b$ moves, and we denote this situation by $W\to_{\gamma}W_{b+1}$. The nodes in $V-W_{b+1}$ are called *free* nodes.
In the *solitaire* version of [[Honey-Bee]{}]{}, the goal is to conquer the entire playing field with the smallest possible number of moves. Note that [[[Honey-Bee-Solitaire]{}]{}]{} is trivial in the case of only two colors. But as we will see in Section \[s\_one\], the case of three colors can already be difficult.
In the *two-player* version of [[[Honey-Bee]{}]{}]{}, the two players $A$ and $B$ start from two distinct nodes $a_0$ and $b_0$ and then extend their regions step by step by alternately calling colors. Player $A$ makes the first move. One round of the game consists of a move of $A$ followed by a move of $B$. Consider a round, where at the beginning the two players control node sets $W_A$ and $W_B$, respectively. If player $A$ calls color $c$, then he extends his region $W_A$ to $W^\prime_A=W_A\cup(\Gamma(W_A,c)-W_B)$. If afterwards player $B$ calls color $d$, then he extends his region $W_B$ to $W^\prime_B=W_B\cup(\Gamma(W_B,c)-W^\prime_A)$. Note that once a player controls a node, he can never lose it again.
The game terminates as soon as one player controls more than half of all nodes. This player wins the game. To avoid draws, we require that the number of nodes is odd. There are three important rules that constrain the colors that a player is allowed to call.
1. A player must never call the color that has just been called by the other player.
2. A player must never call the color that he has called in his previous move.
3. A player must always call a color that strictly enlarges his territory, unless rules R1 and R2 prevent him from doing so.
r2.tex
What is the motivation for these three rules? Rule R1 is a technical condition that arises from the graphical implementation [@Bor09] of the game: Whenever a player calls a color $c$, his current territory is entirely recolored to color $c$. This makes it visually easier to recognize the territories controlled by both players. Rule R2 prevents the players from permanently blocking some color for the opponent. Fig. \[fig\_rule\_R2\] shows a situation where rule R2 actually prevents the game from stalling. Rule R3 is quite delicate, and is justified by situations as depicted in Fig. \[fig\_rule\_R3\]. Rule R3 guarantees that every game must terminate with either a win for player A or a win for player B. Note that rule R2 is redundant except in the case when a player has no move to gain territory (see Fig. \[fig\_rule\_R2\].
r3.tex
Note that [[[Honey-Bee-2-Players]{}]{}]{} is trivial in the case of only three colors: The players do not have the slightest freedom in choosing their next color, and always must call the unique color allowed by rules R1 and R2. However we will see in Section \[s\_two\] that the case of four colors can already be difficult.
Finally we observe that calling a color $c$ always conquers all connected components induced by $V_c$ that are adjacent to the current territory. Hence an equivalent definition of the game could use a graph with node weights (that specify the size of the corresponding connected component) and a *proper* coloring of the nodes. Any instance under the original definition can be transformed into an equivalent instance under the new definition by contracting each connected component of $V_c$, for some $c$, into a single node of weight $|V_c|$. However, we are interested in restrictions of the game to particular graph classes, some of which are not closed under edge contractions (as for instance the hex-grid graph of the original [[[Honey-Bee]{}]{}]{} game).
The Solitaire Game {#s_one}
==================
In this section we study the complexity of finding optimally short color sequences for [[[Honey-Bee-Solitaire]{}]{}]{}. We will show that this is easy for co-comparability graphs, while it is NP-hard for trees and split graphs. Since the family of co-comparability graphs contains interval graphs, permutation graphs, and co-graphs as sub-families, our positive result for co-comparability graphs implies all other positive results in Fig. \[fig\_results\].
A first straightforward observation is that [[[Honey-Bee-Solitaire]{}]{}]{} lies in NP: Any connected graph $G=(V,E)$ can be conquered in at most $|V|$ moves, and hence such a sequence of polynomially many moves can serve as an NP-certificate.
The Solitaire Game on Co-Comparability Graphs {#sec1:cocomparability}
---------------------------------------------
A *co-comparability graph* $G=(V,E)$ is an undirected graph whose nodes $V$correspond to the elements of some partial order $<$ and whose edges $E$ connect any two elements that are incomparable in that partial order, i.e., $[u,v]\in E$ if neither $u<v$ nor $v<u$ holds. For simplicity, we identify the nodes with the elements of the partial order. Golumbic [@GoRoUr83] showed that co-comparability graphs are exactly the intersection graphs of continuous real-valued functions over some interval $I$. If two function curves intersect, the corresponding elements are incomparable in the partial order; otherwise, the curve that lies complete above the other one corresponds to the larger element in the partial order. The function graph representation readily implies that the class of co-comparability graphs is closed under edge contractions. Therefore, we may w.l.o.g. restrict our analysis of [[[Honey-Bee-Solitaire]{}]{}]{} to co-comparability graphs with a proper node coloring, i.e., adjacent nodes have distinct colors (in the solitaire game we do not care about the weight of a node after an edge contraction). In this case, every color class is totally ordered because incomparable node pairs have been contracted.
Consider an instance of [[[Honey-Bee-Solitaire]{}]{}]{} with a minimal start node $v_0$ (in the partial order on $V$); a maximal start node could be handled similarly. The function graph representation implies the following observation.
\[obs\_smaller\] Conquering a node will simultaneously conquer all smaller nodes of the same color.
For any color $c$, let $Max(c)$ denote the largest node of color $c$. By Obs. \[obs\_smaller\], it suffices to find the shortest color sequence conquering all nodes $Max(c)$, for all colors $c$. We can do that by a simple shortest path computation. We assign every node $Max(c)$ weight $0$, and all other nodes weight $1$. Then we compute a shortest path (with respect to the node-weights) from $v_0$ to every node $Max(c)$ that is a *maximal element* in the partial order (which is actually exactly the set of all maximal elements). Let $OPT$ denote the smallest cost over all such paths.
For a color sequence $\gamma={\langle \gamma_1,\ldots,\gamma_b\rangle}$, we define the *length* of $\gamma$ as $|\gamma|=b$. We also define the *essential length* $ess(\gamma)$ of $\gamma$ as $|\gamma|$ minus the number of steps where $\gamma$ conquers a maximal node $Max(c)$ of some color class $c$. Obviously, $|\gamma|=ess(\gamma)+k$. Note that $OPT$ is the minimal essential cost of any color sequence conquering one of the maximal nodes.
\[thm\_opt\] The optimal solution for [[[Honey-Bee-Solitaire]{}]{}]{} has cost $OPT+k$.
Let $\gamma$ be a shortest color sequence conquering the entire graph starting at $v_0$. After conquering $v$, $\gamma$ only needs to conquer all free nodes $Max(c)$ to conquer the entire graph. Thus, $|\gamma| = ess(\gamma) + k \ge OPT+k$.
\[th\_cocomp\] [[[Honey-Bee-Solitaire]{}]{}]{} starting at an extremal node $v_0$ can be solved in polynomial time on co-comparability graphs.
Given the co-comparability graph $G$, we can compute the underlying partial order $<$ in polynomial time [@GoRoUr83]. Assigning the weights and solving one single source shortest path problem starting at $v_0$ also takes polynomial time.
We can also formulate this algorithm as a dynamic program. For any node $v$, let $D(v)$ denote the essential length of the shortest color sequence $\gamma$ that can conquer $v$ when starting at $v_0$. For any color $c$, let $min_v(c)$ denote the smallest node of color $c$ connected to $v$, if such nodes exist. Then we can compute $D(v)$ recursively as follows: $$D(v_0) = 0$$ and $$D(v) = \min_{c} (D(min_v(c)) + \delta_v) \>,$$ where $D(min_v(c))=\infty$ if $min_v(c)$ is undefined, and $\delta_v=0$ ($1$) if $v$ is (not) a maximal node for some color class.
Clearly, this dynamic program simulates the shortest path computation of our first algorithm and we have $OPT = \min_{v}(D(v)+k)$, where we minimize over all maximal nodes $v$. We now extend the dynamic program to the case that $v_0$ is not an extremal element. The problem is that we now must extend our territory in two directions. If we choose a move that makes good progress upwards it may make little progress downwards, or vice versa. In particular, the optimal strategy cannot be decomposed into two independent optimal strategies, one conquering upwards and one conquering downwards. Analogously to the algorithm above, for a clor $c$ define $Min(c)$ as the smallest node of color $c$, and $max_v(c)$ as the largest node of color $c$ connected to a node $v$.
Unfortunately, we must now redefine the essential length of a color sequence $\gamma$. In our original definition, we did not count coloring steps that conquered maximal elements of some color class. This is intuitively justified by the fact that these steps must be done by any color sequence conquering the entire graph at some time, therefore it is advantageous to do them as early as possible (which is guaranteed by giving these moves cost 0). But now we must also consider the minimal nodes of each color class. An optimal sequence conquering the entire graph will at some time have conquered a minimal node and a maximal node. Afterwards, it will only call extremal nodes for some color class. If both extremal nodes of a color class are still free, we only need *one* move to conquer both simultaneously. If one of them had been captured earlier, we still need to conquer the other one. This indicates that we should charge 1 for the first extremal node conquered while the second one should be charged 0, as before. If both nodes are conquered in the same move, we should also charge 0. Therefore, we now define the *essential length* $ess(\gamma)$ of $\gamma$ as $|\gamma|$ minus the number of steps where $\gamma$ conquers the second extremal node of some color class.
For a node $v$ below $v_0$ or incomparable to $v_0$ and a node $w$ above $v_0$ or incomparable to $v_0$ let $D(v,w)$ denote the essential length of the shortest color sequence $\gamma$ that can conquer $v$ and $w$ when starting at $v_0$. Note that we do not need to keep track of which first extremal nodes of a color class have been conquered because we can deduce this from the two nodes $v$ and $w$ currently under consideration. In particular, we can compute $D(v,w)$ recursively as follows: $$D(v_0,v_0) = 0$$ and $$D(v,w) = \min_{c} (D(v,min_w(c)) + \delta_w(v),
D(max_v(c),w) + \delta_v(w)) \>,$$ where $\delta_v(w)=0$ if and only if $w$ is an extremal node of some color class $c$ and the other extremal node of color class $c$ is either between $v$ and $w$, or incomparable to either $v$ or $w$, or both (it was either conquered earlier, or it will be conquered in this step); otherwise, $\delta_v(w)=1$. Obviously, $|\gamma|=ess(\gamma)+k$.
\[thm\_opt\_general\] The optimal solution for [[[Honey-Bee-Solitaire]{}]{}]{} has cost $\min_{v,w}(D(v,w)+k)$, where we minimize over all minimal nodes $v$ and all maximal nodes $w$.
Let $\gamma$ be a shortest color sequence conquering the entire graph starting at $v_0$. Let $v$ be the first minimal node conquered by $\gamma$ and $w$ the first maximal node. After conquering $v$ and $w$, $\gamma$ only needs to conquer all free nodes $Max(c)$ to conquer the entire graph. Thus, $|\gamma| \ge D(v,w) + k$.
\[th\_cocomp\_general\] [[[Honey-Bee-Solitaire]{}]{}]{} can be solved in polynomial time on co-comparability graphs.
The Solitaire Game on Split Graphs {#ss_split}
----------------------------------
A *split graph* is a graph whose node set can be partitioned into an induced clique and into an induced independent set. We will show that [[[Honey-Bee-Solitaire]{}]{}]{} is NP-hard on split graphs. Our reduction is from the NP-hard [Feedback Vertex Set]{} ([[FVS]{}]{}) problem in directed graphs; see for instance Garey and Johnson [@GaJo79].
\[thm\_split\] [[[Honey-Bee-Solitaire]{}]{}]{} on split graphs is NP-hard.
Consider an instance $(X,A,t)$ of [[FVS]{}]{}. To construct an instance $(V,E,b)$ of [[Honey-Bee-Solitaire]{}]{}, we first build a clique from the nodes in $X$ together with a new node $v_0$, the start node of [[Honey-Bee-Solitaire]{}]{}, where each node $x\in X+v_0$ has a different color $c_x$. Next, we build the independent set. For every arc $(x,y)\in A$, we introduce a corresponding node $v(x,y)$ of color $c_y$ which is only connected to node $x$ in the clique, i.e., it has degree one. Finally, we set $b=|X|+t$. We claim that the constructed instance of [[[Honey-Bee-Solitaire]{}]{}]{} has answer YES, if and only if the instance of [[[FVS]{}]{}]{} has answer YES.
Assume that the [[[FVS]{}]{}]{} instance has answer YES. Let $X^\prime$ be a smallest feedback set whose removal makes $(X,A)$ acyclic. Let $\pi$ be a topological order of the nodes in $X-X^\prime$, and let $\tau$ be an arbitrary ordering of the nodes in $X^\prime$. Consider the color sequence $\gamma$ of length $|X|+t$ that starts with $\tau$, followed by $\pi$, and followed by $\tau$ again. We claim that $\{v_0\}\to_{\gamma}V$. Indeed, $\gamma$ first runs through $\tau$ and $\pi$ and thereby conquers all clique nodes. Every independent set node $v(x,y)$ with $y\in X^\prime$ is conquered during the second transversal of $\tau$. Every independent set node $v(x,y)$ with $y\in X-X^\prime$ is conquered during the transversal of $\pi$, since $\pi$ first conquers $x$ with color $c_x$, and afterwards $v(x,y)$ with color $y$.
Next assume that the instance of [[[Honey-Bee-Solitaire]{}]{}]{} has answer YES. Let $\gamma$ be a color sequence of length $b=|X|+t$ conquering $V$. Define $X^\prime$ as the set of nodes $x$ such that color $c_x$ occurs at least twice in $\gamma$; clearly, $|X^\prime|\le t$. Consider an arc $(x,y)\in A$ with $x,y\in X-X^\prime$. Since $\gamma$ contains color $c_y$ only once, it must conquer node $v(x,y)$ of color $c_y$ after node $v(x)$ of color $c_x$. Hence, $\gamma$ induces a topological order of $X-X^\prime$.
The construction in the proof above uses linearly many colors. What about the case of few colors? On split graphs, [[[Honey-Bee-Solitaire]{}]{}]{} can always be solved by traversing the color set $C$ twice; the first traversal conquers all clique nodes, and the second traversal conquers all remaining free independent set nodes. Thus, every split graph can be completely conquered in at most $2|C|$ steps. If there are only few colors, we can simply check all color sequences of this length $2|C|$.
\[thm\_split\_const\] If the number of colors is bounded by a fixed constant, [[[Honey-Bee-Solitaire]{}]{}]{} on split graphs is polynomial-time solvable.
The Solitaire Game on Trees {#ss_tree}
---------------------------
In this section we will show that [[[Honey-Bee-Solitaire]{}]{}]{} is NP-hard on trees, even if there are at only three colors. We reduce [[[Honey-Bee-Solitaire]{}]{}]{} from a variant of the [Shortest Common Supersequence]{} ([[SCS]{}]{}) problem which is know to be NP-complete (see Middendorf [@Mid94]).
Middendorf’s hardness result also implies the hardness of the following variant of [[SCS]{}]{}:
\[thm\_mscs\] [[[MSCS]{}]{}]{} is NP-complete.
Here is a reduction from [[[SCS]{}]{}]{} to [[[MSCS]{}]{}]{}. Consider an arbitrary sequence $\tau$ with elements from $\{0,1\}$. We define $f(\tau)$ as the sequence we obtain from replacing every occurrence of the element 0 in $\tau$ by two consecutive elements 0 and 2. Now consider an instance $(\sigma_1,\ldots,\sigma_s,t)$ of [[SCS]{}]{}. We construct an instance $(\sigma_1^\prime,\ldots,\sigma_s^\prime,t^\prime)$ of [[[MSCS]{}]{}]{} by setting $\sigma^\prime_i=f(\sigma_i)$, for $1\le i\le s$. Then, for any sequence $\sigma$ with elements from $\{0,1\}$, $\sigma$ is a common supersequence of $\sigma_1,\ldots,\sigma_s$ if and only if $f(\sigma)$ is a common supersequence of $\sigma^\prime_1,\ldots,\sigma^\prime_s$. This implies the NP-hardness of [[MSCS]{}]{}.
\[thm\_tree\] [[[Honey-Bee-Solitaire]{}]{}]{} is NP-hard on trees, even in case of only three colors.
We reduce [[[MSCS]{}]{}]{} to [[[Honey-Bee-Solitaire]{}]{}]{} on trees. Consider an instance $(\sigma_1,\ldots,\sigma_s,t)$ of [[MSCS]{}]{}. We use color set $C=\{0,1,2\}$. We first construct a root $v_0$ of color $2$. Then we attach a path of length $|\sigma_i|$ to $v_0$ for each sequence $\sigma_i$, where an element $j$ is colored $j$. See the left half of Fig. \[fig\_sp\] for an example. Finally, we set $b=t$. It its straightforward to see that the constructed instance of [[[Honey-Bee-Solitaire]{}]{}]{} has answer YES if and only if the instance of [[[MSCS]{}]{}]{} has answer YES.
The Two-Player Game {#s_two}
===================
In this section we study the complexity of the two-player game. While on outerplanar graphs the players can compute their winning strategies in polynomial time, this problem is NP-hard for series-parallel graphs with four colors, and PSPACE-complete with four colors on arbitrary graphs. Our positive result for outerplanar graphs works for an arbitrary number of colors. Our negative results work for four colors, which is the strongest possible type of result (recall that instances with three colors are trivial to solve).
The Two-Player Game on Outer-Planar Graphs {#ss_outerplanar}
------------------------------------------
A graph is *outer-planar* if it contains neither $K_4$ nor $K_{2,3}$ as a minor. Outer-planar graphs have a planar embedding in which every node lies on the boundary of the so-called *outer face*. For example, every tree is an outer-planar graph.
Consider an outer-planar graph $G=(V,E)$ as an instance of [[[Honey-Bee-2-Players]{}]{}]{} with starting nodes $a_0$ and $b_0$ in $V$, respectively. The starting nodes divide the nodes on the boundary of the outer face $F$ into an upper chain $u_1,\ldots,u_s$ and a lower chain $\ell_1,\ldots,\ell_t$, where $u_1$ and $\ell_1$ are the two neighbors of $a_0$ on $F$, while $u_s$ and $\ell_t$ are the two neighbors of $b_0$ on $F$. We stress that this upper and lower chain are not necessarily disjoint (for instance, articulation nodes will occur in both chains).
Now consider an arbitrary situation in the middle of the game. Let $U$ (respectively $L$) denote the largest index $k$ such that player $A$ has conquered node $u_k$ (respectively node $\ell_k$). See Fig. \[fig\_outerplanar\] to illustrate these definitions and the following lemma.
\[thm\_outerplanar\_conquer\] Let $X$ denote the set of nodes among $u_1,\ldots,u_U$ and $\ell_1,\ldots,\ell_L$ that currently do neither belong to $A$ nor to $B$. Then no node in $X$ can have a neighbor among $u_{U+1},\ldots,u_s,b_0,\ell_t,\ldots,\ell_{L+1}$.
The existence of such a node in $X$ would lead to a $K_4$-minor in the outer-planar graph.
outerplanar.tex
\[thm\_outerplanar\] [[[Honey-Bee-2-Players]{}]{}]{} on outer-planar graphs is polynomial-time solvable.
The two indices $U$ and $L$ encode all necessary information on the future behavior of player $A$. Eventually, he will own all nodes $u_1,\ldots,u_U$ and $\ell_1,\ldots,\ell_L$, and the possible future expansions of his area beyond $u_U$ and $\ell_L$ only depend on $U$ and $L$. Symmetric observations hold true for player $B$.
As every game situation can be concisely described by just four indices, there is only a polynomial number $O(|V|^4)$ of relevant game situations. The rest is routine work in combinatorial game theory: We first determine the winner for every end-situation, and then by working backwards in time we can determine the winners for the remaining game situations.
The Two-Player Game on Series-Parallel Graphs {#ss_sp}
---------------------------------------------
A graph is *series-parallel* if it does not contain $K_4$ as a minor. Equivalently, a series-parallel graph can be constructed from a single edge by repeatedly doubling edges, or removing edges, or replacing edges by a path of two edges with a new node in the middle of the path. We stress that we do not know whether the two-player game on series-parallel graphs is contained in the class NP (and we actually see no reason why it should lie in NP); therefore the following theorem only states NP-hardness.
\[thm\_sp\] For four (or more) colors, problem [[[Honey-Bee-2-Players]{}]{}]{} on series-parallel graphs is NP-hard.
We use the color set $C=\{0,1,2,3\}$. A central feature of our construction is that player $B$ will have no real decision power, but will only follow the moves of player $A$: If player $A$ starts a round by calling color $0$ or $1$, then player $B$ must follow by calling the other color in $\{0,1\}$ (or waste his move). And if player $A$ starts a round by calling color $2$ or $3$, then player $B$ must call the other color in $\{2,3\}$ (or waste his move). In the even rounds the players will call the colors in $\{0,1\}$ and in the odd rounds they will call the colors in $\{2,3\}$. Both players are competing for a set of honey pots in the middle of the battlefield, and need to get there as quickly as possible. If a player deviates from the even-odd pattern indicated above, he might perhaps waste his move and delay the game by one round (in which neither player comes closer to the honey pots), but this remains without further impact on the outcome of the game.
The proof is by reduction from the supersequence problem [[[SCS]{}]{}]{} with binary sequences; see Section \[ss\_tree\]. Consider an instance $(\sigma_1,\ldots,\sigma_s,t)$ of [[[SCS]{}]{}]{}, and let $n$ denote the common length of all sequences $\sigma_i$. We first construct two start nodes $a_0$ and $b_0$ of colors $2$ and $3$, respectively. For each sequence $\sigma_i$ with $1\le i\le s$ we do the following:
- We construct a path $P_i$ that consists of $2n-1$ nodes and that is attached to $a_0$: The $n$ nodes with odd numbers mimic sequence $\sigma_i$, while the $n-1$ nodes with even numbers along the path all receive color $2$. The first node of $P_i$ is adjacent to $a_0$, and its last node is connected to a so-called honey pot $H_i$.
- The *honey pot* $H_i$ is a long path consisting of $4st$ nodes of color $3$. Intuitively, we may think of a honey pot as a single node of large weight, because conquering one of the nodes will simultaneously conquer the entire path.
- Every honey pot $H_i$ can also be reached from $b_0$ by another path $Q_i$ that consists of $2t-1$ nodes. Nodes with odd numbers get color $0$, and nodes with even numbers get color $3$. The first node of $Q_i$ is adjacent to $b_0$, and its last node is connected to $H_i$. Furthermore, we create for each odd-numbered node (of color $0$) a new twin node of color $1$ that has the same two neighbors as the color $0$ node. Note that for every path $Q_i$ there are $t$ twin pairs.
Finally we create a private honey pot $H_B$ for player $B$, that is connected to node $b_0$ and that consists of $4s(s-1)t+(2n-1)s$ nodes of color $2$. This completes the construction; see Fig. \[fig\_sp\] for an example.
Assume that the [[[SCS]{}]{}]{} instance has answer YES. During his first $2t-1$ steps, player $B$ can only conquer the paths $Q_i$ and his private honey pot $H_B$. At the same time, player $A$ can conquer all paths $P_i$ by calling color $2$ in his even moves and by following a shortest 0-1 supersequence in his odd moves. Then, in round $2t$ player $A$ will simultaneously conquer all the honey pots $H_i$ with $1\le i\le s$. This gives $A$ a territory of at least $1+(2n-1)s+4s^2t$ nodes, and $B$ a smaller territory of at most $1+(3t-1)s+4s(s-1)t+(2n-1)s$ nodes. Hence $A$ can enforce a win.
Next assume that player $A$ has a winning strategy. Player $B$ can always conquer his starting node $b_0$ and his private honey pot $H_B$. If $B$ also manages to conquer one of the pots $H_i$, then he gets a territory of at least $1+4s(s-1)t+(2n-1)s+4st$ nodes and surely wins the game. Hence player $A$ can only win if he conquers all $s$ honey pots $H_i$. To reach them before player $B$ does, player $A$ must conquer them within his first $2t$ moves. In every odd round, player $A$ will call a color $0$ or $1$ and player $B$ will call the other color in $\{0,1\}$. Hence, in the even rounds, colors $0$ and $1$ are forbidden for player $A$, and the only reasonable move is to call color $2$. Note that the slightest deviation of these forced moves would give player $B$ a deadly advantage. In order to win, the odd moves of player $A$ must induce a supersequence of length at most $t$ for all sequences $\sigma_i$. Therefore, the [[[SCS]{}]{}]{} instance has answer YES.
sp.tex
The Two-Player Game on Arbitrary Graphs {#ss_pspace}
---------------------------------------
In this section we will show that problem [[[Honey-Bee-2-Players]{}]{}]{} is PSPACE-complete on arbitrary graphs. Our reduction is from the PSPACE-complete [Quantified Boolean Formula]{} ([[QBF]{}]{}) problem; see for instance Garey & Johnson [@GaJo79].
\[thm\_pspace\] For four (or more) colors, problem [[[Honey-Bee-2-Players]{}]{}]{} on arbitrary graphs is PSPACE-complete.
We reduce from [[[QBF]{}]{}]{}. Let $F=\exists x_1\forall x_2\cdots\exists x_{2n-1}\forall x_{2n}
\bigwedge_j C_j$ be an instance of [[QBF]{}]{}. We construct a bee graph $G_F=(V,E)$ with four colors (white, light-gray, dark-gray, and black) such that player $A$ has a winning strategy if and only if $F$ is true. Let $a_0$ (colored light-gray) and $b_0$ (colored dark-gray) denote the start nodes of players $A$ and $B$, respectively.
Each player controls a *pseudo-path*, that is, a path where some nodes may be duplicated as parallel nodes in a diamond-shaped structure; see Fig. \[fig\_var\]. A so-called *choice pair* consists of a node on a pseudo-path together with some duplicated node in parallel. The start nodes are at one end of the respective pseudo-paths, and the players can conquer the nodes on their own path without interference from the other player. However, they must do so in a timely manner because either path ends at a humongous *honey pot*, denoted respectively by $H_A$ and $H_B$. A honey pot is a large clique of identically-colored nodes (we may think of it as a single node of large weight, because conquering one node will simultaneously conquer the entire clique). Both honey pots have the same size but different colors, namely black ($H_A$) and white ($H_B$), and they are connected to each other by an edge. Consequently, both players must rush along their pseudo-paths as quickly as possible to reach their honey pot before the opponent can reach it and to prevent the opponent from winning by conquering both honey pots. The last nodes before the honey pots are denoted by $a_f$ and $b_f$, respectively. They separate the last variable gadgets (described below) from the honey pots.
var.tex
Fig. \[fig\_var\] shows an overview of the pseudo-paths and one *variable gadget* in detail. A variable gadget is a part of the two pseudo-paths corresponding to a pair of variables $\exists x_{2i-1} \forall x_{2i}$, for some $i\ge1$. For player $A$, the gadget starts at node $a_{i-1}$ with a choice pair $a_{2i-1}^F$ and $a_{2i-1}^T$, colored white and black, respectively. The first node conquered by $A$ will determine the truth value for variable $x_{2i-1}$. In the same round, player $B$ has a choice on his pseudo-path $P_B$ between nodes $b_{2i-1}^F$ and $b_{2i-1}^T$. Since these nodes have the same color as $A$’s choices in the same round, $B$ actually does not have a choice but must select the other color not chosen by $A$.
Three rounds later, player $B$ has a choice pair $b_{2i}^F$ and $b_{2i}^T$, assigning a truth value to variable $x_{2i}$. In the next step (which is in the next round), player $A$ has a choice pair $a_{2i}^F$ and $a_{2i}^T$ with the same colors as $B$’s choice pair for $x_{2i}$. Again, this means that $A$ does not really have a choice but must select the color not chosen by $B$ in the previous step. Since we want $A$ to conquer those clauses containing a literal set to true by player $B$, the colors in $B$’s choice pair have been switched, i.e., $b_{2i}^F$ is black and $b_{2i}^T$ is white.
Note that all the nodes $a_0,a_1,\ldots,a_n$ are light-gray and all the nodes $b_0,b_1,\ldots,b_n$ are dark-gray. This allows us to concatenate as many variable gadgets as needed. Further note that $a_f$ is white, while $b_f$ is light-gray.
The *clause* gadgets are very simple. Each clause $C_j$ corresponds to a small honey pot $H_j$ of color white. The size of the small honey pots is smaller than the size of the large honey pots $H_A$ and $H_B$, but large enough such that player $A$ loses if he misses one of them. Player $A$ should conquer $H_j$ if and only if $C_j$ is true in the assignment chosen by the players while conquering their respective pseudo-paths. We could connect $a_{2i-1}^T$ directly with $H_j$ if $C_j$ contains literal $x_{2i-1}$, however then player $A$ could in subsequent rounds shortcut his pseudo-path by entering variable gadgets for the other variables in $C_j$ from $H_j$. To prevent this from happening, we place waiting gadgets between the variable gadgets and the clauses.
Let $a_{k}^\star$ denote the node on $P_A$ right after the choice pair $a_k^F$ and $a_k^T$, for $k=1,\ldots,2n$; similarly, $b_k^\star$ are the nodes on $P_B$ right after $B$’s choice pairs. A *waiting gadget* $W_k$ consists of two copies $W_k^F$ and $W_k^T$ of the sub-path of $P_A$ starting at $a_k^\star$ and ending at $a_n$, see Fig. \[fig\_wait\]. If clause $C_j$ contains literal $x_k$, $H_j$ is connected to the node $w_n^T$ corresponding to $a_n$ in $W_k^T$; if $C_j$ contains literal $\overline{x_k}$, $H_j$ is connected to the node $w_n^F$ corresponding to $a_n$ in $W_k^F$. If $k=2i-1$ (i.e., we have an existential variable $x_{2i-1}$ whose value is assigned by player $A$), then $a_{2i-1}^F$ and $b_{2i-1}^F$ are connected to $w_{2i-1}^{\star F}$, and $a_{2i-1}^T$ and $b_{2i-1}^T$ are connected to $w_{2i-1}^{\star T}$. If $k=2i$ (i.e., we have a universal variable $x_{2i}$ whose value is assigned by player $B$), then $a_{2i}^F$ and $b_{2i}^\star$ are connected to $w_{2i}^{\star F}$, and $a_{2i}^T$ and $b_{2i-1}^\star$ are connected to $w_{2i}^{\star T}$.
wait.tex
Finally, we connect $b_f$ with all clause honey pots $H_j$ to give player $B$ the opportunity to conquer all those clauses that contain no true literal. This completes the construction of $G_F$. Fig. \[fig\_example\] shows the complete graph $G_F$ for a small example formula $F$.
We claim that player $A$ has a winning strategy on $G_F$ if and only if formula $F$ is true. It is easy to verify that player $A$ can indeed win if $F$ is true. All he has to do is to conquer those nodes in his existential choice pairs corresponding to the variable values in a satisfying assignment for $F$. For the existential variables, he has full control to select any value, and for the universal variables he must pick the opposite color as selected by player $B$ in the previous step, which corresponds to setting the variable to exactly the value that player $B$ has selected. Hence player $B$ can block a move of player $A$ by appropriately selecting a value for a universal variable. Note that no other blocking moves of player $B$ are advantageous: If $B$ blocks $A$’s next move by choosing a color that does not make progress on his own pseudo-path, then $A$ will simply make an arbitrary waiting move and then in the next round $B$ cannot block $A$ again. When player $A$ conquers node $a_n$, he will simultaneously conquer the last nodes in all waiting gadgets corresponding to true literals. Since every clause contains a true literal for a satisfying assignment, player $A$ can then in the next round conquer $a_f$ together with all clause honey pots (which all have color white). Player $B$ will respond by conquering $b_f$, and the game ends with both players conquering their own large honey pots $H_A$ and $H_B$, respectively. Since player $A$ got all clause honey pots, he wins.
To make this argument work, we must carefully chose the sizes of the honey pots. Each pseudo-path contains $9n+1$ nodes, of which at most $n$ can be conquered by the other player. The waiting gadgets contain two paths of length $9k+6$ for existential variables and $9k+1$ for universal variables. At the end, player $A$ will have conquered one of the two paths completely and maybe some parts of the sibling path, that is, we do not know exactly the final owner of less than $n^2$ nodes. The clause honey pots should be large enough to absorb this fuzzyness, which means it is sufficient to give them $2n^2$ nodes. The honey pots $H_A$ and $H_B$ should be large enough to punish any foul play by the players, that is, when they do not strictly follow their pseudo-paths. It is sufficient to give them $2n^3$ nodes.
To see that $F$ is true if player $A$ has a winning strategy note that player $A$ must strictly follow his pseudo-path, as otherwise player $B$ could beat him by reaching the large honey pots first. Thus player $A$’s strategy induces a truth assignment for the existential variables. Similarly, player $B$’s strategy induces a truth assignment for the universal variables. Player $A$ can only win if he also conquers all clause honey pots, and hence the players must haven chosen truth values that make at least one literal per clause true. This means that formula $F$ is satisfiable.
Conclusions {#s_conclusion}
===========
We have modeled the Honey Bee game as a combinatorial game on colored graphs. For the solitaire version, we have analyzed the complexity on many classes of perfect graphs. For the two player version, we have shown that even the highly restricted case of series-parallel graphs is hard to tackle. Our results draw a clear separating line between easy and hard variants of these problems.
Acknowledgements
================
Part of this research was done while G. Woeginger visited Fudan University in 2009.
[4]{}
A. Born. Flash application for the computer game *“Biene” (Honey-Bee)*, 2009.\
<http://www.ursulinen.asn-graz.ac.at/Bugs/htm/games/biene.htm>.
M. R. Garey and D. S. Johnson. . W. H. Freeman and Company, New York, 1979.
M. C. Golumbic, D. Rotem, and J. Urrutia. Comparability graphs and intersection graphs. , 43(1):37–46, 1983.
M. Middendorf. More on the complexity of common superstring and supersequence problems. , 125:205–228, 1994.
[^1]: School of Computer Science, IIPL, Fudan University, Shanghai 200433, China. Email: [[email protected]]{}.
[^2]: [[email protected]]{}. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, Netherlands.
[^3]: RF acknowledges support by the National Natural Science Foundation of China (No. 60973026), the Shanghai Leading Academic Discipline Project (project number B114), the Shanghai Committee of Science and Technology of China (09DZ2272800), and the Robert Bosch Foundation (Science Bridge China 32.5.8003.0040.0). GJW acknowledges support by the Netherlands Organisation for Scientific Research (NWO grant 639.033.403), and by BSIK grant 03018 (BRICKS: Basic Research in Informatics for Creating the Knowledge Society).
| ArXiv |
=1
Introduction
============
We recently introduced a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs and studied the associated discrete integrable systems [@f14-3; @f17-2]. Many well known examples belong to that scheme for $N=2$, so, for $N\geq 3$, some of our systems may be regarded as generalisations of these.
In this paper we give a short review of our considerations and discuss the general framework for the derivation of continuous flows compatible with our discrete Lax pairs. These derivations lead to differential-difference equations which define generalised symmetries of our systems [@f14-3]. Here we are interested in the particular subclass of self-dual discrete integrable systems, which exist only for $N$ odd [@f17-2], and derive their lowest order generalised symmetries which are of order one. We also derive corresponding master symmetries which allow us to construct infinite hierarchies of symmetries of increasing orders.
These self-dual systems also have the interesting property that they can be [*reduced*]{} from $N-1$ to $\frac{N-1}{2}$ components, still with an $N\times N$ Lax pair. However not all symmetries of our original systems are compatible with this reduction. From the infinite hierarchies of generalised symmetries only the even indexed ones are reduced to corresponding symmetries of the reduced systems. Thus the lowest order symmetries of our reduced systems are of order two.
Another interesting property of these differential-difference equations is that they can be brought to a polynomial form through a Miura transformation. In the lowest dimensional case ($N=3$) this polynomial equation is directly related to the Bogoyavlensky equation (see (\[eq:Bog\]) below), whilst the higher dimensional cases can be regarded as multicomponent generalisations of the Bogoyavlensky lattice.
Our paper is organised as follows. Section \[sec:ZN-LP\] contains a short review of our framework, the fully discrete Lax pairs along with the corresponding systems of difference equations, and Section \[continuous-defs\] discusses continuous flows and symmetries. The following section discusses the self-dual case and the reduction of these systems. It also presents the systems and corresponding reductions for $N=3$, $5$ and $7$. Section \[sec:Miura\] presents the Miura transformations for the reduced systems in $N=3$ and $5$, and discusses the general formulation of these transformations for any dimension $N$.
$\boldsymbol{{\mathbb{Z}}_N}$-graded Lax pairs {#sec:ZN-LP}
==============================================
We now consider the specific discrete Lax pairs, which we introduced in [@f14-3; @f17-2]. Consider a pair of matrix equations of the form
\[eq:dLP-gen\] $$\begin{gathered}
\Psi_{m+1,n} = L_{m,n} \Psi_{m,n} \equiv \big( U_{m,n} + \lambda \Omega^{\ell_1}\big) \Psi_{m,n}, \label{eq:dLP-gen-L} \\
\Psi_{m,n+1} = M_{m,n} \Psi_{m,n} \equiv \big( V_{m,n} + \lambda \Omega^{\ell_2}\big) \Psi_{m,n}, \label{eq:dLP-gen-M}\end{gathered}$$ where $$\begin{gathered}
\label{eq:A-B-entries}
U_{m,n} = \operatorname{diag}\big(u^{(0)}_{m,n},\dots,u^{(N-1)}_{m,n}\big) \Omega^{k_1},\qquad
V_{m,n} = \operatorname{diag}\big(v^{(0)}_{m,n},\dots,v^{(N-1)}_{m,n}\big) \Omega^{k_2},\end{gathered}$$ and $$\begin{gathered}
(\Omega)_{i,j} = \delta_{j-i,1} + \delta_{i-j,N-1}.\end{gathered}$$
The matrix $\Omega$ defines a grading and the four matrices of (\[eq:dLP-gen\]) are said to be of respective levels $k_i$, $\ell_i$, with $\ell_i\neq k_i$ (for each $i$). The Lax pair is characterised by the quadruple $(k_1,\ell_1;k_2,\ell_2)$, which we refer to as [*the level structure*]{} of the system, and for consistency, we require $$\begin{gathered}
k_1 + \ell_2 \equiv k_2 + \ell_1 \quad (\bmod N).\end{gathered}$$ Since matrices $U$, $V$ and $\Omega$ are independent of $\lambda$, the compatibility condition of (\[eq:dLP-gen\]), $$\begin{gathered}
L_{m,n+1} M_{m,n} = M_{m+1,n} L_{m,n},\end{gathered}$$ splits into the system
\[eq:dLP-gen-scc\] $$\begin{gathered}
U_{m,n+1} V_{m,n} = V_{m+1,n} U_{m,n} , \label{eq:dLP-gen-scc-1}\\
U_{m,n+1} \Omega^{\ell_2} - \Omega^{\ell_2} U_{m,n} = V_{m+1,n} \Omega^{\ell_1} - \Omega^{\ell_1} V_{m,n}, \label{eq:dLP-gen-scc-2}\end{gathered}$$
which can be written explicitly as
\[eq:dLP-ex-cc\] $$\begin{gathered}
u^{(i)}_{m,n+1} v_{m,n}^{(i+k_1)} = v^{(i)}_{m+1,n} u^{(i+k_2)}_{m,n} , \label{eq:dLP-ex-cc-1}\\
u^{(i)}_{m,n+1} - u_{m,n}^{(i+\ell_2)} = v^{(i)}_{m+1,n} - v^{(i+\ell_1)}_{m,n} , \label{eq:dLP-ex-cc-2}\end{gathered}$$
for $i \in {\mathbb{Z}}_N$.
Quotient potentials {#sect:coprime-quotient}
-------------------
Equations (\[eq:dLP-ex-cc-1\]) hold identically if we set $$\begin{gathered}
\label{eq:dLP-gen-ph-1}
u^{(i)}_{m,n} = \alpha \frac{\phi^{(i)}_{m+1,n}}{\phi^{(i+k_1)}_{m,n}} ,\qquad v^{(i)}_{m,n} = \beta \frac{\phi^{(i)}_{m,n+1}}{\phi^{(i+k_2)}_{m,n}} ,\qquad i \in {\mathbb{Z}}_N,\end{gathered}$$ after which (\[eq:dLP-ex-cc-2\]) takes the form $$\begin{gathered}
\label{eq:dLP-gen-sys-1}
\alpha \left(\frac{\phi^{(i)}_{m+1,n+1}}{\phi^{(i+k_1)}_{m,n+1}} - \frac{\phi^{(i+\ell_2)}_{m+1,n}}{\phi^{(i+\ell_2+k_1)}_{m,n}} \right) =
\beta \left(\frac{\phi^{(i)}_{m+1,n+1}}{\phi^{(i+k_2)}_{m+1,n}} - \frac{\phi^{(i+\ell_1)}_{m,n+1}}{\phi^{(i+\ell_1+k_2)}_{m,n}} \right) ,
\qquad i \in {\mathbb{Z}}_N,\end{gathered}$$ defined on a square lattice. These equations can be explicitly solved for the variables on any of the four vertices and, in particular, $$\begin{gathered}
\label{eq:dLP-gen-sys-1-a}
\phi^{(i)}_{m+1,n+1} = \frac{\phi_{m,n+1}^{(i+k_1)} \phi_{m+1,n}^{(i+k_2)}}{\phi_{m,n}^{(i+k_1+\ell_2)}} \left(
\frac{\alpha \phi_{m+1,n}^{(i+\ell_2)}- \beta \phi_{m,n+1}^{(i+\ell_1)}}{\alpha \phi_{m+1,n}^{(i+k_2)}- \beta \phi_{m,n+1}^{(i+k_1)}}
\right) ,\qquad i \in {\mathbb{Z}}_N.\end{gathered}$$ In this potential form, the Lax pair (\[eq:dLP-gen\]) can be written
\[eq:LP-ir-g-rat\] $$\begin{gathered}
\Psi_{m+1,n} = \big( \alpha {\boldsymbol{\phi}}_{m+1,n} \Omega^{k_1} {\boldsymbol{\phi}}_{m,n}^{-1} + \lambda \Omega^{\ell_1}\big) \Psi_{m,n},\nonumber\\
\Psi_{m,n+1} = \big( \beta {\boldsymbol{\phi}}_{m,n+1} \Omega^{k_2} {\boldsymbol{\phi}}_{m,n}^{-1} + \lambda \Omega^{\ell_2}\big) \Psi_{m,n}, \label{eq:LP-ir-g-rat-1}\end{gathered}$$ where $$\begin{gathered}
\label{eq:LP-ir-g-rat-2}
{\boldsymbol{\phi}}_{m,n} := \operatorname{diag}\big(\phi^{(0)}_{m,n},\dots,\phi^{(N-1)}_{m,n}\big) \qquad {\mbox{and}} \qquad \det\left({\boldsymbol{\phi}}_{m,n}\right) = \prod_{i=0}^{N-1}\phi^{(i)}_{m,n}=1.\end{gathered}$$
We can then show that the Lax pair (\[eq:LP-ir-g-rat\]) is compatible if and only if the system (\[eq:dLP-gen-sys-1\]) holds.
Differential-difference equations as symmetries {#continuous-defs}
===============================================
Here we briefly outline the construction of [*continuous*]{} isospectral flows of the Lax equations (\[eq:dLP-gen\]), since these define continuous symmetries for the systems (\[eq:dLP-ex-cc\]). The most important formula for us is (\[eq:phi-sys-sym\]), which gives the explicit form of the symmetries in potential form.
We seek continuous time evolutions of the form $$\begin{gathered}
\label{psit}
\partial_{t} \Psi_{m,n} = S_{m,n} \Psi_{m,n},\end{gathered}$$ which are compatible with each of the discrete shifts defined by (\[eq:dLP-gen\]), if $$\begin{gathered}
\partial_t L_{m,n} = S_{m+1,n} L_{m,n} - L_{m,n}S_{m,n}, \nonumber\\
\partial_t M_{m,n} = S_{m,n+1} M_{m,n} - M_{m,n}S_{m,n}. $$ Since $$\begin{gathered}
\partial_t (L_{m,n+1} M_{m,n} - M_{m+1,n} L_{m,n}) \\
\qquad {}= S_{m+1,n+1} (L_{m,n+1} M_{m,n} - M_{m+1,n} L_{m,n}) - (L_{m,n+1} M_{m,n} - M_{m+1,n} L_{m,n}) S_{m,n},\end{gathered}$$ we have compatibility on solutions of the fully discrete system (\[eq:dLP-gen-scc\]).
If we define $S_{mn}$ by $$\begin{gathered}
\label{Q=LS}
S_{m,n} = L_{m,n}^{-1}Q_{m,n},\qquad\mbox{where}\quad Q_{m,n}=\operatorname{diag}\big(q^{(0)}_{m,n},q^{(1)}_{m,n},\dots ,q^{(N-1)}_{m,n}\big) \Omega^{k_1},\end{gathered}$$ then $$\begin{gathered}
Q_{m,n}U_{m-1,n} - U_{m,n} \Omega^{-\ell_1} Q_{m,n}\Omega^{\ell_1} = 0, \qquad \partial_{t} U_{m,n} = \Omega^{-\ell_1} Q_{m+1,n}\Omega^{\ell_1} - Q_{m,n},\end{gathered}$$ which are written explicitly as
\[X1\] $$\begin{gathered}
q^{(i)}_{m,n} u^{(i+k_1)}_{m-1,n} = u^{(i)}_{m,n} q^{(i+k_1-\ell_1)}_{m,n}, \label{q-eqs} \\
\partial_t u^{(i)}_{m,n} = q^{(i-\ell_1)}_{m+1,n} - q^{(i)}_{m,n}. \label{eq:gen-eq-sym}\end{gathered}$$ It is also possible to prove (see [@f14-3]) that $$\begin{gathered}
\label{tracecon1}
\sum_{i=0}^{N-1} \frac{q_{m,n}^{(i)}}{u_{m,n}^{(i)}} = \frac{1}{\alpha^N},\end{gathered}$$
for an autonomous symmetry.
Equations (\[q-eqs\]) and (\[tracecon1\]), fully determine the functions $q^{(i)}_{m,n}$ in terms of ${\bf u}_{m,n}$ and ${\bf u}_{m-1,n}$.
The formula (\[Q=LS\]) defines a symmetry which (before prolongation) only involves shifts in the $m$-direction. There is an analogous symmetry in the $n$-direction, defined by $$\begin{gathered}
\label{n-sym}
\partial_s \Psi_{m,n} = \big(V_{m,n}+\lambda \Omega^{\ell_2}\big)^{-1} R_{m,n} \Psi_{m,n},\qquad\mbox{with}\quad R_{m,n} = \operatorname{diag}\big(r^{(0)}_{m,n},\dots,r^{(N-1)}_{m,n}\big) \Omega^{k_2}.\!\!\!\end{gathered}$$
### Master symmetry and the hierarchy of symmetries {#master-symmetry-and-the-hierarchy-of-symmetries .unnumbered}
The vector field $X^M$, defined by the evolution $$\begin{gathered}
\partial_{\tau} u^{(i)}_{m,n} = (m+1) q^{(i-\ell)}_{m+1,n} - m q^{(i)}_{m,n},\qquad \partial_\tau \alpha = 1/\big(N \alpha^{N-1}\big),\end{gathered}$$ with $q^{(i)}$ being the solution of (\[q-eqs\]) and (\[tracecon1\]), is a [*master symmetry*]{} of $X^1$, satisfying $$\begin{gathered}
\big[\big[X^M,X^1\big],X^1\big]=0, \qquad\mbox{with}\quad \big[X^M,X^1\big]\neq 0.\end{gathered}$$ We then define $X^k$ recursively by $X^{k+1}=[X^M,X^k]$. We have
Given the sequence of vector fields $X^k$, defined above, we suppose that, for some $\ell\ge 2$, $\{X^1,\dots ,X^\ell\}$ pairwise commute. Then $[X^i,X^{\ell+1}]=0$, for $1\leq i\leq \ell-1$.
This follows from an application of the Jacobi identity.
We [*cannot*]{} deduce that $[[X^M,X^\ell],X^\ell]=0$ by using the Jacobi identity. Since we are [*given*]{} this equality for $\ell=2$, we [*can*]{} deduce that $[X^1,X^3]=0$ (see the discussion around Theorem 19 of [@Y]). Nevertheless it [*is*]{} possible to check this by hand for low values of $\ell$, for all the examples given in this paper.
Symmetries in potentials variables
----------------------------------
If we write (\[X1\]) and the corresponding $n$-direction symmetry in the potential variables (\[eq:dLP-gen-ph-1\]), we obtain
\[eq:phi-sys-sym\] $$\begin{gathered}
\partial_t \phi^{(i)}_{m,n} = \alpha^{-1} q^{(i-\ell_1)}_{m,n} \phi_{m-1,n}^{(i+k_1)} -\frac{\phi^{(i)}_{m,n}}{N\alpha^{N}} , \label{eq:phi-sys-sym-1} \\
\partial_s \phi^{(i)}_{m,n} = \beta^{-1} q^{(i-\ell_2)}_{m,n} \phi_{m,n-1}^{(i+k_2)} -\frac{\phi^{(i)}_{m,n}}{N\beta^{N}}. \label{eq:phi-sys-sym-2}\end{gathered}$$
The symmetry (\[eq:phi-sys-sym-1\]) is a combination of the “generalised symmetry” (\[X1\]) and a simple scaling symmetry, with coefficient chosen so that the vector field is [*tangent*]{} to the level surfaces $\prod\limits_{i=0}^{N-1} \phi^{(i)}_{m,n}=\mbox{const}$, so this symmetry survives the reduction to $N-1$ components, which we always make in our examples. The symmetry (\[eq:phi-sys-sym-2\]) is similarly related to (\[n-sym\]) and also survives the reduction to $N-1$ components.
The [*master symmetries*]{} are similarly adjusted, to give
\[eq:phi-sys-msym\] $$\begin{gathered}
\partial_{\tau} \phi^{(i)}_{m,n} = m \alpha^{-1} q^{(i-\ell_1)}_{m,n} \phi_{m-1,n}^{(i+k_1)}
-\frac{m \phi^{(i)}_{m,n}}{N\alpha^{N}} , \label{dtauphim} \\
\partial_{\sigma} \phi^{(i)}_{m,n} = n \beta^{-1} q^{(i-\ell_2)}_{m,n} \phi_{m,n-1}^{(i+k_2)}
-\frac{n \phi^{(i)}_{m,n}}{N\beta^{N}}, \label{dsigmaphim}\end{gathered}$$ where $\partial_\tau \alpha = 1/\big(N \alpha^{N-1}\big)$ and $\partial_\sigma \beta = 1/\big(N \beta^{N-1}\big)$.
The self-dual case {#sect:selfdual}
==================
In [@f17-2] we give a number of equivalence relations for our general discrete system. For the case with $(k_2,\ell_2)=(k_1,\ell_1)=(k,\ell)$ the mapping
\[sd\] $$\begin{gathered}
\label{sd-kl}
(k,\ell)\mapsto \big(\tilde k,\tilde \ell\big) = (N-\ell,N-k)\end{gathered}$$ is an involution on the parameters, so we refer to such systems as [*dual*]{}. The [*self-dual*]{} case is when $(\tilde k,\tilde \ell)=(k,\ell)$, giving $k+\ell=N$. In particular, we consider the case with $$\begin{gathered}
\label{s-dual}
k+\ell=N,\qquad \ell-k =1 \quad\Rightarrow\quad N=2k+1,\end{gathered}$$ so we require that $N$ is [*odd*]{}. In this case, we have that Equations (\[eq:dLP-gen-sys-1\]) are invariant under the change $$\begin{gathered}
\label{sd-phi}
\big(\phi^{(i)}_{m,n},\alpha,\beta\big) \mapsto \big(\widetilde\phi^{(i)}_{m,n},\widetilde\alpha,\widetilde\beta\big),\qquad\mbox{where}\quad \widetilde\alpha \alpha =1,\quad \widetilde\beta \beta =1,\quad \widetilde{\phi}^{(i)}_{m,n} \phi^{(2k-1-i)}_{m,n} = 1.\end{gathered}$$
The self-dual case admits the reduction $\widetilde{\phi}^{(i)}_{m,n}=\phi^{(i)}_{m,n}$, when $\alpha=-\beta$ ($=1$, without loss of generality), which we write as $$\begin{gathered}
\phi^{(i+k)}_{m,n} \phi^{(k-1-i)}_{m,n} = 1,\qquad i=0,\dots ,k-1.\end{gathered}$$ The condition $\prod\limits_{i=0}^{N-1} \phi^{(i)}_{m,n} = 1$ then implies $\phi^{(N-1)}_{m,n}=1$. Therefore the matrices $U_{m,n}$ and $V_{m,n}$ are built from $k$ components: $$\begin{gathered}
U_{m,n} = \operatorname{diag}\Bigg(\phi^{(0)}_{m+1,n}\phi^{(k-1)}_{m,n},\dots,\phi^{(k-1)}_{m+1,n}\phi^{(0)}_{m,n},
\frac{1}{\phi^{(k-1)}_{m+1,n}},\frac{1}{\phi^{(0)}_{m,n}\phi^{(k-2)}_{m+1,n}},\dots \\
\hphantom{U_{m,n} = \operatorname{diag}\bigg(}{}\dots ,\frac{1}{\phi^{(k-2)}_{m,n}\phi^{(0)}_{m+1,n}}, \frac{1}{\phi^{(k-1)}_{m,n}}\Bigg)\Omega^k ,\end{gathered}$$ with $V_{m,n}$ given by the same formula, but with $(m+1,n)$ replaced by $(m,n+1)$. In this case the system (\[eq:dLP-gen-sys-1-a\]) reduces to $$\begin{gathered}
\label{self-dual-equn}
\phi^{(i)}_{m+1,n+1} \phi^{(i)}_{m,n} = \frac{1}{\phi^{(k-i-2)}_{m+1,n}\phi^{(k-i-2)}_{m,n+1}} \left(\frac{\phi^{(k-i-2)}_{m+1,n}+\phi^{(k-i-2)}_{m,n+1}}{\phi^{(k-i-1)}_{m+1,n}+\phi^{(k-i-1)}_{m,n+1}}\right),
\quad\mbox{for}\quad i=0,1,\dots , k-1.\!\!\!\!\end{gathered}$$
This reduction has $\frac{N-1}{2}$ components and is represented by an $N\times N$ Lax pair, but is [*not*]{} $3D$ consistent.
### Symmetries {#symmetries .unnumbered}
Below we give the explicit forms of the self-dual case for $N=3$, $N=5$ and $N=7$. In each case, we give the lowest order symmetry $X^1$. However, this symmetry does [*not*]{} reduce to the case of (\[self-dual-equn\]), but the second symmetry, $X^2$, of the hierarchy generated by the master symmetries (\[eq:phi-sys-msym\]), is a symmetry of the reduced system.
The case $\boldsymbol{N=3}$, with level structure $\boldsymbol{(1,2;1,2)}$
--------------------------------------------------------------------------
After the transformation $\phi^{(0)}_{m,n} \rightarrow 1/\phi^{(0)}_{m,n}$, this system becomes
\[eq:3D-1212\] $$\begin{gathered}
\phi^{(0)}_{m+1,n+1} = \frac{\alpha \phi_{m+1,n}^{(1)} - \beta \phi^{(1)}_{m,n+1}}{\alpha \phi_{m+1,n}^{(0)}\phi^{(1)}_{m,n+1} - \beta \phi_{m,n+1}^{(0)}\phi^{(1)}_{m+1,n}} \frac{1}{\phi^{(0)}_{m,n}} ,\\
\phi^{(1)}_{m+1,n+1} = \frac{\alpha \phi_{m,n+1}^{(0)} - \beta \phi^{(0)}_{m+1,n}}{\alpha \phi_{m+1,n}^{(0)}\phi^{(1)}_{m,n+1} - \beta \phi_{m,n+1}^{(0)}\phi^{(1)}_{m+1,n}} \frac{1}{\phi^{(1)}_{m,n}} .
\end{gathered}$$
System (\[eq:3D-1212\]) admits two point symmetries generated by $$\begin{gathered}
\begin{cases} \partial_\epsilon \phi^{(0)}_{m,n} = \omega^{n+m} \phi^{(0)}_{m,n}, \\ \partial_\epsilon \phi^{(1)}_{m,n} = 0,\end{cases} \qquad \begin{cases} \partial_\eta \phi^{(0)}_{m,n} =0,\\ \partial_\eta \phi^{(1)}_{m,n} = \omega^{n+m} \phi^{(1)}_{m,n},\end{cases}\qquad \omega^2+\omega+1=0,\end{gathered}$$ and two local generalized symmetries. Here we present the symmetry for the $m$-direction whereas the ones in the $n$-direction follow by changing $\phi^{(i)}_{m+j,n} \rightarrow \phi^{(i)}_{m,n+j}$ $$\begin{gathered}
\partial_{t_1} \phi^{(0)}_{m,n} = \phi^{(0)}_{m,n} \frac{1+\phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} - 2 \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}}{1+\phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} + \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}} ,\\
\partial_{t_1} \phi^{(1)}_{m,n} = -\phi^{(1)}_{m,n} \frac{1-2 \phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} + \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}}{1+\phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} + \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}} .
\end{gathered}$$ We also have the master symmetry (\[eq:phi-sys-msym\]), which can be written $$\begin{gathered}
\partial_\tau \phi^{(0)}_{m,n} = m \partial_{t^1} \phi^{(0)}_{m,n} ,\qquad \partial_\tau \phi^{(1)}_{m,n} = m \partial_{t^1} \phi^{(1)}_{m,n} ,\qquad \partial_\tau \alpha = \alpha,\end{gathered}$$ which allows us to construct a hierarchy of symmetries of system (\[eq:3D-1212\]) in the $m$-direction. For instance, the second symmetry is
\[eq:3D-1212-sym-2\] $$\begin{gathered}
\partial_{t_2} \phi^{(0)}_{m,n} = \frac{\phi^{(0)}_{m,n} \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}}{{\cal{F}}_{m,n}} ({\cal{S}}_m+1)\left( \frac{({\cal{S}}_m- 1)\big(\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} \phi^{(0)}_{m-2,n} \big)}{{\cal{F}}_{m,n} {\cal{F}}_{m-1,n}} \right),\\
\partial_{t_2} \phi^{(1)}_{m,n} = \frac{\phi^{(1)}_{m,n} \phi^{(0)}_{m+1,n} \phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n}}{{\cal{F}}_{m,n}} ({\cal{S}}_m+1)\left( \frac{({\cal{S}}_m- 1)\big(\phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n} \phi^{(1)}_{m-2,n} \big)}{{\cal{F}}_{m,n} {\cal{F}}_{m-1,n}} \right),\end{gathered}$$ where $$\begin{gathered}
{\cal{F}}_{m,n} := 1+\phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} + \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n} ,\end{gathered}$$ and ${\cal{S}}_m$ denotes the shift operator in the $m$-direction.
### The reduced system {#the-reduced-system .unnumbered}
The reduced system (\[self-dual-equn\]) takes the explicit form (first introduced in [@14-6]) $$\begin{gathered}
\label{eq:MX2}
\phi_{m,n} \phi_{m+1,n+1} ( \phi_{m+1,n} + \phi_{m,n+1} ) = 2,\end{gathered}$$ where $$\begin{gathered}
\phi^{(0)}_{m,n} = \phi^{(1)}_{m,n} = \frac{1}{\phi_{m,n}} ,\qquad \beta = - \alpha.\end{gathered}$$ With this coordinate, the second symmetry (\[eq:3D-1212-sym-2\]) takes the form $$\begin{gathered}
\partial_{t_2} \phi_{m,n} = \phi_{m,n} \frac{1}{P^{(1)}_{m,n}} \left( \frac{1}{P^{(1)}_{m+1,n}}-\frac{1}{P^{(1)}_{m-1,n}}\right),\end{gathered}$$ where $$\begin{gathered}
\label{eq:N3-P-G}
P^{(0)}_{m,n} = \phi_{m+1,n} \phi_{m,n} \phi_{m-1,n} , \qquad P^{(1)}_{m,n} = 2 + P^{(0)}_{m,n},\end{gathered}$$ first given in [@14-6]. Despite the $t_2$ notation, this is the [*first*]{} of the hierarchy of symmetries of the reduction (\[eq:MX2\]).
The case $\boldsymbol{N=5}$, with level structure $\boldsymbol{(2,3;2,3)}$
--------------------------------------------------------------------------
In this case, equations (\[eq:dLP-gen-sys-1-a\]) take the form
\[eq:N5-sys\] $$\begin{gathered}
\phi ^{(0)}_{m+1,n+1}= \frac{\phi ^{(2)}_{m+1,n} \phi ^{(2)}_{m,n+1}}{\phi^{(0)}_{m,n}} \frac{\alpha \phi ^{(3)}_{m+1,n}-\beta \phi ^{(3)}_{m,n+1}}{\alpha \phi ^{(2)}_{m+1,n}- \beta \phi ^{(2)}_{m,n+1}}, \\
\phi ^{(1)}_{m+1,n+1} = \frac{1}{\phi ^{(1)}_{m,n} \big(\alpha \phi ^{(3)}_{m+1,n}-\beta \phi ^{(3)}_{m,n+1}\big)} \left(\frac{\alpha \phi ^{(3)}_{m,n+1}}{\phi ^{(0)}_{m+1,n}\phi ^{(1)}_{m+1,n} \phi^{(2)}_{m+1,n}} -\frac{\beta \phi ^{(3)}_{m+1,n}}{\phi ^{(0)}_{m,n+1} \phi ^{(1)}_{m,n+1} \phi ^{(2)}_{m,n+1}}\right), \nonumber \\
\phi ^{(2)}_{m+1,n+1} = \frac{\alpha \phi ^{(0)}_{m+1,n}-\beta \phi ^{(0)}_{m,n+1}}{\phi ^{(2)}_{m,n}
\big(\alpha \phi ^{(0)}_{m,n+1} \phi ^{(1)}_{m,n+1} \phi ^{(2)}_{m,n+1} \phi ^{(3)}_{m,n+1}-\beta \phi ^{(0)}_{m+1,n} \phi ^{(1)}_{m+1,n} \phi ^{(2)}_{m+1,n} \phi^{(3)}_{m+1,n}\big)}, \!\!\!\\
\phi ^{(3)}_{m+1,n+1} = \frac{\phi ^{(0)}_{m+1,n} \phi ^{(0)}_{m,n+1}}{ \phi ^{(3)}_{m,n}} \frac{\alpha \phi^{(1)}_{m+1,n} - \beta \phi ^{(1)}_{m,n+1}}{\alpha \phi ^{(0)}_{m+1,n}-\beta\phi ^{(0)}_{m,n+1}}.\end{gathered}$$
Under the transformation (\[sd\]), the first and last of these interchange, as do the middle pair.
The lowest order generalised symmetry in the $m$-direction is generated by $$\begin{gathered}
\partial_{t_1} \phi^{(i)}_{m,n} = \phi^{(i)}_{m,n} \left(\frac{5 A^{(i)}_{m,n}}{B_{m,n}}-1\right),\qquad i=0,\ldots,3,\end{gathered}$$ where, if we denote $F^{(i)}_{m,n} =\phi^{(i)}_{m-1,n} \phi^{(i)}_{m,n} \phi^{(i)}_{m+1,n} $, $$\begin{gathered}
A^{(0)}_{m,n} = F^{(0)}_{m,n} F^{(1)}_{m,n} F^{(2)}_{m,n} \phi^{(1)}_{m,n}\phi^{(2)}_{m,n}\phi^{(3)}_{m,n} ,\qquad
A^{(1)}_{m,n} = F^{(0)}_{m,n} F^{(1)}_{m,n} F^{(2)}_{m,n} F^{(3)}_{m,n} \frac{\phi^{(2)}_{m,n}}{\phi^{(0)}_{n,m}},\\
A^{(2)}_{m,n} = \phi^{(2)}_{m,n} \phi^{(3)}_{m,n},\qquad A^{(3)}_{m,n} = \phi^{(0)}_{m-1,n} \phi^{(0)}_{m+1,n},\\
B_{m,n} = \sum_{j=0}^{3} A^{(j)}_{m,n} + F^{(0)}_{m,n} F^{(1)}_{m,n}\phi^{(2)}_{m,n}.\end{gathered}$$ The corresponding master symmetry is $$\begin{gathered}
\partial_\tau \phi^{(i)}_{m,n} = m \partial_{t_1} \phi^{(i)}_{m,n}, \qquad i= 0,\ldots,3,\end{gathered}$$ along =-1 with $\partial_\tau \alpha = 1$. This is used to construct a hierarchy of symmetries for the system (\[eq:N5-sys\]). We omit here the second symmetry as the expressions become cumbersome for the unreduced case.
### The reduced system {#the-reduced-system-1 .unnumbered}
The reduction (\[self-dual-equn\]) now has components $\phi^{(0)}_{m,n}$, $\phi^{(1)}_{m,n}$, with $\phi^{(2)}_{m,n}=\frac{1}{\phi^{(1)}_{m,n}}$, $\phi^{(3)}_{m,n}=\frac{1}{\phi^{(0)}_{m,n}}$, $\phi^{(4)}_{m,n}= 1$, and the $2$-component system takes the form
\[eq:sys-5-red\] $$\begin{gathered}
\phi^{(0)}_{m+1,n+1} \phi^{(0)}_{m,n} = \frac{1}{\phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n+1}} \left(\frac{\phi^{(0)}_{m+1,n}+\phi^{(0)}_{m,n+1}}{\phi^{(1)}_{m+1,n}+\phi^{(1)}_{m,n+1}}\right),\\[3mm]
\phi^{(1)}_{m+1,n+1} \phi^{(1)}_{m,n} (\phi^{(0)}_{m+1,n}+\phi^{(0)}_{m,n+1}) = 2.\end{gathered}$$
Only the even indexed generalised symmetries of the system (\[eq:N5-sys\]) are consistent with this reduction. This means that the lowest order generalised symmetry is
\[eq:sym-self-dual-5a\] $$\begin{gathered}
\partial_{t_2} \phi^{(0)}_{m,n} = \phi^{(0)}_{m,n} \frac{P^{(0)}_{m,n}}{ P^{(2)}_{m,n}} \left( \frac{1}{P^{(2)}_{m+1,n}} - \frac{1}{P^{(2)}_{m-1,n}}\right),\\
\partial_{t_2} \phi^{(1)}_{m,n} = \phi^{(1)}_{m,n} \frac{1}{ P^{(2)}_{m,n}} \left(\frac{1 + P^{(0)}_{m+1,n}}{P^{(2)}_{m+1,n}} - \frac{1+P^{(0)}_{m-1,n}}{P^{(2)}_{m-1,n}} \right),\end{gathered}$$
where
\[eq:N5-P-G\] $$\begin{gathered}
P^{(0)}_{m,n} = \phi^{(0)}_{m-1,n} \phi^{(0)}_{m,n} \phi^{(0)}_{m+1,n} \phi^{(1)}_{m,n}, \\
P^{(1)}_{m,n} = \phi^{(0)}_{m-1,n} \big(\phi^{(0)}_{m,n}\big)^2 \phi^{(0)}_{m+1,n} \phi^{(1)}_{m-1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m+1,n},\\
P^{(2)}_{m,n} = 2 +2 P^{(0)}_{m,n} + P^{(1)}_{m,n}.\label{eq:G-5}\end{gathered}$$
The case $\boldsymbol{N=7}$, with level structure $\boldsymbol{(3,4;3,4)}$
--------------------------------------------------------------------------
=-1 The fully discrete system (\[eq:dLP-gen-sys-1-a\]) and its lower order symmetries (\[eq:phi-sys-sym\]) and master symmetries (\[eq:phi-sys-msym\]) can be easily adapted to our choices $N=7$ and $(k,\ell)=(3,4)$. In the same way the corresponding reduced system follows from (\[self-dual-equn\]) with $k=3$. Thus we omit all these systems here and present only the lowest order symmetry of the reduced system which takes the following form $$\begin{gathered}
\partial_{t_2} \phi^{(0)}_{m,n} = \phi^{(0)}_{m,n} \frac{ P^{(1)}_{m,n}}{P^{(3)}_{m,n}} \left(\frac{1}{P^{(3)}_{m+1,n}} - \frac{1}{P^{(3)}_{m-1,n}}\right), \nonumber \\
\partial_{t_2} \phi^{(1)}_{m,n} = \phi^{(1)}_{m,n} \frac{P^{(0)}_{m,n}}{P^{(3)}_{m,n}} \left(\frac{1+ P^{(0)}_{m+1,n}}{P^{(3)}_{m+1,n}} - \frac{1+ P^{(0)}_{m-1,n}}{P^{(3)}_{m-1,n}}\right), \label{eq:N7-red-dd} \\
\partial_{t_2} \phi^{(2)}_{m,n}= \phi^{(2)}_{m,n} \frac{1}{P^{(3)}_{m,n}} \left(\frac{1+ P^{(0)}_{m+1,n}+ P^{(1)}_{m+1,n}}{P^{(3)}_{m+1,n}} - \frac{1+ P^{(0)}_{m-1,n}+ P^{(1)}_{m-1,n}}{P^{(3)}_{m-1,n}}\right), \nonumber\end{gathered}$$ where
\[eq:N7-P-G\] $$\begin{gathered}
P^{(0)}_{m,n} = \phi^{(0)}_{m-1,n} \phi^{(0)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(2)}_{m,n}, \\
P^{(1)}_{m,n} = \phi^{(0)}_{m-1,n} \phi^{(0)}_{m,n} \phi^{(0)}_{m+1,n} \phi^{(1)}_{m-1,n} \big(\phi^{(1)}_{m,n}\big)^2 \phi^{(1)}_{m+1,n} \phi^{(2)}_{m,n}, \\
P^{(2)}_{m,n} =\phi^{(0)}_{m-1,n} \big(\phi^{(0)}_{m,n}\big)^2 \phi^{(0)}_{m+1,n} \phi^{(1)}_{m-1,n} \big(\phi^{(1)}_{m,n}\big)^2 \phi^{(1)}_{m+1,n} \phi^{(2)}_{m-1,n} \phi^{(2)}_{m,n} \phi^{(2)}_{m+1,n},\\
P^{(3)}_{m,n} = 2 + 2 P^{(0)}_{m,n} + 2 P^{(1)}_{m,n}+ P^{(2)}_{m,n}.
\end{gathered}$$
Miura transformations and relation to Bogoyavlensky lattices {#sec:Miura}
============================================================
In this section we discuss Miura transformations for the reduced systems and their symmetries which bring the latter to polynomial form. In the lowest dimensional case ($N=3$) the polynomial system is directly related to the Bogoyavlensky lattice (see (\[eq:Bog\]) below), whereas the higher dimensional ones result in systems which generalise (\[eq:Bog\]) to $k$ component systems.
The reduced system in $\boldsymbol{N=3}$ {#the-reduced-system-in-boldsymboln3 .unnumbered}
----------------------------------------
The Miura transformation [@14-6] $$\begin{gathered}
\psi_{m,n} = \frac{P^{(0)}_{m,n}}{P^{(1)}_{m,n}} - 1,\end{gathered}$$ where $P^{(0)}_{m,n}$ and $P^{(1)}_{m,n}$ are given in (\[eq:N3-P-G\]), maps equation (\[eq:MX2\]) to $$\begin{gathered}
\label{eq:MX2a}
\frac{\psi_{m+1,n+1}+1}{\psi_{m,n}+\psi_{m,n+1}+1} + \frac{\psi_{m+1,n}}{\psi_{m,n+1}} = 0,\end{gathered}$$ and its symmetry to $$\begin{gathered}
\label{eq:Bog}
\partial_{t_2} \psi_{m,n} = \psi_{m,n} (\psi_{m,n}+1) (\psi_{m+2,n} \psi_{m+1,n} - \psi_{m-1,n} \psi_{m-2,n}),\end{gathered}$$ which is related to the Bogoyavlensky lattice [@B] $$\begin{gathered}
\partial_{t_2} \chi_{m,n} = \chi_{m,n}(\chi_{m+2,n} + \chi_{m+1,n} - \chi_{m-1,n} - \chi_{m-2,n}),\end{gathered}$$ through the Miura transformation $$\begin{gathered}
\chi_{m,n} = \psi_{m+1,n} \psi_{m,n} (\psi_{m-1,n}+1).\end{gathered}$$
The reduced system in $\boldsymbol{N=5}$ {#the-reduced-system-in-boldsymboln5 .unnumbered}
----------------------------------------
The Miura transformation $$\begin{gathered}
\psi_{m,n}^{(0)} = \frac{2 P^{(0)}_{m,n}}{P^{(2)}_{m,n}},\qquad \psi_{m,n}^{(1)} = \frac{P^{(1)}_{m,n}}{P^{(2)}_{m,n}} - 1,\end{gathered}$$ where $P^{(i)}_{m,n}$ are given in (\[eq:N5-P-G\]), maps system (\[eq:sys-5-red\]) to $$\begin{gathered}
\frac{\psi^{(0)}_{m,n+1}\psi^{(0)}_{m+1,n+1}+\psi^{(0)}_{m+1,n} \big(\psi^{(0)}_{m,n}+\psi^{(1)}_{m,n+1}\big)}{\psi^{(1)}_{m,n}+1} = \frac{\psi^{(0)}_{m,n+1}\psi^{(1)}_{m+1,n}-\psi^{(0)}_{m+1,n}\psi^{(1)}_{m,n+1}}{\psi^{(0)}_{m,n+1} + \psi^{(1)}_{m,n+1}}, \nonumber\\
\frac{\psi^{(1)}_{m+1,n+1}+1}{\psi^{(1)}_{m,n}+\psi^{(1)}_{m,n+1}+\psi^{(0)}_{m,n+1}+1} + \frac{\psi^{(0)}_{m+1,n}+\psi^{(1)}_{m+1,n}}{\psi^{(0)}_{m,n+1}+\psi^{(1)}_{m,n+1}} = 0,\label{eq:M-red-sys-5}\end{gathered}$$ and its symmetry (\[eq:sym-self-dual-5a\]) to the following system of polynomial equations in which we have suppressed the dependence on the second index $n$: $$\begin{gathered}
\frac{\partial_{t_2} \psi^{(0)}_m}{\psi^{(0)}_m} = \big(\psi^{(0)}_m+\psi^{(1)}_m\big) \big(\psi^{(0)}_{m+2} \psi^{(0)}_{m+1} - \psi^{(0)}_{m-1} \psi^{(0)}_{m-2} + \psi^{(0)}_{m+1}-\psi^{(0)}_{m-1} + \psi^{(1)}_{m+1}-\psi^{(1)}_{m-1}\big) \nonumber \\
\hphantom{\frac{\partial_{t_2} \psi^{(0)}_m}{\psi^{(0)}_m} =}{} - \big(\psi^{(1)}_m+1\big) \big(\psi^{(1)}_{m+2} \psi^{(1)}_{m+1} - \psi^{(1)}_{m-1} \psi^{(1)}_{m-2}\big) + \big(\psi^{(0)}_m-1\big) \big(\psi^{(1)}_{m+2} \psi^{(0)}_{m+1}-\psi^{(0)}_{m-1} \psi^{(1)}_{m-2}\big), \nonumber\\
\frac{\partial_{t_2} \psi^{(1)}_m}{\psi^{(1)}_m+1} = \big(\psi^{(0)}_m+\psi^{(1)}_m\big) \big(\psi^{(0)}_{m+2} \psi^{(0)}_{m+1} - \psi^{(0)}_{m-1} \psi^{(0)}_{m-2}\big) - \psi^{(1)}_m \big(\psi^{(1)}_{m+2} \psi^{(1)}_{m+1} - \psi^{(1)}_{m-1} \psi^{(1)}_{m-2}\big) \nonumber \\
\hphantom{\frac{\partial_{t_2} \psi^{(1)}_m}{\psi^{(1)}_m+1} =}{} + \psi^{(0)}_m \big(\psi^{(1)}_{m+2} \psi^{(0)}_{m+1}-\psi^{(0)}_{m-1} \psi^{(1)}_{m-2}\big).\label{eq:M-sys-5}
\end{gathered}$$ The above system and its symmetry can be considered as a two-component generalisation of the equation (\[eq:MX2a\]) and its symmetry (\[eq:Bog\]) in the following sense. If we set, $\psi^{(0)}_{m,n} = 0$ and $\psi^{(1)}_{m,n} = \psi_{m,n}$ in (\[eq:M-red-sys-5\]) and (\[eq:M-sys-5\]), then they will reduce to equations (\[eq:MX2a\]) and (\[eq:Bog\]), respectively.
The reduced systems for $\boldsymbol{N>5}$ {#the-reduced-systems-for-boldsymboln5 .unnumbered}
------------------------------------------
It can be easily checked that for each $k$ ($N=2 k+1$), the lowest order symmetry of the reduced system (\[self-dual-equn\]) involves certain functions $P^{(i)}_{m,n}$, $i=0,\ldots,k$, with $$\begin{gathered}
P^{(k)}_{m,n} = 2 + 2 \sum_{i=0}^{k-2} P^{(i)}_{m,n} + P^{(k-1)}_{m,n},\end{gathered}$$ which are given in terms of $\phi^{(i)}_{m,n}$ and their shifts (see relations (\[eq:N5-P-G\]) and (\[eq:N7-P-G\])). Then, the Miura transformation $$\begin{gathered}
\label{eq:gen-Miura}
\psi^{(i)}_{m,n} = \frac{2 P^{(i)}_{m,n}}{P^{(k)}_{m,n}}, \qquad i=0,\ldots,k-2, \qquad
\psi^{(k-1)}_{m,n} = \frac{P^{(k-1)}_{m,n}}{P^{(k)}_{m,n}} - 1,\end{gathered}$$ brings the symmetries of the reduced system to polynomial form. One could derive the polynomial system corresponding to $N=7$ ($k=3$) starting with system (\[eq:N7-red-dd\]), the functions given in (\[eq:N7-P-G\]) and using the corresponding Miura transformation (\[eq:gen-Miura\]). The system of differential-difference equations is omitted here because of its length but it can be easily checked that if we set $\psi^{(0)}_{m,n}=0$ and then rename the remaining two variables as $\psi^{(i)}_{m,n} \mapsto \psi^{(i-1)}_{m,n}$, then we will end up with system (\[eq:M-sys-5\]).
This indicates that every $k$ component system is a generalisation of all the lower order ones, and thus of the Bogoyavlensky lattice (\[eq:Bog\]). To be more precise, if we consider the case $N=2 k+1$ along with the $k$-component system, set variable $\psi^{(0)}_{m,n}=0$ and then rename the remaining ones as $\psi^{(i)}_{m,n} \mapsto \psi^{(i-1)}_{m,n}$, then the resulting $(k-1)$-component system is the reduced system corresponding to $N = 2 k-1$. Recursively, this means that it also reduces to the $N=3$ system, i.e., equation (\[eq:Bog\]). Other systems with similar behaviour have been presented in [@BW].
Acknowledgements {#acknowledgements .unnumbered}
----------------
PX acknowledges support from the EPSRC grant [*Structure of partial difference equations with continuous symmetries and conservation laws*]{}, EP/I038675/1.
[99]{} =-1pt
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Fordy A.P., Xenitidis P., [$\mathbb{Z}_N$]{} graded discrete [Lax]{} pairs and discrete integrable systems, [arXiv:1411.6059](https://arxiv.org/abs/1411.6059).
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| ArXiv |
---
abstract: 'The recent results of 13 TeV ATLAS and CMS di-photon searches show an excess at di-photon invariant mass of 750 GeV. We look for possible explanation of this within minimal left right symmetric model (MLRSM). The possible candidate is a neutral Higgs of mass 750 GeV that can decay to di-photon via charged Higgs and right handed gauge boson loop. However, the cross-section is not consistent with the ATLAS and CMS results. We then discuss one possible variation of this model with universal seesaw for fermion masses that can explain this excess.'
author:
- Arnab Dasgupta
- Manimala Mitra
- Debasish Borah
title: 'Minimal Left-Right Symmetry Confronted with the 750 GeV Di-photon Excess at LHC'
---
Introduction
============
The recently reported 13 TeV center of mass energy data of the large hadron collider (LHC) experiment have pointed towards the existence of a resonance of mass about 750 GeV and width around 45 GeV decaying into two photons [@lhcrun2a; @atlasconf; @CMS:2015dxe]. The ATLAS collaboration has reported the presence of this 750 GeV resonance from their $3.2 \; \text{fb}^{-1}$ data with a statistical significance of $3.9\sigma$ ($2.3\sigma$ including look-elsewhere effects) whereas CMS collaboration has reported the same with a significance of $2.6\sigma$ from their $2.6\; \text{fb}^{-1}$ data. Apart from the mass and decay width, these two experiments have also measured the cross section $\sigma(pp\rightarrow \gamma \gamma)$ to be $10\pm3$ fb (ATLAS) and $6\pm3$ fb (CMS). The large decay width as well as the sizeable cross section have made it a challenging task to come up with beyond standard model (BSM) frameworks which can accommodate it.
Although the reported signal could well be a statistical fluctuation, it has drawn significant attention from the particle physics community leading to a large number of interesting possible explanations including two Higgs doublet models, additional scalars coupling to vector like fermions, extra dimensions, dark matter among others as well as the implications of this signal reported in the works [@750pheno16Dec; @750GeVsinglet; @750pheno17Dec; @750pheno18Dec; @750pheno21Dec; @750pheno22Dec; @750pheno23Dec; @750pheno24Dec; @LR750GeV; @750pheno25Dec; @750pheno29Dec]. In this work, we try to scrutinize one of the very popular BSM framework, known as the left right symmetric model [@lrsm; @lrsmpot] in the light of the reported ATLAS and CMS results. Very recently, this model have been analyzed in the context of several other excesses: such as 2.8$\sigma$ excess in the $eejj$ final state reported by CMS [@Khachatryan:2014dka; @Gluza:2015goa], $3.4 \sigma$ diboson excess reported by ATLAS [@lrdiboson; @Dobrescu:2015qna] and [@ATLASdiboson] and the dijet results [@lrdijet] and [@CMSdijet].
The MLRSM model has few additional Higgs states, where few of the scalars can have lower than TeV scale masses. The neutral Higgs state which has 750 GeV mass, can decay to di-photon via charged Higgs and gauge boson loop. We compute its production cross section at LHC and the branching ratio into two photons. We do a parameter scan of the model by varying the different parameters of the potential, with a fixed symmetry breaking scale. The computed cross section $\sigma(pp \to H^0_2 \to \gamma \gamma)$ is way below the observed one with a few fb. We show that the minimal version of this model with the neutral Higgs states as 750 GeV resonance can not explain the observed signal.
We then consider the possible modification to the MLRSM in order to explain the observed signal. The easiest modification is the addition of vector like quarks which can couple directly to the 750 GeV neutral scalar. This will not only enhance the production cross section but also the partial decay width into two photons. This scenario has already been explored within several different models mentioned above. Instead of arbitrarily adding vector like fermions into the MLRSM to explain the observed signal, we try to focus on the possibility of having these exotic fermions to serve another purposes. One possibility could be to realize these fermions within some higher fermion representations of grand unified theories like SO(10). Since TeV scale MLRSM does not give rise to gauge coupling unification at high energy scale, it will be very natural to have these exotic fermions at TeV scale which not only arise naturally within SO(10) framework but also can help to achieve gauge coupling unification. Another interesting motivation for additional vector like fermions is their role in generating masses of the known standard model fermions. This can happen for example, if the standard model Higgs can not directly couple to the left and right handed fermions of the model, but can do so only through additional heavy fermions. There is one such a version of LRSM where such vector like fermions have to be incorporated in order to generate observed fermion masses [@VLQlr; @univSeesawLR]. This model was studied later in the context of cosmology [@gulr] and neutrinoless double beta decay [@lr0nu2beta]. The model gives rise to fermion masses through a universal seesaw framework where the standard model fermion masses arise after integrating out heavy fermions. Leaving the possibility of having new physics source within a grand unified theory framework to a future work, here we focus on the LRSM with universal seesaw for fermion masses. We consider a 750 GeV neutral singlet scalar which can couple to vector like fermions and can give rise to the observed signal.
This paper is organized as follows. In section \[sec1\], we briefly discuss the MLRSM and then discuss the possibility of a 750 GeV neutral scalar in view of the LHC signal in section \[sec2\]. In section \[sec3\], we briefly discuss the LRSM with universal seesaw and in section \[sec4\] we study the possibility of explaining the LHC signal of a 750 GeV resonance decaying into two photons. We finally conclude in section \[sec5\].
Minimal Left-Right Symmetric Model (MSLRM) {#sec1}
==========================================
Left-Right Symmetric Model [@lrsm; @lrsmpot] is one of the very well motivated BSM frameworks where the gauge symmetry of the electroweak theory is extended to $SU(3)_c \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$. The right handed fermions which are singlets under the $SU(2)_L$ of SM, transform as doublets under $SU(2)_R$, making the presence of right handed neutrinos natural in this model. The Higgs doublet of the SM is replaced by a Higgs bidoublet to allow couplings between left and right handed fermions, both of which are doublets under $SU(2)_L$ and $SU(2)_R$ respectively. The enhanced gauge symmetry of the model $SU(2)_R \times U(1)_{B-L}$ is broken down to the $U(1)_Y$ of SM by the vacuum expectation value (vev) of additional Higgs scalar, transforming as triplet under $SU(2)_R$ and having non-zero $U(1)_{B-L}$ charge. This triplet gives rise to the Majorana masses of the right handed neutrinos through symmetry breaking. The heavy right handed neutrinos participate in the seesaw mechanism and generate the Majorana masses of the light neutrinos. On the other hand, the left handed Higgs triplet generates Majorana masses of the light neutrinos through type II.
The fermion content of the minimal LRSM is $$Q_L=
\left(\begin{array}{c}
\ u_L \\
\ d_L
\end{array}\right)
\sim (3,2,1,\frac{1}{3}),\hspace*{0.8cm}
Q_R=
\left(\begin{array}{c}
\ u_R \\
\ d_R
\end{array}\right)
\sim (3^*,1,2,\frac{1}{3}),\nonumber$$ $$\ell_L =
\left(\begin{array}{c}
\ \nu_L \\
\ e_L
\end{array}\right)
\sim (1,2,1,-1), \quad
\ell_R=
\left(\begin{array}{c}
\ \nu_R \\
\ e_R
\end{array}\right)
\sim (1,1,2,-1) \nonumber$$ Similarly, the Higgs content of the minimal LRSM is $$\Phi=
\left(\begin{array}{cc}
\ \phi^0_{11} & \phi^+_{11} \\
\ \phi^-_{12} & \phi^0_{12}
\end{array}\right)
\sim (1,2,2,0)
\nonumber$$ $$\Delta_L =
\left(\begin{array}{cc}
\ \delta^+_L/\surd 2 & \delta^{++}_L \\
\ \delta^0_L & -\delta^+_L/\surd 2
\end{array}\right)
\sim (1,3,1,2), \hspace*{0.2cm}
\Delta_R =
\left(\begin{array}{cc}
\ \delta^+_R/\surd 2 & \delta^{++}_R \\
\ \delta^0_R & -\delta^+_R/\surd 2
\end{array}\right)
\sim (1,1,3,2) \nonumber$$ where the numbers in brackets correspond to the quantum numbers with respect to the gauge group $SU(3)_c\times SU(2)_L\times SU(2)_R \times U(1)_{B-L}$. In the symmetry breaking pattern, the neutral component of the Higgs triplet $\Delta_R$ acquires a vev to break the gauge symmetry of the LRSM into that of the SM and then to the $U(1)$ of electromagnetism by the vev of the neutral component of Higgs bidoublet $\Phi$: $$SU(2)_L \times SU(2)_R \times U(1)_{B-L} \quad \underrightarrow{\langle
\Delta_R \rangle} \quad SU(2)_L\times U(1)_Y \quad \underrightarrow{\langle \Phi \rangle} \quad U(1)_{em}$$ The symmetry breaking of $SU(2)_R \times U(1)_{B-L}$ into the $U(1)_Y$ of standard model can also be achieved at two stages by choosing a non-minimal scalar sector. [[We]{}]{} denote the vev of the two neutral components of the bidoublet as $k_1, k_2$ and that of triplets $\Delta_{L, R}$ as $v_{L, R}$. Considering $g_L=g_R$, $k_2 \sim v_L \approx 0$ and $v_R \gg k_1$, the gauge boson masses after symmetry breaking can be written as $$M^2_{W_L} = \frac{g^2}{4} k^2_1, \;\;\; M^2_{W_R} = \frac{g^2}{2}v^2_R$$ $$M^2_{Z_L} = \frac{g^2 k^2_1}{4\cos^2{\theta_w}} \left ( 1-\frac{\cos^2{2\theta_w}}{2\cos^4{\theta_w}}\frac{k^2_1}{v^2_R} \right), \;\;\; M^2_{Z_R} = \frac{g^2 v^2_R \cos^2{\theta_w}}{\cos{2\theta_w}}$$ where $\theta_w$ is the Weinberg angle. After the symmetry breaking, four neutral scalars emerge, two from the bidoublet $(H^0_0, H^0_1)$, one from right handed triplet $(H^0_2)$ and another from left handed triplet $(H^0_3)$. Similarly there are two neutral pseudoscalars, one from the bidoublet $(A^0_1)$ and another from the left handed triplet $(A^0_2)$. Among the charged scalars, there are two singly charged ones $(H^{\pm}_1, H^{\pm}_2)$ and two doubly charged ones $(H^{\pm \pm}_1, H^{\pm \pm}_2)$.
Under the approximations made above, the scalar masses are given by $$M^2_{H^0_0} = 2 \lambda_1 k^2_1, \, \, \, M^2_{H^0_1} = \frac{1}{2}\alpha_3 v^2_R, \, \, M^2_{H^0_2} = 2\rho_1 v^2_R, \, \, M^2_{H^0_3} = \frac{1}{2}(\rho_3-2\rho_1)v^2_R$$ $$M^2_{A^0_1} = \frac{1}{2}\alpha_3 v^2_R - 2(2\lambda_2-\lambda_3)k^2_1, \, \, M^2_{A^0_2} = \frac{1}{2}v^2_R (\rho_3-2\rho_1), \, \,
M^2_{H^{\pm}_1} = \frac{1}{2} (\rho_3-2\rho_1)v^2_R +\frac{1}{4} \alpha_3 k^2_1$$ $$M^2_{H^{\pm}_2} = \frac{1}{2}\alpha_3 v^2_R + \frac{1}{4} \alpha_3 k^2_1, \, \,
M^2_{H^{\pm \pm}_1} = \frac{1}{2} (\rho_3-2\rho_1)v^2_R+\frac{1}{2}\alpha_3 k^2_1, \, \,
M^2_{H^{\pm \pm}_2} = 2\rho_2 v^2_R +\frac{1}{2} \alpha_3 k^2_1$$ where $\lambda_i, \alpha_i, \rho_i$ are dimensionless couplings of the scalar potential of this model [@lrsmpot]. We take into account the following results and experimental searches that fix the dimensionless parameters.
- In the above, $H^0_0$ can be identified as SM like Higgs of mass 125 GeV, that fixes the coupling $\lambda_1$. Few of the other couplings can be constrained after taking into account the experimental limits on the scalar masses.
- We demand that the Higgs $H^0_2$ has a mass 750 GeV that explain the di-photon bump, that fixes the dimensionless coupling $\rho_1$.
- In order to avoid the flavor changing neutral currents (FCNC) processes, the neutral scalars from bi-doublet $H^0_1, A^0_1$ have to be heavier than 10 TeV [@fcnc], which puts further constraint on $\alpha_3$ for a fixed $v_R$. This also constrain the charged Higgs mass $H^{\pm}_2$ to be heavy.
- The lower bound on the mass of doubly charged scalar $H^{\pm \pm}_1 $ from the multilepton search [@Aad:2014hja] fixes the coupling $\rho_3$. This automatically fixes the mass of singly charged scalar $H^{\pm}_1$ as well.
This leaves only two free parameters $ (2\lambda_2-\lambda_3)$ and $\rho_2$ in the expressions for scalar masses to be varied arbitrarily. The other couplings in the full scalar potential are also free to be varied and are not affected by the chosen spectrum of scalar masses.
750 GeV neutral scalar in MLRSM {#sec2}
===============================
Among the neutral physical scalars in MLRSM, the $H^0_0$ is the standard model like Higgs with mass $125$ GeV. The other two neutral (pseudo) scalars from the bidoublet namely $H^0_1, A^0_1$ are heavier than at least 10 TeV due to tight constraints from FCNC. However, the other three neutral scalars $(H^0_2, H^0_3, A^0_2)$ originating from the scalar triplets can lie at 750 GeV. We consider $H^0_2 \equiv \Delta^0_R$ as a possible candidate for a 750 GeV neutral scalar decaying into two photons.
[ As the heavy Higgs $H^0_2$ does not directly couple with gluons or quarks, hence its production will be governed by the mixing with the SM Higgs. The production cross-section of the $750$ GeV heavy Higgs at 13 TeV LHC is [@physrepdjouadi] $$\sigma (p p \to H^0_2) \sim \theta^2 \times 0.85 \,\rm{pb},$$ ]{} where $\theta \sim \alpha \frac{k_1}{M_{H^0_2}}$ is the mixing between SM Higgs state and the $H^0_2$, and we have considered the dimensionless parameters $\alpha_1=\alpha_2=\alpha$.
As the neutral scalar $H^0_2$ does not couple to a pair of charged fermions at the tree level, the only way it can decay into two photons is through a charged gauge or scalar boson loop. This can happen through a loop containing $W_R$ bosons or one of the charged scalars $H^{\pm}_1, H^{\pm}_2, H^{\pm \pm}_1, H^{\pm \pm}_2$. The total production cross-section of $p p \to H^0_2 \to \gamma \gamma$ is $$\sigma(p p \to H^0_2 \to \gamma \gamma) \sim \theta^2 \times 0.85 \times \rm{Br}(H^0_2 \to \gamma \gamma) \, \rm{pb}.$$
We first consider the following benchmark values of the scalar masses and compute the decay widths. Following this we will provide a full parameter scan. $$m_{H^0_0} = 125 \; \text{GeV}, \;\; m_{H^0_2} = 750 \; \text{GeV}, \;\; m_{H^{\pm}_1} = 380 \; \text{GeV}, \;\; m_{H^0_1} = m_{A^0_1} = m_{H^{\pm}_2} = 10\; \text{TeV}.$$ $$m_{H^{\pm \pm}_1} = 465 \; \text{GeV}, \;\; m_{H^{\pm \pm}_2} = 380 \; \text{GeV}.$$ The values are chosen in such a way to satisfy the current experimental bounds. We show the partial decay width of $H^0_2$ to $\gamma \gamma$ and di-Higgs modes for these illustrative points in the parameter space. Few comments are in order.
- We show the partial decay width to two photons through the $W_R$ loop in Fig. \[fig2\], where the $W_R$ mass has been set to be 3 TeV-in agreement with the collider constraint [@Khachatryan:2014dka]. The partial decay width through gauge boson loop is extremely suppressed and can not explain the diphoton signal.
- We show the partial decay width of $H^0_2 \rightarrow \gamma \gamma$ through charged Higgs loop in Fig. \[fig1\] as a function of the parameter $\alpha$. This also decides the decay mode $H^0_2 \rightarrow H^0_0 H^0_0$. We have set the charged Higgs masses to the lowest possible value, which is consistent with collider searches. The masses of the neutral scalars have been set to 10 TeV and $W_R$ mass as 3 TeV.
- The mixing parameter $\theta$ can be constrained from di-Higgs search [@dihiggscms], which is $\mathcal{O}(0.3)$ for 750 GeV scalar resonance decaying into di-Higgs with 100$\%$ branching ratio.
- From Fig. \[fig1\], it is evident that for the parameter $\alpha > 0.001$, the partial decay width to di-Higgs will be larger than the di-photon. Including both the gauge boson $W_R$ and charged Higgs, this limit goes to $\alpha \sim 0.01$.
- It is straightforward to see from Fig. \[fig2\] and Fig. \[fig1\] that the partial width of $H^0_2 \rightarrow \gamma \gamma$ through all available charged particles in loop falls below 1 GeV for small $\alpha$ and can not explain the large decay width preferred by the ATLAS and CMS data.
- In the above, we have considered large mass > 1 TeV for the heavy neutrinos, which is consistent with the collider constraint [@Khachatryan:2014dka]. Therefore, the decay of $H^0_2 \to N_R N_R$ is absent. However, we have checked that even with lighter heavy neutrino masses, the total decay width of $H^0_2$ is not in agreement with the di-photon results.
- The partial decay width to $\gamma \gamma$ via top loop is extremely suppressed $10^{-7}$ GeV, while its decay to $t-\bar{t}$ and $W-W$ is 2.78 GeV and 11.62 GeV for mixing $\theta \sim 0.3$.
[[ The total cross section $\sigma (pp\rightarrow H^0_2 \rightarrow \gamma \gamma) = \sigma(pp\rightarrow H^0_2) \text{BR}(H^0_2 \rightarrow \gamma \gamma)$ for the above mentioned parameter values is $ < $ 1 fb for smaller branching ratio, and clearly can not take into account the required cross-section $\sim $ 10 fb to fit the ATLAS and CMS data. ]{}]{} Considering the other two neutral scalars $H^0_3$ and $A^0_2$ as potential 750 GeV candidate will not significantly improve the situation. This requires beyond MLRSM physics to explain the recently observed di-photon excess at 750 GeV by ATLAS and CMS.
[ [Following the above discussion with a particular set of parameters, we now provide a full parameter scan where we vary the parameters $ \alpha_1=\alpha_2 , \alpha_3, \rho_3, \rho_2, 2\lambda_2-\lambda_3$ in the following ranges.]{}]{} $$\alpha_1 , \alpha_3, \rho_3, \rho_2, 2\lambda_2-\lambda_3 \equiv 10^{-4} - 4 \pi.$$
In Fig. \[figx\] and Fig. \[figdw\], we show the total cross-section and decay width for these ranges of parameters with $\alpha_1=\alpha_2=\alpha$. The blue region corresponds to the following mass cut on the scalars: $$m_{H^{\pm}_{1}}, m_{H^{\pm \pm}_{2}} \ge 380 \; \text{GeV}, \;\; m_{H^0_1} = m_{A^0_1} = m_{H^{\pm}_2} > 10\; \text{TeV}, \, \, m_{H^{\pm \pm}_1} \ge 465 \; \text{GeV}.
\label{masscut}$$ Note that, with the cut on the decay width of $H^0_2 \le 50 $ GeV, the total cross-section is extremely small and can not explain the di-photon result. In addition to this, although for large value of $\alpha$, the cross-section increases, however the branching ratio to di-Higgs also increases. Hence to avoid any constraint from di-Higgs channel, $\alpha$ should be small.
LRSM with Universal Seesaw (LRSM-US) {#sec3}
====================================
The fermion content of the LRSM with universal seesaw is the extension of the MLRSM fermion content by the following vector like fermions $$U_{L} (3, 1, 1, \frac{4}{3}), \;\; U_{R} (3^*, 1, 1, \frac{4}{3})\;\; D_{L} (3, 1, 1, -\frac{2}{3}), \;\;D_{R} (3^*, 1, 1, -\frac{2}{3})$$ $$E_{L,R} (1,1,1, -2), \;\; N_{L,R} (1,1,1,0)$$ each of which comes in three different copies corresponding to the three fermion generations of the MLRSM or the standard model. The presence of these extra fermions is necessary due to the fact that the usual scalar sector of the MSLRM is replaced by the following scalar fields $$H_L (1, 2, 1, -1), \;\;H_R (1,1,2,-1), \;\; \sigma (1,1,1,0)$$ Due to the absence of the usual bidoublet, the left and right handed fermion doublets of the MSLRM can not directly couple to each other. However, they can couple to the scalar fields $H_{L,R}$ via the additional vector like fermions. $$\begin{aligned}
\mathcal{L} & \supset Y_U (\bar{Q_L} H^{\dagger}_L U_L+\bar{Q_R} H^{\dagger}_R U_R) + Y_D (\bar{Q_L} H_L D_L+\bar{Q_R} H_R D_R) +M_U \bar{U_L} U_R+ M_D \bar{D_L} D_R\nonumber \\
& Y_E (\bar{\ell_L} H_L E_L+\bar{\ell_R} H_R E_R) +M_E \bar{E_L} E_R+ \text{h.c.}\end{aligned}$$ where we have ignored the terms corresponding to neutrino masses. For details of the origin of neutrino masses, one may refer to the discussions in [@gulr]. After integrating out the heavy fermions, the charged fermions of the standard model develop Yukawa couplings to the scalar doublet $H_L$ as follows $$y_u = Y_U \frac{v_R}{M_U} Y^T_U, \;\;y_d = Y_D \frac{v_R}{M_D} Y^T_D, \;\;y_e = Y_E \frac{v_R}{M_E} Y^T_E$$ where $v_R$ is the vev of the neutral component of $H_R$. The apparent seesaw then can explain the observed mass hierarchies among the three generations of fermions. The non-zero vev of the neutral component of $H_R$ also breaks the $SU(2)_R\times U(1)_{B-L}$ symmetry of the model into $U(1)_Y$ of the standard model. The left handed Higgs doublet can acquire a non-zero vev $v_L$ at a lower energy to induce electroweak symmetry breaking. However, the left-right symmetry of the theory forces one to have the same vev for both $H_L$ and $H_R$ that is, $v_L=v_R$ which is unacceptable from phenomenological point of view. To decouple these two symmetry breaking scales, the extra singlet scalar $\sigma$ is introduced into the model. This field is odd under the discrete left-right symmetry and hence couple to the two scalar doublets with a opposite sign. After this singlet acquires a non-zero vev at high scale, this generates a difference between the effective mass squared of $H_L$ and $H_R$ which ultimately decouples the symmetry breaking scales.
750 GeV neutral scalar in LRSM-US
=================================
[^1] \[sec4\] The LRSM with universal seesaw has three neutral scalars: one from $H_L$, one from $H_R$ and one from the singlet $\sigma$. Now the neutral scalar part of $H_L$ is the standard model like Higgs with mass 125 GeV. On the other hand the neutral scalars from $H_R$ is supposed to be heavy in order to avoid dangerous flavor changing neutral currents. The scalar $\sigma$ is naturally heavy as it is responsible for discrete left-right symmetry breaking at a scale above $v_R$. To allow the possibility of a neutral 750 GeV scalar, we add another singlet $\zeta$ into the model which can couple to the vector like quarks and leptons as $Y_{f\zeta} \zeta \bar{f_L}f_R$ where $f$ is the vector like fermion. This essentially boils down to the singlet scalar resonance coupled to additional vector like fermions as an explanation of 750 GeV di-photon excess put forward by [@750GeVsinglet].
The singlet scalar can be produced dominantly in pp collisions by two different ways: (a) through mixing with the standard model Higgs and (b) through gluon gluon fusion via new vector like quarks. The singlet scalar can decay into two photons through the vector like fermions in a loop. Since the mixing with the standard model Higgs is constrained, we assume the corresponding production channel to be negligible. We then consider the production of the singlet scalar $\zeta$ in proton proton collisions dominantly through gluon gluon fusion with the vector like quarks in loop. This singlet scalar can decay either into two photons or two gluons or one photon, one Z boson at one loop level whereas the tree level decay into a pair of standard model Higgs can be neglected assuming small mixing. Using the loop level production cross section and decay width expressions given in [@physrepdjouadi], we calculate for what values of vector like fermion masses $m_f$ and their Yukawa couplings $Y_f$, the desired cross section $\sigma(pp\rightarrow \zeta \rightarrow \gamma \gamma)$ can be obtained. For simplicity we consider all the quark and lepton masses and their Yukawa couplings degenerate. Since the masses of vector like leptons are less constrained than that of vector like quarks, we consider vector like lepton masses to be half of vector like quark masses. It should be noted that vector like quark masses are restricted to be $m_q \geq 750-920$ GeV depending on the particular channel of decay [@VLQconstraint] whereas this bound gets relaxed to $m_q \geq 400$ GeV [@VLQconstraint2] for long lived vector like quarks. Further constraints on vector like quarks can be found in [@vlqhandbook]. The constraints on vector like leptons are much weaker $m_l \geq 114-176$ GeV and allows the possibility of the 750 GeV scalar to decay into them at tree level [@VLLconstraint]. We however, do not allow tree level decay of $\zeta$ into vector like leptons which will reduce the branching ratio $\text{BR} (\zeta \rightarrow \gamma \gamma)$. Considering $m_l = m_q/2$, we then constrain the corresponding Yukawa couplings $Y_q, Y_l$ from the requirement of producing the observed signal. The restricted Yukawa couplings for some benchmark values of quark masses are shown in figure \[figyukawa\].
Further constraints on the model comes in terms of the Yukawa couplings involved in the seesaw relations for fermion masses discussed in the previous section. For standard model fermion mass $m_f$, the Yukawa couplings are constrained as $$\frac{y^2 v_R}{M} = \frac{m_f}{v_L}$$ where $M$ is the heavy vector like fermion mass, $v_R$ is the $SU(2)_R$ breaking scale and $v_L=246$ GeV is the electroweak symmetry breaking scale. If $m_f$ is top quark mass and $v_R \approx 6$ TeV, then for $M=1500$ GeV, the corresponding Yukawa couplings are constrained to be $y \approx 0.42$. However, fitting with all the fermion masses will require non-degenerate heavy vector like fermion masses. Another constraints comes from the mixing of these heavy fermions with the standard model fermions. For the case of vector like quarks, such mixings with standard model quarks are constrained to be small $\theta \leq 0.1$ from precision measurements of electroweak parameters [@vlqhandbook]. Parametrising the heavy-light quark mixing from the seesaw relations as $\sin^2\theta \approx \frac{y^2 v_R v_L}{M^2}$ and using the constraints $\theta \leq 0.1, \frac{y^2 v_R v_L}{M} = m_f$, we get the constraints on the heavy quark masses as $M \geq 100 m_f$. This is possible to achieve for $M=1500$ GeV in case of bottom and lighter quark masses. But for top quark seesaw, the corresponding heavy vector like quark has to be much heavier than 1500 GeV considered in this simple analysis.
It should be noted that there are 6 vector like quarks and 3 vector like charged leptons in the model. However, the singlet scalar can not decay into them at tree level. These exotic fermions only appear at one loop to allow the scalar to decay into $gg, \gamma \gamma$ or $\gamma Z$. However, such loop level decay is not enough to generate the total decay width observed by the LHC. Similar observations were also made by [@750GeVsinglet]. If the LHC confirms the measured decay width in future, this will invite further modification to the left-right model considered in this work.
Conclusion {#sec5}
===========
We have studied the minimal left right symmetric extension of the standard model in view of the latest LHC observations of a 750 GeV neutral resonance decaying into two photons with a cross section of around 10 fb. Since the extra neutral scalars (in addition to the 125 GeV Higgs boson) from the bidoublet of MLRSM are very heavy to be in agreement with flavor constraints, we consider the neutral scalar $H^0_2$ from one of the triplet scalars of the model namely, $\Delta_R$. The discussion will be similar for the neutral component of $\Delta_L$. Since the triplet does not couple to quarks, we consider the production of this scalar only through its mixing with the standard model Higgs. We then consider the possible decay of $H^0_2$ and calculate the total as well as partial decay widths. After incorporating the LHC constraints on neutral as well as charged scalar masses, we find that the total cross section $\sigma(pp\rightarrow H^0_0 \rightarrow \gamma \gamma)$ remain below the observed 10 fb signal, after putting constrain on the decay width of $H^0_2 \le 50$ GeV. The cross section is maximal, close to 1 fb only for very high values of the dimensionless parameter $\alpha = \alpha_1 = \alpha_2$ of the scalar potential. This parameter also decides the size of the $H^0_2$ mixing with the standard model like Higgs $H^0_0$ and hence constrained to be $\alpha \leq 0.3 M_{H^0_2}/k_1$, to avoid collider constraint. Thus, the di-photon cross section will be much smaller than 1 fb after taking the constraint on $H^0_0-H^0_2$ mixing into account. It is observed from Fig. \[figx\], that after taking the experimental lower bounds on neutral and charged scalar masses into account, the total cross section gets shifted to higher side. This is due to the fact that, experimental lower bounds on scalar masses also restricts relevant dimensionless parameters to high values for fixed $v_R$. As the same parameters also appear in the decay widths, they increases the total cross section. We also observe that the neutral scalar $H^0_2$ can have a sizeable total decay width $\approx 50$ GeV as observed by the LHC for allowed parameter space though it can not give rise to the 10 fb di-photon signal simultaneously.
We then briefly mention another possible left right model with universal seesaw for fermion masses. Due to the existence of additional vector like fermions, the production of a neutral scalar and its decay into two photons can he enhanced at the same time. By taking some benchmark values of additional fermion masses, we show how their couplings to a neutral 750 GeV scalar get restricted from the requirement of producing a 10 fb di-photon signal. The neutral scalar in such a scenario however, fails to give rise to the large decay width observed by experiment. Thus, if the di-photon cross section as well as the decay width are both confirmed by future LHC data, then further improvement of the left right symmetric models discussed in this work will be required. We leave such an investigation to future work.
The work of M.M was supported by the DST-INSPIRE Faculty grant. The authors would like to thank Ketan M. Patel, IISER Mohali for very useful discussions.
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[^1]: While preparing this manuscript, we found a similar model proposed by [@LR750GeV] with additional field content.
| ArXiv |
---
abstract: 'Analogy to Cherenkov radiation, when a particle moves faster than the propagation velocity of gravitational wave in matter ($v>c_{\rm{g}}$), we expect gravitational wave-Cherenkov radiation (GWCR). In the situation that a photon travels across diffuse dark matters, the GWCR condition is always satisfied, photon will thence loss its energy all the path. This effect is long been ignored in the practice of astrophysics and cosmology, without justification with serious calculation. We study this effect for the first time, and shows that this energy loss time of the photon is far longer than the Hubble time, therefore justify the practice of ignoring this effect in astrophysics context.'
address: |
Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong.\
[email protected]
author:
- 'Shu-Xu Yi'
title: 'On gravitational wave-Cherenkov radiation from photons when passing through diffused dark matters'
---
*Keywords:* Gravitational waves, Cherenkov radiation, diffused dark matters PACS Nos.: 04.30.Db
Introduction to gravitational wave-Cherenkov radiation
======================================================
Generally speaking, when a source of perturbation travels faster than the speed of propagation of that perturbation in medium, the front surfaces of influence from the source at different instances will coherently add up to form a shock wave. In the case that a stream of particles rams into another bulk of gas, if the velocity of the stream is larger than the sound speed in the gas, shock wave will occur; In the case of an electromagnetic field source, e.g. a charged particle, if the velocity of the particle is faster than the speed of light in the medium, an electromagnetic shock wave, i.e., Cherenkov radiation will arise.
When specific to gravitational perturbation, the source is a package of energy-momentum tensor and the propagation speed is the speed of gravitational wave (GW) $c_{\rm{g}}$. If some how the speed of the source can surpass $c_{\rm{g}}$, we would also expect the GW-shock wave, also known as the GW Cherenkov radiation (GWCR). However since the speed of GW is generally believed to be the speed of light in vacuum ($c_{\rm{g}}=c$), GWCR was thought impossible to occur in our ordinary physical world. Some researchers [@1972NPhS..235....6L; @1972NPhS..236...79W; @Schwartz2011] considered GWCR emitted by particles faster than $c$ (Tachyons). Since the existence of Tachyons is not wildly believed, those work receive limited attention. Another way to realize GWCR is to consider slower $c_{\rm{g}}$. Pioneered by ref. , and recently ref. reconsidered the case of possible GWCR from ultra-energetic cosmic rays when $c_{\rm{g}}<c$ due to 4-D Lorentz violation (Rosen’s bimetric theory of gravity in ref. ). They argued that in this case, GWCR sets a maximum travel time of a particle with given momentum. With the observation of energetic protons from $>10$ kpc, they set the upper bound of the difference between $c_g$ and $c$.
In fact, even in ordinary framework, $c_{\rm{g}}$ is less than $c$ when passing through matters due to dispersion. It is interesting to study the GWCR in diffused dark matters, which dominates mass of the universe in large scale. For $c_{\rm{g}}$ in dark matters, we use the formula for dust [@1980grg2.conf..393G]: $$\frac{c^2}{c^2_{\rm{g}}}=1+\frac{4\pi G\rho}{3\omega_{\rm{g}}^2}, (\omega_{\rm{g}}\gg\sqrt{\frac{4\pi G\rho}{3}}).
\label{eqn:1}$$ where $\omega_{\rm{g}}$ is the circular frequency of the GW. For a particle with mass $m$, Lorentz factor $\Gamma$ and energy $E$, the range of the GWCR spectrum is limited by two conditions:
1. [the particle’s velocity is larger than the phase velocity of GW at $\omega_{\rm{g}}$, which gives:]{} $$\omega_{\rm{g}}<\sqrt{\Gamma\frac{4\pi G\rho}{3}}$$ and
2. [the energy of each graviton $\hbar\omega_{\rm{g}}<E$, where $\hbar$ is the Plank constant divided by $2\pi$]{}.
Since $\Gamma=E/mc^2$, for a given $E$, $m\rightarrow0$ gives $\Gamma\rightarrow\infty$. In this case, condition 1 is always satisfied and therefore the GWCR power is max.
Therefore, GWCR from photons traveling through diffused dark matters is the most important scenario to study this effect. This effect is completely ignored in the practice of astrophysics and cosmology, but without serious discussion.
Photons as sources of GWCR
==========================
The energy spectrum of the Cherenkov Gravitational Wave Radiation is [@1994PhLB..336..362P]: $$%p(\omega_g)=\frac{G\omega_gE^2_{\rm{\gamma}}}{c(1+\frac{4\pi G\rho}{\omega_g^2})}\big(\frac{4\pi G\rho}{\omega_g^2}\big)^2,
P(\omega_g)=\frac{G\omega_gE^2_{\rm{\gamma}}}{n^2c^5}(n^2-1)^2.\label{eqn:3}$$ where $n\equiv c/c_{\rm{g}}$ is the refractive index of the GW. Although this formula is derived with non-zero mass particles, we assume it also applies to photons [@2012Optik.123..814G]. The energy losing rate of the photon is: $$\frac{dE}{dt}=-KE_{\rm{\gamma}}^2,$$ where $$\begin{aligned}
K&\equiv&\frac{G}{c^5}\int_{\omega_{g,-}}^{\omega_{g,+}}\frac{\omega_g}{n^2}(n^2-1)^2d\omega_g\nonumber\\
&=&\frac{G}{2c^5}\int_{\omega_{g,-}}^{\omega_{g,+}}\frac{(n^2-1)^2}{n^2}d\omega_g^2.
\label{eqn:2}\end{aligned}$$ From equation (\[eqn:1\]) we know that: $$\omega^2_{\rm{g}}=\frac{2\pi G\rho}{3(n-1)}.$$ take above formula into the integration in equation (\[eqn:2\]): $$\begin{aligned}
K&=&\frac{G^2\rho\pi}{3c^5}\int^{n_+}_{n_-}\frac{(n^2-1)^2}{n^2(n-1)^2}dn\nonumber\\
&\approx&\frac{2G^2\rho\pi}{3c^5}(n_+-n_-)\nonumber\\
&<&\frac{2G^2\rho\pi}{3c^5}.\end{aligned}$$ In the universe, galactic clusters are places with most ambient dark matters, where the average density of dark matters can be up to $\rho=10^{12}\,M_\odot/\rm{Mpc}^3$, where $M_\odot$ is the mass of the sun. Thus the upper limit of $K$ is: $$K<4\times10^{-107}\,\rm{eV^{-1}s^{-1}}.$$ The time life of the photon is: $$\begin{aligned}
\tau&\equiv&E/\dot{E}\nonumber\\
&>&10^{106}\big(\frac{\text{eV}}{E}\big)\,\text{s}.\end{aligned}$$ From the limit of photon-photon scattering process with the cosmology microwave back ground (CMB) and Extragalactic background light (EBL), the energy of photons cannot access $\sim10^{14}\text{eV}$. Therefore the life time of the photon under the GWCR is much longer than Hubble time, i.e., the age of our universe. As a conclusion, a lthough GWCR from photons when passing through diffuse dark matters is nonzero, it hardly leaves clues for its existence.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author appreciates the support from the department of physics, university of Hong Kong. The author thanks helpful discussion with Prof. Wu Kinwah and comments from anonymous reviewer.
[0]{} Caves, C. M. 1980, Annals of Physics, 125, 35 Lapedes, A. S., & Jacobs, K. C. 1972, *Nature Physical Science*, 235, 6 Wimmel, H. K. 1972, *Nature Physical Science*, 236, 79 Schwartz, C. 2011, *Modern Physics Letters A*, 26, 2223 Moore, G. D., & Nelson, A. E. 2001, *Journal of High Energy Physics*, 9, 023 Grado-Caffaro, M. A., & Grado-Caffaro, M. 2012, [*Optik*]{}, 123, 814 Grishchuk, L. P., & Polnarev, A. G. 1980, *General Relativity and Gravitation II*, 2, 393 Pardy, M. 1994, *Physics Letters B*, 336, 362
| ArXiv |
\
[ **On Sudakov and Soft resummations in QCD**]{}\
[ **V. Ravindran** ]{}\
[ *Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad, India.\
*]{}
[**ABSTRACT**]{}
In this article we extract soft distribution functions for Drell-Yan and Higgs production processes using mass factorisation theorem and the perturbative results that are known upto three loop level. We find that they are maximally non-abelien. We show that these functions satisfy Sudakov type integro differential equations. The formal solutions to such equations and also to the mass factorisation kernel upto four loop level are presented. Using the soft distribution function extracted from Drell-Yan production, we show how the soft plus virtual cross section for the Higgs production can be obtained. We determine the threshold resummation exponents upto three loop using the soft distribution function.
0.3 cm
The Drell-Yan(DY) production of di-leptons and Higgs boson production play crucial role in the hadronic colliders. The di-lepton production can not only serve as a luminosity monitor but also provide vital information on physics beyond standard model at present collider Tevatron at Fermi-Lab and future Large Hadron Collider (LHC) which is going to be up at CERN in few years. Higgs production at such colliders will establish the Standard Model(SM) as well as beyond SM Higgs [@Djouadi:2005gi; @Djouadi:2005gj]. From the theoretical side, the DY production of di-leptons and Higgs boson production are known upto Next to Next to leading order(NNLO) level in QCD. For DY at NLO level, see [@Altarelli:1978id] and for the Higgs production at NLO level, see [@Dawson:1990zj; @Djouadi:1991tk; @Spira:1995rr]. The NNLO contribution to DY can be found in [@Matsuura:1987wt; @Matsuura:1988sm; @Hamberg:1990np]. Beyond NLO, the Higgs production cross sections are known only in the large top quark mass limit. For the NNLO soft plus virtual part of the Higgs production, see [@Harlander:2001is; @Catani:2001ic] and the full NNLO for the Higgs production can be found in [@Harlander:2002wh; @Anastasiou:2002yz; @Ravindran:2003um]. Apart from these fixed order results, the resummation programs for the threshold corrections to both DY and Higgs productions have also been very successful [@Sterman:1986aj; @Catani:1989ne]. For next to next to leading logarithmic (NNLL) resummation, see [@Vogt:2000ci; @Catani:2003zt]. Due to several important results at three loop level that are available in recent times [@Moch:2004pa]-[@Blumlein:2004xt], the resummation upto $N^3LL$ has also become reality [@Moch:2005ky; @Laenen:2005uz; @Idilbi:2005ni].
With all these new results in both fixed order as well as resummed calculations, one is now able to unravel the interesting structures in the perturbative results (for example: [@Blumlein:2000wh; @Blumlein:2005im; @Dokshitzer:2005bf]). Along this line, in this paper, we extract the soft distribution functions of Drell-Yan and Higgs production cross sections in perturbative QCD and show that they do not depend on the process under consideration. By that we mean that the soft distribution function of Drell-Yan production can be got entirely from the Higgs production by a simple multiplication of the colour factor $C_F/C_A$. We prove this for the pole parts upto three loop level and for the finite part we could show only to those terms that are not proportional to $\delta(1-z)$ because the three loop finite part proportional to $\delta(1-z)$ is not available yet and can be obtained only from the explicit fixed order computation of bremsstrahlung contribution. The extraction of the soft distribution function is achieved with the help of mass factorisation theorem supplemented by the recent developments in the computation of three loop anomalous dimensions, three loop form factors of quark and gluon operators and two loop bremsstrahlung contributions to Drell-Yan and Higgs productions. We discuss the consequences of our observation in the determination of soft plus virtual cross sections and the threshold resummation exponents. A brief account on the soft and jet distribution functions and the resummation exponents relevant for deep inelastic scattering (DIS) is given.
We start by writing the partonic cross section as $$\begin{aligned}
\hspace{-1cm}
\hat \sigma^{sv}_I(z,q^2,\mu_R^2)&=&
\Big(Z^I(\hat a_s,\mu_R^2,\mu^2)\Big)^2~
|\hat F^I\left(\hat a_s,Q^2,\mu^2\right)|^2~
\delta(1-z)\otimes {\cal C} e^{\displaystyle{2 ~
\Phi^I\left(\hat a_s, q^2,\mu^2,z\right)}},
\nonumber\\[2ex]
&&
\quad \quad \quad
\hspace{9cm}
I=q,g\end{aligned}$$ with the normalisation, $\hat \sigma^{sv}_{I,born}=\delta(1-z)$. The symbol $sv$ means that we restrict to only the soft and virtual contributions to the partonic cross sections $\hat \sigma^{sv}_I$. In the above equation we have introduced a “${\cal C}$ ordered exponential” which has the following expansion: $$\begin{aligned}
{\cal C}e^{\displaystyle f(z) }= \delta(1-z) + {1 \over 1!} f(z)
+{1 \over 2!} f(z) \otimes f(z) + {1 \over 3!} f(z) \otimes f(z) \otimes f(z)
+ \cdot \cdot \cdot\end{aligned}$$ The function $f(z)$ is a distribution of the kind $\delta(1-z)$ and ${\cal D}_i$, where $$\begin{aligned}
{\cal D}_i=\Bigg[{\ln^i(1-z) \over (1-z)}\Bigg]_+
\quad \quad \quad i=0,1,\cdot\cdot\cdot\end{aligned}$$ and the symbol $\otimes$ means the Mellin convolution. The letters $q$ and $g$ stand for Drell-Yan(DY) and Higgs(H) productions respectively. $q^2$($=-Q^2$) is the invariant mass of the final state (di-lepton pair in the case of DY and single Higgs boson for the Higgs production). $z$ is the scaling variable defined as the ratio of $q^2$ over $\hat s$, where $\hat s$ is the center of mass of the partonic system. $F^I(\hat a_s,Q^2,\mu^2)$ are the form factors that enter in the Drell-Yan(for $I=q$) and Higgs(for $I=g$) production cross sections. The functions $\Phi^I(\hat a_s,q^2,\mu^2,z)$ are called the soft distribution functions. The unrenormalised(bare) strong coupling constant $\hat a_s$ is defined as $$\begin{aligned}
\hat a_s={\hat g^2_s \over 16 \pi^2}\end{aligned}$$ where $\hat g_s$ is the strong coupling constant which is dimensionless in $n=4+{\mbox{$\varepsilon$}}$, with $n$ being the number of space time dimensions. The scale $\mu$ comes from the dimensional regularisation in order to make the bare coupling constant $\hat g_s$ dimensionless in $n$ dimensions.
The bare coupling constant $\hat a_s$ is related to renormalised one by the following relation: $$\begin{aligned}
S_{{\mbox{$\varepsilon$}}} \hat a_s = Z(\mu_R^2) a_s(\mu_R^2) \left(\mu^2 \over \mu_R^2\right)^{{\mbox{$\varepsilon$}}\over 2}
\label{renas}\end{aligned}$$ The scale $\mu_R$ is the renormalisation scale at which the renormalised strong coupling constant $a_s(\mu_R)$ is defined. $$\begin{aligned}
S_{{\mbox{$\varepsilon$}}}=exp\left\{{{\mbox{$\varepsilon$}}\over 2} [\gamma_E-\ln 4\pi]\right\}\end{aligned}$$ is the spherical factor characteristic of $n$-dimensional regularisation.
The fact that $\hat a_s$ is independent of the choice of $\mu_R$ leads to the following renormalisation group equation (RGE) for the coupling constant: $$\begin{aligned}
\mu_R^2 {d \ln a_s(\mu_R^2) \over d \mu_R^2}
={{\mbox{$\varepsilon$}}\over 2} + {1 \over a_s(\mu_R^2)}~ \beta(a_s(\mu_R^2))\end{aligned}$$ where $$\begin{aligned}
\beta(a_s(\mu_R^2)) &=& -a_s(\mu_R^2)~\mu_R^2 {d \ln Z(\mu_R^2) \over d \mu_R^2}
=-\sum_{i=0}^\infty a_s^{i+2}(\mu_R^2)~ \beta_i\end{aligned}$$ The solution to the above equation is given by $$\begin{aligned}
Z(\mu_R^2)= 1+ a_s(\mu_R^2) {2 \beta_0 \over {\mbox{$\varepsilon$}}}
+ a_s^2(\mu_R^2) \Bigg({4 \beta_0^2 \over {\mbox{$\varepsilon$}}^2 }+
{\beta_1 \over {\mbox{$\varepsilon$}}} \Bigg)
+ a_s^3(\mu_R^2) \Bigg( {8 \beta_0^3 \over {\mbox{$\varepsilon$}}^3}
+{14 \beta_0 \beta_1 \over 3 {\mbox{$\varepsilon$}}^2}
+{2 \beta_2 \over 3 {\mbox{$\varepsilon$}}}\Bigg)\end{aligned}$$ The renormalisation constant $Z(\mu_R^2)$ relates the bare coupling constant $\hat a_s$ to the renormalised one $a_s(\mu_R^2)$ through the eqn.(\[renas\]). The coefficients $\beta_0$ and $\beta_1$ are $$\begin{aligned}
\beta_0&=&{11 \over 3 } C_A - {4 \over 3 } T_F n_f
\nonumber \\
\beta_1&=&{34 \over 3 } C_A^2-4 T_F n_f C_F -{20 \over 3} T_F n_f C_A\end{aligned}$$ where the color factors for $SU(N)$ QCD are given by $$\begin{aligned}
C_A=N,\quad \quad \quad C_F={N^2-1 \over 2 N} , \quad \quad \quad
T_F={1 \over 2}\end{aligned}$$ and $n_f$ is the number of active flavours. In the case of the Higgs production, the number of active flavours is five because the top degrees of freedom is integrated out in the large $m_{top}$ limit.
The factors $Z^I(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})$ are the overall operator renormalisation constants. For the vector current $Z^q(\hat a_s,\mu_R^2,\mu^2)=1$, but the gluon operator gets overall renormalisation [@Chetyrkin:1997un] given by $$\begin{aligned}
Z^g(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})&=&
1+\hat a_s \left({\mu_R^2 \over \mu^2}\right)^{{\mbox{$\varepsilon$}}\over 2}
S_{{\mbox{$\varepsilon$}}} ~\Bigg[{2 \beta_0 \over {\mbox{$\varepsilon$}}}\Bigg]
+\hat a_s^2 \left({\mu_R^2 \over \mu^2}\right)^{{\mbox{$\varepsilon$}}}
S_{{\mbox{$\varepsilon$}}}^2 ~\Bigg[{2 \beta_1 \over {\mbox{$\varepsilon$}}} \Bigg]
\nonumber\\[2ex]
&& +\hat a_s^3 \left ({\mu_R^2 \over \mu^2}\right)^{3{{\mbox{$\varepsilon$}}\over 2}}
S_{{\mbox{$\varepsilon$}}}^3~ \Bigg[
{1 \over {\mbox{$\varepsilon$}}^2}\Big(-2 \beta_0 \beta_1 \Big)
+{2 \beta_2 \over {\mbox{$\varepsilon$}}}\Bigg]\end{aligned}$$
The bare form factors $\hat F^I(\hat a_s,Q^2,\mu^2)$ (before performing overall renormalisation) of both fermionic and gluonic operators satisfy the following integro differential equation that follows from the gauge as well as renormalisation group invariances [@Sudakov:1954sw; @Mueller:1979ih; @Collins:1980ih; @Sen:1981sd]. In dimensional regularisation, $$\begin{aligned}
Q^2{d \over dQ^2} \ln \hat {F^I}\left(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}}\right)&=&
{1 \over 2 }
\Bigg[K^I\left(\hat a_s,{\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\right)
+ G^I\left(\hat a_s,{Q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\right)
\Bigg]
\label{sud1}\end{aligned}$$ where $K^I$ contains all the poles in ${\mbox{$\varepsilon$}}$. On the other hand, $G^I$ collects rest of the terms that are finite as ${\mbox{$\varepsilon$}}$ becomes zero. In other words $G^I$ contains only non-negative powers of ${\mbox{$\varepsilon$}}$. Since $\hat F^I$ is RG invariant, we find $$\begin{aligned}
\mu_R^2 {d \over d\mu_R^2}
K^I\Bigg(\hat a_s,{\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\Bigg)=-A^I(a_s(\mu_R^2))
\nonumber\\[2ex]
\mu_R^2 {d \over d\mu_R^2}
G^I\Bigg(\hat a_s,{Q^2\over \mu_R^2},
{\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\Bigg)=A^I(a_s(\mu_R^2))\end{aligned}$$ The quantities $A^I$ are the standard cusp anomalous dimensions and they are expanded in powers of renormalised strong coupling constant $a_s(\mu_R^2)$ as $$\begin{aligned}
A^I(\mu_R^2)=\sum_{i=1}^\infty a_s^{i}(\mu_R^2)~ A_i^I\end{aligned}$$ The total derivative is given by $$\begin{aligned}
\mu_R^2 {d \over d\mu_R^2} = \mu_R^2 {\partial \over \partial \mu_R^2}
+{d a_s(\mu_R^2) \over d\mu_R^2} {\partial \over \partial a_s(\mu_R^2)}\end{aligned}$$
The RGE of $K^I$ can be solved in powers of bare coupling constant $\hat a_s$ as $$\begin{aligned}
K^I\left(\hat a_s,{\mu_R^2\over \mu^2},{\mbox{$\varepsilon$}}\right)
=\sum_{i=1}^\infty \hat a_s^i
\left({\mu_R^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}S^i_{{\mbox{$\varepsilon$}}}~ K^{I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ where, $$\begin{aligned}
K^{I,{1}}({\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}} \Bigg(- 2 A_1^I\Bigg)
\nonumber\\[2ex]
K^{I,{2}}({\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}^2} \Bigg(2 \beta_0 A_1^I\Bigg)
+{1 \over {\mbox{$\varepsilon$}}}\Bigg(- A_2^I\Bigg)
\nonumber\\[2ex]
K^{I,{3}}({\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}^3} \Bigg(-{8 \over 3} \beta_0^2 A_1^I\Bigg)
+{1 \over {\mbox{$\varepsilon$}}^2} \Bigg({2 \over 3} \beta_1 A_1^I
+{8 \over 3} \beta_0 A_2^I \Bigg)
+{1 \over {\mbox{$\varepsilon$}}} \Bigg(-{2 \over 3} A_3^I \Bigg)
\nonumber\\[2ex]
K^{I,{4}}({\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}^4} \Bigg(4 \beta_0^3 A_1^I\Bigg)
+{1 \over {\mbox{$\varepsilon$}}^3} \Bigg(-{8 \over 3} \beta_0 \beta_1 A_1^I
- 6 \beta_0^2 A_2^I \Bigg)
\nonumber\\[2ex]
&& +{1 \over {\mbox{$\varepsilon$}}^2} \Bigg({1 \over 3} \beta_2 A_1^I
+\beta_1 A_2^I + 3 \beta_0 A_3^I \Bigg)
+{1 \over {\mbox{$\varepsilon$}}} \Bigg(-{1 \over 2} A_4^I\Bigg)\end{aligned}$$
Similarly RGE for $G^I$ can also be solved and the solution is found to be $$\begin{aligned}
G^I\left(\hat a_s,{Q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\right)
&=& G^I \left(a_s(\mu_R^2),{Q^2 \over \mu_R^2},{\mbox{$\varepsilon$}}\right)
\nonumber\\[2ex]
&=&G^I\left(a_s(Q^2),1,{\mbox{$\varepsilon$}}\right)+ \int_{Q^2 \over \mu_R^2}^1
{d\lambda^2 \over \lambda^2} A^I\left(a_s(\lambda^2 \mu_R^2)\right)\end{aligned}$$ The integral in the above equation can be performed and it is found to be $$\begin{aligned}
\int_{Q^2 \over \mu_R^2}^1 {d \lambda^2 \over \lambda^2}
A^I\left(a_s(\lambda^2 \mu_R^2)\right)
&=& \sum_{i=1}^\infty \hat a_s^i
\left({\mu_R^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}
\left[\left({Q^2 \over \mu_R^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}-1\right]
S^i_{{\mbox{$\varepsilon$}}}~ K^{I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ The finite function $G^I(a_s(Q^2),1,{\mbox{$\varepsilon$}})$ can also be expanded in powers of $a_s(Q^2)$ as $$\begin{aligned}
G^I(a_s(Q^2),1,{\mbox{$\varepsilon$}})=\sum_{i=1}^\infty a_s^i(Q^2)~ G^{I}_i({\mbox{$\varepsilon$}})\end{aligned}$$ After substituting these solutions in the eqn.(\[sud1\]) and performing the final integration, we obtain the following solution $$\begin{aligned}
\ln \hat F^I(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}})
=\sum_{i=1}^\infty \hat a_s^i
\left({Q^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}S^i_{{\mbox{$\varepsilon$}}}~ \hat {\cal L}_F^{I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ where $$\begin{aligned}
\hat {\cal L}_ F^{I,(1)}&=&{1\over {\mbox{$\varepsilon$}}^2} \Bigg(-2 A_1^I\Bigg)
+{1 \over {\mbox{$\varepsilon$}}} \Bigg(G_1^I({\mbox{$\varepsilon$}})\Bigg)
\nonumber\\[2ex]
\hat {\cal L}_ F^{I,(2)}&=&{1\over {\mbox{$\varepsilon$}}^3} \Bigg(\beta_0 A_1^I\Bigg)
+{1\over {\mbox{$\varepsilon$}}^2} \Bigg(-{1 \over 2} A_2^I
- \beta_0 G_1^I({\mbox{$\varepsilon$}})\Bigg)
+{1 \over 2 {\mbox{$\varepsilon$}}} G_2^I({\mbox{$\varepsilon$}})
\nonumber\\[2ex]
\hat {\cal L}_ F^{I,(3)}&=& {1\over {\mbox{$\varepsilon$}}^4} \Bigg(-{8 \over 9}\beta_0^2 A_1^I\Bigg)
+ {1\over {\mbox{$\varepsilon$}}^3} \Bigg({2 \over 9} \beta_1 A_1^I
+{8 \over 9} \beta_0 A_2^I +{4 \over 3}
\beta_0^2 G_1^I({\mbox{$\varepsilon$}})\Bigg)
\nonumber\\[2ex]
&& +{1\over {\mbox{$\varepsilon$}}^2} \Bigg(-{2 \over 9} A_3^I
-{1 \over 3} \beta_1 G_1^I({\mbox{$\varepsilon$}})
-{4 \over 3}\beta_0 G_2^I({\mbox{$\varepsilon$}})\Bigg)
+{1 \over {\mbox{$\varepsilon$}}}\Bigg({1 \over 3} G_3^I({\mbox{$\varepsilon$}})\Bigg)
\nonumber\\[2ex]
\hat {\cal L}_F^{I,(4)}&=& {1\over {\mbox{$\varepsilon$}}^5} \Bigg(\beta_0^3 A_1^I\Bigg)
+{1 \over {\mbox{$\varepsilon$}}^4} \Bigg(-{2 \over 3} \beta_0 \beta_1 A_1^I
-{3 \over 2}\beta_0^2 A_2^I -2 \beta_0^3 G_1^I({\mbox{$\varepsilon$}})\Bigg)
\nonumber\\[2ex]
&& +{1 \over {\mbox{$\varepsilon$}}^3} \Bigg({1 \over 12} \beta_2 A_1^I
+{1 \over 4} \beta_1 A_2^I
+ {3 \over 4}\beta_0 A_3^I +{4 \over 3}
\beta_0 \beta_1 G_1^I({\mbox{$\varepsilon$}})
+3\beta_0^2 G_2^I({\mbox{$\varepsilon$}})\Bigg)
\nonumber\\[2ex]
&& +{1 \over {\mbox{$\varepsilon$}}^2} \Bigg(-{1 \over 8} A_4^I
-{1 \over 6} \beta_2 G_1^I({\mbox{$\varepsilon$}})
-{1 \over 2} \beta_1 G_2^I({\mbox{$\varepsilon$}})
-{3\over 2} \beta_0 G_3^I({\mbox{$\varepsilon$}})\Bigg)
+{1 \over {\mbox{$\varepsilon$}}} \Bigg({1 \over 4} G_4^I({\mbox{$\varepsilon$}})\Bigg)\end{aligned}$$ The above result is in agreement with [@Moch:2005id], which was evaluated using various algorithms designed for solving nested sums. The cusp anomalous dimensions $A^I_i$ and $G^{I}_i({\mbox{$\varepsilon$}})$ are known upto order $a_s^3$. The cusp anomalous dimensions are maximally non-abelien and hence satisfy the following relation: $$\begin{aligned}
A^q={ C_F \over C_A }~A^g \end{aligned}$$
The coefficients $G^{I}_i({\mbox{$\varepsilon$}})$ can be found for both $I=q$ and $I=g$ in [@Moch:2005tm] to the required accuracy in ${\mbox{$\varepsilon$}}$. They satisfy $$\begin{aligned}
G^{I}_1({\mbox{$\varepsilon$}})&=&
2 (B^I_1 - \delta_{I,g} \beta_0) + f_1^I
+\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k g^{~I,k}_1
\nonumber \\[2ex]
G^{I}_2({\mbox{$\varepsilon$}})&=&
2(B_2^I-2 \delta_{I,g} \beta_1) + f_2^I
-2 \beta_0 g^{~I,1}_1
+\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k g^{~I,k}_2
\nonumber \\[2ex]
G^{I}_3({\mbox{$\varepsilon$}})&=&
2 (B_3^I - 3\delta_{I,g} \beta_2) + f_3^I
-2 \beta_1 g^{~I,1}_1
-2 \beta_0 \Big(g^{~I,1}_2+2 \beta_0 g^{~I,2}_1\Big)
\nonumber \\[2ex]
&& +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k g^{~I,k}_3\end{aligned}$$ The constants $B_i^I$ are also known upto order $a_s^3$ thanks to the recent computation of three loop anomalous dimensions/splitting functions [@Moch:2004pa; @Vogt:2004mw].
The constants $f_i^I$ are analogous to the cusp anomalous dimensions $A_i^I$ that enter the form factors. It was first noticed in [@Ravindran:2004mb] that the single pole (in ${\mbox{$\varepsilon$}}$) of the logarithm of form factors upto two loop level ($a_s^2$) can be predicted due the presence of these constants $f_i^I$ because they are found to be maximally non-abelien obeying the relation $$\begin{aligned}
f_i^q={C_F\over C_A} f_i^g\end{aligned}$$ similar to $A_i^I$. In [@Moch:2005tm], this relation has been found to hold even at the three loop level.
The partonic cross sections $\hat \sigma^{sv}_I(z,q^2,\mu_R^2)$ is UV finite after the coupling constant and overall operator renormalisations are performed using $Z(\mu_R^2)$ and $Z^I(\mu_R^2)$. But they still require mass factorisation in order to remove the collinear divergences: $$\begin{aligned}
\hat \sigma^{sv}_I(z,q^2,\mu_R^2,{\mbox{$\varepsilon$}})=
\Gamma^T(z,\mu_F^2,{\mbox{$\varepsilon$}})\otimes
\Delta^{sv}_{I}\left(z,q^2,\mu_R^2,\mu_F^2\right) \otimes\Gamma(z,\mu_F^2,{\mbox{$\varepsilon$}})\end{aligned}$$ with $\mu_F$ being the factorisation scale. The resulting coefficient functions $\Delta^{sv}_{I}(z,q^2,\mu_R^2,\mu_F^2)$ are finite and free of collinear singularities. $$\begin{aligned}
\Delta^{sv}_{I}(z,q^2,\mu_R^2,\mu_F^2)=
\delta(1-z)+\sum_{i=1}^{\infty}a_s^i(\mu_R^2)~ \Delta_{I}^{sv,(i)}
\left(z,q^2,\mu_R^2,\mu_F^2\right)\end{aligned}$$ The coefficient functions $\Delta_{I}^{sv,(i)}$ for $i=1,2$ are known(see [@Dawson:1990zj] to [@Ravindran:2003um]). The partial result for $\Delta_{I}^{sv,(3)}$ (i.e., all ${\cal D}_i$ except $\delta(1-z)$ are known for $i=3$) is also available(see [@Moch:2005ky]).
The kernel $\Gamma(z,\mu_F^2,{\mbox{$\varepsilon$}})$ satisfies the following renormalisation group equation: $$\begin{aligned}
\mu_F^2 {d \over d\mu_F^2}\Gamma(z,\mu_F^2,{\mbox{$\varepsilon$}})={1 \over 2} P
\left(z,\mu_F^2\right)
\otimes \Gamma \left(z,\mu_F^2,{\mbox{$\varepsilon$}}\right)\end{aligned}$$ The $P(z,\mu_F^2)$ are well known Altarelli-Parisi splitting functions(matrix valued) known upto three loop level [@Moch:2004pa; @Vogt:2004mw]: $$\begin{aligned}
P(z,\mu_F^2)=
\sum_{i=1}^{\infty}a_s^i(\mu_F^2) P^{(i-1)}(z)\end{aligned}$$ The diagonal terms of splitting functions $P^{(i)}(z)$ have the following structure $$\begin{aligned}
P^{(i)}_{II}(z) = 2\Bigg[ B^I_{i+1} \delta(1-z)
+ A^I_{i+1} {\cal D}_0\Bigg] + P_{reg,II}^{(i)}(z)\end{aligned}$$ where $P_{reg,II}^{(i)}$ are regular when the argument takes the kinematic limit(here $z \rightarrow 1$). The RGE of the kernel can be solved in dimensional regularisation in powers of strong coupling constant. Since we are interested only in the soft plus virtual part of the cross section, only the diagonal parts of the kernels contribute. In the $\overline{MS}$ scheme, the kernel contains only poles in ${\mbox{$\varepsilon$}}$. Expanding the kernel in powers of bare coupling $\hat a_s$, $$\begin{aligned}
\Gamma(z,\mu_F^2,{\mbox{$\varepsilon$}})=\delta(1-z)+\sum_{i=1}^\infty \hat a_s^i
\left({\mu_F^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}S^i_{{\mbox{$\varepsilon$}}}
\Gamma^{(i)}(z,{\mbox{$\varepsilon$}})\end{aligned}$$ we can solve the RGE for the kernel. The solutions in the $\overline{MS}$ scheme are given by $$\begin{aligned}
\Gamma_{II}^{(1)}(z,{\mbox{$\varepsilon$}})&=&{1 \over {\mbox{$\varepsilon$}}} {P_{II}^{(0)}(z)}\nonumber\\[2ex]
\Gamma_{II}^{(2)}(z,{\mbox{$\varepsilon$}})&=&
{1 \over {\mbox{$\varepsilon$}}^2}\Bigg({1 \over 2} {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}-\beta_0 {P_{II}^{(0)}(z)}\Bigg)
+{1 \over {\mbox{$\varepsilon$}}} \Bigg({1 \over 2} {\mbox{$P_{II}^{(1)}(z)$}}\Bigg)
\nonumber\\[2ex]
\Gamma_{II}^{(3)}(z,{\mbox{$\varepsilon$}})&=&
{1 \over {\mbox{$\varepsilon$}}^3}\Bigg(
{4 \over 3} \beta_0^2 {P_{II}^{(0)}(z)}-\beta_0 {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\nonumber\\[2ex]
&& +{1 \over 6} {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\Bigg) +
{1 \over {\mbox{$\varepsilon$}}^2} \Bigg( {1 \over 2} {P_{II}^{(0)}(z)}\otimes {\mbox{$P_{II}^{(1)}(z)$}}\nonumber\\[2ex]
&& -{1 \over 3} \beta_1 {P_{II}^{(0)}(z)}-{4 \over 3} \beta_0 {\mbox{$P_{II}^{(1)}(z)$}}\Bigg)
+{1 \over {\mbox{$\varepsilon$}}} \Bigg({1 \over 3} {\mbox{$P_{II}^{(2)}(z)$}}\Bigg)
\nonumber\\[2ex]
\Gamma_{II}^{(4)}(z,{\mbox{$\varepsilon$}})&=&
{1 \over {\mbox{$\varepsilon$}}^4}\Bigg(
{1 \over 24} {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\nonumber\\[2ex]
&& -{1 \over 2} \beta_0 {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}+{11 \over 6} \beta_0^2 {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\nonumber\\[2ex]
&& -2 \beta_0^3 {P_{II}^{(0)}(z)}\Bigg)
\nonumber\\[2ex]
&& +{1 \over {\mbox{$\varepsilon$}}^3}\Bigg(
{1 \over 4} {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {\mbox{$P_{II}^{(1)}(z)$}}-{1 \over 3} \beta_1 {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\nonumber\\[2ex]
&& -{11 \over 6} \beta_0 {P_{II}^{(0)}(z)}{\mbox{$P_{II}^{(1)}(z)$}}+{4 \over 3} \beta_0 \beta_1 {P_{II}^{(0)}(z)}+3 \beta_0^2 {\mbox{$P_{II}^{(1)}(z)$}}\Bigg)
\nonumber\\[2ex]
&& +{1 \over {\mbox{$\varepsilon$}}^2}\Bigg(
{1\over 3}{P_{II}^{(0)}(z)}\otimes {\mbox{$P_{II}^{(2)}(z)$}}+{1 \over 8} {\mbox{$P_{II}^{(1)}(z)$}}\otimes {\mbox{$P_{II}^{(1)}(z)$}}-{1 \over 6} \beta_2 {P_{II}^{(0)}(z)}\nonumber\\[2ex]
&& -{1 \over 2} \beta_1 {\mbox{$P_{II}^{(1)}(z)$}}-{3 \over 2} \beta_0 {\mbox{$P_{II}^{(2)}(z)$}}\Bigg)
+{1 \over {\mbox{$\varepsilon$}}} \Bigg({1 \over 4} {\mbox{$P_{II}^{(3)}(z)$}}\Bigg)\end{aligned}$$ It is now straightforward to obtain the soft distribution functions $\Phi^I(\hat a_s,q^2,\mu^2,z)$ from the available results known upto three loop level for the form factors $\hat F^I$, the kernels $\Gamma_{II}$ and the coefficient functions $\Delta^{sv}_I$(the $\delta(1-z)$ function part of $\Delta^{sv,(3)}_I$ is still unknown). The fact that $\Delta^{sv}_I$ are finite in the limit ${\mbox{$\varepsilon$}}\rightarrow 0$ implies that the soft distribution functions have pole structure in ${\mbox{$\varepsilon$}}$ similar to that of $\hat F^I$ and $\Gamma_{II}$. Also, $\Phi^I(\hat a_s,q^2,\mu^2,z)$ satisfy the renormalisation group equation: $$\begin{aligned}
\mu_R^2 {d \over d\mu_R^2}\Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})=0\end{aligned}$$ From the above observations, it is natural to expect that the soft distribution functions also satisfy Sudakov type integro differential equation that the form factors $\hat F^I(Q^2)$ satisfy(see eqn.(\[sud1\])). Hence, $$\begin{aligned}
q^2 {d \over dq^2}\Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}}) =
{1 \over 2 }
\Bigg[\overline K^{~I}\left(\hat a_s,{\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\right)
+ \overline G^{~I}\left(\hat a_s,{q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\right)
\Bigg]
\label{sud2}\end{aligned}$$ where again $\overline K^{~I}$ contains all the singular terms and $\overline G^{~I}$ are finite functions of ${\mbox{$\varepsilon$}}$. The renormalisation group invariance leads to $$\begin{aligned}
\mu_R^2 {d\over d\mu_R^2} \overline K^{~I}
\Bigg(\hat a_s, {\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\Bigg)=
-\overline A^{~I}(a_s(\mu_R^2)) \delta(1-z)
\nonumber\\[2ex]
\mu_R^2 {d \over d\mu_R^2} \overline G^{~I}
\Bigg(\hat a_s,{q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\Bigg)
=\overline A^{~I}(a_s(\mu_R^2)) \delta(1-z)\end{aligned}$$ If $\Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})$ have to contain the right poles to cancel the poles coming from $\hat F^I$,$Z^I$ and $\Gamma_{II}$ in order to make $\Delta^{sv}_I$ finite, then $\overline A^{~I}$ have to satisfy $$\begin{aligned}
\overline A^{~I}=-A^I\end{aligned}$$ The above relation along with the renormalisation group invariance implies that $$\begin{aligned}
\overline G^{~I}\left(\hat a_s,{q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\right)
&=&\overline G^{~I} \left(a_s(\mu_R^2),{q^2 \over \mu_R^2},z,{\mbox{$\varepsilon$}}\right)
\nonumber\\[2ex]
&=&\overline G^{~I}\left(a_s(q^2),1,z,{\mbox{$\varepsilon$}}\right)
- \delta(1-z) \int_{q^2 \over \mu_R^2}^1
{d\lambda^2 \over \lambda^2} A^I\left(a_s(\lambda^2 \mu_R^2)\right)\end{aligned}$$
Now it is now straight forward to determine all $\overline G^{~I}(a_s(q^2),1,z,{\mbox{$\varepsilon$}})$ from the available informations. The functions $\overline G^{~I}(a_s(q^2),1,z,{\mbox{$\varepsilon$}})$ can be expanding in powers of $a_s(q^2)$ as $$\begin{aligned}
\overline G^{~I}(a_s(q^2),1,z,{\mbox{$\varepsilon$}})=\sum_{i=1}^{\infty} a_s^i(q^2)
~\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})\end{aligned}$$ The solution to the eqn(\[sud2\]) can be obtained in the way we obtained $\ln \hat F^I(Q^2)$. Expanding the soft distribution functions in powers of bare coupling $\hat a_s$ as $$\begin{aligned}
\Phi^I\left(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}}\right)=\sum_{i=1}^\infty \hat a_s^i
\left({q^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}S^i_{{\mbox{$\varepsilon$}}}
~\hat \Phi^{I,(i)}(z,{\mbox{$\varepsilon$}})\end{aligned}$$ we find the solution: $$\begin{aligned}
\hat \Phi^{I,(i)}(z,{\mbox{$\varepsilon$}}) = \hat {\cal L}_F^{I,(i)}({\mbox{$\varepsilon$}})
\Bigg(A^I\rightarrow -\delta(1-z)~A^I,~~
G^I({\mbox{$\varepsilon$}}) \rightarrow \overline G^{~I}(z,{\mbox{$\varepsilon$}})\Bigg)\end{aligned}$$ The finite functions $\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})$ can be obtained using the mass factorisation formula by demanding the finiteness of the coefficient functions $\Delta^{sv,(i)}_I$. The RG invariance of theses soft functions and the simple rescaling $q \rightarrow (1-z) q$ imply that the following expansion is also the solution to the integro differential equation: $$\begin{aligned}
\Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}}) &=& \Phi^I(\hat a_s,q^2 (1-z)^2,\mu^2,{\mbox{$\varepsilon$}})
\nonumber\\[2ex]
&=&\sum_{i=1}^\infty \hat a_s^i \left({q^2 (1-z)^2 \over \mu^2}\right)^{i
{{\mbox{$\varepsilon$}}\over 2}} S_{{\mbox{$\varepsilon$}}}^i \left({i~ {\mbox{$\varepsilon$}}\over 1-z} \right)\hat \phi^{~I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ where $$\begin{aligned}
\hat \phi^{I,(i)}({\mbox{$\varepsilon$}})=\hat {\cal L}_F^{I,(i)}({\mbox{$\varepsilon$}})
\Bigg( A^I \rightarrow - A^I, G^I({\mbox{$\varepsilon$}}) \rightarrow \overline {\cal G}^I({\mbox{$\varepsilon$}})
\Bigg)\end{aligned}$$ The $z$ independent constants $\overline {\cal G}^I({\mbox{$\varepsilon$}})$ in $\hat \phi^{~I,(i)}({\mbox{$\varepsilon$}})$ can be obtained using the form factors, mass factorisation kernels and coefficient functions $\Delta^{sv,(i-1)}_I$ expanded in powers of ${\mbox{$\varepsilon$}}$ to the desired accuracy. This is achieved by comparing the poles as well as non-pole terms in ${\mbox{$\varepsilon$}}$ of $\hat \phi^{~I,(i)}({\mbox{$\varepsilon$}})$ with those coming from the form factors, overall renormalisation constants and splitting functions and the lower order $\Delta^{sv,(i-1)}_I$. We find $$\begin{aligned}
\overline {\cal G}^{~I}_1({\mbox{$\varepsilon$}})&=&-f_1^I+
\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \overline {\cal G}^{~I,(k)}_1
\nonumber\\[2ex]
\overline {\cal G}^{~I}_2({\mbox{$\varepsilon$}})&=&-f_2^I
-2 \beta_0 \overline{\cal G}_1^{~I,(1)}
+\sum_{k=1}^\infty{\mbox{$\varepsilon$}}^k \overline {\cal G}^{~I,(k)}_2
\nonumber\\[2ex]
\overline {\cal G}^{~I}_3({\mbox{$\varepsilon$}})&=&-f_3^I
-2 \beta_1 \overline{\cal G}_1^{~I,(1)}
-2 \beta_0 \left(\overline{\cal G}_2^{~I,(1)}
+2 \beta_0 \overline{\cal G}_1^{~I,(2)}\right)
+\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \overline {\cal G}^{~I,(k)}_3\end{aligned}$$ with $$\begin{aligned}
\overline{\cal G}^{I,(1)}_1&=& C_I~ \overline{\cal G}^{~(1)}_1
\nonumber\\[2ex]
&=&C_I~ \Big(-3 \zeta_2\Big)
\nonumber\\[2ex]
\overline{\cal G}^{I,(2)}_1&=&C_I~ \overline{\cal G}^{~(2)}_1
\nonumber\\[2ex]
&=& C_I~ \Bigg({7 \over 3} \zeta_3\Bigg)
\nonumber\\[2ex]
\overline{\cal G}^{I,(1)}_2&=&C_I~ \overline{\cal G}^{~(1)}_2
\nonumber\\[2ex]
&=& C_I C_A~ \Bigg({2428 \over 81} -{469 \over 9} \zeta_2
+4 \zeta_2^2 -{176 \over 3} \zeta_3\Bigg)
\nonumber\\[2ex]
&& +C_I n_f~ \Bigg(-{328 \over 81} + {70 \over 9} \zeta_2
+{32 \over 3} \zeta_3 \Bigg) \end{aligned}$$ where $C_I=C_F$ for $I=q$(DY) and $C_I=C_A$ for $I=g$(Higgs).
Using such compensating $\hat \phi^{I,(i)}({\mbox{$\varepsilon$}})$ and the following expansion, $$\begin{aligned}
{1 \over 1-z} \big[(1-z)^2\big]^{i {{\mbox{$\varepsilon$}}\over 2}}
={1 \over i {\mbox{$\varepsilon$}}}\delta(1-z)
+ \sum_{j=0}^{\infty} { (i {\mbox{$\varepsilon$}})^j \over j!} {\cal D}_j\end{aligned}$$ we obtain $\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})$ upto three loop level. We find that the finite functions $\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})$ have the following decomposition in terms of cusp anomalous dimension $A^I_i$ and $f^I_i$ that appear in the form factors: $$\begin{aligned}
\overline G^{{~I}}_1&=& -f_1^I ~\delta(1-z)+2 A_1^I~ {\cal D}_0
+\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \overline g^{~I,k}_1(z)
\nonumber \\[2ex]
\overline G^{{~I}}_2&=& -f_2^I ~\delta(1-z)+2 A_2^I~ {\cal D}_0
-2 \beta_0 ~\overline g^{~I,1}_1(z)
+\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k ~\overline g^{~I,k}_2(z)
\nonumber \\[2ex]
\overline G^{{~I}}_3&=& -f_3^I ~\delta(1-z)+2 A_3^I~ {\cal D}_0
-2 \beta_1 ~\overline g^{~I,1}_1(z)
-2 \beta_0 \Big(\overline g^{~I,1}_2(z)+2 \beta_0 ~\overline g^{~I,2}_1(z)\Big)
\nonumber \\[2ex]
&& +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k ~\overline g^{~I,k}_3(z)\end{aligned}$$ where $$\begin{aligned}
\overline g^{~I,1}_1&=&C_I \Bigg( 8 {\cal D}_1 -3 \zeta_2 \delta(1-z)\Bigg)
\nonumber\\[2ex]
\overline g^{~I,2}_1&=&C_I \Bigg( -3 \zeta_2 {\cal D}_0 +
4 {\cal D}_2 +{7 \over 3} \zeta_3 \delta(1-z)\Bigg)
\nonumber\\[2ex]
\overline g^{~I,3}_1&=&C_I \Bigg( {7 \over 3} \zeta_3 {\cal D}_0 -
3 \zeta_2 {\cal D}_1
+{4 \over 3} {\cal D}_3
-{3 \over 16} \zeta_2^2 \delta(1-z)\Bigg)
\nonumber\\[2ex]
\overline g^{~I,1}_2&=&C_I C_A \Bigg(
\Bigg(-{1616 \over 27} +{242 \over 3} \zeta_2
+56 \zeta_3 \Bigg) {\cal D}_0
+\Bigg({1072 \over 9} -32 \zeta_2\Bigg) {\cal D}_1
+\Big(-88\Big) {\cal D}_2
\nonumber\\[2ex]
&& +\Bigg({2428 \over 81} -{469 \over 9} \zeta_2
+4 \zeta_2^2 -{176 \over 3} \zeta_3\Bigg) \delta(1-z)\Bigg)
+C_I n_f \Bigg(\Bigg({224 \over 27}
-{44 \over 3} \zeta_2\Bigg) {\cal D}_0
\nonumber\\[2ex]
&& +\Bigg(-{160\over 9}\Bigg) {\cal D}_1
+16 {\cal D}_2
+\Bigg(-{328 \over 81} + {70 \over 9} \zeta_2
+{32 \over 3} \zeta_3 \Bigg) \delta(1-z)\Bigg)
\Bigg)
\nonumber\\[2ex]
\overline g^{~I,2}_2&=&C_I C_A \Bigg( \Bigg(
{4856 \over 81} - {938 \over 9} \zeta_2
+8 \zeta_2^2 -{1210 \over 9} \zeta_3
\Bigg) {\cal D}_0
+\Bigg(-{3232 \over 27}+ {550 \over 3} \zeta_2
+112 \zeta_3\Bigg) {\cal D}_1
\nonumber\\[2ex]
&& +\Bigg({1072 \over 9} -32 \zeta_2\Bigg) {\cal D}_2
+\Bigg(-{616 \over 9} \Bigg){\cal D}_3 \Bigg)
+C_I n_f \Bigg(
\Bigg(-{656 \over 81} +{140 \over 9} \zeta_2
+{220 \over 9} \zeta_3 \Bigg) {\cal D}_0
\nonumber\\[2ex]
&& +\Bigg({448 \over 27} -{100 \over 3} \zeta_2\Bigg) {\cal D}_1
+\Bigg( -{160 \over 9} \Bigg){\cal D}_2
+\Bigg( {112 \over 9}\Bigg) {\cal D}_3\Bigg)
\Bigg)
\nonumber\\[2ex]
&& + \delta(1-z) \delta \overline g^{g,2}_2
\nonumber\\[2ex]
\overline g^{~I,1}_3&=&
C_I C_A^2 \Bigg(
\Bigg(
-{403861 \over 243}-176 \zeta_2 \zeta_3
+{71584 \over 27} \zeta_2
-{5368 \over 15} \zeta_2^2 +{9272\over 3} \zeta_3
-576 \zeta_5
\Bigg){\cal D}_0
\nonumber\\[2ex]
&& + \Bigg(
{257140 \over 81} -{28696 \over 9} \zeta_2
+{1056 \over 5} \zeta_2^2
-{5632 \over 3} \zeta_3
\Bigg){\cal D}_1
+ \Bigg(
-{68752 \over 27} +{1760 \over 3} \zeta_2
\Bigg){\cal D}_2
\nonumber\\[2ex]
&& + \Bigg(
{7744 \over 9}
\Bigg){\cal D}_3
\Bigg)
+C_I C_A n_f\Bigg(
\Bigg(
{96482 \over 243} -{2452 \over 3} \zeta_2
+{1264 \over 15} \zeta_2^2
-{6536 \over 9} \zeta_3
\Bigg){\cal D}_0
\nonumber\\[2ex]
&&+\Bigg(
-{72008 \over 81}+{9056 \over 9} \zeta_2
+{448 \over 3} \zeta_3
\Bigg){\cal D}_1
+\Bigg(
{22400 \over 27}-{320 \over 3} \zeta_2
\Bigg){\cal D}_2
+\Bigg(
-{2816 \over 9}
\Bigg){\cal D}_3\Bigg)
\nonumber\\[2ex]
&&+C_I n_f^2\Bigg(
\Bigg(
-{4480 \over 243}+{1520 \over 27} \zeta_2
+{416 \over 9} \zeta_3
\Bigg){\cal D}_0
+\Bigg(
{4192 \over 81}-{736 \over 9} \zeta_2
\Bigg){\cal D}_1
\nonumber\\[2ex]
&&+\Bigg(
-{1600 \over 27}
\Bigg){\cal D}_2
+\Bigg(
{256 \over 9}
\Bigg){\cal D}_3
\Bigg)
+C_I C_F n_f\Bigg(
\Bigg(
{1711 \over 9} -60 \zeta_2
-{96 \over 5}\zeta_2^2-{304 \over 3} \zeta_3
\Bigg){\cal D}_0
\nonumber\\[2ex]
&&+\Bigg(
-220 +192 \zeta_3
\Bigg){\cal D}_1
+\Bigg(
64
\Bigg){\cal D}_2
\Bigg)+\delta\overline g^{g,1}_3 \delta(1-z)\end{aligned}$$ In the above equation $\delta \overline g^{g,2}_2,\delta\overline g^{g,1}_3$ are not known because the full fixed order $N^3LO$ computation for the soft part of the cross section is not available yet.
With the available informations(ignoring $\delta \overline g^{g,2}_2,\delta \overline g^{g,1}_3$), we find that the soft distribution functions for DY and Higgs productions are maximally non-abelien: $$\begin{aligned}
\Phi^q(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})={C_F \over C_A}~
\Phi^g(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}}) \end{aligned}$$ upto three loop level. At the cross section level($\Delta_I^{sv}$), this property does not show up because of the form factors which do not have this property. The overall factors $C_F$ and $C_A$ ordinate from the leading order contributions to the soft distribution functions. Hence if you factor out this colour factor ($C_F$ for the DY and $C_A$ for the Higgs) we find that the soft distribution functions are universal. $$\begin{aligned}
\Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})=C_I \Phi(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})\end{aligned}$$ The universality of the soft distribution functions can be understood if you notice that the soft part of the cross section is always independent of the spin, colour, flavour or any other quantum numbers after factoring out the born level cross section. It depends only on the gauge interaction, here it is $SU(N)$. This universal property can be utilised to compute soft part of the any new cross section where incoming particles carry any spin,colour,flavour or other quantum numbers. For example, if we know $\Phi(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})$ extracted from the Drell-Yan production results, we can predict the $\Delta^{sv}_g(z,q^2,\mu_R^2,\mu_F^2)$ for the Higgs production using mass factorisation formula provided we know the gluon form factor $F^g(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}})$ and the overall renormalisation constant $Z^g(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})$.
The soft plus virtual part of the cross section ($\Delta^{sv}_I(z,q^2,\mu_R^2,\mu_F^2)$) using mass factorisation formula is found to be $$\begin{aligned}
\Delta^{sv}_I(z,q^2,\mu_R^2,\mu_F^2)={\cal C} \exp
\Bigg({\Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}})}\Bigg)\Bigg|_{{\mbox{$\varepsilon$}}=0}\end{aligned}$$ where $\Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}})$ is a finite distribution. The $\mu_R$ dependence comes from the coupling constant and operator renormalisation: $$\begin{aligned}
\Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}})&=&
\Bigg(
\ln \Big(Z^I(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})\Big)^2
+\ln \big|\hat F^I(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}})\big|^2
\Bigg)
\delta(1-z)
\nonumber\\[2ex]
&&+2 C_I \Phi(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})
-2~ {\cal C}\ln \Gamma_{II}(\hat a_s,\mu^2,\mu_F^2,z,{\mbox{$\varepsilon$}})\end{aligned}$$ In the above equation “${\cal C} \ln$” means the “convolution ordered” logarithm. All the products of distributions in the logarithmic expansion are understood as Mellin convolutions. The distribution $\Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}})$ is regular as ${\mbox{$\varepsilon$}}\rightarrow 0$. The soft plus virtual cross section can be obtained by expanding $\Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}}=0)$ as $$\begin{aligned}
\Psi^I(z,q^2,\mu_F^2,{\mbox{$\varepsilon$}})
=\sum_{i=1} a_s^i(\mu_F^2) \Psi^{~I,(i)}(z,q^2,\mu_F^2) \end{aligned}$$ where we have set $\mu_R=\mu_F$ and expressing $a_s(\mu_F^2)$ in terms of $a_s(\mu_R^2)$ is straightforward. We find that the cross sections $\Delta^{sv}_I(z,q^2,\mu_F^2)$ can be obtained using $$\begin{aligned}
\Delta^{sv,(0)}_I(z,q^2,\mu_F^2)&=&C_I \delta(1-z)
\nonumber\\[2ex]
\Delta^{sv,(1)}_I(z,q^2,\mu_F^2)&=&\Psi^{~I,(1)}(z,q^2,\mu_F^2)
\nonumber\\[2ex]
\Delta^{sv,(1)}_I(z,q^2,\mu_F^2)&=&\Psi^{~I,(2)}(z,q^2,\mu_F^2)
+{1 \over 2} \Psi^{~I,(1)}(z,q^2,\mu_F^2)
\otimes \Psi^{~I,(1)}(z,q^2,\mu_F^2)
\nonumber\\[2ex]
\Delta^{sv,(3)}_I(z,q^2,\mu_F^2)&=&\Psi^{~I,(3)}(z,q^2,\mu_F^2)+\Psi^{~I,(1)}
(z,q^2,\mu_F^2)\otimes \Psi^{~I,(2)}(z,q^2,\mu_F^2)
\nonumber\\[2ex]
&&+{1 \over 6} \Psi^{~I,(1)}(z,q^2,\mu_F^2) \otimes \Psi^{~I,(1)}(z,q^2,\mu_F^2)
\otimes \Psi^{~I,(1)}(z,q^2,\mu_F^2)\end{aligned}$$ where $$\begin{aligned}
\Psi^{~I,(1)}&=&
\Big((-2 \beta_0 \delta_{I,g}+ 2 B^I_1)\delta(1-z)
+2 A^I_1 {\cal D}_0 \Big)\ln\left({q^2 \over \mu_F^2}\right)
+\Big(3 \zeta_2 A^I_1 +2 g_1^{~I,1}
\nonumber\\[2ex]
&&+ 2 C_I \overline{\cal G}_1^{~(1)}\Big)\delta(1-z)
+ \Big(4 A^I_1\Big){\cal D}_1
\nonumber\\[2ex]
\Psi^{~I,(2)}&=&
\Bigg[6 \zeta_2 \beta_0 B^I_1
-6 \zeta_2 \beta_0^2 \delta_{I,g}
+2 \beta_0 g_1^{~I,2}
+g_2^{~I,1}
+3 \zeta_2 A^I_2
+2 \beta_0 C_I \overline {\cal G}_1^{~(2)}
+C_I \overline {\cal G}_2^{~(1)}
\nonumber\\[2ex]
&&
+\Bigg(
-4 \beta_1 \delta_{I,g}
-2 \beta_0 g_1^{(1)}
+2 B^I_2
-3\zeta_2 \beta_0 A^I_1
-2 \beta_0 C_I \overline {\cal G}_1^{~(1)}\Bigg)
\ln\left({q^2 \over \mu_F^2}\right)
\nonumber\\[2ex]
&&+\Big(-\beta_0 B^I_1
+\beta_0^2 \delta_{I,g}\Big) \ln^2\left({q^2 \over \mu_F^2}\right)
\Bigg]\delta(1-z)
+\Bigg[
-4 \beta_0 C_I \overline {\cal G}_1^{~(1)}
-2 f_2^I
+2 A^I_2 \ln\left({q^2 \over \mu_F^2}\right)
\nonumber\\[2ex]
&&-\beta_0 A^I_1 \ln^2\left({q^2 \over \mu_F^2}\right)
\Bigg] {\cal D}_0
+\Bigg[4 A^I_2
-4 \beta_0 A^I_1 \ln\left({q^2 \over \mu_F^2}\right)
\Bigg]{\cal D}_1
+\Bigg[-4 \beta_0 A^I_1 \Bigg]{\cal D}_2
\nonumber\\[2ex]
\Psi^{~I,(3)}&=&
\Bigg[-30 \zeta_2 \beta_0 \beta_1\delta_{I,g}
+12\zeta_2 \beta_0 B^I_2
-12 \zeta_2 \beta_0^2 g_1^{~I,(1)}
+ 6 \zeta_2 \beta_1 B^I_1
+{4 \over 3} \beta_0 g_2^{~I,(2)}
+{8 \over 3} \beta_0^2 g_1^{~I,(3)}
\nonumber\\[2ex]
&&+{4 \over 3} \beta_1 g_1^{~I,(2)}
+{2 \over 3} g_3^{~I,(1)}
+3 \zeta_2 A^I_3
-3 \zeta_2^2 \beta_0^2 A^I_1
+{4 \over 3} \beta_0 C_I \overline {\cal G}_2^{(2)}
+{8 \over 3} \beta_0^2 C_I \overline {\cal G}_1^{~(3)}
\nonumber\\[2ex]
&&
+{4 \over 3} \beta_1 C_I \overline {\cal G}_1^{~(2)}
+{2 \over 3} C_I \overline {\cal G}_3^{~(1)}
+6 \zeta_2 \beta_0 f^I_2
+
\Bigg(
12 \zeta_2 \beta_0^3\delta_{I,g}
-6 \beta_2\delta_{I,g}
+2 B^I_3
-12 \zeta_2 \beta_0^2 B^I_1
\nonumber\\[2ex]
&&-2 \beta_0 g_2^{~I,(1)}
-4 \beta_0^2 g_1^{~I,(2)}
-2 \beta_1 g_1^{~I,(1)}
-6 \zeta_2\beta_0 A^I_2
-3 \zeta_2 \beta_1 A_1^I
-2 \beta_0 C_I \overline {\cal G}_2^{~(1)}
\nonumber\\[2ex]
&&-4 \beta_0^2 C_I \overline {\cal G}_1^{~(2)}
-2 \beta_1 C_I \overline {\cal G}_1^{~(1)}
\Bigg)\ln\left({q^2 \over \mu_F^2}\right)
+
\Bigg(
5 \beta_0 \beta_1 \delta_{I,g}
-2 \beta_0 B^I_2
-\beta_1 B^I_1
+2 \beta_0^2 g_1^{~I,(1)}
\nonumber\\[2ex]
&&+3 \zeta_2 \beta_0^2 A^I_1
+2 \beta_0^2 C_I \overline {\cal G}_1^{~(1)}
\Bigg)\ln^2 \left({q^2 \over \mu_F^2}\right)
+\Bigg({2 \over 3 } \beta_0^2 B^I_1
-{2 \over 3}\beta_0^3 \delta_{I,g}\Bigg)
\log^3\left({q^2 \over \mu_F^2}\right)
\Bigg]
\delta(1-z)
\nonumber\\[2ex]
&&+\Bigg[
-4 \beta_0 C_I \overline {\cal G}_2^{~(1)}
-8 \beta_0^2 C_I \overline {\cal G}_1^{~(2)}
-4 \beta_1 C_I \overline {\cal G}_1^{~(1)}
-2 f_3^I
+\Bigg(4 \beta_0 f_2^I
+8 \beta_0^2 C_I \overline {\cal G}_1^{~(1)}
\nonumber\\[2ex]
&&+2 A^I_3 \Bigg) \ln\left({q^2 \over \mu_F^2}\right)
+\Bigg(-2 \beta_0 A^I_2 -\beta_1 A^I_1 \Bigg)
\ln^2 \left({q^2 \over \mu_F^2}\right)
+\Bigg({2 \over 3} \beta_0^2 A^I_1 \Bigg)
\ln^3 \left({q^2 \over \mu_F^2}\right)\Bigg]{\cal D}_0
\nonumber\\[2ex]
&&+\Bigg[
8 \beta_0 f_2^I
+16 \beta_0^2 C_I \overline {\cal G}_1^{~(1)}
+4 A_3^I
+\Big(-8 \beta_2 A_2^I
-4 \beta_1 A^I_1\Big)\ln\left({q^2 \over \mu_F^2}\right)
\nonumber\\[2ex]
&&+\Big(4 \beta_0^2 A^I_1\Big) \ln^2 \left({q^2 \over \mu_F^2}\right)
\Bigg] {\cal D}_1
+\Bigg[
-8 \beta_0 A^I_2
-4 \beta_1 A^I_1
+\Big(8 \beta_0^2 A^I_1\Big) \ln\left({q^2 \over \mu_F^2}\right)
\Bigg]{\cal D}_2
\nonumber\\[2ex]
&&+\Bigg[
\Bigg({16 \over 3} \beta_0^2 A^I_1 \Bigg)
\Bigg]{\cal D}_3\end{aligned}$$ Notice that $\Psi^{~I,(1)}$ and $\Psi^{~I,(2)}$ are completely known. Using our $\Psi^{~I,(1)}$, and $\Psi^{~I,(2)}$ we could successfully reproduce soft plus virtual cross section $\Delta_g^{sv,(i)}(z,q^2,\mu_F^2)$ (see [@Harlander:2001is; @Catani:2001ic]) ($i=1,2$) of the Higgs production from that of DY [@Matsuura:1988sm] and vice versa. To compute $\Delta_I^{sv,(3)}(z,q^2,\mu_F^2)$ (equivalently $\Psi^{~I,(3)}$) we need to know $\overline {\cal G}_2^{~I,(2)}$ and $\overline {\cal G}_3^{~I,(1)}$ either from DY or Higgs production because these constants are maximally non-abelien. Notice that these constants appear only in the coefficient of $\delta(1-z)$ part of $\Psi^{~I,(3)}$. Since the coefficients of ${\cal D}_i$($i=0,1,2,3$) in $\Psi^{~I,(3)}$ do not depend on these unknown constants $\overline {\cal G}_3^{~I,(1)}$ and $\overline {\cal G}_2^{~I,(2)}$, we can predict these coefficients(say for the Higgs production) by using the universal soft distribution function extracted from a process(say DY), the three loop form factors and the renormalisation constants. Our prediction agrees with the partial $N^3LO$ soft plus virtual results [@Moch:2005ky] for DY and Higgs productions.
With these available informations one can also determine all the quantities upto $N^3LL$ level in the threshold resummation. To do this, we first recollect that the soft distribution function is renormalisation group invariant. Its UV divergence can be removed by the coupling constant renormalisation. This introduces a renormalisation scale $\mu_R$ which is arbitrary to all orders in perturbation theory. In order to compute various quantities in the threshold resummation formula from the soft distribution function, we choose $\mu_R=\mu_F$. With this choice, one can express the soft distribution function as a sum of pole and finite parts in ${\mbox{$\varepsilon$}}$ as ${\mbox{$\varepsilon$}}\rightarrow 0$, that is $$\begin{aligned}
\Phi^I\left(a_s(\mu_F^2),{q^2 \over \mu_F^2},z,{\mbox{$\varepsilon$}}\right)
=\Phi^I_{pole}\Bigg(a_s(\mu_F^2),{q^2 \over \mu_F^2},z,{1 \over {\mbox{$\varepsilon$}}}\Bigg)
+\Phi^I_{fin}\Bigg(a_s(\mu_F^2),{q^2\over \mu_F^2},z,{\mbox{$\varepsilon$}}\Bigg) \end{aligned}$$ With this decomposition, it is now straightforward to identify the finite part $\Phi^I_{fin}$ with the threshold resummation formula as $$\begin{aligned}
2 \int_0^1 dz ~z^{N-1} \Phi^I_{fin}\Bigg(a_s(\mu_F^2),
{q^2 \over \mu_F^2},z,{\mbox{$\varepsilon$}}=0\Bigg)
&=&\int_0^1 dz {z^{N-1}- 1\over 1-z}
\Bigg[ D^I\Big(a_s\Big(q^2(1-z)^2\Big)\Big)
\nonumber\\[2ex]
&& + 2 \int_{\mu_F^2}^{q^2 (1-z)^2} {d \lambda^2 \over \lambda^2}
A^I\Big(a_s(\lambda^2)\Big) \Bigg]
\nonumber\\[2ex]
&& +H^I_S\Bigg(a_s(\mu_F^2),{q^2 \over \mu_F^2}\Bigg) \end{aligned}$$ where the subscript $S$ in $H^I_S$ indicates that it comes from the soft part of the cross section. The remaining contribution comes from the form factor. $D^I(a_s(q^2(1-z)^2))$ can be expanded in powers of bare coupling constant $\hat a_s$ as follows: $$\begin{aligned}
D^I\Big(a_s\Big(q^2 (1-z)^2\Big)\Big)=\sum_{i=1}^\infty
\hat a_s^i \Bigg({q^2 (1-z)^2 \over \mu^2}\Bigg)^{i {{\mbox{$\varepsilon$}}\over 2}}
S_{{\mbox{$\varepsilon$}}}^i~ \hat D^{I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ The finiteness of $D^I$ after coupling constant renormalisation demands that it satisfies the following expansion in ${\mbox{$\varepsilon$}}$: $$\begin{aligned}
\hat D^{I,(i)}({\mbox{$\varepsilon$}})=\sum_{j=1-i}^\infty \hat d^{~I,(i)}_{j} {\mbox{$\varepsilon$}}^{j}\end{aligned}$$ Using RG invariance, the coefficients of negative powers of ${\mbox{$\varepsilon$}}$ can be evaluated as $$\begin{aligned}
\hat d^{~g,(2)}_{-1}&=& -2 \beta_0 ~\hat d^{~g,(1)}_0
\nonumber\\[2ex]
\hat d^{~g,(3)}_{-2}&=& 4 \beta_0^2 ~\hat d^{~g,(1)}_0
\nonumber\\[2ex]
\hat d^{~g,(3)}_{-1}&=& -4 \beta_0 ~\hat d^{~g,(2)}_0
-4 \beta_0^2 ~\hat d^{~g,(1)}_1
-\beta_1 ~\hat d^{~g,(1)}_0\end{aligned}$$ We find that for non-negative powers of ${\mbox{$\varepsilon$}}$($j\ge 0$), $$\begin{aligned}
\hat d_j^{~I,(1)}&=&2 \overline g_1^{~I,j+1}\Big|_{{\cal D}_0}
\nonumber\\[2ex]
\hat d_j^{~I,(2)}&=&\Bigg(\overline g_2^{~I,j+1}
-2 \beta_0 ~\overline g_1^{~I,j+2} \Bigg)\Big|_{{\cal D}_0}
\nonumber\\[2ex]
\hat d_j^{~I,(3)}&=&\Bigg({2 \over 3} ~\overline g_3^{~I,j+1}
-{8 \over 3} \beta_0 ~\overline g_2^{~I,j+2}
-{2 \over 3} \beta_1 ~\overline g_1^{~I,j+2}
+{8 \over 3} \beta_0^2
~\overline g_1^{~I,j+3}\Bigg)\Big|_{{\cal D}_0}\end{aligned}$$ Using the above equations, we find explicitly $$\begin{aligned}
\hat d^{~g,(1)}_0&=&0
\nonumber\\[2ex]
\hat d^{~g,(1)}_1&=&C_A \Big(- 6 \zeta_2\Big)
\nonumber\\[2ex]
\hat d^{~g,(1)}_2&=&C_A \Bigg({14 \over 3} \zeta_3 \Bigg)
\nonumber\\[2ex]
\hat d^{~g,(2)}_0&=&C_A^2 \Bigg(
-{1616 \over 27 } +{308 \over 3} \zeta_2 +56 \zeta_3
\Bigg)
+ C_A n_f \Bigg(
{224 \over 27} -{56 \over 3} \zeta_2 \Bigg)
\nonumber\\[2ex]
\hat d^{~g,(2)}_1&=&C_A^2 \Bigg({4856 \over81} -{938 \over 9} \zeta_2
+8 \zeta_2^2 -{1364 \over 9} \zeta_3 \Bigg)
\nonumber\\[2ex]
&& +C_A n_f\Bigg(
-{656 \over 81} +{140 \over 9} \zeta_2
+{248 \over 9} \zeta_3 \Bigg)
\nonumber\\[2ex]
\hat d^{~g,(3)}_0&=&C_A^3 \Bigg(
-{1235050 \over 729}-{352 \over 3} \zeta_2 \zeta_3
+{227548 \over 81} \zeta_2 -{1584 \over 5} \zeta_2^2
+{10376 \over 3} \zeta_3 -384 \zeta_5
\Bigg)
\nonumber\\[2ex]
&& +C_A^2 n_f \Bigg(
{328388 \over 729} -{72004 \over 81} \zeta_2
+{352 \over 5} \zeta_2^2 -{26800 \over 27} \zeta_3
\Bigg)
\nonumber\\[2ex]
&& +C_A C_F n_f \Bigg(
{3422 \over 27} -44 \zeta_2 -{64 \over 5} \zeta_2^2
-{608 \over 9} \zeta_3
\Bigg)
\nonumber\\[2ex]
&& +C_A n_f^2 \Bigg(
-{19456 \over 729} +{1760 \over 27} \zeta_2
+{2080 \over 27} \zeta_3
\Bigg)\end{aligned}$$ The coefficients $\hat d^{~q,(i)}_{j}$ for the DY can be obtained using $$\begin{aligned}
\hat d^{~q,(i)}_{j}={C_F \over C_A} ~\hat d^{~g,(i)}_{j}\end{aligned}$$ because the soft distributions functions are maximally non-abelien. Also, the coefficients of $a_s^i (q^2) {\cal D}_0$ in the soft distribution function $\Phi^I_{fin}$ are related to the coefficients $D_i^I$ that appear in threshold resummation formula. Hence it is straightforward to obtain $D^I_i$ from the soft distribution function $\Phi^I_{fin}$. We find that $D^I_i$ are related to $\hat d^{~I,(i)}_k$ and hence $\overline g_i^{~I,k}$ as follows: $$\begin{aligned}
D^I_1&=& \hat d^{~I,(1)}_0 =2 ~\overline g_1^{~I,1} \Big|_{{\cal D}_0}
\nonumber\\[2ex]
&=&2 \overline{\cal G}_1^I({\mbox{$\varepsilon$}}=0)
\nonumber\\[2ex]
D^I_2&=&\hat d^{~I,(2)}_0+2 \beta_0 ~\hat d^{~I,(1)}_1
=\Bigg(\overline g_2^{~I,1}
+2 \beta_0 ~\overline g_1^{~I,2}\Bigg)\Big|_{{\cal D}_0}
\nonumber\\[2ex]
&=&2 \overline{\cal G}_2^I({\mbox{$\varepsilon$}}=0)
\nonumber\\[2ex]
D^I_3 &=&\hat d^{~I,(3)}_0
+4 \beta_0 ~\hat d^{~I,(2)}_1
+\beta_1 ~\hat d^{~I,(1)}_1
+4 \beta_0^2 ~\hat d^{~I,(1)}_2
\nonumber\\[2ex]
&=&\Bigg({2 \over 3} ~\overline g_3^{~I,1}
+{4 \over 3} \beta_0 ~\overline g_2^{~I,2}
+{4 \over 3} \beta_1 ~\overline g_1^{~I,2}
+{8 \over 3} \beta_0^2 ~\overline g_1^{~I,3}
\Bigg)\Big|_{{\cal D}_0}
\nonumber\\[2ex]
&=&2 \overline{\cal G}_3^I({\mbox{$\varepsilon$}}=0)\end{aligned}$$ From the available informations on three loop results, we find that the following result holds $$\begin{aligned}
D_i^I&=&2 ~\overline {\cal G}_i^I({\mbox{$\varepsilon$}}=0)
\quad \quad \quad \quad i=1,2,3\end{aligned}$$ The fact that $D^I_i$ can be expressed entirely in terms of $\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})$ (i.e., in terms of $\overline g^{~I,k}_i$ or $\overline {\cal G}_i^I({\mbox{$\varepsilon$}}=0)$) which are maximally non-abelien, implies that $D^I_i$ are also maximally non-abelien. Hence, $$\begin{aligned}
D^q_i={C_F \over C_A}~ D^g_i \end{aligned}$$ with $$\begin{aligned}
D^g_1 & =&0
\nonumber\\[2ex]
D^g_2 & =&
C_A^2 \left( - {1616\over 27} + {176\over 3}\,\zeta_2 + 56\,\zeta_3 \right)
+ C_A n_f \left( {224\over 27} - {32\over 3}\,\zeta_2 \right)
\nonumber\\[2ex]
D^g_3 & =&
C_A^3 \Bigg( - {594058 \over 729} + {98224 \over 81}\,\zeta_2
+ {40144 \over 27}\,\zeta_3 - {2992 \over 15}\,\zeta_2^2
- {352 \over 3}\,\zeta_2\zeta_3 - 384\,\zeta_5 \Bigg)
\nonumber \\[2ex]
&& + C_A^2 n_f \Bigg( {125252 \over 729} - {29392 \over 81}\,\zeta_2
- {2480 \over 9}\,\zeta_3 + {736 \over 15}\,\zeta_2^2 \Bigg)
\nonumber \\[2ex]
&& + C_A C_F n_f \Bigg( {3422 \over 27} - 32\,\zeta_2
- {608 \over 9}\,\zeta_3 - {64 \over 5}\,\zeta_2^2 \Bigg)
+ C_A n_f^2 \Bigg( - {3712 \over 729} + {640 \over 27} \,\zeta_2
+ {320 \over 27}\,\zeta_3 \Bigg)\end{aligned}$$ The above results are in agreement with [@Moch:2005ky; @Laenen:2005uz; @Idilbi:2005ni]. We also find the resummation exponents $D^I_i$ can be extracted by using the following relations: $$\begin{aligned}
D^I_1&=&
\Delta_I^{sv,(1)}\Big|_{{\cal D}_0}
\nonumber\\[2ex]
D^I_2&=&
\Bigg(\Delta_I^{sv,(2)} -{1 \over 2} \Delta_I^{sv,(1)}\otimes
\Delta_I^{sv,(1)}\Bigg) \Big|_{{\cal D}_0}
\nonumber\\[2ex]
D^I_3&=&
\Bigg(\Delta_I^{sv,(3)} -\Delta_I^{sv,(1)} \otimes \Delta_I^{sv,(2)}
+{1 \over 3} \Delta_I^{sv,(1)} \otimes \Delta_I^{sv,(1)} \otimes
\Delta_I^{sv,(1)}\Bigg) \Big|_{{\cal D}_0}\end{aligned}$$ where $\Delta_I^{sv,(i)}$ in the above are computed at the scale $\mu_F^2=\mu_R^2=q^2$. From the following convolution identity[@vanNeerven:2001pe] upto irrelevant regular terms(denoted by $\cdot \cdot \cdot$) $$\begin{aligned}
{\cal D}_i \otimes {\cal D}_j = d_{ij} \delta(1-z)
+\sum_{l=0}^{i+j+1} c_{ij,l} {\cal D}_l
+\cdot \cdot \cdot\end{aligned}$$ it is interesting to notice that in order to obtain $D^I_{i}$, it is sufficient to know the coefficients of all ${\cal D}_{l}$ ($l=l_{max}$ to $0$) (that means, we need not know the information on the coefficient of $\delta(1-z)$ function and the regular part of $\Delta_I^{sv,(i)}$) and the complete soft information of $\Delta_I^{sv,(i-1)}$(i.e., the coefficients of all ${\cal D}_i$ and $\delta(1-z)$ are needed).
Finally, the coefficient of $\delta(1-z)$ in the resummation formula can be obtained from $\Phi^I_{fin}$ by defining the coupling constant at the scale $\mu_F^2$. The result is $$\begin{aligned}
H^I_S\left(a_s(\mu_F^2),{q^2 \over \mu_F^2}\right)
=\sum_{i=1}^\infty a_s^i(\mu_F^2) H^{I}_{S,i}\end{aligned}$$ where $$\begin{aligned}
H^{g}_{S,1}&=&-3 \zeta_2 + \ln^2\left({q^2 \over \mu_F^2}\right)
\nonumber\\[2ex]
H^{g}_{S,2}&=& C_A^2 \Bigg(-{164 \over 81} +{35 \over 9} \zeta_2
+{34 \over 9} \zeta_3
+\Bigg(-{8 \over 3} \zeta_2 +{56 \over 27}
\Bigg)\ln\left({q^2 \over \mu_F^2}\right)
\nonumber\\[2ex]&&
-{10 \over 9}\ln^2\left({q^2 \over \mu_F^2}\right)
+{2 \over 9}\ln^3\left({q^2 \over \mu_F^2}\right)\Bigg)
+C_A n_f \Bigg({1214 \over 81} -{469 \over 18} \zeta_2
+2 \zeta_2^2-{187 \over 9} \zeta_3
\nonumber\\[2ex]&&
+\Bigg({44 \over 3} \zeta_2 +14 \zeta_3 -
{404 \over 27}\Bigg)\ln\left({q^2 \over \mu_F^2}\right)
+\Bigg(-2 \zeta_2 +{67 \over 9}\Bigg)
\ln^2\left({q^2 \over \mu_F^2}\right)
\nonumber\\[2ex]&&
-{11\over 9}
\ln^3\left({q^2 \over \mu_F^2}\right)\Bigg)\end{aligned}$$ and $$\begin{aligned}
H^{q,(i)}_S={C_F \over C_A}~ H^{g,(i)}_S\end{aligned}$$ The remaining contribution to the exponent comes from the the finite part of form factor.
We conclude our discussion on this subject with a brief discussion on the corresponding soft as well as jet distribution functions that appear in deep inelastic scattering. The soft plus virtual coefficient function $c^{sv}_{I,2}(Q^2,z)$ that appear in the hadronic structure function $F_2$ can be expressed as $$\begin{aligned}
{\cal C}\ln c_{I,2}^{sv}(Q^2,z)&=&
\Bigg(
\ln \Big(Z^I(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})\Big)^2
+\ln \big|\hat F^I(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}})\big|^2
\Bigg)
\delta(1-z)
\nonumber\\[2ex]
&&+ 2 \Phi^I_{SJ}(\hat a_s,Q^2,\mu^2,z,{\mbox{$\varepsilon$}})
-{\cal C}\ln \Gamma_{II}(\hat a_s,\mu^2,\mu_F^2,z,{\mbox{$\varepsilon$}})\end{aligned}$$ where $\Phi^I_{SJ}(\hat a_s,Q^2,\mu^2,z,{\mbox{$\varepsilon$}})$ is sum of soft and jet distribution functions. We find that this soft plus jet distribution function also satisfies Sudakov type integro differential equation (see eqn.(\[sud2\])) which can be solved in the same way we solved soft distribution functions. We find that this soft plus jet distribution function $\Phi^I_{SJ}$ can be expressed as $$\begin{aligned}
\Phi^I_{SJ}(\hat a_s,Q^2,\mu^2,z,{\mbox{$\varepsilon$}}) &
=& \Phi^I_{SJ}(\hat a_s,Q^2 (1-z),\mu^2,{\mbox{$\varepsilon$}})
\nonumber\\[2ex]
&=&\sum_{i=1}^\infty \hat a_s^i \left({Q^2 (1-z) \over \mu^2}\right)^{i
{{\mbox{$\varepsilon$}}\over 2}} S_{{\mbox{$\varepsilon$}}}^i \left({i~ {\mbox{$\varepsilon$}}\over 2(1-z)} \right)
\hat \xi^{~I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ where $$\begin{aligned}
\hat \xi^{I,(i)}({\mbox{$\varepsilon$}})=\hat {\cal L}_F^{I,(i)}({\mbox{$\varepsilon$}})
\Bigg( A^I \rightarrow - A^I, G^I({\mbox{$\varepsilon$}}) \rightarrow \widetilde {\cal G}^I({\mbox{$\varepsilon$}})
\Bigg)\end{aligned}$$ We find that the constants $\widetilde {\cal G}^I({\mbox{$\varepsilon$}})$ have the following expansion in terms of $B_i^I$, $f^I_i$ and the ${\mbox{$\varepsilon$}}$ dependent part of lower order coefficient functions. $$\begin{aligned}
\widetilde {\cal G}^{~q}_1({\mbox{$\varepsilon$}})&=&-(B_1^q+f_1^q)+
\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \widetilde {\cal G}^{~q,(k)}_1
\nonumber\\[2ex]
\widetilde {\cal G}^{~q}_2({\mbox{$\varepsilon$}})&=&-(B_2^q+f_2^q)
-2 \beta_0 \widetilde{\cal G}_1^{~q,(1)}
+\sum_{k=1}^\infty{\mbox{$\varepsilon$}}^k \widetilde {\cal G}^{~q,(k)}_2
\nonumber\\[2ex]
\widetilde {\cal G}^{~q}_3({\mbox{$\varepsilon$}})&=&-(B_3^q+f_3^q)
-2 \beta_1 \widetilde{\cal G}_1^{~q,(1)}
-2 \beta_0 \left(\widetilde{\cal G}_2^{~q,(1)}
+2 \beta_0 \widetilde{\cal G}_1^{~q,(2)}\right)
+\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \widetilde {\cal G}^{~q,(k)}_3\end{aligned}$$ The $z$ independent constants $\widetilde {\cal G}^{q,(k)}_i$ are computed using the coefficient functions $c^{sv}_{q,2}(z,Q^2)$ known upto three loop level [@vanNeerven:1991nn; @Moch:2005ba]. Recollect that the three loop form factors were obtained from these coefficient functions by demanding the finiteness of the partonic cross sections after mass factorisation and also notice that the method used there is very different from the method presented in this paper. We obtain $$\begin{aligned}
\widetilde{\cal G}^{q,(1)}_1&=&
C_F~ \Big({7 \over 2}-3 \zeta_2\Big)
\nonumber\\[2ex]
\widetilde{\cal G}^{q,(2)}_1&=&
C_F~ \Bigg(-{7 \over 2} +{9 \over 8} \zeta_2 +{7 \over 3} \zeta_3\Bigg)
\nonumber\\[2ex]
\widetilde{\cal G}^{q,(1)}_2&=&
C_F^2 \Bigg(
{9 \over 8} -{41 \over 2}\zeta_2
+{82 \over 5} \zeta_2^2 -6 \zeta_3\Bigg)
\nonumber\\[2ex]
&&+C_F C_A~ \Bigg({69761 \over 648}-{1961 \over 36} \zeta_2
-{17 \over 5} \zeta_2^2 -40 \zeta_3\Bigg)
\nonumber\\[2ex]
&&+ C_F n_f\Bigg(
-{5569 \over 324} +{163 \over 18} \zeta_2 +4 \zeta_3\Bigg) \end{aligned}$$ Using the following decomposition, $$\begin{aligned}
\Phi^I_{SJ}\left(a_s(\mu_F^2),{Q^2 \over \mu_F^2},z,{\mbox{$\varepsilon$}}\right)
=\Phi^I_{SJ,pole}\Bigg(a_s(\mu_F^2),{Q^2 \over \mu_F^2},z,{1 \over {\mbox{$\varepsilon$}}}\Bigg)
+\Phi^I_{SJ,fin}\Bigg(a_s(\mu_F^2),{Q^2\over \mu_F^2},z,{\mbox{$\varepsilon$}}\Bigg)\end{aligned}$$ it is now straightforward to identify the finite part $\Phi^I_{SJ,fin}$ with the DIS threshold resummation formula as $$\begin{aligned}
2 \int_0^1 dz ~z^{N-1} \Phi^I_{SJ,fin}\Bigg(a_s(\mu_F^2),
{Q^2 \over \mu_F^2},z,{\mbox{$\varepsilon$}}=0\Bigg)
&=&\int_0^1 dz {z^{N-1}- 1\over 1-z}
\Bigg[ B_{DIS}^I\Big(a_s\Big(Q^2(1-z)\Big)\Big)
\nonumber\\[2ex]
&& + \int_{\mu_F^2}^{Q^2 (1-z)} {d \lambda^2 \over \lambda^2}
A^I\Big(a_s(\lambda^2)\Big) \Bigg]
\nonumber\\[2ex]
&& +H^I_{SJ,S}\Bigg(a_s(\mu_F^2),{Q^2 \over \mu_F^2}\Bigg)\end{aligned}$$ Using the above equation, we find that the resummation constants $B^q_{DIS,i}$ satisfy the following relation $$\begin{aligned}
B^q_{DIS,i}= \widetilde {\cal G}^q_i({\mbox{$\varepsilon$}}=0) \quad \quad \quad i=1,2,3\end{aligned}$$ The resulting $B^q_{DIS,i}$s agree with those given in [@Moch:2005ba].
To summarise, we have extracted the soft distribution function $\Phi^I$ using mass factorisation formula for both Drell-Yan as well as Higgs productions within the framework of perturbative QCD. This is possible now thanks to various three loop results available for the form factors and splitting functions. The $\Phi^I$ is known completely upto two loop level. Except the soft bremsstrahlung contributions proportional to $\delta(1-z)$ (at three loop level), all the other soft terms$({\cal D}_i)$ are known for the soft distribution functions $\Phi^I$ upto three loop level. We have also shown that the soft distribution functions satisfy Sudakov type integro-differential equation that the quark and gluon form factors satisfy. We found that they are process independent. In other words, knowing the soft distribution function of the Drell-Yan process, one can obtain the same for the Higgs production by simply multiplying the colour factor combination $C_A/C_F$. Hence, unlike the cross sections $\Delta^{sv}_I$, these soft distribution functions are maximally non-abelien. Notice that the form factors have this property only at the pole level. Using the universal soft distribution function extracted from DY production, we could reproduce the soft plus virtual cross section for the Higgs production using the gluon form factor and its renormalisation constant. We have also shown how these soft distribution functions are related to threshold resummation formula. We have extracted various coefficients appearing in threshold resummation formula upto three loop level using the soft distribution function. We have also discussed the DIS soft and jet distribution functions in the present context.
Acknowledgments: The author would like to thank S. Moch and A. Vogt for providing the constants ${\cal S}_i$ of DY that are computed in their paper [@Moch:2005ky].
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| ArXiv |
---
abstract: 'While model selection is a well-studied topic in parametric and nonparametric regression or density estimation, model selection of possibly high dimensional nuisance parameters in semiparametric problems is far less developed. In this paper, we propose a new model selection framework for making inferences about a finite dimensional functional defined on a semiparametric model, when the latter admits a doubly robust estimating function. The class of such doubly robust functionals is quite large, including many missing data and causal inference problems. Under double robustness, the estimated functional should incur no bias if either of two nuisance parameters is evaluated at the truth while the other spans a large collection of candidate models. We introduce two model selection criteria for bias reduction of functional of interest, each based on a novel definition of pseudo-risk for the functional that embodies this double robustness property and thus may be used to select the candidate model that is nearest to fulfilling this property even when all models are wrong. Both selection criteria have a bias awareness property that selection of one nuisance parameter can be made to compensate for excessive bias due to poor learning of the other nuisance parameter. We establish an oracle property for a multi-fold cross-validation version of the new model selection criteria which states that our empirical criteria perform nearly as well as an oracle with a priori knowledge of the pseudo-risk for each candidate model. We also describe a smooth approximation to the selection criteria which allows for valid post-selection inference. Finally, we apply the approach to perform model selection of a semiparametric estimator of average treatment effect given an ensemble of candidate machine learning methods to account for confounding in a study of right heart catheterization in the intensive care unit of critically ill patients.'
author:
- |
Yifan Cui, Eric Tchetgen Tchetgen\
Department of Statistics, The Wharton School, University of Pennsylvania
bibliography:
- 'causal.bib'
- 'survtrees.bib'
title: 'Bias-aware model selection for machine learning of doubly robust functionals'
---
[***Keywords:*** Model Selection, Machine Learning, Doubly Robust, Influence Function, Average Treatment Effect, Cross-validation]{}
Introduction {#sec:intro}
============
Model selection is a well-studied topic in statistics, econometrics and machine learning. In fact, methods for model selection and corresponding theory abound in these disciplines, although primarily in settings of parametric and nonparametric regression and density estimation ([@akaike1974new; @BIC1978; @vuong1989likelihood; @zhang1993model; @wand1994kernel; @fan1995jrssb; @ruppert1995; @HALL1996165; @lasso; @yang2000; @birge2001gaussian; @fan2001variable; @rao; @wegkamp2003; @ruppert2003semiparametric; @efron2004least; @zhao2006lasso; @BIRGE2006497; @candes2007dantzig; @celisse2014; @belloni2014] and many others). Model selection methods are far less developed in settings where one aims to make inferences about a finite dimensional, pathwise differentiable functional defined on a semiparametric model. Model selection for the purpose of estimating such a functional may involve selection of an infinite dimensional parameter, say a nonparametric regression for the purpose of more accurate estimation of functional in view, which can be considerably more challenging than selecting a regression model strictly for the purpose of prediction. This is because whereas the latter admits a risk, e.g., mean squared error loss, that can be estimated unbiasedly and therefore can be minimized with small error, the risk of a semiparametric functional will typically not admit an unbiased estimator and therefore may not be minimized without excessive error. This is an important gap in both model selection and semiparametric theory which this paper aims to address.
Specifically, we propose a novel approach for model selection of a functional defined on a semiparametric model, in settings where inferences about the targeted functional involves infinite dimensional nuisance parameters, and the functional of scientific interest admits a doubly robust estimating function. Doubly robust inference has received considerable interest in the past few years across multiple disciplines including Statistics, Epidemiology and Econometrics [@robins1994; @Rotnitzky1998; @Scharfstein1999; @robins2000; @robins2001comment; @unified; @Lunceford2004; @bang2005; @tan2006; @cao2009; @ett2010dr; @funk2011; @rotnitzky2012dr; @han2013; @FARRELL20151; @2015biasreduce; @vermeulen2016adaptive; @Chernozhukov2018; @rotnitzky2019mix; @tan2019; @fulcher2017robust]. An estimator is said to be doubly robust if it remains consistent if one of two nuisance parameters needed for estimation is consistent, even if both are not necessarily consistent. The class of functionals that admit doubly robust estimators is quite rich, and includes estimation of pathwise differentiable functionals in missing data problems under missing at random assumptions, and also in more complex settings where missingness process might be not at random. Several problems in causal inference also admit doubly robust estimating equations, the most prominent of which is the average treatment effect under assumptions that include positivity, consistency and no unmeasured confounding [@Scharfstein1999; @robins2000]. All of these functionals are members of a large class of doubly robust functionals recently studied by [@robins2008HOIF] in a unified theory of first and higher order influence functions.
The literature on double robustness combined with machine learning methods is rapidly expanding [@van2010collaborative; @van2011targeted; @belloni2014res; @FARRELL20151; @belloni2017Econometrica; @robins2017; @Chernozhukov2018; @athey2018jrssb; @van2018targeted; @dukes2018high; @rotnitzky2019mix; @tan2019model]. A well-documented advantage of using doubly robust influence functions is that flexible machine learning or other nonparametric data adaptive methods may generally be used to estimate high dimensional nuisance parameters such that valid inferences may be obtained about the functional of interest provided that estimated nuisance parameters have mean squared error of order smaller than $n^{-1/4}$, which can be considerably slower than converge rates attained by parametric models [@robins2017; @Chernozhukov2018]. As in practice, one cannot be certain that any model is either correctly specified or estimated with small bias, model selection remains important even in the advent of doubly robust estimation including for methods that leverage machine learning. Clearly, the performance of doubly robust semiparametric estimators is intimately related to the performance of estimators of its nuisance parameters, a task towards which model selection is paramount.
This paper aims at the selection of an optimal estimator for the functional $\psi(\theta)$ in a class of doubly robust functionals where $\theta$ is a parameter (possibly infinite dimensional) indexing the observed data law within a semiparametric/nonparametric model. Given a large collection of doubly robust estimators $\Psi_{K}=\{\widehat \psi_k: k=1,\ldots,K\}$ of size $K$ (which may grow with sample size) indexed by candidate estimators of nuisance parameters, we ultimately wish to identify an estimator that minimizes the risk associated with a measurable loss function. A natural choice would be to try to select the estimator that minimizes the mean squared error $E(\widehat \psi_{k,\tk}-\psi)^2$. However, it is clear that this cannot be done empirically in a straightforward fashion, as an unbiased estimator of the mean squared error (even up to a constant shift) is generally not available, so that model selection becomes challenging.
In this paper, we propose two novel model selectors each based on minimization of a certain cross-validated quadratic pseudo-risk for a large class of doubly robust functionals. The proposed pseudo-risk embodies the idea of double robustness: The first kind of pseudo-risk is given by the overall maximum squared bias (i.e., change in the estimated functional) at a given candidate estimator, induced by perturbing one nuisance parameter at the time over candidate models holding the other one fixed; The second proposed pseudo-risk is given by the sum of two maximum squared bias quantities, each capturing the bias induced by perturbing a single nuisance parameter only. As we establish both procedures are guaranteed to recover a consistent estimator for the functional whenever consistent estimators of nuisance parameters are available, with corresponding pseudo-risk converging to zero. However, even when all models are wrong, as in many practical settings where parametric models are used, and therefore all candidate estimators are inconsistent with pseudo-risk bounded away from zero, a minimizer of pseudo-risk nevertheless corresponds to a choice of models that is least sensitive to perturbations, i.e., misspecification of either nuisance parameter. Both selection criteria have a bias awareness property that selection of one nuisance parameter is aware and therefore may be made to compensate for excessive bias due to poor learning of the other nuisance parameter. We find such awareness may be key to bias reduction in context of machine learning of doubly robust functionals.
Our cross-validation scheme is akin to that of [@vanderlann2003cross] and [@vaart2006cv; @vaart2006oracle], who formally established that such a scheme can perform nearly as well as an oracle with access to underlying data generating mechanism, in selecting an optimal estimator in settings such as nonparametric regression or density estimation. In contrast, we aim to perform model selection for a pathwise differentiable functional of such nonparametric regression or density function, and therefore to minimize a risk function for the functional; a different task which generally proves to be more challenging. For each split of the observed sample, a training sample is used to estimate each candidate model of the nuisance parameters. The validation subsample is then used to construct corresponding candidate estimators of functional $\psi$, and subsequently, to estimate the pseudo-risk of each candidate estimator conditional on the training sample. The optimal model is selected by minimizing multi-fold cross-validated pseudo-risk over the set of candidate nuisance models. To our knowledge, this is the first model selection result for doubly robust functionals which aims directly at bias reduction of the functional. Significant amounts of work have been devoted to improving performance of doubly robust estimators [@bang2005; @tan2006; @cao2009; @tan2010; @rotnitzky2012dr; @FARRELL20151; @2015biasreduce; @vermeulen2016adaptive; @van2018targeted; @dukes2018high; @smucler2019unifying; @rotnitzky2019mix; @tan2019; @bradic2019sparsity] from a variety of perspectives, however, none have considered model selection for the underlying functional, over a generic collection of candidate nuisance parameter models that may include classical parametric, semiparametric and nonparametric estimators, as well as modern highly data adaptive machine learning estimators. The task of model selection of parametric nuisance models for specific semiparametric doubly robust problems was recently considered by [@han2013; @chan2013; @HAN2014101; @han2014jasa; @chan2014; @duan2017; @chen2017; @liu2019], although, their goal differs from ours as they aim to select parametric nuisance models that best approximate each nuisance model, which may generally conflict with selecting the nuisance models that minimize a well-defined pseudo-risk of the targeted functional, especially when as often the case in practice, all candidate models are wrong.
A related targeted maximum likelihood learning approach for model selection in functional estimation, known as cross-validated targeted maximum likelihood estimation [@zheng2010asymptotic; @van2011targeted; @van2018targeted] can provide notable improvements on the above methods by allowing the use of an ensemble of semiparametric or nonparametric methods, including modern machine learning for flexible estimation of nuisance parameters, still the ensemble learning is targeted at optimal estimation of nuisance parameters, not bias reduction of the functional ultimately of interest. Another state of the art approach recently proposed incorporates modern machine learning in functional estimation via double debiased machine learning (DDML) [@Chernozhukov2018]; however the approach uses a single machine learning algorithm for estimating each nuisance parameter, and does not leverage model selection targeted at the functional of interest. In comparison, as we will show, our approach ensures that selection of one nuisance model is made to minimize bias due to possible misspecification of the other, such bias awareness for the functional of interest endows the proposed model selection procedure with additional robustness.
The proposed approach is generic, in the sense that it allows the space of candidate models/learners to be quite large ($K_1\times K_2$ of order $c^{n^\gamma}$ for any constants $c>0$ and $\gamma<1$), and arbitrary in the sense of including parametric, semiparametric, nonparametric, as well as modern machine learning highly adaptive estimators. Importantly, our results are completely agnostic as to whether the collection of models includes a correct model for nuisance parameters, in the sense that our procedure will select the nuisance models that optimize our doubly robust criteria. Another aspect in which our approach is generic is that it does not depend on a particular choice of doubly robust estimator of a given functional. In the sense, the approach may be used with say doubly robust targeted maximum likelihood learning to construct an ensemble of doubly robust targeted maximum likelihood estimators, each of which based on different estimators of nuisance parameters. As discussed in Section \[sec:dr\], a very general class of doubly robust functionals are considered here. The purpose of considering a broad class is to demonstrate the flexibility of our method for various functionals that are generally of interest. Several functionals, such as the expected conditional covariance, marginal mean of an outcome subject to missingness as well as the closely related marginal mean of a counterfactual outcome are within our class. As a running example to develop the proposed methodology, throughout we consider the average treatment effect under unconfoundedness as the target of inference.
In settings where all candidate estimators of the functional are regular and asymptotically linear, although not necessarily consistent, we propose a smooth approximation of the proposed criteria which allows for valid post-selection inference. In case the selected model fails to be consistent for the functional of interest, because all candidate models fail to consistently estimate nuisance parameters, valid inference can nevertheless be obtained for the approximate functional that minimizes a population version of the proposed doubly-robust-inspired pseudo-risk function, whenever such approximate functional is well-defined. Confidence intervals can then be constructed either using an estimate of asymptotic variance of smooth selected estimator of the functional based on a standard application of the delta method, or via the nonparametric bootstrap.
The paper is organized as follows: in Section \[sec:dr\], we introduce the general class of doubly robust functionals and give specific examples of interest within this class. In Section \[sec:prelim\], we introduce the problem setting, and demonstrate the main challenge of model selection. Section \[sec:selection\] is devoted to developing our proposed selection criteria. Utilizing these criteria, we propose a general cross-validation scheme to construct empirical minimax functional selectors in Section \[sec:cf\]. In Section \[sec:theory\], we use powerful exponential inequalities for tails of extreme of second order U-statistics to establish a risk bound for the cross-validated minimax and mixed-minimax criteria. The risk bound firmly establishes that our empirical criteria select a pair of nuisance models which performs nearly as well as the pair of models selected by an oracle with access to the law that generated the data. In Section \[sec:simulations\], we present simulation studies to evaluate the performance of the proposed approach in a range of settings. In Section \[sec:softmax\], we describe a smooth approximation to the cross-validated pseudo-risk minimizer which allows for post-selection inferences. In Section \[sec:real\], we illustrate the proposed methods by studying the effectiveness of right heart catheterization in the insentive care unit (ICU) of critically ill patients. Details of proofs are given in the appendices.
A class of doubly robust functionals\[sec:dr\]
==============================================
Suppose we observe $n$ i.i.d. samples $\mathcal O \equiv \{O_i,i=1,\cdots,n\}$ from a law $F_0$, belonging to a model $\mathcal M=\{F_\theta: \theta \in \Theta\}$, where $\Theta$ may be infinite dimensional. We are interested in inference about a functional $\psi(\theta)=\psi(F_\theta)$ on $\cal M$ for a large class of functionals known to admit a doubly robust first order influence function as defined in [@robins2016tr].
Suppose that $\theta = \theta_1 \times \theta_2$, where $\times$ denotes Cartesian product, $\theta_1 \in \Theta_1$ governs the marginal law of $X$ which is a $d$-dimensional subset of variables in $O$, and $\theta_2 \in \Theta_2$ governs the conditional distribution of $O|X$. \[as1\]
An influence function is a fundamental object of statistical theory that allows one to characterize a wide range of estimators and their efficiency. The influence function of a regular and asymptotically linear estimator $\widehat \psi$ of $\psi (\theta)$, $\theta \in \cal M$, is a random variable $IF(\theta)\equiv IF(O;\theta)$ which captures the first order asymptotic behavior of $\widehat \psi$, such that ${n}^{1/2}\{\widehat \psi-\psi(\theta)\}=n^{-1/2} \sum_{i=1}^n IF(O_i;\theta) + o_p(1)$. The set of influence functions of all regular and asymptotically linear estimators of a given functional $\psi(\theta)$ on $\cal M$ is contained in the Hilbert subspace of mean zero random variables $U\equiv u(O;\theta)$ that solve the following equation, $$d(\psi(\theta_t)/dt|_{t=0} =E\{US\},$$ for all regular parametric submodels of $\cal M$, $F_{\theta_t}$, $t \in (-\epsilon,\epsilon)$ with $F(\theta_0)=F_0$, and $S$ the score function of $f(O; \theta_t)$ at $t = 0$ [@newey1990semiparametric; @bickel1993efficient; @van2000asymptotic; @tsiatis2007semiparametric]. Once one has identified the influence function of a given estimator, one knows its asymptotic distribution, and can construct corresponding confidence intervals for the target parameter. We now describe a large class of doubly robust influence functions.
The parameter $\theta_2$ contains components $b:\mathbbm{R}^d\rightarrow \mathbbm{R}$ and $p:\mathbbm{R}^d\rightarrow \mathbbm{R}$, such that the functional $\psi(\theta)$ of interest has a first order influence function $IF(\theta)= H(b,p)-\psi(\theta)$, where $$\begin{aligned}
\label{eq:H}
H(b,p) \equiv b(X)p(X)h_1(O) + b(X)h_2(O) + p(X)h_3(O) + h_4(O),\end{aligned}$$ and $h_1,\ldots,h_4$ are measurable functions of $O$. \[as2\]
$\Theta_{2b} \times \Theta_{2p} \subseteq \Theta_2$, where $\Theta_{2b}$ and $\Theta_{2p}$ are the parameter spaces for the functions $b$ and $p$. Furthermore, the sets $\Theta_{2b}$ and $\Theta_{2p}$ are dense in $L_2(F_0)$ at each $\theta_1\in \Theta_1$, where $L_2(F_0)$ is the Hilbert space of all functions with finite variance. \[as3\]
[@robins2016tr] point out that Assumptions \[as1\]-\[as3\] imply the following double robustness property, $$\begin{aligned}
\label{eq:dr0}
E_\theta[H(b^*,p^*)]-E_\theta[H(b,p)]=E[(b(X)- b^*(X))(p(X)- p^*(X))h_1(O)],\end{aligned}$$ for all $(b^*,p^*)\in \Theta_{2b} \times \Theta_{2p}$. In which case $E[H(b^*,p^*)]=\psi$ if either $b^*=b$ or $p^*=p$. Examples of functionals within this class include:
(Expected product of conditional expectations) Suppose we observe $O=(A,Y,X)$, where $A$ and $Y$ are univariate random variables. Let $\psi(\theta)= E_\theta[p(X)b(X)] $, where $b(X)=E_\theta[Y|X]$ and $p(X)=E_\theta[A|X]$ are a priori unrestricted. In this nonparametric model, the first order influence function of $\psi$ is given by $$IF(\theta)= p(X)b(X)+p(X)\{Y-b(X)\} +b(X)\{A-p(X)\}-\psi(\theta),$$ so $h_1(O)=-1, h_2(O)=A, h_3(O)=Y, h_4(O)=0$.
(Expected conditional covariance) Suppose $O=(A,Y,X)$, where $A$ and $Y$ are univariate random variables. Let $\psi(\theta)= E_\theta [Cov_\theta (Y,A|X)]= E_\theta[AY]- E_\theta[p(X)b(X)]$, where $b(X)=E_\theta[Y|X]$ and $p(X)=E_\theta[A|X]$. In this model, the first order influence function is $$IF(\theta)= AY- \big[ p(X)b(X)-\psi(\theta)+p(X)\{Y-b(X)\} +b(X)\{A-p(X)\} \big]-\psi(\theta),$$ so $h_1(O)=1, h_2(O)=-A, h_3(O)=-Y, h_4(O)=AY$.
As pointed out by [@robins2008HOIF; @robins2016tr; @robins2017], inference about expected conditional covariance is key to obtaining valid inferences about $\beta$ in the widely used semiparametric regression model $E(Y|A,X)=\beta A+b(X)$, where $b(X)$ is unrestricted [@robins2008HOIF].
(Missing at random) Suppose $O=(AY,A,X)$, where $A$ is the binary missing indicator, and $X$ is a $d$-dimensional vector of fully observed continuous covariates. We assume $Y$ is missing at random, i.e., $A{\rotatebox[origin=c]{90}{$\models$}}Y|X$. Let $b(X) = E(Y|A=1,X)$ be the outcome model and $\Pr(A = 1|X)>0$. The parameter of interest $\psi(\theta)$ is the marginal mean of $Y$. In this model, the first order influence function is $$IF(\theta)= Ap(X)\{Y-b(X)\}+b(X)-\psi(\theta),$$ where $p(X) = 1/\Pr(A = 1|X)$. So $h_1(O)=-A, h_2(O)=1, h_3(O)=AY, h_4(O)=0$.
(Missing not at random) We consider the setting in the last example allowing for missing not at random. We assume that $\Pr(A=1|X,Y)=[1+\exp\{-[\gamma(X)+\alpha Y]\} ]^{-1}$, where $\gamma(X)$ is an unknown function and $\alpha$ is a known constant. The marginal mean of $Y$ is again of interest and given by $\psi(\theta)=E_\theta (AY[1+\exp\{-[\gamma(X)+\alpha Y]\} ])$. [@robins2001comment] derived the first order influence function of $\psi$, $$IF(\theta)=A[1+\exp\{-\alpha Y\}p(X)][Y-b(X)]+b(X)-\psi(\theta),$$ where $b(X)=E[ Y\exp\{ -\alpha Y\} |A=1,X]/E[ \exp\{-\alpha Y\}|A=1,X]$ and $p(X)=\exp\{-\gamma(X)\}$. So $h_1(O)= -A \exp\{-\alpha Y\}, h_2(O)=1-A, h_3(O)=AY\exp\{-\alpha Y\}$, $h_4(O)=AY. $
(Average treatment effect) Suppose we observe $O=(A,Y,X)$, where $A$ is a binary treatment taking values in $\{0,1\}$, $Y$ is a univariate response, and $X$ is a collection of covariates. We wish to make inferences about the average treatment effect $E\left\{ Y_{1}-Y_{0}\right\}$, where $Y_{1}$ and $Y_{0}$ are potential outcomes.
Three important assumptions are sufficient for identification of the average treatment effect from the observed data. First, we make the consistency assumption that $Y = Y_A$ almost surely. This assumption essentially states that one observes $Y_a$ only if the treatment $a$ is equal to a subject’s actual treatment assignment $A$. The next assumption is known as ignorability [@10.2307/2335942], which requires that there are no unmeasured confounders for the effects of $A$ on $Y$, i.e., for both $a\in \{0,1\}$, $Y_a {\rotatebox[origin=c]{90}{$\models$}}A|X$. Finally, we assume that $\pi(a|X=x)=\Pr(A = a|X=x)>0$ for $a\in \{0,1\}$ if $f(x)>0$. This positivity assumption basically states that any subject with an observed value of $x$ has a positive probability of receiving both values of the treatment.
Under these three identifying conditions, functional $\psi_0(\theta) = E[E(Y |A = 1, X) - E(Y |A = 0, X)]$ is the average effect of treatment on the outcome. The first order influence function of this functional is $$\begin{aligned}
IF(\theta)=\frac{\left( -1\right) ^{1-A}}{\pi\left( A|X\right) }Y
-\left\{ \frac{\left( -1\right) ^{1-A}}{\pi\left( A|X\right) }E(Y|A,X)-\sum_{a}\left( -1\right) ^{1-a}E(Y|a,X)\right\}-\psi_0(\theta).
\label{eq:dr}\end{aligned}$$ In fact, this model has four nuisance parameters. Note that we can rewrite the influence function as $$\frac{A}{\pi(1|X)}\{Y-E(Y|1,X)\}+E(Y|1,X) - \left[\frac{1-A}{\pi(0|X)}\{Y-E(Y|0,X)\}+E(Y|0,X) \right]-\psi_0(\theta).$$ Then $IF(\theta)$ can be viewed as a difference of two influence functions of similar form as the missing at random (MAR), where $p^{(1)}(X)=1/\pi(1|X),$ $b^{(1)}(X)= E(Y|1,X)$, $p^{(2)}(X)=1/\pi(0|X),$ $b^{(2)}(X)= E(Y|0,X)$, $h^{(1)}_1(O)=-A, h^{(1)}_2(O)=1, h^{(1)}_3(O)=AY, h^{(1)}_4(O)=0$, $h^{(2)}_1(O)=-(1-A), h^{(2)}_2(O)=1, h^{(2)}_3(O)=(1-A)Y, h^{(2)}_4(O)=0$.
\[remark:rm2\] [@rotnitzky2019mix] study a more general class of doubly robust influence functions, which admit the following “mixed bias property”: For each $\theta$, there exist functions $c(X;\theta)$ and $d(X;\theta)$ such that for any $\theta'$, $$\begin{aligned}
\psi(\theta') - \psi(\theta) + E_\theta (IF(\theta')) = E_\theta[ s_{ab}(O) \{ c(X,\theta')-c(X,\theta) \}\{ d(X,\theta') - d(X,\theta) \} ],
\label{eq:mix}\end{aligned}$$ where $s_{ab}$ is a known function not depending on $\theta$. Note that the selection procedure proposed in Section \[sec:select\] extends to this richer class of doubly robust influence functions which includes both the classes of [@riesz2018] and of doubly robust functionals described above [@robins2016tr]. In fact, all that is required by the proposed approach is the influence function has mean zero when either nuisance parameter is evaluated at the truth. As will be discussed later, the approach can readily be extended to multiply robust influence functions in the sense of [@tchetgentchetgen2012; @wang2018bounded; @Caleb2019; @Shi2019MultiplyRC; @sun2019multiple].
The practical implication of double robustness is that the asymptotic bias of an estimator obtained by solving $\PP_n \widehat {IF}(\widehat \psi)=\PP_n IF(\widehat p, \widehat b,\widehat \psi) = 0$ is guaranteed to be zero provided either but not necessarily both $\widehat p$ is consistent for $p$ or $\widehat b$ is consistent for $b$. Despite this local robustness property, in practice one may be unable to ensure that either model is consistent, and even when using nonparametric models, that the resulting bias is small. For this reason, model selection over a class of candidate estimators may be essential to optimize performance in practical settings.
Challenges of model selection for doubly robust inference\[sec:prelim\]
=======================================================================
Hereinafter, in order to ground ideas, we focus the presentation to the average treatment effect functional of Example 2.5. It is well known that $$\begin{aligned}
\psi_0 &=&E\left[ E\left( Y|A=1,X\right) -E\left( Y|A=0,X\right) \right] \\
&=&E\left( \frac{\left( -1\right) ^{1-A}}{\pi\left( A|X\right) }Y\right) \\
&=&E\left(
\begin{array}{c}
\frac{\left( -1\right) ^{1-A}}{\pi\left( A|X\right) }Y
-\left\{ \frac{\left( -1\right) ^{1-A}}{\pi\left( A|X\right) }E(Y|A,X)-\sum_{a}\left( -1\right) ^{1-a}E(Y|a,X)\right\}
\end{array}\right).\end{aligned}$$
The first representation is known as outcome regression as it depends on the regression of $Y$ on $\left( A,X\right) $; the second is inverse probability weighting with weights depending on the propensity score [@10.2307/2335942]; the third representation is known as doubly robust as it relies on outcome regression or propensity score model to be correct but not necessarily both. In fact, the doubly robust representation based on the efficient influence function of $\psi _{0}$ is given by Equation , which will be used as our estimating equation for $\psi_0$ for the proposed model selection.
In order to describe the inherent challenges of performing model selection for $\psi_0$, consider the sample splitting scheme whereby a random half of the sample is used to construct $\widehat \pi_k(A|X)$ and $\widehat E_\tk(Y|A,X)$, while the other half is used to obtain the doubly robust estimator $\widehat \psi_{k,\tk}$. Consider the goal of selecting a pair of models $(k, \tk)$ that minimizes the mean squared error $E[(\widehat \psi_{k, \tk}-\psi_0)^2|\text{Training sample}]$ = bias$^2(\widehat \psi_{k,\tk})$ + variance$(\widehat \psi_{k,\tk})$, where bias$^2(\widehat \psi_{k,\tk})$ is given by Equation As we expect the variance term to be of order $1/n$ conditional on training sample, we may focus primarily on minimizing the squared bias. As no unbiased estimator of $\text{bias}^2(\widehat \psi_{k,\tk})$ exists, minimizing $\text{bias}^2(\widehat \psi_{k,\tk})$ will generally not be possible without incurring excessive bias. Hereafter, for a given split of sample we shall refer to $\arg \min_{k,\tk} \text{bias}^2(\widehat \psi_{k,\tk})$ as “squared bias minimizer”, which depends on the true data generating law (through $\pi(A|X)$ and $E(Y|A,X)$), and therefore may not be accurately estimated even in large samples. In the next section, we propose alternative criteria for selecting an estimator with a certain optimality condition that is nearly attainable empirically.
Recall that consistent estimators of the propensity score and outcome regression are not necessarily contained as candidates for model selection, so the minimal squared bias may not necessarily converge to zero asymptotically; nevertheless, it will do so when at least one nuisance parameter is estimated consistently. Furthermore, as we formally establish in Sections \[sec:theory1\] and \[sec:theory2\] and illustrate in our simulations, when a library of flexible machine learning estimators is used to estimate nuisance parameters, the approach proposed in the next section behaves nearly as well as an oracle that selects the estimator with smallest average squared bias, which vanishes at least as fast as any given choice of machine learners. This is quite remarkable as the proposed approach avoids directly estimating the squared bias.
Model selection via a minimax cross-validation\[sec:select\]
============================================================
Minimax criteria for model selection \[sec:selection\]
------------------------------------------------------
In this section, we consider alternative selection criteria which avoid estimating and directly minimizing Equation . Suppose that we have candidate models $\pi_k =\left( A|X\right), k \in \mathcal K_1 \equiv \{1,\cdots,K_1\}$ for the propensity score and $E_\tk\left( Y|A,X\right), \tk \in \mathcal K_2 \equiv \{1,\cdots,K_2\}$ for the outcome model, respectively. We begin by describing the population version of our minimax criteria, i.e., we focus on $\pi_{k}\left( A|X\right)$ and $E_{\tk}(Y|a,X)$, the asymptotic limits of $\widehat \pi_{k}\left( A|X\right)$ and $\widehat E_{\tk}(Y|a,X)$. We will introduce the cross-validated estimator in Section \[sec:cf\]. For each pair of candidate models $(k_1, \tk_1)$, we have $$\begin{aligned}
\psi _{k_1,\tk_1} = E\left(\frac{\left( -1\right) ^{1-A}}{\pi_{k_1}\left( A|X\right) }Y
-\left\{ \frac{\left( -1\right) ^{1-A}}{\pi_{k_1}\left( A|X\right) }E_{\tk_1}(Y|A,X)-\sum_{a}\left( -1\right) ^{1-a}E_{\tk_1}(Y|a,X)\right\}\right).
\label{eq:if2}\end{aligned}$$ The working models for the propensity score and outcome model could be parametric, semiparametric or nonparametric. A simple parametric case may entail positing that $\pi_k \left( A|X\right)$ and $E_\tk\left( Y|A,X\right)$ are chosen to be the following regression models $$\begin{aligned}
\text{logit}\Pr \left( A=1|X\right) &=&\alpha _{k,0}+\alpha _{k,1}^{T}h_{k}(X),\label{model1}\\
E\left( Y|A,X\right) &=&\beta _{\tk,0}+ \beta _{\tk,1}^{T}h_{\tk}(X)+\beta _{\tk,2}^{T}g_{\tk}(X)A+\beta _{\tk,3}A, \label{model2}\end{aligned}$$ for dictionary $\{h_k, h_\tk, g_\tk: k \in \mathcal K_1;\tk \in \mathcal K_2\}$. Subsequently, [$$\begin{aligned}
\psi_{k,\tk}=E \bigg( \frac{\left( -1\right) ^{1-A}}{\pi \left( A|X;\alpha _{k}\right) }\Big\{
Y -\beta _{\tk,0}-\beta _{\tk,1}^{T} h_{\tk}(X)
- \beta _{\tk,2}^{T} g_{\tk}(X) A -\beta _{\tk,3} A
\Big\}
+ \beta _{\tk,2}^{T} g_{\tk}(X)+ \beta _{\tk,3} \bigg).\end{aligned}$$]{}
Recall that the doubly robust estimator which depends on both unknown functions has zero bias if either one contains the truth. Motivated by this observation, we define the following perturbation of a fixed index pair $(k_1,\tk_1)$, $$\begin{aligned}
\per(k,\tk; k_1,\tk_1) \equiv (\psi_{k,\tk}- \psi_{k_1,\tk_1})^2. \label{definition:per}\end{aligned}$$
The perturbations defined above have the following forms.
\[lemma:1\] $$\begin{aligned}
\per( k_1,\tk; k_1,\tk_1) = E\left[\sum_{a}\left( -1\right) ^{1-a} (\frac{\pi\left(
a|X\right) }{\pi_{k_1}\left( a|X\right) }-1) (E_{\tk_1}(Y|a,X)-E_{\tk}(Y|a,X))\right]^2,\end{aligned}$$
$$\begin{aligned}
\per( k,\tk_1; k_1,\tk_1) = E\left[\sum_{a}\left( -1\right) ^{1-a} (\frac{\pi\left(
a|X\right) }{\pi_k\left( a|X\right) }- \frac{\pi\left(
a|X\right) }{\pi_{k_1}\left( a|X\right) } ) (E(Y|a,X)-E_{\tk_1}(Y|a,X))\right]^2.\end{aligned}$$
Subsequently, for each fixed pair $(k_1,\tk_1)$, we only consider perturbations over pairs $(k,\tk)$ with either $k=k_1$ or $\tk=\tk_1$, and evaluate the perturbation of $ \psi_{k,\tk}$ at $\psi_{k_1,\tk_1}$ as $$\text{per}(k,\tk; k_1,\tk_1)=
\begin{cases}
\text{per}(k_1,\tk; k_1,\tk_1) & \text{if}\ k=k_1, \\
\text{per}(k,\tk_1; k_1,\tk_1) & \text{if}\ \tk=\tk_1,\\
0 & \text{otherwise}.
\end{cases}
\label{bias}$$ We may define the pseudo-risk $$\begin{aligned}
B^{(1)}_{k_1,\tk_1}=\max_{\substack{k=k_1,\tk \in \mathcal K_2;\\ \tk=\tk_1, k\in \mathcal K_1}} \text{per}(k,\tk; k_1,\tk_1),\end{aligned}$$ which measures the maximum change of underlying functional at a candidate selected model $(k_1,\tk_1)$ induced by perturbing one of the nuisance parameters at the time, and holding the other fixed. We call this a pseudo-risk because unlike a standard definition of risk (e.g., mean squared error) which is typically defined in terms of the data generating mechanism and a given candidate model/estimator, the proposed definition is in terms of all candidate models/estimators. We also consider the following pseudo-risk, $$\begin{aligned}
B^{(2)}_{k_1,\tk_1}=\max_{\tk_0 \in \mathcal K_2} \max_{\tk \in \mathcal K_2} \text{per}(k_1,\tk; k_1,\tk_0) + \max_{k_0\in \mathcal K_1} \max_{k\in \mathcal K_1} \text{per}(k,\tk_1; k_0,\tk_1).\end{aligned}$$ Evaluating the above perturbation for each pair $(k_1,\tk_1)$ gives $K_1 \times K_2$ pseudo-risk values $B_{k_1,\tk_1}, k_1\in \mathcal K_1;\tk_2\in \mathcal K_2$. Finally, we define $$\begin{aligned}
\arg\min_{(k_1,\tk_1)} B^{(1)}_{k_1,\tk_1},~~ \arg\min_{(k_1,\tk_1)} B^{(2)}_{k_1,\tk_1},\end{aligned}$$ as population version of selected models, respectively. we refer to $B^{(1)}_{k_1,\tk_1}$ as population minimax pseudo-risk and $B^{(2)}_{k_1,\tk_1}$ as population mixed-minimax pseudo-risk.
One may also consider the following alternative criterion: $$\begin{aligned}
\begin{cases}
\dot{k} &=\arg\min_{k_1} \max_{\tilde k} \text{per}( k_1,\tk; k_1,\dot{\tilde k}),\\
\dot{\tilde k} &= \arg\min_{\tilde k_1} \max_{k} \text{per}( k,\tk_1; \dot{k},\tk_1).\\
\end{cases}\end{aligned}$$ However, there may not exist such pair $(\dot{k},\dot{\tilde k})$. The proposed minimax criteria solve the optimization jointly and avoid this difficulty.
There may be different ways to define pseudo-risk using different norms, e.g., the first kind $B^{(1)}_{k_1,\tk_1}$ can also be defined as $$\max_{\tk \in \mathcal K_2} \text{per}(k_1,\tk; k_1,\tk_1) + \max_{ k\in \mathcal K_1} \text{per}(k,\tk_1; k_1,\tk_1);$$ the second kind $B^{(2)}_{k_1,\tk_1}$ can also be defined as $$\sum_{\tk_0 \in \mathcal K_2} \max_{\tk \in \mathcal K_2} \text{per}(k_1,\tk; k_1,\tk_0) + \sum_{k_0\in \mathcal K_1} \max_{k\in \mathcal K_1} \text{per}(k,\tk_1; k_0,\tk_1).$$ The proposed two criteria are representatives of two novel ideas: The first type is given by the overall maximum squared bias (i.e., change in the estimated functional) at a given candidate estimator, induced by perturbing one nuisance parameter at the time over candidate models holding the other one fixed; The second type is given by the sum of two maximum squared bias terms, each capturing the bias induced by perturbing a single nuisance parameter only.
The second mixed-minimax criterion has a doubly robust property, i.e., $\psi_{\arg\min_{(k_1,\tk_1)} B^{(2)}_{k_1,\tk_1}}$ has zero bias if either nuisance model is consistently estimated by at least one candidate learner.
Multi-fold cross-validated estimator\[sec:cf\]
----------------------------------------------
Following [@vaart2006oracle], we avoid overfitting in implementing an empirical minimax selector by considering a multi-fold cross-validation scheme which repeatedly randomly splits the data $\mathcal O$ into two subsamples: a training set $\mathcal O^{0s}$ and a test set $\mathcal O^{1s}$, where $s$ refers to $s$-th split. The splits may be either deterministic or random without loss of generality. In the following, we consider random splits, whereby we let $T^s = (T_1^s,\cdots,T_n^s)\in \{0,1\}^n$ denote a random vector independent of $O_1,\cdots, O_n$. If $T_i^s=0$, $O_i$ belongs to the $s$-th training sample $\mathcal O^{0s}$; otherwise it belongs to the $s$-th test sample $\mathcal O^{1s}$, $s=1, \ldots,S$. For each $s$ and $(k,\tk)$, our construction uses the training samples to construct estimators $\widehat \pi_k^s$ and $\widehat E_\tk^s(Y|A,X)$. The validation sample is then used to estimate the perturbation defined in Equation , $$\begin{aligned}
\widehat {\text{per}}(k,\tk; k_1,\tk_1) =& \frac{1}{S}\sum_{s=1}^S \left[ (\widehat \psi_{k,\tk}^s - \widehat \psi_{k_1,\tk_1}^s )
\right]^{2},
$$ where $$\widehat \psi_{k,\tk}^s = \PP^1_s
\left\{ \frac{\left( -1\right) ^{1-A}}{\widehat \pi^s_k\left( A|X\right) }\left\{
\begin{array}{c}
Y - \widehat E^s_{\tk}(Y| X,A)
\end{array}\right\} +\sum_a (-1)^{1-a}\widehat E^s_\tk(Y|a,X) \right\},$$ $$\widehat \psi_{k_1,\tk_1}^s = \PP^1_s
\left\{ \frac{\left( -1\right) ^{1-A}}{\widehat \pi^s_{k_1}\left( A|X\right) }\left\{
\begin{array}{c}
Y - \widehat E^s_{\tk_1}(Y| X,A)
\end{array}\right\} +\sum_a (-1)^{1-a}\widehat E^s_{\tk_1}(Y|a,X) \right\},$$ $$\PP^j_s= \frac{1}{\#\{1\leq i\leq n:T_i^s=j\}}\sum_{i:T_i^s=j} \delta_{X_i}, \quad j=0,1,$$ and $\delta_X$ is the Dirac measure. We then select the empirical minimizers of $$\widehat B^{(1)}_{k_1,\tk_1} = \max_{\substack{k=k_1,\tk \in \mathcal K_2;\\ \tk=\tk_1, k\in \mathcal K_1}} \widehat {\text{per}}(k,\tk; k_1,\tk_1)$$ and $$\widehat B^{(2)}_{k_1,\tk_1} = \max_{\tk_0 \in \mathcal K_2} \max_{\tk \in \mathcal K_2} \widehat {\text{per}}(k_1,\tk; k_1,\tk_0) + \max_{k_0\in \mathcal K_1} \max_{k\in \mathcal K_1} \widehat {\text{per}}(k,\tk_1; k_0,\tk_1)$$ among all candidate pairs $(k_1,\tk_1)$ as our final models, i.e., $$\begin{aligned}
(k^\dagger,\tk^\dagger) = \arg \min_{(k_1,\tk_1)}\widehat B^{(1)}_{k_1,\tk_1}, \label{eq:dagger} \\
(k^\diamond,\tk^\diamond) = \arg \min_{(k_1,\tk_1)}\widehat B^{(2)}_{k_1,\tk_1}.
\end{aligned}$$ The final estimators are $$\widehat \psi_{k^\dagger,\tk^\dagger}=\frac{1}{S}\sum_{s=1}^S \widehat \psi^s_{k^\dagger,\tk^\dagger},~~\widehat \psi_{k^\diamond,\tk^\diamond}=\frac{1}{S}\sum_{s=1}^S \widehat \psi^s_{k^\diamond,\tk^\diamond}.$$ We provide a high-level Algorithm 1 for the proposed selection procedure. We also define the cross-validated oracle selectors $$\begin{aligned}
(k^\star,\tk^\star) = \arg \min_{(k_1,\tk_1)} B^{(1)}_{k_1,\tk_1}, \label{eq:defstar1}\\
(k^\circ,\tk^\circ) = \arg \min_{(k_1,\tk_1)} B^{(2)}_{k_1,\tk_1}, \label{eq:defstar2}
\end{aligned}$$ where in a slight abuse of notation, we define the cross-validated pseudo-risk $$B^{(1)}_{k_1,\tk_1} = \max_{\substack{k=k_1,\tk \in \mathcal K_2;\\ \tk=\tk_1, k\in \mathcal K_1}} \frac{1}{S}\sum_{s=1}^S \left[ (\psi_{k,\tk}^s - \psi_{k_1,\tk_1}^s )
\right]^{2},$$ $$B^{(2)}_{k_1,\tk_1} = \max_{\tk_0 \in \mathcal K_2} \max_{\tk \in \mathcal K_2} \frac{1}{S}\sum_{s=1}^S \left[ (\psi_{k_1,\tk}^s - \psi_{k_1,\tk_0}^s )
\right]^{2} + \max_{k_0\in \mathcal K_1} \max_{k\in \mathcal K_1} \frac{1}{S}\sum_{s=1}^S \left[ (\psi_{k,\tk_1}^s - \psi_{k_0,\tk_1}^s )
\right]^{2},$$ $$\psi_{k,\tk}^s = \PP^1
\left\{ \frac{\left( -1\right) ^{1-A}}{\widehat \pi^s_k\left( A|X\right) }\left\{
\begin{array}{c}
Y - \widehat E^s_{\tk}(Y| X,A)
\end{array}\right\} +\sum_a (-1)^{1-a}\widehat E^s_\tk(Y|a,X) \right\},$$ $$\psi_{k_1,\tk_1}^s = \PP^1
\left\{ \frac{\left( -1\right) ^{1-A}}{\widehat \pi^s_{k_1}\left( A|X\right) }\left\{
\begin{array}{c}
Y - \widehat E^s_{\tk_1}(Y| X,A)
\end{array}\right\} +\sum_a (-1)^{1-a}\widehat E^s_{\tk_1}(Y|a,X) \right\},$$ and $\PP^1$ denotes the true measure of $\PP_s^1$.
For each pair $(k_1,\tk_1)$, average the perturbations over the splits and obtain $$\widehat {\text{per}}(k,\tk; k_1,\tk_1) = \frac{1}{S}\sum_{s=1}^S \left[ (\widehat \psi_{k,\tk}^s - \widehat \psi_{k_1,\tk_1}^s )
\right]^{2},$$ where $k=k_1,\tk \in \mathcal K_2$ or $\tk=\tk_1, k\in \mathcal K_1$ Calculate $$\widehat B^{(1)}_{k_1,\tk_1}=\max_{\substack{k=k_1,\tk \in \mathcal K_2;\\ \tk=\tk_1, k\in \mathcal K_1}} \wper(k,\tk,k_1,\tk_1),$$ $$\widehat B^{(2)}_{k_1,\tk_1} = \max_{\tk_0 \in \mathcal K_2} \max_{\tk \in \mathcal K_2} \widehat {\text{per}}(k_1,\tk; k_1,\tk_0) + \max_{k_0\in \mathcal K_1} \max_{k\in \mathcal K_1} \widehat {\text{per}}(k,\tk_1; k_0,\tk_1)$$ for each pair $(k_1,\tk_1)$ Pick $(k^\dagger,\tk^\dagger)=\arg\min_{(k,\tk)} \widehat B^{(1)}_{k,\tk}$, $(k^\diamond,\tk^\diamond)=\arg\min_{(k,\tk)} \widehat B^{(2)}_{k,\tk}$ as our selected models, and obtain the estimations of the parameter over the splits $$\widehat \psi_{k^\dagger,\tk^\dagger}=\frac{1}{S}\sum_{s=1}^S \widehat \psi^s_{k^\dagger,\tk^\dagger},~~\widehat \psi_{k^\diamond,\tk^\diamond}=\frac{1}{S}\sum_{s=1}^S \widehat \psi^s_{k^\diamond,\tk^\diamond};$$\
**Return** $(k^\dagger,\tk^\dagger)$, $(k^\diamond,\tk^\diamond)$ and $\widehat \psi_{k^\dagger,\tk^\dagger}$, $\widehat \psi_{k^\diamond,\tk^\diamond}$
Theoretical results\[sec:theory\]
=================================
Optimality of oracle selectors\[sec:theory1\]
---------------------------------------------
In this section, we establish certain optimality properties of the minimax and mixed-minimax oracle pseudo-risk selectors defined by Equations and respectively. As we will later show by establishing excess risk bounds relating empirical selectors to their oracle counterparts, these optimality results imply near optimal behavior of the corresponding empirical (cross-validated) selector. In this vein, focusing on a functional of the doubly robust class with property , consider the collections of learners for $p$ and $b$ obtained from an independent sample of size $n$: $$\begin{aligned}
\mathcal{C}_{p} = \left\{ \widehat{p}_{1},\ldots,\widehat{p}_{K_{1}}\right\};~~
\mathcal{C}_{b} = \left\{ \widehat{b}_{1},\ldots,\widehat{b}_{K_{2}}\right\}.\end{aligned}$$For the purpose of inference, in the following, our analysis is conditional on $\mathcal{C}_{p}$ and $\mathcal{C}_{b}$. Suppose further that these learners satisfy the following assumptions.
\[asm:1\] Given any $\epsilon>0$, there exist constants $C_p, C_b>1$ and sufficiently large $n_0$ such that for $n>n_0$, $$\begin{aligned}
\frac{1}{C_{p}}\nu _{j}\leq \left\vert \widehat{p}_{j}(x)-p\left( x\right)
\right\vert \leq C_{p}\nu _{j},~~ j\in \mathcal K_1,\\
\frac{1}{C_{b}}\omega _{j}\leq \left\vert \widehat{b}_{j}(x)-b\left(
x\right) \right\vert \leq C_{b}\omega _{j},~~ j\in \mathcal K_2,\end{aligned}$$for any $x$ with probability $1-\epsilon$, where $\nu _{j}$ and $\omega_{j}$ depend on $n$.
In the following we write $a_n \lesssim b_n$ when there exists a constant $C>0$ such that $a_n \leq Cb_n$ for sufficiently large $n$. Without loss of generality, suppose that$$\nu _{\min }=\min_{j}\left\{ \nu _{j}:j \in \mathcal K_1\right\} \lesssim \omega
_{\min }=\min_{j}\left\{ \omega _{j}:j \in \mathcal K_2\right\}.$$ Let $$\begin{aligned}
\nu _{\max } =\max_{j}\left\{ \nu _{j}:j \in \mathcal K_1 \right\},~~ \omega _{\max } =\max_{j}\left\{ \omega _{j}:j \in \mathcal K_2\right\}.\end{aligned}$$
\[asm:2\] We assume that $\lim_{n\rightarrow \infty }v_{\max } < \infty ; ~~
\lim_{n\rightarrow \infty }\omega _{\max } < \infty .$
\[asm:3\] Suppose $$(p(X)- \widehat p_i(X))(b(X)- \widehat b_j(X))E[h_1(O)|X],$$ is continuous with respect to $X$ for $i\in \mathcal K_1; j\in \mathcal K_2$. Furthermore, suppose that the support of $X$ is closed and bounded.
Assumption 5.1 essentially states that the bias of $\widehat p_j$ and $\widehat b_j$ is eventually exactly of order $v_j$ and $w_j$ with large probability. Note that $\widehat p_j$ and $\widehat b_j$ may not necessarily be consistent i.e., $v_j$ and $w_j$ may converge to a positive constant. Assumption \[asm:2\] guarantees the bias of each learner does not diverge. Note also that Assumption \[asm:3\] need only hold for $i$ and $j$ such that $v_i=v_{\text{min}}$ and $w_j=w_{\text{max}}$ for Lemma \[lemma:rate\] given below to hold for the minimax bias selector. Let $\psi_{k^\star,\tk^\star} =\PP^1 \{H(\widehat{p}_{k^\star},\widehat{b}_{\tk^\star})\}$ and $\psi_{k^\circ,\tk^\circ} =\PP^1 \{H(\widehat{p}_{k^\circ},\widehat{b}_{\tk^\circ})\}$, where $H(\cdot,\cdot)$ is defined in Equation , $(k^\star,\tk^\star)$ and $(k^\circ,\tk^\circ)$ are defined in Equations and . We have the following lemma.
\[lemma:rate\] Under Assumptions \[asm:1\]-\[asm:3\], we have that the bias of the minimax oracle selector is of the order of:$$\left\vert \psi _{k^\star,\tk^\star}-\psi_0 \right\vert =O_{P}\left( \frac{\nu _{\max }}{\omega _{\max }}\omega _{\min }^{2}\right),$$while the bias of the mixed-minimax oracle selector is of the order of: $$\left\vert \psi _{k^\circ,\tk^\circ}-\psi_0
\right\vert =O_{P}\left( \nu _{\min }\omega _{\min }\right).$$
The above lemma implies that in the event $\nu _{\max }/\omega _{\max }~ {\rightarrow }$ as $n \rightarrow \infty$ in probability, as would be the case, say in a setting where at least one machine learner of both $p$ and $b$ fails to be consistent, the bias of the oracle minimax selector $\left\vert \psi _{k^\star,\tk^{\star}}-\psi_0
\right\vert $ is of order of the maximum (comparing learners of $b$ to those of $p)$ of the minimum (across learners for $b$ and $p,$ respectively) squared bias, that is the maximin squared bias of learners of $b$ and $p$ which under our assumptions is equal to $\omega _{\min }^{2}$. In this case, the minimax selector provides a guaranty for adaptation only up to the least favorable optimal learner across nuisance parameters, and therefore may fail to fully leverage the fact that a fast learner of one nuisance parameter may compensate for a slower learner of another. In contrast, the mixed-minimax selector can leverage such a gap to improve estimation rate, so that in the above scenario, its rate of estimation would be $\nu _{\min }\omega _{\min
}\leq \omega _{\min }^{2}$ with equality only if $\nu _{\min }=\omega _{\min
},$ that is if one can learn $b$ and $p$ equally well.
Excess risk bound of the proposed minimax selector\[sec:theory2\]
-----------------------------------------------------------------
In this section, we focus on our first estimator, however, analogous results hold for the second estimator. We first introduce some notation used to study the excess risk bound of $\widehat \psi_{k^\dagger,\tk^\dagger}$. Define $U^s_{(k,\tk)}(k_1,\tk_1)$ as $$\begin{aligned}
&\frac{\left( -1\right) ^{1-A}}{\widehat \pi_k^s \left(
A|X\right) }Y -\left\{ \frac{\left( -1\right) ^{1-A}}{\widehat \pi_k^s \left( A|X\right) }\widehat E_\tk^s(Y|A,X)-\sum_{a}\left( -1\right) ^{1-a}\widehat E_\tk^s(Y|a,X)\right\} \nonumber \\
-& \frac{\left( -1\right) ^{1-A}}{\widehat \pi_{k_1}^s \left(
A|X\right) }Y -\left\{ \frac{\left( -1\right) ^{1-A}}{\widehat \pi_{k_1}^s \left( A|X\right) }\widehat E_{\tk_1}^s(Y|A,X)-\sum_{a}\left( -1\right) ^{1-a}\widehat E_{\tk_1}^s(Y|a,X)\right\}.
\label{eq:U}\end{aligned}$$ Based on our minimax selection criterion, Equation and are equivalently expressed as $$(k^\dagger,\tk^\dagger) = \arg\min_{(k_1,\tk_1)} \max_{\substack{k=k_1,\tk \in \mathcal K_2;\\ \tk=\tk_1, k\in \mathcal K_1}} \frac{1}{S}\sum_{s=1}^{S} [\PP^1_s\{U^s_{(k,\tk)}(k_1,\tk_1) \}]^2,$$ $$(k^\star,\tk^\star) = \arg\min_{(k_1,\tk_1)} \max_{\substack{k=k_1,\tk \in \mathcal K_2;\\ \tk=\tk_1, k\in \mathcal K_1}} \frac{1}{S} \sum_{s=1}^{S} [\PP^1 \{U^s_{(k,\tk)}(k_1,\tk_1) \} ]^2.$$
Next, we derive a risk bound for empirically selected model $(k^\dagger,\tk^\dagger)$ which states that its risk is not much bigger than the risk provided by the oracle selected model $(k^\star,\tk^\star)$. For this purpose, it is convenient to make the following boundedness assumption.
\[asm:positivity\] (1) $\pi(a|X)\geq M_1$ and $\widehat \pi_{k}(a|X)\geq M_1$ almost surely for $a=0,1$, $k \in \{1,\ldots,K_1\}$, and some $0<M_1<1$. (2) $|Y| \leq M_2$ and $|\widehat E_{\tk}(Y|A,X)| \leq M_2$ almost surely for $\tk \in \{1,\ldots,K_2\}$, and some $M_2>0$.
Suppose Assumptions \[asm:positivity\] holds, then we have that $$\begin{aligned}
& \frac{1}{S}\sum_{s=1}^S \PP^0[\PP^1 \{U^s_{(k^\ddagger,\tk^\ddagger)}(k^\dagger,\tk^\dagger)\}]^2 \\
\leq & \frac{1+2\delta}{S}\sum_{s=1}^S \PP^0[\PP^1 \{U^s_{(k^\sstar,\tk^\sstar)}(k^\star,\tk^\star)\}]^2 + O\left(\frac{(1+\delta)\log(1+K_1^2 \times K_2^2)}{n^{1/p}} \left(\frac{1+\delta}{\delta}\right)^{(2-p)/p} \right),\end{aligned}$$ for any $\delta>0$, and $1\leq p\leq 2$, where $$(k^\ddagger,\tk^\ddagger)=\arg\max_{\substack{k=k^\dagger,\tk \in \mathcal K_2;\\ \tk=\tk^\dagger, k\in \mathcal K_1}} \frac{1}{S} \sum_{s=1}^S [\PP^1_s\{U^s_{(k,\tk)}(k^\dagger,\tk^\dagger)\}]^2,$$ $$(k^\sstar,\tk^\sstar)=\arg\max_{\substack{k=k^\star,\tk \in \mathcal K_2;\\ \tk=\tk^\star, k\in \mathcal K_1}} \frac{1}{S}\sum_{s=1}^S [\PP^1 \{U^s_{(k,\tk)}(k^\star,\tk^\star)\}]^2,$$ and $\PP^0$ denotes the true measure of $\PP_s^0$. \[thm:2\]
The proof of this result is based on a finite sample inequality for $$\begin{aligned}
\frac{1}{S}\sum_{s=1}^S \PP^0[\PP^1 \{U^s_{(k^\ddagger,\tk^\ddagger)}(k^\dagger,\tk^\dagger)\}]^2 - \frac{1}{S}\sum_{s=1}^S \PP^0[\PP^1 \{U^s_{(k^\sstar,\tk^\sstar)}(k^\star,\tk^\star)\}]^2,\end{aligned}$$ the excess pseudo-risk of our selected model compared to the oracle’s, which requires an extension of Lemma 8.1 of [@vaart2006oracle] to handle second order U-statistics. We obtained such an extension by making use of a powerful exponential inequality for the tail probability of the maximum of a large number of second order U-statistics derived by [@10.1007/978-1-4612-1358-1_2]. Note that Theorem \[thm:2\] generalizes to the doubly robust functionals in the class of [@rotnitzky2019mix], with Equation replaced by $IF^s_{k,\tk}(\widehat \psi^s_{k_1,\tk_1})-IF^s_{k_1,\tk_1}(\widehat \psi^s_{k_1,\tk_1})$ in the definition of $U^s_{(k,\tk)}(k_1,\tk_1)$, where $IF^s_{k,\tk}(\widehat \psi^s_{k_1,\tk_1})$ is an influence function of $\psi$ evaluated at nuisance parameters $(k,\tk)$ and $\widehat \psi^s_{k_1,\tk_1}$ solves $\PP_1^s IF^s_{k_1,\tk_1}(\widehat \psi^s_{k_1,\tk_1})=0$ (see Algorithm 2 in the Appendix for details).
The bound given in Theorem \[thm:2\] is valid for any $\delta>0$, such that the error incurred by empirical risk is of order $n^{-1}$ for any fixed $\delta$ if $p=1$, suggesting in this case that our cross-validated selector performs nearly as well as a oracle selector with access to the true pseudo-risk. Theorem \[thm:2\] is most of interest in nonparametric/machine learning setting where pseudo-risk can be of order substantially larger than $O(n^{-1})$ in which can the error made in selecting optimal learner is negligible relative to risk. By allowing $\delta_n \rightarrow 0$, the choice $p=2$ gives an error of order $n^{-1/2}$, which may be substantially larger. If we write the bound in the form of [$$\begin{aligned}
\frac{\sum_{s=1}^S \PP^0[\PP^1 \{U^s_{(k^\ddagger,\tk^\ddagger)}(k^\dagger,\tk^\dagger)\}]^2}{\sum_{s=1}^S \PP^0[\PP^1 \{U^s_{(k^\sstar,\tk^\sstar)}(k^\star,\tk^\star)\}]^2} \leq (1+2\delta) + O\left(\frac{n^{-1/p}S(1+\delta)^{2/p}\log(1+K_1^2 \times K_2^2)}{{\sum_{s=1}^S \PP^0[\PP^1 \{U^s_{(k^\sstar,\tk^\sstar)}(k^\star,\tk^\star)\}]^2 \delta^{(2-p)/p}}} \right),\end{aligned}$$ ]{}the risk ratio converges to 1 with the remainder of order $n^{-1/2}$. Furthermore, the derived excess risk bound holds for as many as $c^{n^\gamma}$ models ($K_1\times K_2$) for any $\gamma<1$ and $c>0$.
Suppose that $\widehat \pi_k(X)\rightarrow \pi(X)$ and $\widehat E_\tk(Y|A,X) \rightarrow E(Y|A,X)$ in probability for some pair $(k,\tk)$, $[\PP^1 \{U^s_{(k^\sstar,\tk^\sstar)}(k^\star,\tk^\star)\}]^2$ converges to zero as $n \rightarrow \infty$ by Lemma \[lemma:1\], otherwise $\lim_{n\rightarrow \infty}[\PP^1 \{U^s_{(k^\sstar,\tk^\sstar)}(k^\star,\tk^\star)\}]^2>0$ and we select model/estimator that is nearest to satisfying the double robustness property.
An theorem analogous to Theorem \[thm:2\] holds for the mixed-minimax selector. Although details are omitted, the proof is essentially the same.
Simulation studies\[sec:simulations\]
=====================================
In Section \[sec:simu1\], we first consider two different settings in context of (possibly misspecified) parametric models as illustrative examples of the proposed approach. Next, in Section \[sec:simu3\], we evaluate the proposed approach in the context of primary interest, where modern machine learning methods are used to estimate nuisance parameters; and compare to double debiased machine learning (DDML) [@Chernozhukov2018] for each candidate machine learner of nuisance parameters. For each setting, we use three-fold cross-validation, i.e., $S=3$ with $T_i$’s from Bernoulli(0.5).
Illustrative examples with parametric candidate models {#sec:simu1}
------------------------------------------------------
Consider the following five-variate functional forms [@zhao2017selective] as potential candidate parametric models, $$\begin{aligned}
f_1(x) &= \Big((x_1-0.5)^2, (x_2-0.5)^2, \ldots, (x_{5}-0.5)^2\Big)^T,\\
f_2(x) &= \Big((x_1-0.5)^3, (x_2-0.5)^3, \ldots, (x_{5}-0.5)^3\Big)^T,\\
f_3(x) &= \Big(x_1,x_2, \ldots, x_{5}\Big)^T,\\
f_4(x) &= \Big(\frac{1}{1+\exp(-20(x_1-0.5))}, \ldots, \frac{1}{1+\exp(-20(x_5-0.5))}\Big)^T.\end{aligned}$$
In the first simulation setting, the true model is not included as candidate model and all working models are misspecified, whereas in the second setting, the true model is included as a candidate model. Each simulation was repeated 500 times. For each setting, covariates $X_i$’s were independently generated from a uniform distribution, and the noise of outcome was normal with mean 0 and standard deviation $0.1$. In the first scenario, the data were generated from $$\begin{aligned}
\text{logit}\Pr \left( A=1|X\right) &=&(1,-1,1,-1,1)f_1(X),\\
E\left( Y|A,X\right) &=&1 + \mathbbm{1}^Tf_1(X)+ \mathbbm{1}^Tf_1(X) A+ A,\end{aligned}$$ where $ \mathbbm{1}=(1,1,1,1,1)^T$. In the second scenario, the data were generated from $$\begin{aligned}
\text{logit}\Pr \left( A=1|X\right) & =& (1,-1,1,-1,1)f_2(X),\\
E\left( Y|A,X\right) &= & 1 + \mathbbm{1}^Tf_2(X)+ \mathbbm{1}^Tf_2(X) A+ A.\end{aligned}$$ For both scenarios, we used $\{f_2,f_3,f_4\}$ as candidate models of $g$ and $h$ specified in Equations and .
The squared bias of $\widehat \psi$ for both scenarios is shown in Figures \[fig:1\]-\[fig:3\], respectively. (bias$^2 \times 10^{-4}$) is displayed. “Minimizer” refers to “squared bias minimizer” given by Equation ; “Oracle1” and “Oracle2” refer to the oracle minimax and mixed-minimax selectors evaluated by Lemma \[lemma:1\], respectively; “Proposed1” and “Proposed2” refer to the proposed minimax and mixed-minimax selectors, respectively; “Separate” refers to the more conventional practical approach which performs model selection separately for each nuisance parameter via AIC [@akaike1974new]; “Truth” refers to using the underlying true candidate models for estimation.
In the first scenario, because the true model is not a candidate model, there is a notable gap between the squared bias of estimators obtained from working models and those estimated directly from true models. In addition, because the “squared bias minimizer” minimizes the squared bias, it naturally has smaller Monte Carlo squared bias than the proposed criteria. Note that we do not expect “squared bias minimizer” and the proposed approach to perform similarly even asymptotically because they minimize different objective functions, and recall that the former may not be attainable in this specific setting. The proposed method has smaller bias than selecting models separately. However, this may not always be the case because the proposed method does not necessarily minimize the squared bias directly. Additional simulations in the Appendix illustrate this point. In the second scenario, the gap observed in the first scenario vanishes asymptotically, and both proposed methods perform nearly as well as both oracle and “squared bias minimizer” selectors in large samples as illustrated in Figure \[fig:3\]. Mixed-minimax selector appears to perform somewhat better than minimax selector in small to moderate samples.
![Squared bias of Scenario 1[]{data-label="fig:1"}](biasplot1.pdf "fig:"){width="4in" height="2.5in"}\
![Squared bias of Scenario 2[]{data-label="fig:3"}](biasplot2.pdf "fig:"){width="4in" height="2.5in"}\
Model selection with machine learners {#sec:simu3}
-------------------------------------
Finally, we report simulation results for the setting of primarily interest, where various machine learners are used to form candidate estimators of nuisance parameters. We considered the following machine learning methods. For the propensity score model: 1. Logistic regression with $l_1$ regularization [@lasso; @friedman2010regularization]; 2. Classification random forests [@Breiman2001; @randomForest]; 3. Gradient boosting trees [@friedman2001greedy; @gbm2019]. For the outcome model: 1. Lasso [@lasso; @friedman2010regularization]; 2. Regression random forests [@Breiman2001; @randomForest]; 3. Gradient boosting trees. Data were generated from $$\begin{aligned}
\text{logit}\Pr \left( A=1|X\right) &=&(1,-1,1,-1,1)^Tf_4(X),\\
E\left( Y|A,X\right) &=&1 + \mathbbm{1}^Tf_4(X)+ \mathbbm{1}^Tf_4(X) A+ A.\end{aligned}$$ The outcome error term was normal with mean 0 and standard deviation $0.25$.
Implementing candidate estimators required selecting corresponding tuning parameters: The penalty $\lambda_n$ for Lasso was chosen using 10-fold cross-validation over the pre-specified grid $[10^{10},\ldots,10^{-2}]$; For gradient boosting trees [@gbm2019], all parameters were tuned using a 4-fold cross-validation over the following grid: `ntrees`=\[100,300\], `depth`=\[1,2,3,4\], `shrinkag`=\[0.001,0.01,0.1\]; We used the default values of minimum node size (1 for classification, 5 for regression), and number of trees (500) for random forest [@randomForest], while the number of variables randomly sampled at each split, i.e., `mtry`, was tuned by `tuneRF` function [@randomForest].
We compared the proposed estimators with three DDML estimators using Lasso, random forests, and gradient boosting trees to estimate nuisance parameters respectively [@Chernozhukov2018]. Each DDML was estimated by cross-fitting [@Chernozhukov2018], i.e., 1) using training data (random half of sample) to estimate nuisance parameters and validation data to obtain $\widehat \psi_1$; 2) swaping the role of training and validation dataset to obtain $\widehat \psi_2$; 3) computing the estimator as the average $\widehat \psi_{\text{CF}} = (\widehat \psi_1 + \widehat \psi_2)/2$. The squared bias of $\widehat \psi$ of different methods are shown in Figure \[fig:ml\]. (bias$^2 \times 10^{-4}$) is displayed. “LASSO” refers to using logistic regression with $l_1$ regularization for propensity score model, and standard Lasso for outcome model; “RF” refers to using classification forests for propensity score model, and regression forests for outcome model; “GBT” refers to using gradient boosting classification tree for propensity score model, and gradient boosting regression tree for outcome model. Both proposed estimators have smallest bias across sample sizes, and there is a notable gap between the proposed estimators and those estimated by DDML. It is not surprising that Lasso has the largest bias because the working models are not correctly specified. This confirms our earlier claim that combined with flexible nonparametric/machine learning methods, our proposed approach can in finite samples yield smaller squared bias than any given machine learning estimator, without directly estimating the corresponding squared bias.
![Squared bias of the proposed estimator and different machine learners[]{data-label="fig:ml"}](bias1.pdf "fig:"){width="4in" height="2.5in"}\
A smooth-max approach to post-selection approximate inference {#sec:softmax}
=============================================================
In this section, we propose a novel smooth-max approach as smooth approximation to proposed minimax and mixed-minimax model selection criteria. Such smooth approximation provides a genuine opportunity to perform valid post-selection inference, appropriately accounting for uncertainty in both selecting and estimating nuisance parameters. It is well known that the following smooth-max function $$\begin{aligned}
\Gamma(\tau) = \frac{1}{\tau} \log \sum_{i=1}^{m} \exp(\tau z_i),\end{aligned}$$ converges to $ \max(z_1,\ldots,z_m)$ as $\tau \rightarrow \infty$, where $z_1,\ldots,z_m$ are positive real numbers. Similarly, we define $$\begin{aligned}
\Gamma_{k_1,\tk_1}(\tau) = \frac{1}{\tau} \log \sum_{\substack{k=k_1,\tk \in \mathcal K_2;\\ \tk=\tk_1, k\in \mathcal K_1}} \exp\{\frac{\tau}{S}\sum_{s=1}^{S}(\widehat \psi^s_{k,\tk} -\widehat \psi^s_{k_1,\tk_1})^2 \},\end{aligned}$$ for minimax selector and $$\begin{aligned}
\Gamma_{k_1,\tk_1}(\tau) =& \frac{1}{\tau} \log \sum_{\tk,\tk_0 \in \mathcal K_2} \exp\{\frac{\tau}{S}\sum_{s=1}^{S}(\widehat \psi^s_{k_1,\tk} -\widehat \psi^s_{k_1,\tk_0})^2 \} \\+& \frac{1}{\tau} \log \sum_{ k,k_0 \in \mathcal K_1} \exp\{\frac{\tau}{S}\sum_{s=1}^{S}(\widehat \psi^s_{k,\tk_1} -\widehat \psi^s_{k_0,\tk_1})^2 \},\end{aligned}$$ for mixed-minimax selector. Note that $$\begin{aligned}
\Gamma_{k_1,\tk_1}(\tau) \rightarrow \max_{\substack{k=k_1,\tk \in \mathcal K_2;\\ \tk=\tk_1, k\in \mathcal K_1}} \frac{1}{S}\sum_{s=1}^S (\widehat \psi^s_{k,\tk} -\widehat \psi^s_{k_1,\tk_1})^2,
\label{eq:limit1}
\end{aligned}$$ and $$\begin{aligned}
\Gamma_{k_1,\tk_1}(\tau) \rightarrow \max_{\tk,\tk_0 \in \mathcal K_2} \frac{1}{S}\sum_{s=1}^S (\widehat \psi^s_{k_1,\tk} -\widehat \psi^s_{k_1,\tk_0})^2 + \max_{ k,k_0 \in \mathcal K_1} \frac{1}{S}\sum_{s=1}^S (\widehat \psi^s_{k,\tk_1} -\widehat \psi^s_{k_0,\tk_1})^2,
\label{eq:limit2}
\end{aligned}$$ as $\tau \rightarrow \infty$.
Recall that our goal is to select the model $(k_1,\tk_1)$ minimizing the right hand side of Equations or , which is then used to estimate $\psi_0$. A smooth-max approximation to this selection process is given by $$\begin{aligned}
\widehat \psi(\tau) = \sum_{(k_1,\tk_1)} p_{k_1,\tk_1}(\tau) \widehat \psi_{k_1,\tk_1},\end{aligned}$$ where $$\begin{aligned}
p_{k_1,\tk_1}(\tau)
= \frac{\exp\{ \tau [\sum_{(k,\tk)} \Gamma_{k,\tk}(\tau)-\Gamma_{k_1,\tk_1}(\tau)] \}}{ \sum_{(k',\tk')} \exp\{ \tau [\sum_{(k,\tk)} \Gamma_{k,\tk}(\tau)-\Gamma_{k',\tk'}(\tau)] \} },\end{aligned}$$ and $$\begin{aligned}
\widehat \psi_{k_1,\tk_1}=\frac{1}{S}\sum_{s=1}^S \widehat \psi^s_{k_1,\tk_1}.\end{aligned}$$ Because $\widehat \psi(\tau)$ is a smooth transformation of $\widehat \psi_{k_1,\tk_1}, k_1=1,\ldots,K_1, \tk_1=1,\ldots,K_2$, justification of this smooth approximation is that $p_{k_1, \tk_1}(\tau) \rightarrow 1$ as $\tau \rightarrow \infty$ if $(k_1, \tk_1)=\arg\min_{k,\tk}\Gamma_{k,\tk}(\tau)$, otherwise $p_{k_1, \tk_1}(\tau) \rightarrow 0$. Given a fixed $\tau$, inference can be carried out with the nonparametric bootstrap or sandwich asymptotic variance estimators using the delta method. In following section, we use this smooth approximation for post-selection inference in a data application.
The above claim certainly holds when $K_1$ and $K_2$ are bounded, and may be established even when $K_1$ and $K_2$ grow with $n$ by high-dimensional central limit theorem [@chernozhukov2017] although we do not formally prove this here. Assuming $K_1$ and $K_2$ are bounded is not a real limitation, particularly in machine learning settings, as most practical applications similar to that in the next section will likely be limited to a small to moderate number of machine learners. Furthermore, unlike in previous section, our proposed approach for post-selection inference technically requires that each $\widehat \psi_{k,\tk}$ admits a first order influence function, which is a condition that as previously mentioned may still hold even when candidate estimators include nonparametric regression or flexible machine learning methods provided that these estimators are consistent at rates faster than $n^{-1/4}$ [@robins2017; @Chernozhukov2018].
To conclude this section, we briefly discuss selection of $\tau$. The choice of $\tau$ essentially determines how well the smooth-max function approximates the minimax estimator as captured by the following inequality, $$\begin{aligned}
\max\{z_1,\ldots,z_m\}\leq \frac{1}{\tau} \log \sum_{i=1}^m \exp(\tau z_i) \leq \frac{1}{\tau} \log m + \max\{z_1,\ldots,z_m\},\end{aligned}$$ which holds for any positive real numbers $z_1,\ldots,z_m$. Thus, the approximation error of $\Gamma_{k,\tk}(\tau)$ is controlled by $\epsilon = \log m/\tau$.
Data Analysis\[sec:real\]
=========================
In this section, similarly to [@tan2006; @2015biasreduce; @tan2019model; @tan2019regularized], we reanalyze data from the Study to Understand Prognoses and Preferences for Outcomes and Risks of Treatments (SUPPORT) to evaluate the effectiveness of right heart catheterization (RHC) in the intensive care unit of critically ill patients. At the time of the study by [@5c6af36c0fb64cfcbb482d75c2bc7ff1], RHC was thought to lead to better patient outcomes by many physicians. [@5c6af36c0fb64cfcbb482d75c2bc7ff1] found that RHC leads to lower survival compared to not performing RHC.
We consider the effect of RHC on 30-day survival. Data are available on 5735 individuals, 2184 treated and 3551 controls. In total, 3817 patients survived and 1918 died within 30 days. To estimate the additive treatment effect $\psi_0 = E\{Y_1-Y_0\}$, 72 covariates were used to adjust for potential confounding [@Hirano2001]. We posited $K_1=3$ candidate models/estimators for the propensity score model including all 72 covariates: 1. Logistic regression with $l_1$ regularization; 2. Classification random forests; 3. Gradient boosting trees. We posited $K_2=6$ candidate estimators for the outcome model $E_\tk\left( Y|A,X\right)$ with 72 covariates: 1. Linear regression; 2. Logistic regression; 3. Lasso; 4. Logistic regression with $l_1$ regularization; 5. Regression random forests; 6. Classification random forests. The proposed selection procedure was implemented with three-fold cross-validation. Tuning parameters were selected as in Section \[sec:simu3\].
The proposed minimax selection selected gradient boosting trees estimator of propensity score model and logistic regression with $l_1$ regularization for the outcome model. The proposed mixed-minimax selection selected classification random forest estimator of propensity score model and regression random forest for the outcome model. The estimated causal effect of RHC was $-0.0548$ and $-0.0476$ for the minimax and mixed-minimax criteria, respectively, while the point estimate obtained by smooth-max approach was $-0.0528$ and $-0.0483$ which, as expected, is close to the minimax point estimates, respectively. Results were somewhat smaller than other improved estimators considered by [@2015biasreduce], who did not perform model selection. Specifically, the TMLE with default super learner [@van2011targeted] gave $\widehat \psi_{\text{TMLE-SL}} = - 0.0586$; the bias reduced doubly robust estimation with linear and logit link gave $\widehat \psi_{\text{BR},\text{lin}}=-0.0612$ and $\widehat \psi_{\text{BR},\text{logit}}=-0.0610$, respectively; the calibrated likelihood estimator [@tan2010] gave $\widehat \psi_{\text{TAN}}=-0.0622$. Our estimates suggest that previous estimates may still be subject to a small amount of misspecification bias. To obtain valid confidence intervals, we applied the proposed smooth-max approach with error tolerance $\epsilon =0.002$. Smooth-max based 95% confidence intervals by nonparametric bootstrap with 200 replications were estimated as $(-0.1041,-0.0301)$ and $(-0.1083,-0.0277)$ for the minimax and mixed-minimax criteria, respectively, which are slightly wider than other improved estimators considered in [@2015biasreduce], e.g., the targeted maximum likelihood estimation with default super learner gave $(-0.0877, -0.0295)$; the bias reduced doubly robust estimator with linear and logit link gave $(-0.0889,-0.0335)$ and $(-0.0879,-0.0340)$, respectively; the calibrated likelihood estimator gave $(-0.0924,-0.0319)$. This is not surprising because we consider a richer class of models and formally account for such selection step, potentially resulting in smaller bias and more accurate confidence intervals.
Discussion\[sec:discussion\]
============================
We have proposed a general model selection approach to estimate a functional $\psi(\theta)$ in a general class of doubly robust functionals which admit an estimating equation that is unbiased if at least one of two nuisance parameters is correctly specified. The proposed method works by selecting the candidate model based on minimax or mixed-minimax criterion of pseudo-risk defined in terms of the doubly robust estimating equation. A straightforward, cross-validation scheme was proposed to estimate the pseudo-risk. While, throughout the paper, we have described and evaluated the proposed selection procedure primarily in estimating average treatment effect as a running example, in the appendix, all results are extended to the more general class of mixed-bias functionals of [@rotnitzky2019mix]. Extensive simulation studies and a real data example on the effectiveness of right heart catheterization in the intensive care unit of critically ill patients were also presented to illustrate the proposed approach.
As mentioned in Remark \[remark:rm2\], our selection criteria extend to multiply robust influence functions in the sense of [@tchetgentchetgen2012; @wang2018bounded; @Caleb2019; @Shi2019MultiplyRC; @sun2019multiple], where three or more nuisance parameters are needed to evaluate the influence function, however, the influence function remains unbiased if all but one of the nuisance parameters are evaluated at the truth. Briefly, in such setting, the minimax criterion entails the maximum squared change of the functional over all perturbations of one nuisance parameter holding the others fixed. The mixed-minimax selector likewise generalizes. We expect our theoretical results to extend to this setting, an application of which is in progress [@sun2019multiple].
The proposed methods may be improved or extended in multiple ways. The choice of the criterion could be more flexible and one may use a different norm rather than $L_\infty$ norm, e.g., $L_2$ or $L_1$ norm. Another potential extension of our method is in the direction of statistical inference. It would be both interesting and important to derive the exact asymptotic distribution of the proposed estimators, as originally described in Section \[sec:select\], instead of relying on a smooth approximation. Finally, in principle one could develop a stacked generalization of our proposed approach by forming linear combinations of various candidate estimators of nuisance parameters [@wolpert1992stacked; @breiman1996bagging]. An optimal estimator could then be obtained by minimizing the pseudo-risk for the functional of interest with respect to the weights. Clearly, the current minimax approach explores only a small set of possible values for such weights, i.e., all values that have unit mass at one candidate model and zero elsewhere, and therefore may be sup-optimal relative to the stacked generalization. Because candidate learners may yield estimates that are highly correlated, one may need a form of regularization to ensure good performance, such as restricting set of weights to a finite support (in the spirit of [@van2007super]), or alternatively penalizing the pseudo-risk. We are currently investigating theoretical properties of such stacked minimax functional learning which we plan to report elsewhere.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank James Robins, Andrea Rotnitzky, and Weijie Su for helpful discussions and suggestions. We thank Karel Vermeulen and Stijn Vansteelandt for providing the dataset. This research is supported in part by U.S. National Institutes of Health grants.
Proofs
======
In this section, we present proofs of the theoretical results.
[**Proof of Lemma \[lemma:1\].**]{}
We have that
$$\begin{aligned}
\left[ \psi_{k_1,\tk} - \psi_{k_1,\tk_1}\right]^2 &= E \Bigg[\frac{\left( -1\right) ^{1-A}}{\pi_{k_1}\left(
A|X\right) }Y -\left\{ \frac{\left( -1\right) ^{1-A}}{\pi_{k_1} \left( A|X\right) }
E_\tk(Y|A,X)-\sum_{a}\left( -1\right) ^{1-a}E_\tk(Y|a,X)\right\} \\
&- \frac{\left( -1\right) ^{1-A}}{\pi_{k_1}\left(
A|X\right) }Y + \left\{ \frac{\left( -1\right) ^{1-A}}{\pi_{k_1}\left( A|X\right) }
E_{\tk_1}(Y|A,X)-\sum_{a}\left( -1\right) ^{1-a}E_{\tk_1}(Y|a,X)\right\} \Bigg]^2\\
&= E\Bigg[\frac{\left( -1\right) ^{1-A}}{\pi_{k_1}\left(
A|X\right) } \left\{ E_{\tk_1}(Y|A,X)-E_{\tk}(Y|A,X) \right\} \\
& + \sum_{a}\left( -1\right) ^{1-a}\left\{ E_\tk(Y|a,X)-E_{\tk_1}(Y|a,X)\right\} \Bigg]^2\\
& = E\left[\sum_{a}\left( -1\right) ^{1-a} \left\{\frac{\pi\left(
a|X\right) }{\pi_{k_1}\left( a|X\right) }-1\right\} \left\{E_{\tk_1}(Y|a,X)-E_{\tk}(Y|a,X)\right\}\right]^2.\end{aligned}$$
$$\begin{aligned}
\left[\psi_{k,\tk_1} - \psi_{k_1,\tk_1}\right]^2 &= E \Bigg[\frac{\left( -1\right) ^{1-A}}{\pi_{k}\left(
A|X\right) }Y -\left\{ \frac{\left( -1\right) ^{1-A}}{\pi_{k} \left( A|X\right) }
E_{\tk_1}(Y|A,X)-\sum_{a}\left( -1\right) ^{1-a}E_{\tk_1}(Y|a,X)\right\} \\
&- \frac{\left( -1\right) ^{1-A}}{\pi_{k_1}\left(
A|X\right) }Y + \left\{ \frac{\left( -1\right) ^{1-A}}{\pi_{k_1}\left( A|X\right) }
E_{\tk_1}(Y|A,X)-\sum_{a}\left( -1\right) ^{1-a}E_{\tk_1}(Y|a,X)\right\} \Bigg]^2\\
&= E\Bigg[\left\{\frac{\left( -1\right) ^{1-A}}{\pi_{k}\left(
A|X\right)}-\frac{\left( -1\right)^{1-A}}{\pi_{k_1}\left(
A|X\right)}\right\} E(Y|A,X) \\
&- \left\{\frac{\left( -1\right) ^{1-A}}{\pi_{k}\left(
A|X\right)}-\frac{\left( -1\right)^{1-A}}{\pi_{k_1}\left(
A|X\right)}\right\} E_{\tk_1}(Y|A,X)\Bigg]^2\\
&= E\left[\sum_{a}\left( -1\right) ^{1-a} \left\{\frac{\pi\left(
a|X\right) }{\pi_k\left( a|X\right) }- \frac{\pi\left(
a|X\right) }{\pi_{k_1}\left( a|X\right) } \right\} \left\{E(Y|a,X)-E_{\tk_1}(Y|a,X)\right\}\right]^2.\quad \Box\end{aligned}$$
[**Proof of Lemma \[lemma:rate\]**]{}
Without loss of generality, we focus on $S=1$ and the proof of cross-validated oracle selector follows similarly. Let $$k^{\max }=\arg \max_{k\in \mathcal K_1} \text{per}\left( k,\tk^\star,k^\star,\tk^\star\right) ^{1/2},$$ $$\tk^{\max}=\arg \max_{\tk\in \mathcal K_2} \text{per}\left( k^\star,\tk,k^\star,\tk^\star\right) ^{1/2}.$$ By Assumptions \[asm:1\]-\[asm:3\] and the mean value theorem, there exist constants $C_{0},C_{1},$ and a value $x_{0}$ in the support of $X$ such that$$\begin{aligned}
&&\max_{\substack{ k=k^\star,\tk\in \mathcal K_2 \\ \tk = \tk^\star,k\in \mathcal K_1}}\text{per}\left(
k,\tk,k^\star,\tk^\star\right) ^{1/2}\\
&=&\max_{\tk=\tk^\star,k\in \mathcal K_1}\text{per}\left(
k,\tk,k^\star,\tk^\star \right) ^{1/2} \\
&=&\left\vert \left\{ \int \left( \widehat{p}_{k^\star}(x)-\widehat{p}_{k^{\max }}\left( x\right) \right) \left( \widehat{b}_{\tk^\star}(x)-b(x)\right) E[h_1(O)|X=x] \right\} dF\left( x\right) \right\vert \\
&=&\left\vert \left\{ \int \left( \widehat{p}_{k^\star}(x)-\widehat{p}_{k^{\max }}\left( x\right) \right) \left( \widehat{b}_{1}(x)-b(x)\right)E[h_1(O)|X=x] \right\} dF\left( x\right) \right\vert \\
&=&\left\vert \left( \widehat{p}_{k^\star}(x_{0})-\widehat{p}_{k^{\max
}}\left( x_0 \right) \right) \left( \widehat{b}_{1}(x_{0})-b(x_{0})\right) E[h_1(O)|X=x_0]
\right\vert \\
&=&C_{0}\nu _{\max }\omega _{\min } \\
&\geq &
\max_{k=k^\star,\tk\in \mathcal K_2}\text{per}\left(
k,\tk,k^\star,\tk^\star\right) ^{1/2}\\
&=&\left\vert \left\{ \int
\left( \widehat{p}_{k^\star}(x)-p\left( x\right) \right) \left( \widehat{b}_{1}(x)-\widehat{b}_{\tk^{\max }}(x)\right) E[h_1(O)|X=x] \right\} dF\left( x\right)
\right\vert \\
&=&C_{1}\nu _{k^\star}\omega _{\max }.\end{aligned}$$Therefore $$\nu _{k^\star}\leq \frac{C_{0}}{C_{1}}\nu _{\max }\frac{\omega _{\min }}{\omega _{\max }}.$$Implied by Equation and the mean value theorem, there exists a positive constant $C_{2}$ and a value $x^{\ast }$ in the support of $X$ such that $$\begin{aligned}
\left\vert \psi _{k^\star,\tk^\star}-\psi_0 \right\vert &=&\left\vert \int
\left( \widehat{p}_{k^\star}(x)-p\left( x\right) \right) \left( \widehat{b}_{1}(x)-b(x)\right) E[h_1(O)|X=x] dF\left( x\right) \right\vert \\
&=&C_{2}\left\vert \left( \widehat{p}_{k^\star}(x^{\ast })-p\left( x^{\ast
}\right) \right) \left( \widehat{b}_{1}(x^{\ast })-b(x^{\ast })\right) E[h_1(O)|X=x^*]
\right\vert \\
&\lesssim &\nu_{k^\star}\omega _{\min } \\
&\lesssim &\frac{\nu _{\max }}{\omega _{\max }}\omega _{\min }^{2}.\end{aligned}$$For $\psi _{k^\circ,\tk^\circ}$, it is straightforward to verify that, by Equation and the mean value theorem, there exists a positive constant $\overline{C}_{2}$ and a value $\overline{x}$ in the support of $X$ such that $$\begin{aligned}
\left\vert \psi _{k^\circ,\tk^\circ}-\psi_0
\right\vert &=&\overline{C}_{2}\left\vert \left( \widehat{p}_{k^\circ}(\overline{x})-p\left( \overline{x}\right) \right) \left( \widehat{b}_{\tk^\circ}(\overline{x})-b(\overline{x})\right) \right\vert \\
&=&O_{P}\left( \nu _{\min }\omega _{\min }\right).\end{aligned}$$
[**Proof of Theorem \[thm:2\].**]{}
By the definition of our estimator, $$\max_{\substack{k=k^\dagger,\tk \in \mathcal K_2;\\ \tk=\tk^\dagger, k\in \mathcal K_1}} \sum_{s=1}^{S} [\PP^1_s\{U^s_{(k,\tk)}(k^\dagger,\tk^\dagger)\}]^2 \leq \max_{\substack{k=k^\star,\tk \in \mathcal K_2;\\ \tk=\tk^\star, k\in \mathcal K_1}} \sum_{s=1}^{S} [\PP^1_s(U^s_{(k,\tk)}(k^\star,\tk^\star))]^2.$$ So we have that, $$\sum_{s=1}^{S} [\PP^1_s\{U^s_{(k^\ddagger,\tk^\ddagger)}(k^\dagger,\tk^\dagger)\}]^2 \leq \sum_{s=1}^{S} [\PP^1_s(U^s_{(k^\vartriangle,\tk^\vartriangle)}(k^\star,\tk^\star))]^2,$$ where $$(k^\ddagger,\tk^\ddagger)=\arg\max_{\substack{k=k^\dagger,\tk \in \mathcal K_2;\\ \tk=\tk^\dagger, k\in \mathcal K_1}} \sum_{s=1}^{S} [\PP^1_s(U^s_{(k,\tk)}(k^\dagger,\tk^\dagger))]^2,$$ and $$(k^\vartriangle,\tk^\vartriangle)=\arg\max_{\substack{k=k^\star,\tk \in \mathcal K_2;\\ \tk=\tk^\star, k\in \mathcal K_1}} \sum_{s=1}^{S} [\PP^1_s(U^s_{(k,\tk)}(k^\star,\tk^\star))]^2.$$
We denote $n^s_j = \#\{1\leq i\leq n:T_i^s=j\},$ $j=0,1$. For simplicity, in a slight abuse of notation in the following, we use $(k,\tk)$ instead of $(k^\ddagger,\tk^\ddagger), (k^\vartriangle,\tk^\vartriangle)$ for the sub-indices of $U$, i.e., $U^s_{(k,\tk)}(k^\dagger,\tk^\dagger)$ denotes $U^s_{(k^\ddagger,\tk^\ddagger)}(k^\dagger,\tk^\dagger)$; $U^s_{(k,\tk)}(k^\star,\tk^\star)$ denotes $U^s_{(k^\vartriangle,\tk^\vartriangle)}(k^\star,\tk^\star)$. By simple algebra, we have $$\begin{aligned}
& [\PP^1_s (U^s_{(k,\tk)}(k^\dagger,\tk^\dagger))]^2 \\ =& \frac{1}{{n_1^s}^2} \sum_{i,j} [U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j ]\\
= &\frac{1}{{n_1^s}^2} \sum_{i,j} \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \Big) \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j \big] \Big)\\
+ & \frac{1}{{n_1^s}^2} \sum_{i,j} \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j\\
+ & \frac{1}{{n_1^s}^2} \sum_{i,j} \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \Big) \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j \big]\end{aligned}$$ $$\begin{aligned}
= & \frac{1}{{n_1^s}^2} \sum_{i,j} \Big(\PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j \big] \Big) \\
+ & \frac{2}{{n_1^s}^2} \sum_{i,j} \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \Big) \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j \big]\\
+ & \frac{1}{{n_1^s}^2} \sum_{i,j} \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \Big) \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j \big] \Big),\end{aligned}$$ where $U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i$ denotes the estimating equation evaluated at $i$-th observation. Thus, $[\PP^1_s (U^s_{(k,\tk)}(k^\dagger,\tk^\dagger))]^2$ further equals to $$\begin{aligned}
& [\PP^1_s (U^s_{(k,\tk)}(k^\dagger,\tk^\dagger))]^2\\
= & [\PP^1 \big( U^s_{k,\tk} ({k^\dagger,\tk^\dagger}) \big)]^2 \\
+ & \frac{2}{{n_1^s}^2} \sum_{i,j} \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \Big) \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j \big]\\
+ & \frac{1}{{n_1^s}^2} \sum_{i,j} \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \Big) \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j \big] \Big).\end{aligned}$$ A similar decomposition holds for $[\PP_s^1(U^s_{(k,\tk)}({k^\star,\tk^\star}))]^2$. By definition of our estimator, for any $\delta>0$, we have that $$\begin{aligned}
& \sum_{s=1}^S [\PP^1 (U^s_{(k,\tk)}({k^\dagger,\tk^\dagger}))]^2 \\
\leq &(1+2\delta)\sum_{s=1}^S[\PP^1 (U^s_{(k,\tk)}({k^\star,\tk^\star}))]^2\\
+ & \frac{1}{\sqrt{\nn}} \Big\{ (1+\delta)\sqrt{\nn} \sum_{s=1}^S\Big[ \big(\PP_s^1(U^s_{(k,\tk)}({k^\star,\tk^\star})) \big)^2 -\big( \PP^1 (U^s_{(k,\tk)}({k^\star,\tk^\star})) \big)^2 \Big] \\ &- \delta \sqrt{\nn}\sum_{s=1}^S \Big[ \PP^1 (U^s_{(k,\tk)}({k^\star,\tk^\star})) \Big]^2 \Big\}\\
-& \frac{1}{\sqrt{\nn}} \Big\{ (1+\delta)\sqrt{\nn} \sum_{s=1}^S \Big[ \big(\PP_s^1(U^s_{(k,\tk)}({k^\dagger,\tk^\dagger})) \big)^2 -\big( \PP^1(U^s_{(k,\tk)}({k^\dagger,\tk^\dagger})) \big)^2 \Big] \\ &+ \delta \sqrt{\nn} \sum_{s=1}^S \Big[ \PP^1 (U^s_{(k,\tk)}({k^\dagger,\tk^\dagger})) \Big]^2 \Big\}.\\\end{aligned}$$ Combined with the decomposition of $[\PP_s^1(U^s_{(k,\tk)}({k^\dagger,\tk^\dagger}))]^2$ and $[\PP_s^1(U^s_{(k,\tk)}({k^\star,\tk^\star}))]^2$, we further have that $$\begin{aligned}
&\sum_{s=1}^S [\PP^1(U^s_{(k,\tk)}({k^\dagger,\tk^\dagger}))]^2 \\
\leq &(1+2\delta)\sum_{s=1}^S [\PP^1(U^s_{(k,\tk)}({k^\star,\tk^\star}))]^2\\
+ & \frac{1}{\sqrt{n_1^s}} \Big\{ (1+\delta)\sqrt{n^s_1} \sum_{s=1}^S \Big[ \frac{2}{n_1^s} \sum_{i} \Big ( U^s_{k,\tk} ({k^\star,\tk^\star})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\star,\tk^\star})_i \big] \Big) \PP^1 \big[ U^s_{k,\tk} (\widehat \psi_{k^\star,\tk^\star}) \big] \\
+ & \frac{1}{{n_1^s}^2} \sum_{i, j} \Big ( U^s_{k,\tk} ({k^\star,\tk^\star})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\star,\tk^\star})_i \big] \Big) \Big ( U^s_{k,\tk} ({k^\star,\tk^\star})_j- \PP^1 \big[ U^s_{k,\tk} ({k^\star,\tk^\star})_j \big] \Big)\Big] \\ -& \delta \sqrt{n_1^s}\sum_{s=1}^S \Big[ \PP^1(U^s_{(k,\tk)}({k^\star,\tk^\star})) \Big]^2 \Big\}\\
- & \frac{1}{\sqrt{n_1^s}} \Big\{ (1+\delta)\sqrt{n^s_1}\sum_{s=1}^S \Big[ \frac{2}{n_1^s} \sum_{i} \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \Big) \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger}) \big] \\
+ & \frac{1}{{n_1^s}^2} \sum_{i, j} \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \Big) \Big ( U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j - \PP^1 \big[ U^s_{k,\tk} ({k^\dagger,\tk^\dagger})_j \big] \Big)\Big] \\+& \delta \sqrt{n_1^s} \sum_{s=1}^S \Big[ \PP^1(U^s_{(k,\tk)}({k^\dagger,\tk^\dagger})) \Big]^2 \Big\}.
\end{aligned}$$
Note that the only assumption on $s$ is its stochastic independence of the observations, we omit sup-index $s$ hereinafter. Because the maximum of sum is at most the sum of maxima, we deal with the first order and second order terms separately. By Lemma \[lemma:0\], and note that given Assumption \[asm:positivity\], the estimating equation is bounded, we further have the following bound for the first order term, $$\begin{aligned}
& \PP^0 \max_{k,\tk,k_1,\tk_1} \Big\{ \frac{2(1+\delta)\sqrt \nn}{\nn} \sum_{i} \Big ( U_{k,\tk} ({k_1,\tk_1})_i - \PP^1 \big[ U_{k,\tk} ({k_1,\tk_1})_i \big] \Big) \PP^1 \big( U_{k,\tk} ({k_1,\tk_1}) \big) \\
- & \delta \sqrt \nn \Big[ \PP^1 \big( U_{k,\tk} ({k_1,\tk_1}) \big) \Big]^2 \Big\} \\
\leq & \PP^0 \frac{16(1+\delta)}{\nn^{1/p-1/2}} \log(1+K^2_1K^2_2)\max_{k,\tk,k_1,\tk_1}\bigg [\frac{|| U_{k,\tk} ({k_1,\tk_1}) \PP^1 \big[ U_{k,\tk} ({k_1,\tk_1}) \big] ||_\infty}{\nn^{1-1/p}}\\
+& \bigg(\frac{3 \PP^1 \Big(U_{k,\tk} ({k_1,\tk_1})\PP^1 \big[ U_{k,\tk} ({k_1,\tk_1}) \big] \Big)^2 2^{(1-p)}(1+\delta)^{(2-p)}}{\delta^{2-p} \Big[ \PP^1 \big( U_{k,\tk} ({k_1,\tk_1}) \big) \Big]^{4-2p}} \bigg)^{1/p} \bigg ] = (I),\end{aligned}$$ and $$\begin{aligned}
& \PP^0 \max_{k,\tk,k_1,\tk_1} \Big\{-\Big[ \frac{2(1+\delta)\sqrt \nn}{\nn} \sum_{i} \Big ( U_{k,\tk} ({k_1,\tk_1})_i - \PP^1 \big[ U_{k,\tk} ({k_1,\tk_1})_i \big] \Big) \PP^1 \big( U_{k,\tk} ({k_1,\tk_1}) \big)\\
+& \delta \sqrt \nn \Big[ \PP^1 \big( U_{k,\tk} ({k_1,\tk_1}) \big) \Big]^2 \Big] \Big\}
\leq (I),\end{aligned}$$ where $\max$ is taken on the set $\{ k_1\in \{1,\ldots,K_1\},\tk_1 \in \{1,\ldots,K_2\}, k=k_1~\text{or}~ \tk = \tk_1\}$. Thus, $\PP^0 [\PP^1 (U_{(k,\tk)}(k^\dagger,\tk^\dagger))]^2$ is further bounded by $$\begin{aligned}
&\PP^0 [\PP^1 (U_{(k,\tk)}({k^\dagger,\tk^\dagger}))]^2 \\
\leq &(1+2\delta)\PP^0 [\PP^1 (U_{(k,\tk)}({k^\star,\tk^\star}))]^2 + \frac{2}{\sqrt \nn}(I)\\ + & \frac{1+\delta}{\nn^{2}} \PP^0 \sum_{i, j} \Big ( U_{k,\tk} ({k^\star,\tk^\star})_i - \PP^1 \big[ U_{k,\tk} ({k^\star,\tk^\star})_i \big] \Big) \Big ( U_{k,\tk} ({k^\star,\tk^\star})_j - \PP^1 \big[ U_{k,\tk} ({k^\star,\tk^\star})_j \big] \Big) \\
-& \frac{1+\delta}{\nn^{2}} \PP^0 \sum_{i, j} \Big ( U_{k,\tk} ({k^\dagger,\tk^\dagger})_i - \PP^1 \big[ U_{k,\tk} ({k^\dagger,\tk^\dagger})_i \big] \Big) \Big( U_{k,\tk} ({k^\dagger,\tk^\dagger})_j - \PP^1 \big[ U_{k,\tk} ({k^\dagger,\tk^\dagger})_j \big] \Big).\end{aligned}$$
By Lemma \[lemma:2\] and \[lemma:3\], the U-statistics are bounded and we have the bound of the risk of our selector $({k^\dagger,\tk^\dagger})$, $$\begin{aligned}
&\PP^0[\PP^1(U_{(k^\ddagger,\tk^\ddagger)}({k^\dagger,\tk^\dagger}))]^2 \\
\leq &(1+2\delta)\PP^0[\PP^1(U_{(k^\vartriangle,\tk^\vartriangle)}({k^\star,\tk^\star}))]^2 \\
+ & (1+\delta)C \Bigg\{ \left( \frac{2M}{n_1^2} \log(1+\frac{MK^2_1 K^2_2}{2} ) \right)^{1/2} + \frac{2M}{n_1} \log(1+\frac{MK^2_1 K^2_2}{2} ) \\
+ & \frac{4M^{3/2}}{n_1^{3/2}} \log^{3/2}(1+\frac{MK^2_1K^2_2}{2}+D_0)+ \frac{4 M^2}{n_1^2} \log^{2}(1+\frac{MK^2_1K^2_2}{2}+D_1) \Bigg\}\\
+ & \PP^0 \frac{16(1+\delta)}{\nn^{1/p}} \log(1+K^2_1K^2_2)\max_{k,\tk,k_1,\tk_1}\bigg [\frac{|| U_{k,\tk} ({k_1,\tk_1})\PP^1 \big[ U_{k,\tk} ({k_1,\tk_1}) \big] ||_\infty}{\nn^{1-1/p}} \\
+& \bigg(\frac{3\PP^1 \Big(U^2_{k,\tk} ({k_1,\tk_1}) \Big) 2^{(1-p)}(1+\delta)^{(2-p)}}{\delta^{2-p} \Big[ \PP^1 \big( U_{k,\tk} ({k_1,\tk_1}) \big) \Big]^{2-2p}} \bigg)^{1/p} \bigg ],\\
\end{aligned}$$ where $C$, $M$, $D_0$, and $D_1$ are some universal constants.
Finally, recall that for the term $(1+2\delta)\PP^0[\PP^1 (U_{(k^\vartriangle,\tk^\vartriangle)}({k^\star,\tk^\star}))]^2$, $(k^\vartriangle,\tk^\vartriangle)$ is chosen corresponding to $(k^\star,\tk^\star)$ under measure $\PP_s^1$. It is further bounded by $(1+2\delta)\PP^0[\PP^1 (U_{(k^\sstar,\tk^\sstar)}({k^\star,\tk^\star}))]^2$, where $(k^\sstar,\tk^\sstar)$ is chosen corresponding to $(k^\star,\tk^\star)$ under true measure $\PP^1$. $\Box$
[(Lemma 2.2 in [@vaart2006oracle])]{}\[lemma:0\] Assume that $E f\geq 0$ for every $f \in \cal F$. Then for any $1 \leq p \leq 2$ and $\delta>0$, we have that $$\begin{aligned}
E\max_{f \in \mathcal F}( \GG_n - \delta \sqrt{n} E) f \leq \frac{8}{n^{1/p-1/2}} \log(1+\#{\cal F}) \max_{f\in \mathcal F} [ \frac{M(f)}{n^{1-1/p}}+(\frac{v(f)}{(\delta E f)^{2-p}})^{1/p} ],\end{aligned}$$ where $\GG_n$ is the empirical process of the $n$ i.i.d. observations, and $(M(f),v(f))$ is any pair of Bernstein numbers of measurable function $f$ such that $$M(f)^2 E\Big(\exp\{ |f|/M(f)\} -1-|f|/M(f)\Big)\leq 1/2v(f).$$ Furthermore, if $f$ is uniformly bounded, then $(||f||_\infty,1.5E f^2)$ is a pair of Bernstein number.
$$\begin{aligned}
& \Pr\Bigg\{ \frac{1}{n_1^2} \sum_{1\leq i_1, i_2 \leq n_1} h(O_{i_1},O_{i_2})\geq x \Bigg\}\\
\leq & \frac{M}{2}\exp\Bigg\{ -\frac{1}{M} \min\bigg( \frac{x^2 n_1^2}{Eh^2}, \frac{xn_1}{||h||_{L_2\rightarrow L_2}} ,\frac{x^{2/3} n_1}{[( ||E_{O_1}h^2||_\infty + ||E_{O_2}h^2||_\infty)]^{1/3}},\frac{x^{1/2}n_1}{||h||_\infty^{1/2}} \bigg) \Bigg\},\end{aligned}$$
and $$\begin{aligned}
& \Pr \Bigg\{ -\frac{1}{n_1^2} \sum_{1\leq i_1, i_2 \leq n_1} h(O_{i_1},O_{i_2})\geq x \Bigg\}\\
\leq & \frac{M}{2}\exp\Bigg\{ -\frac{1}{M} \min\bigg( \frac{x^2 n_1^2}{Eh^2}, \frac{xn_1}{||h||_{L_2\rightarrow L_2}} ,\frac{x^{2/3} n_1}{[( ||E_{O_1}h^2||_\infty + ||E_{O_2}h^2||_\infty)]^{1/3}},\frac{x^{1/2}n_1}{||h||_\infty^{1/2}} \bigg) \Bigg\}.\end{aligned}$$ where $M$ is some universal constant, and $||h||_{L_2\rightarrow L_2}$ is defined as $$||h||_{L_2\rightarrow L_2}=\sup\{ E [h(O_1,O_2)a(O_1)c(O_2)]: E(a^2(O_1))\leq 1, E(c^2(O_2))\leq 1 \}.$$ \[lemma:2\]
[**Proof of Lemma \[lemma:2\].**]{}
The inequality follows directly from the Corollary 3.4 in [@10.1007/978-1-4612-1358-1_2].
$$\begin{aligned}
& E\max_{h\in \cal H}\Bigg\{ \frac{1}{n_1^2} \sum_{1\leq i_1, i_2 \leq n_1} h(O_{i_1},O_{i_2}) \Bigg\}\\
\leq & \left( 2M \log(1+\frac{M \# \cal H}{2} ) \max_{h} \frac{Eh^2}{ n_1^2} \right)^{1/2} \\
+ & 2M \log(1+\frac{M\# \cal H}{2} ) \max_{h} \frac{||h||_{L_2\rightarrow L_2}}{n_1 } \\
+ & 4 M^{3/2} \log^{3/2}(1+\frac{M\# \cal H}{2}+D_0) \max_{h} \frac{||E_{O_1}h^2||_\infty^{1/2}}{n_1^{3/2}} \\
+ & 4 M^2 \log^{2}(1+\frac{M\# \cal H}{2}+D_1) \max_{h}\frac{ ||h||_\infty}{n_1^2},\end{aligned}$$
and the same result holds for $E\max_{h\in \cal H} \Bigg\{- \frac{1}{n_1^2} \sum_{1\leq i_1, i_2 \leq n_1} h(O_{i_1},O_{i_2}) \Bigg\}$. \[lemma:3\]
[**Proof of Lemma \[lemma:3\].**]{}
Denote $$\begin{aligned}
\omega_{n_1}(x)=\min\bigg( \frac{x^2 n_1^2}{Eh^2}, \frac{xn_1}{||h||_{L_2\rightarrow L_2}} ,\frac{x^{2/3} n_1}{[ 2||E_{O_1}h^2||_\infty]^{1/3}},\frac{x^{1/2}n_1}{||h||_\infty^{1/2}} \bigg),\end{aligned}$$ and the following four sets:
$\Omega_{1,n_1}(h)=\{x: \omega_{n_1}(x)= \frac{ x^2 n_1^2}{ Eh^2} \}$
$\Omega_{2,n_1}(h)=\{x: \omega_{n_1}(x)= \frac{xn_1}{||h||_{L_2\rightarrow L_2}} \}$
$\Omega_{3,n_1}(h)=\{x: \omega_{n_1}(x)= \frac{x^{2/3}n_1}{[ 2||E_{O_1}h^2||_\infty]^{1/3}} \}$
$\Omega_{4,n_1}(h)=\{x: \omega_{n_1}(x)= \frac{ x^{1/2}n_1}{||h||_\infty^{1/2}} \}$
Then $$\begin{aligned}
&\Pr\Bigg\{ \bigg( \frac{1}{n_1^2} \sum_{1\leq i_1, i_2 \leq n_1} h(O_{i_1},O_{i_2}) \bigg) I_{\Omega_{1,n_1}}\geq x \Bigg\}
\leq \frac{1}{2}M\exp\Bigg\{- \frac{1}{M} \frac{ x^2 n_1^2}{ Eh^2} \Bigg\} ,\\
&\Pr\Bigg\{ \bigg( \frac{1}{n_1^2} \sum_{1\leq i_1, i_2 \leq n_1} h(O_{i_1},O_{i_2})\bigg) I_{\Omega_{2,n_1}}\geq x \Bigg\}\leq \frac{1}{2}M\exp\Bigg\{- \frac{1}{M} \frac{xn_1}{||h||_{L_2\rightarrow L_2}} \Bigg\},\\
& \Pr\Bigg\{ \bigg( \frac{1}{n_1^2} \sum_{1\leq i_1, i_2 \leq n_1} h(O_{i_1},O_{i_2}) \bigg) I_{\Omega_{3,n_1}}\geq x \Bigg\}\leq \frac{1}{2}M\exp\Bigg\{- \frac{1}{M} \frac{x^{2/3}n_1}{[ 2||E_{O_1}h^2||_\infty]^{1/3}} \Bigg\},\\
&\Pr\Bigg\{ \bigg( \frac{1}{n_1^2} \sum_{1\leq i_1, i_2 \leq n_1} h(O_{i_1},O_{i_2})\bigg) I_{\Omega_{4,n_1}}\geq x \Bigg\} \leq \frac{1}{2}M\exp\Bigg\{- \frac{1}{M} \frac{x^{1/2}n_1}{||h||_\infty^{1/2}}\Bigg\}.\end{aligned}$$
Then by the above inequalities and Lemma 8.1 in [@vaart2006oracle], $$\begin{aligned}
& E\max_{h}\Bigg\{ \frac{1}{n_1^2} \sum_{1\leq i_1, i_2 \leq n_1} h(O_{i_1},O_{i_2}) \Bigg\}\\
\leq & \left( 2M \log(1+\frac{M\# \cal H}{2} ) \max_{h} \frac{Eh^2}{ n_1^2} \right)^{1/2} \\
+ & 2M \log(1+\frac{M\# \cal H}{2} ) \max_{h} \frac{||h||_{L_2\rightarrow L_2}}{n_1 } \\
+ & 4 M^{3/2} \log^{3/2}(1+\frac{M\# \cal H}{2}+D_0) \max_{h} \frac{||E_{O_1}h^2||_\infty^{1/2}}{n_1^{3/2}} \\
+ & 4 M^2 \log^{2}(1+\frac{M\# \cal H}{2}+D_1) \max_{h}\frac{ ||h||_\infty}{n_1^2},\end{aligned}$$ where $D_0$ and $D_1$ are some universal constants. $\Box$\
For each pair $(k_1,\tk_1)$, average the perturbations over the splits and obtain $$\widehat {\text{per}}(k,\tk; k_1,\tk_1) = \frac{1}{S}\sum_{s=1}^S \left[ \PP_s^1( IF^s_{k,\tk}(\widehat \psi^s_{k_1,\tk_1}) - IF^s_{k_1,\tk_1}(\widehat \psi^s_{k_1,\tk_1}) )
\right]^{2},$$ where $k=k_1,\tk \in \{1\cdots, K_2\}$ or $\tk=\tk_1, k\in \{1\cdots, K_1\}$ Calculate $$\widehat B^{(1)}_{k_1,\tk_1}=\max_{\substack{k=k_1,\tk \in \{1\cdots, K_2\};\\ \tk=\tk_1, k\in \{1\cdots, K_1\}}} \wper(k,\tk,k_1,\tk_1),$$ $$\widehat B^{(2)}_{k_1,\tk_1} = \max_{\tk_0 \in \mathcal K_2} \max_{\tk \in \mathcal K_2} \widehat {\text{per}}(k_1,\tk; k_1,\tk_0) + \max_{k_0\in \mathcal K_1} \max_{k\in \mathcal K_1} \widehat {\text{per}}(k,\tk_1; k_0,\tk_1)$$ for each pair $(k_1,\tk_1)$ Pick $(k^\dagger,\tk^\dagger)=\arg\min_{(k,\tk)} \widehat B^{(1)}_{k,\tk}$, $(k^\diamond,\tk^\diamond)=\arg\min_{(k,\tk)} \widehat B^{(2)}_{k,\tk}$ as our selected models, and obtain the estimations of the parameter over the splits $$\widehat \psi_{k^\dagger,\tk^\dagger}=\frac{1}{S}\sum_{s=1}^S \widehat \psi^s_{k^\dagger,\tk^\dagger},~~\widehat \psi_{k^\diamond,\tk^\diamond}=\frac{1}{S}\sum_{s=1}^S \widehat \psi^s_{k^\diamond,\tk^\diamond};$$\
**Return** $(k^\dagger,\tk^\dagger)$, $(k^\diamond,\tk^\diamond)$ and $\widehat \psi_{k^\dagger,\tk^\dagger}$, $\widehat \psi_{k^\diamond,\tk^\diamond}$
[**Additional simulations.**]{} In this section, we present more simulations in which all models are misspecified. In both scenarios, the data were generated from $$\begin{aligned}
\text{logit}\Pr \left( A=1|X\right) &=&(1,-1,1,-1,1)f_1(X),\\
E\left( Y|A,X\right) &=&1 + \mathbbm{1}^Tf_1(X)+ \mathbbm{1}^Tf_1(X) A+ A.\end{aligned}$$ For the first scenario, we used $\{f_2,f_3,f_4, f_5\}$ as candidate models of $g$ and $h$ specified in Equations and , where $$f_5(x) = \Big(\text{cos}(\pi x_1), \ldots, \text{cos}(\pi x_5)\Big)^T.$$ For the second scenario, we used $\{f_2,f_3,f_4, f_6\}$ as candidate models of $g$ and $h$ specified in Equations and , where $$f_6(x) = \Big(\text{cos}(\pi x_1/2), \ldots, \text{cos}(\pi x_5/2)\Big)^T.$$
The squared bias of $\widehat \psi$ for each scenario is shown in Figures \[fig:11\]-\[fig:33\], respectively. We see that when all models are misspecified, the performance of each method depends on the class of candidate models. While in Scenario 1, Figure \[fig:11\] is similar to Figure \[fig:1\], and the proposed methods have smaller bias; In Scenario 2, conducting model selection separately performs similar to and sometimes better than the proposed methods.
![Squared bias of Scenario 1[]{data-label="fig:11"}](biasplot3.pdf "fig:"){width="4in" height="2.5in"}\
![Squared bias of Scenario 2[]{data-label="fig:33"}](biasplot4.pdf "fig:"){width="4in" height="2.5in"}\
| ArXiv |
---
abstract: 'The IceCube Neutrino Observatory is the world’s largest high energy neutrino telescope, using the Antarctic ice cap as a Cherenkov detector medium. DeepCore, the low energy extension to IceCube, is an infill array with a fiducial volume of around 30 MTon in the deepest, clearest ice, aiming for an energy threshold as low as 10 GeV and extending IceCube’s sensitivity to indirect dark matter searches and atmospheric neutrino oscillation physics. We will discuss the analysis of the first year of DeepCore data, as well as ideas for a further extension of the particle physics program in the ice with a future PINGU detector.'
address: |
Department of Physics, Pennsylvania State University, University Park, PA 16802, U.S.A.\
E-mail: [email protected]
author:
- |
Tyce DeYoung\
[for the IceCube collaboration]{}
title: Particle Physics in Ice with IceCube DeepCore
---
=1
astroparticle physics; neutrino oscillations; dark matter
Introduction {#sec:intro}
============
The IceCube neutrino telescope, now fully operational at depths of 1450-2450 m below the surface of the Antarctic ice cap, was designed to detect high energy neutrinos from astrophysical accelerators of cosmic rays. Although the energy threshold of a large volume neutrino detector is not a sharp function, the original IceCube design focused on efficiency for neutrinos at TeV energies and above. Recently, the IceCube collaboration decided to augment the response of the detector at lower energies with the addition of DeepCore, a fully contained subarray aimed at improving the sensitivity of IceCube to neutrinos with energies in the range of 10’s of GeV to a few hundred GeV. This energy range is of interest for several topics related to particle physics, including measurements of neutrino oscillations and searches for neutrinos produced in the annihilation or decay of dark matter.
DeepCore consists of an additional eight strings of photosensors (Digital Optical Modules, or DOMs) comprising 10” Hamamatsu photomultiplier tubes and associated data acquisition electronics housed in standard IceCube glass pressure vessels. For most of the DeepCore DOMs, the standard IceCube R7081 PMTs were replaced with 7081MOD PMTs with Hamamatsu’s new super-bialkali photocathode. These PMTs provide approximately 35% higher quantum efficiency (averaged over the detected Cherenkov spectrum) than the standard bialkali PMTs.
Sited at the bottom center of the IceCube array, DeepCore benefits from the high optical quality of the ice at depths of 2100-2450 m, with an attenuation length of approximately 50 m in the blue wavelengths at which most Cherenkov photons are detected in ice. DeepCore also benefits from the ability of the standard IceCube sensors to detect atmospheric muons penetrating the ice from cosmic ray air showers above the detector, allowing substantial reduction in the background rate by vetoing events where traces of penetrating muons are seen. Each DeepCore string bears 50 DOMs in the fiducial region, with an additional 10 DOMs deployed at shallower depths to improve the vetoing efficiency for steeply vertical muons. In addition to the new DeepCore strings, the DeepCore fiducial volume for analysis includes 12 standard IceCube strings, chosen so that the fiducial region is shielded on all sides by a veto region consisting of three rows of standard IceCube strings, as shown in Fig. \[fig:layout\].
The random noise rate of IceCube DOMs is quite low (around 500 Hz, on average) due to the low temperatures and radiopurity of the ice cap. This permits DeepCore to be operated with a very low trigger threshold, demanding that 3 DOMs within the DeepCore fiducial region detect light in “local coincidence” within a period of no more than 2500 ns. The local coincidence criterion counts DOMs as being hit (i.e., having detected light) only if one of the four neighboring DOMs on a string (two above and two below) also registers a hit within $\pm1 \;\mu$s. Most of the resulting 185 Hz of triggers are due to stray light from muons which simultaneously satisfy the main IceCube trigger condition of 8 DOMs hit in local coincidence within 5 $\mu$s, but the DeepCore trigger contributes an additional (exclusive) rate of around 10 Hz.
![Schematic layout of DeepCore within IceCube. The shaded region indicates the fiducal volume of DeepCore, at the bottom center of IceCube, plus the extra veto cap of DOMs deployed at shallower depths to reinforce the veto against vertically-downgoing atmospheric muons. This schematic depicts both the DeepCore configuration used in 2010, when 79 IceCube strings were operational, and the final DeepCore layout and fiducial region used in the 2011 run. \[fig:layout\]](icecube_deepcore_pingu_koskinen_new.pdf){width="0.8\columnwidth"}
The vast majority of the events which trigger DeepCore, irrespective of whether they also trigger IceCube, are due to either penetrating atmospheric muons or random coincidences of dark noise. Immediately after data acquisition, events triggering DeepCore are subjected to an online data rejection algorithm which calculates a characteristic time and location for the activity observed in the DeepCore fiducial region, as an initial estimate of the putative neutrino vertex. The estimated location is the average position of the hit DOMs, and the time is determined by subtracting the time of flight $dn/c$ of an unscattered photon emitted from that location from the observed arrival time of the first photon to hit each DOM. After outliers due to dark noise or scattered light are removed, the average inferred emission time is used as the estimated time of the underlying physics event.
Based on this estimated time and location, every locally coincident hit recorded in the veto region prior to the vertex time is examined to determine whether it lies on the light cone connecting it with the estimated event vertex. The distributions of the inferred speed required to connect hits in the veto region to the DeepCore vertex, for both simulated atmospheric muons and simulated neutrinos, is shown in Fig. \[fig:veto\]; positive speeds indicate hits occuring in the veto region prior to the DeepCore vertex time. If any hits are found with inferred speeds between +0.25 and +0.4 m/ns, the event is rejected as being most likely due to an atmospheric muon. This algorithm reduces the event rate by more than two orders of magnitude, to 18 Hz, while retaining over 99% of simulated triggered events due to neutrinos interacting within the fiducial volume. Additional background rejection criteria are applied offline, depending on the goals of each physics analysis making use of these data.
![Distribution of probabilities of observing hits leading to a given inferred particle speed, for simulated atmospheric muons (dashed line) and atmospheric neutrinos (solid). Positive speeds indicate activity in the veto region prior to that in the DeepCore volume, and a peak around $c =$ 0.3 m/ns is visible for penetrating muons. The integral of each distribution corresponds to the mean number of hits observed in the veto region for the given class of events. \[fig:veto\]](ParticleSpeedProbabilities_v22.pdf){width="\columnwidth"}
The effective volume of the DeepCore detector for detection low energy muon neutrinos, accounting for this online data filter, is shown in Fig. \[fig:nuMuVolume\]. It should be stressed that this effective volume curve does *not* include losses due to later background rejection or event quality criteria. The contribution of DeepCore to low energy analysis is evident in the fact that despite its relatively small geometric volume, around 3% that of IceCube, the overall sample of neutrino events below 100 GeV consists primarily of those detected by DeepCore. This energy range is of considerable interest for several topics in particle physics, including searches for dark matter and measurements of neutrino oscillations. While DeepCore does not have a sharp energy threshold, it retains around 7 megatons of effective volume at energies as low as 10 GeV. Further details regarding DeepCore’s instrumentation and performance are available in Ref. [@Collaboration:2011ym].
![Effective volume of DeepCore for muon neutrinos at trigger level (solid) and after application of the online veto algorithm described in the text (dot-dashed line). The effective volume of IceCube as originally proposed is shown for comparison. \[fig:nuMuVolume\]](effectiveVolume_IC86_numu_GENIE_effVolumes_logScale_prelim.pdf){width="\columnwidth"}
Observation of Neutrino-Induced Cascades {#sec:cascades}
========================================
Using the first year of data recorded with DeepCore, from May 2010 to April 2011, we have observed cascades induced by atmospheric neutrinos interacting in the DeepCore volume. These cascades include charged current (CC) interactions of electron neutrinos, as well as neutral current (NC) interactions of neutrinos of all flavors. (The background rejection criteria used in this analysis result in an energy threshold of around 40 GeV, so only a negligible contribution from atmospheric muon neutrinos oscillating to tau is expected.) Previous searches for neutrino-induced cascades in AMANDA and IceCube [@Ahrens:2002wz; @Ackermann:2004zw; @Achterberg:2007qy; @Abbasi:2011zz; @Abbasi:2011ui] have focused on higher energies, to avoid the background of bremsstrahlung produced by atmospheric muons. In this analysis, we instead rely on the active veto provided by IceCube to reduce the background of penetrating muons, and exploit the high flux of atmospheric neutrinos at energies of a few hundred GeV to observe a set of 1,029 cascade-like neutrino events in 281 days of the 2010 data run. One such event is shown in Fig. \[fig:cascadeEvent\].
![Candidate neutrino-induced cascade observed in DeepCore in the 2010 data run. Each black dot indicates a DOM. Colored dots represent DOMs that detected light during the event, with the size of the dot proportional to the amount of light detected. The color indicates the relative arrival time of the first photon detected by that DOM, running through the spectrum from red (earliest) to purple (latest). \[fig:cascadeEvent\]](Event_06_2.pdf){width="\columnwidth"}
For this data set, recorded with the incomplete 79-string configuration of IceCube, the smaller DeepCore fiducial volume shown in Fig. \[fig:layout\] was used. This initial configuration consisted of only the central seven standard strings, plus 6 additional DeepCore strings. Based on Monte Carlo simulations, we estimate that approximately 60% of the 1,029 events in the final sample are truly neutrino-induced cascades, while around 40% are in fact $\nu_\mu$ CC events with muon tracks too short to be distinguished in the current analysis; efforts to further reduce this background are underway. The level of background due to atmospheric muons is still under investigation but appears to be small. The rates of observed neutrinos are consistent with simulations of atmospheric neutrinos using the leading atmospheric neutrino flux models from the Bartol and Honda groups, although we are still in the process of assessing our systematic uncertainties. It should be noted that the predictions based on the two atmospheric flux models differ for this event set by approximately 10%, due mainly to the modeling of production of higher energy electron neutrinos by kaons.
Work is in progress to lower the energy threshold of the analysis, which would permit observation of neutrino oscillations using the atmospheric neutrino flux. For baselines comparable to the Earth’s diameter, the first maximum of the $\nu_\mu \rightarrow \nu_\tau$ oscillation probability occurs at approximately 25 GeV, well within the energy range accessible to DeepCore [@Mena:2008rh].
Searches for Dark Matter
========================
In addition to studies of atmospheric neutrinos, DeepCore’s reduced energy threshold facilitates indirect searches for evidence of dark matter using IceCube. Searches are underway for neutrinos produced in the annihilation or decay or dark matter captured in the gravitational potential wells of the Earth, Sun [@Abbasi:2009uz; @Abbasi:2009vg], and Galaxy [@Abbasi:2011eq]. Because the WIMP mass must be relatively low compared to the energy range of IceCube, additional sensitivity to lower energy neutrinos substantially extends IceCube’s reach, especially for the lower part of the allowed WIMP mass range or for models where the neutrino spectrum produced is relatively soft.
![Limits on the spin-dependent WIMP-nucleon scattering cross section from various direct and indirect search experiments, and the projected sensitivity of IceCube with DeepCore for a “hard” neutrino spectrum arising from neutralino annihilation in the Sun. The shaded region indicates the possible cross sections in supersymmetric models not already ruled out by direct detection experiments’ limits on the spin-independent cross section. \[fig:WIMPs\]](WIMP_limits.pdf){width="\columnwidth"}
The potential of IceCube including DeepCore for detecting evidence of dark matter annihilation in the Sun is shown in Fig. \[fig:WIMPs\]. The shaded region indicates the allowed MSSM parameter space, for models where the WIMP is a neutralino. Direct detection experiments have already probed substantial parts of the allowed supersymmetric parameter space, primarily in regions where there is a substantial spin-independent neutralino-nucleon scattering cross section, so that coherent scattering from heavy nuclei in the detector target enhances the cross section considerably. For models in which the scattering cross section is primarily spin-dependent, indirect searches exploiting the Sun’s mass as a scattering target have an advantage, although the results depend on the branching ratios for neutralino-neutralino annihilation channels. For WIMP masses below roughly 100 GeV, DeepCore provides the bulk of the sensitivity to the neutrinos arising from Solar neutralino annihilation.
Future Prospects: PINGU {#sec:PINGU}
=======================
Encouraged by the initial success of DeepCore, the IceCube collaboration and other participants are developing a proposal for a Phased IceCube Next Generation Upgrade (PINGU), an extension of IceCube and DeepCore which would further increase the density of instrumentation in the central volume and further reduce the energy threshold. The proposal would augment DeepCore with perhaps 18 to 20 additional strings, of which the majority would be similar to those in DeepCore. Several strings might also include specialized prototypes of novel sensors, perhaps similar to those incorporating a number of 3” PMTs rather than a single 10” PMT, now being developed for the proposed KM3NeT detector. One layout of the additional strings under discussion is shown in Fig. \[fig:PINGUlayout\].
![Top view of one PINGU configuration now under study, including 16 strings of DeepCore-like instrumentation. Additional strings of prototype next-generation instrumentation are envisioned but not shown. This layout would significantly improve the effectiveness of DeepCore at energies below a few 10’s of GeV. \[fig:PINGUlayout\]](PINGU_2test_geometry_nolabels.pdf){width="\columnwidth"}
Such an extension would considerably increase the effective volume of DeepCore at energies below about 30 GeV, with the potential to detect neutrinos as low as a few GeV. The effective volume of the detector for events contained within the geometrical volume is shown in Fig. \[fig:PINGUnuEVolume\], as compared with that of the existing DeepCore detector. Improvements of nearly an order of magnitude can be seen for low energy neutrinos. These effective volumes do not include efficiency losses due to event reconstruction and analysis criteria, which will reduce the effective volume achievable in final physics analysis.
![Preliminary estimate of the effective volume of PINGU for electron neutrino events at trigger level, as compared to that of the completed DeepCore configuration. PINGU would retain considerable effective volume down to energies as low as a few GeV. Analysis and reconstruction efficiencies are not included. The geometry used for this estimate is similar to that shown in Figure \[fig:PINGUlayout\] but with a slightly larger mean spacing between strings, so the effective volume at the lowest energies may be underestimated. \[fig:PINGUnuEVolume\]](effectiveVolume_PINGU_nue_Compare_prelim.pdf){width="\columnwidth"}
Summary
=======
The effectiveness of IceCube at energies below 100 GeV has been significantly enhanced by the addition of DeepCore, which extends IceCube’s reach to energies of 10’s of GeV. This range is of interest for observations of neutrino oscillations, as well as searches for dark matter. As a first step toward these studies, we have observed a significant sample of atmospheric neutrino-induced cascades, enabled by the ability of the IceCube detector to identify and veto atmospheric muons penetrating to the DeepCore volume. We are also investigating the potential for a further reduction in the energy threshold of IceCube with an additional extension known as PINGU, which could extend IceCube’s reach to energies as low as a few GeV
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| ArXiv |
---
abstract: |
During the final growth phase of giant planets, accretion is thought to be controlled by a surrounding circumplanetary disk. Current astrophysical accretion disk models rely on hydromagnetic turbulence or gravitoturbulence as the source of effective viscosity within the disk. However, the magnetically-coupled accreting region in these models is so limited that the disk may not support inflow at all radii, or at the required rate.
Here, we examine the conditions needed for self-consistent accretion, in which the disk is susceptible to accretion driven by magnetic fields or gravitational instability. We model the disk as a Shakura-Sunyaev $\alpha$ disk and calculate the level of ionisation, the strength of coupling between the field and disk using Ohmic, Hall and Ambipolar diffusevities for both an MRI and vertical field, and the strength of gravitational instability.
We find that the standard constant-$\alpha$ disk is only coupled to the field by thermal ionisation within $30\,R_J$ with strong magnetic diffusivity prohibiting accretion through the bulk of the midplane. In light of the failure of the constant-$\alpha$ disk to produce accretion consistent with its viscosity we drop the assumption of constant-$\alpha$ and present an alternate model in which $\alpha$ varies radially according to the level magnetic turbulence or gravitoturbulence. We find that a vertical field may drive accretion across the entire disk, whereas MRI can drive accretion out to $\sim200\,R_J$, beyond which Toomre’s $Q=1$ and gravitoturbulence dominates. The disks are relatively hot ($T\gtrsim800\,$K), and consequently massive ($M_{\text{disk}}\sim0.5\,M_J$).
author:
- |
Sarah L. Keith $^{1,2}$[^1] and Mark Wardle$^{1}$\
$^{1}$Department of Physics & Astronomy and MQ Research Centre in Astronomy, Astrophysics & Astrophotonics, Macquarie University,\
NSW 2109, Australia\
$^{2}$Jodrell Bank Centre for Astrophysics, The University of Manchester, Alan Turing Building, Manchester, M13 9PL, United Kingdom
bibliography:
- 'references.bib'
date: 'Accepted Year Month date Day. Received Year Month Day; in original form Year Month Day'
title: Accretion in giant planet circumplanetary disks
---
\[firstpage\]
accretion discs – magnetic fields – MHD – planets and satellites: formation
Introduction
============
Gas giant planets form within a protoplanetary disk surrounding a young star [@1985prpl.conf..981L]. Those orbiting within $\sim100\,$au of the star form through the aggregation of a $\sim15M_{\text{Earth}}$ solid core and subsequent gas capture from the surrounding disk [@1996Icar..124...62P; @2009ApJ...695L..53B]. During the initial slow accretion phase the protoplanet envelope is thermally supported and distended. However, once the envelope mass reaches the core mass gas accretion accelerates rapidly and, unable to maintain thermal equilibrium, the envelope collapses [@1996Icar..124...62P; @2009Icar..199..338L]. This ‘run-away’ gas accretion ends once the planet is massive enough that it accretes faster than gas can be replenished into its vicinity. Infalling gas has too much angular momentum to fall directly onto the contracted planet, and so an accretion disk, the circumplanetary disk, forms around the planet [@1982Icar...52...14L; @2009MNRAS.397..657A].
In contrast to the icy conditions implied by satellite systems around Solar System giant planets, circumplanetary disks are likely initially hot and convective [@1989oeps.book..723C]. Most of the protoplanet’s mass is delivered during run-away accretion and so the circumplanetary disk must support a high inflow rate during this phase. The formation of Jupiter consistent with the giant planet formation time-scale inferred from the life-time of protoplanetary disks (life-time$\sim3\times 10^6 $years; ) suggests an inflow rate of $\dot{M}\sim10^{-6}M_J/$year. Models of the accretion phase of a circumplanetary disk include self-luminous disks , Shakura-Sunyaev $\alpha$ disks (@2002AJ....124.3404C [[email protected]], [[email protected]]; [@2005AA...439.1205A; @2013arXiv1306.2276T]), time-dependent disks with MRI-Gravitational instability limit cycles [@2011ApJ...740L...6M; @2012ApJ...749L..37L], and hydrodynamical simulations ([@1999ApJ...526.1001L], [@2002AA...385..647D], [@2003ApJ...599..548D], ). The evolution of the disk associated with the contraction of the proto-planetary envelope and changes in the mode of accretion from the protoplanetary disk have also been addressed [@2010AJ....140.1168W].
The angular momentum transport mechanism is key in determining the disk structure and evolution, however little work has been done to model the disk self-consistently with the accretion mechanism. The $\alpha$-model invokes a source of viscosity (typically hydromagnetic turbulence is suggested) however there is no guarantee that the resulting disk complies with the conditions required for viscosity, hydromagnetic or otherwise. An exception is the time-dependent gravo-magneto outbursting cycles modelled by @2012ApJ...749L..37L, however numerical simulations suggest disks rapidly evolve away from a gravitationally unstable state.
There are a variety of candidates for the accretion mechanism, including magnetic forces, gravitational instability, thermally-driven hydrodynamical instabilities, torque from spiral waves generated by satellitesimals \[see and @TurnerPPVI (in preparation) for a review\], and stellar forcing . Magnetic fields and gravitational instability are generally considered the most promising mechanisms within the protoplanetary disk. Magnetically-driven accretion may result from hydromagnetic turbulence produced by the magnetorotational instability (MRI; @1991ApJ...376..214B [@1995ApJ...440..742H]), centrifugally driven disk winds associated with large-scale vertical fields [@1982MNRAS.199..883B; @1993ApJ...410..218W], magnetic braking [@2004ApJ...616..266M], or large-scale toroidal fields [@2000prpl.conf..589S]. MRI turbulence has been modelled extensively (e.g., [@1996ApJ...457..355G; @2004ApJ...605..321S; @2007ApJ...659..729T; @2012MNRAS.420.2419F; @2012MNRAS.422.2737W; @2013ApJ...763...99P]) and simulations of MRI transport in protoplanetary disks indicate $\alpha\sim10^{-3}$, where $\alpha$ is the Shakura-Sunyaev viscosity parameter . Gravitational instability occurs in massive disks and may cause fragmentation or gravitoturbulence [@1964ApJ...139.1217T; @2001ApJ...553..174G].
Certain conditions are required for these mechanisms to be effective. For example, magnetic processes can only act in sufficiently ionised ‘active’ regions, where the evolution of the magnetic field is coupled to the motion of the disk. If the ionisation fraction is too low, magnetic diffusivity decouples their motion (e.g. ). In protoplanetary disks, magnetic coupling is strong enough to permit MRI accretion in two regions: (i) layers above the midplane where cosmic rays, and stellar X-rays and UV photons penetrate, and (ii) close to the star where the disk is hot and thermally ionised [@1996ApJ...457..355G]. Gravitational instability requires strong self-gravity such that Toomre’s stability parameter $Q\lesssim 1$, and quasi-steady gravitoturbulent accretion further requires a cooling time-scale in excess of $\sim30$ orbital time-scales ([@2012MNRAS.427.2022M], [@2012MNRAS.421.3286P]).
Existing steady-state model circumplanetary disks are not massive enough for gravitational instability, and so testing for self-consistent accretion has focussed on identifying regions which are susceptible to the MRI. @2011ApJ...743...53F determined the thickness of the magnetically-uncoupled Ohmic midplane ‘dead zone’ of an $\alpha$ disk for the ionisation by cosmic rays. They find that the dead zone extends up at least $2.5$ scale heights (for plasma $\beta=10^4$) with the presence of grains extending this region to even greater heights. These results agree with the recent paper by @2013arXiv1306.2276T which includes ionisation from X-rays, radioactive decay, turbulent mixing, thermal ionisation as well as cosmic rays and accounting for Ambipolar and Ohmic diffusion. They find that $\alpha$ disks are magnetically coupled in surface layers above $\sim3$ scale heights unless the disk is dusty and is shielded from X-rays. They also consider magnetic coupling in the Jovian analogue to the Minimum Mass Solar Nebula-the Minimum Mass Jovian Nebula (MMJN; [@2003Icar..163..198M]), finding that dust must be removed for magnetically coupled surface layers. They find that thermal ionisation in actively supplied disks may permit coupling within the inner $4\,R_J$ of the midplane, although suggest a larger thermally ionised region ($r\lesssim65\,R_J$). Either way, we conclude that current $\alpha$ models of circumplanetary disks are not necessarily susceptible to the magnetically driven accretion assumed at all radii, and that magnetically active surface layers may be too high above the midplane to carry the required accreting column.
In this paper, we probe the viability of self-consistent steady-state accretion through the circumplanetary disk midplane, with accretion driven by magnetic fields and gravitoturbulence. We model the disk as Shakura-Sunyaev $\alpha$ disk and solve for the disk structure self-consistently with the opacity using the @2009ApJ...694.1045Z opacity-law (§\[sec:disk\_structure\]). In §\[sec:thermal\_ionisation\] we calculate the ionisation level produced by thermal ionisation, cosmic rays, and radioactive decay, and also consider the effectiveness of turbulent mixing , and Joule heating in resistive MRI regions [@2005ApJ...628L.155I; @2012ApJ...760...56M]. We determine the magnetic field strength needed for accretion by an MRI or large-scale vertical field (§\[sec:B\_field\]), and calculate Ohmic, Hall and Ambipolar diffusivities to determine the strength of magnetic coupling (§\[sec:magnetic\_diffusivity\]). Motivated by the failure of the standard constant-$\alpha$ disk (§\[sec:const\_alpha\_model\]) to produce magnetic coupling consistent with the assumed viscosity profile we present an alternate $\alpha$ disk (§\[sec:thermally\_ionised\_model\]) in which the level of magnetic transport (i.e., $\alpha$) varies radially consistent with the level of viscosity proceed by either magnetic forces or gravitational instability, as per the @2002ApJ...577..534S prescription for $\alpha$ for non-ideal magnetic transport. We present the results in §\[sec:results\], with a summary and discussion of findings in §\[sec:discussion\].
Disk structure {#sec:disk_structure}
==============
We model a circumplanetary disk as an axisymmetric, cylindrical, radiative, thin disk surrounding a protoplanet of mass $M$, in orbit around a star of mass $M_*$, at an orbital distance $d$. The disk extends out to a radius $r=R_H/3$ around the planet, where $$\begin{aligned}
R_H&=&d\left(\frac{M}{3 M_*}\right)^{\frac{1}{3}}\nonumber\\
&\approx&743\,R_J \left(\frac{d}{5.2\,\text{au}}\right) \left(\frac{M}{M_J}\right)^{\frac{1}{3}}\left(\frac{M_*}{M_\odot}\right)^{-\frac{1}{3}}\end{aligned}$$ is the Hill radius, $R_J$ is the radius of Jupiter, $M_J$ is the mass of Jupiter, and $M_\odot$ is the mass of the Sun [@1998ApJ...508..707Q; @2011MNRAS.413.1447M].
The scale height, $H$, is determined by a balance between thermal pressure, the planet’s gravity, and self-gravity of the disk. Toomre’s $Q$ quantifies the strength of self-gravity, [@1964ApJ...139.1217T] $$\begin{aligned}
Q&=&\frac{c_s \Omega}{\pi G \Sigma}\nonumber\\
&\approx&5.3\times10^{3}\,\left(\frac{T}{10^3\,\text{K}}\right)^{\frac{1}{2}}\left(\frac{\Sigma}{10^2\text{g\,cm}^{-2}}\right)^{-1}\left(\frac{M}{M_J}\right)^{\frac{1}{2}}\nonumber\\
&&\times\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{2}},
\label{eq:Q}\end{aligned}$$ with $Q\gg1$ for negligible self-gravity and $Q\ll1$ for strong self-gravity. Here, $\Sigma$ is the column density, $\Omega$ is the Keplerian angular velocity, $$\label{eq:keplerian}
\Omega=\sqrt{\frac{G M}{r^3}}\approx5.9\times10^{-7}\,\text{s}^{-1}\,\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{2}}\left(\frac{M}{M_J}\right)^{\frac{1}{2}},$$ $c_s=\sqrt{kT/m_n}\approx1.9$kms$^{-1}\sqrt{T/1000\,\text{K}}$ is the isothermal sound speed with $m_n=2.34 m_p$ the mean neutral particle mass for a H/He gas at temperature $T$, $m_p$ the proton mass, and $k$ is Boltzmann’s constant. Solving for the scale height for arbitrary $Q$ is complex \[e.g, see [@1978AcA....28...91P]\], and so we adopt the simplified equation of vertical equilibrium (c.f., [@2002ApJ...580..987K]) $$\Omega^2 H^2 + \pi G H \Sigma - c_s^2 = 0,
\label{eq:Hrelation_withselfgravity}$$ with solution $$H=\frac{2Q}{1+\sqrt{1+4Q^2}}\frac{c_s}{\Omega}.
\label{eq:h_self_gravity}$$ This reduces to the standard approximations $$\begin{aligned}
\frac{H}{r} &=& \frac{c_s}{r\Omega}\nonumber\\
&\approx&0.45\,\left(\frac{T}{10^3\,\text{K}}\right)^{\frac{1}{2}}\left(\frac{r}{10^2\,R_J}\right)^{\frac{1}{2}}\left(\frac{M}{M_J}\right)^{-\frac{1}{2}}
\label{eq:H_noselfgravity}\end{aligned}$$ for low mass disks (i.e., $M_{\text{disk}}\ll M_J$) where self-gravity is negligible, and $$\begin{aligned}
\frac{H}{r}&=&\frac{c_s^2}{\pi G\Sigma r}=Q\frac{c_s}{r\Omega}\nonumber\\
&\approx&2.4\times10^{-2}\,\left(\frac{T}{10^2\,\text{K}}\right)\left(\frac{\Sigma}{10^6\,\text{g}\,\text{cm}^{-2}}\right)^{-1}\nonumber\\
&&\times\left(\frac{r}{10^2\,R_J}\right)^{-1}\end{aligned}$$ for massive, cool, self-gravitating disks. From this we estimate the vertically-averaged neutral mass density $$\begin{aligned}
\rho&=&\frac{\Sigma}{2H},\nonumber\\
&\approx&6.2\times10^{-9}\,\text{g cm}^{-3}\, \left(\frac{\Sigma}{10^2\,\text{g cm}^{-2}}\right)\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{1}{2}}\nonumber\\
&&\times\left(\frac{r}{10^2\,R_J}\right)\left(\frac{M}{M_J}\right)^{-\frac{1}{2}},
\label{eq:density}\end{aligned}$$ and the associated number density, $ n=\rho/m_n\approx 2.6\times10^{15}\,\text{cm}^{-3}\,\left(\rho/10^{-8}\,\text{g\,cm}^{-3}\right)$.
The thermal structure of the disk is governed by dissipation driven by the inflow. We use the standard plane-parallel stellar atmosphere model [@1990ApJ...351..632H], $$\sigma T^4=\frac{3}{8} \tau \sigma T_s^{4},
\label{eq:radiative_transfer}$$ to calculate the midplane temperature $T$ from the surface temperature $T_s$ and optical depth, $\tau$, from the midplane to the surface. Gravitational binding energy released during infall results in a surface temperature $$\begin{aligned}
T_s&=& \left( \frac{3 \dot{M}\Omega^2}{8\pi \sigma} \right)^{\frac{1}{4}}\nonumber\\
&\approx&82\,\text{K}\left(\frac{\dot{M}}{10^{-6}\,M_J/\text{year}}\right)^{\frac{1}{4}}\left(\frac{M}{M_J}\right)^{\frac{1}{4}}\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{4}},
\label{eq:surface_temp}\end{aligned}$$ where $\dot{M}$ is the inflow rate, and $\sigma$ the Stefan-Boltzmann constant. We consider a uniform, steady, inward mass flux throughout the disk.
Shock heating of infalling material colliding with the disk contributes additional heating, however it is negligible compared to that of the viscous dissipation \[i.e., flux ratio: $F_\text{infall}/F_\text{viscous}<10^{-4}$; @1981Icar...48..353C\]. Similarly, irradiation from the hot young planet \[$T_J=500$K determined from pure contraction of the young planet; e.g. \] and the accretion hot spot \[$T_{\text{hotspot}}=3300$K calculated using equation (3.3) in @1977MNRAS.178..195P\] is also negligible with $F_\text{planet}/F_\text{viscous}<10^{-4}$ and $F_\text{hotspot}/F_\text{viscous}<10^{-2}$ determined using equation (21) from @2013arXiv1306.2276T.
Equations (\[eq:radiative\_transfer\]) and (\[eq:surface\_temp\]) are applicable in optically-thick regions of the disk (i.e., where optical depth $\tau\gg1$). This is appropriate for the midplane, as the high column density favours a large optical depth: $$\label{eq:optical_depth}
\tau=\kappa\Sigma/2\gg1.$$
@1994APJ...427..987B @2009ApJ...694.1045Z
----------------------------- -------------------- -------- -------- -- ---------------------------- --------------------- -------- --------
Opacity Regime $\kappa_i$ $a$ $b$ Opacity Regime $\kappa_i$ $a$ $b$
Ice grains $2\times 10^{-4}$ 0 2 Grains $5.3\times10^{-2}$ $0$ $0.74$
Evaporation of ice grains $2\times10^{16}$ 0 $-7$ Grain evaporation $1.0\times10^{145}$ $1.3$ $-42$
Metal grains 0.1 0 $1/2$ Water vapour $1.0\times10^{-15}$ $0$ $4.1$
Evaporation of metal grains $2 \times 10^{81}$ 1 $-24$ $1.1\times10^{64}$ $0.68$ $-18$
Molecules $10^{-8}$ $2/3 $ 3 Molecules $5.1\times10^{-11}$ $0.50$ $3.4$
H scattering $10^{-36}$ $1/3$ 10 H scattering $8.9\times10^{-39}$ $0.38$ $11$
Bound–free and free–free $1.5\times10^{20}$ 1 $-5/2$ Bound–free and free–free $1.1\times10^{19}$ $0.93$ $-2.4$
Electron scattering 0.348 0 0 Electron scattering $0.33$ $0$ $0$
Molecules and H scattering 1.4 0 3.6
This regime is given in the footnote of Table 1 in @2009ApJ...694.1045Z. The dominant sources of opacity in this regime are molecular lines and H scattering (Z. Zhu 2013, private communication).
To calculate the opacity, $\kappa$, we use the analytic Rosseland mean opacity law presented in @2009ApJ...694.1045Z. This is a piecewise power-law fit to the @2007ApJ...669..483Z [@2008ApJ...684.1281Z] opacity law. We give this in Table \[table:opacity\_law\], re-expressed as a function of temperature and density, using the ideal gas law[^2]. This model features nine opacity regimes, incorporating the effects of dust grains, molecules, atoms, ions and electrons. The transition temperature $T_{j\rightarrow k}$ between regimes $j$ and $k$, as a function of density, is obtained by equating the opacity in neighbouring regimes (i.e., $\kappa_j=\kappa_{k}$), and is $$T_{j\rightarrow k}=\left(\frac{\kappa_{i,j}}{\kappa_{i,k}}\right)^{\frac{1}{b_{k}-{b_j}}}\rho^{\frac{a_j-a_{k}}{b_{k}-b_j}}
\label{eq:transition_temperatures}$$ with two additional constraints:
1. use Grains opacity for $T<794\,K$, and
2. use Molecules and H scattering opacity for $2.34\times10^4\kappa^{0.279}\,K<T<10^4\,K$.
As the opacity law is complex we show the temperature and density boundaries for each opacity regime in Fig. \[fig:opacity\_boundaries\].
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![\[fig:opacity\_boundaries\] Temperature and density boundaries of the @2009ApJ...694.1045Z opacity regimes, given in Table \[table:opacity\_law\], calculated with equation (\[eq:transition\_temperatures\]).](fig1.pdf "fig:"){width="48.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
For comparison, we also give the @1994APJ...427..987B opacity law in Table \[table:opacity\_law\]. This opacity law underestimates the opacity for temperatures $T\sim$1500–3000K because it neglects contributions from TiO and water lines longward of 5$\mu$m . The discrepancy is greatest at $\sim1700$K where the @1994APJ...427..987B opacity is a factor $\sim500$ too low, as compared with the @2009ApJ...694.1045Z model.
We solve for the local structure (i.e., $\Sigma$ and $T$) simultaneously with the opacity, at each radius. Following @1997ApJ...486..372B, we solve for the radial temperature profile by combining equations (\[eq:Q\]), (\[eq:keplerian\]), (\[eq:h\_self\_gravity\]), (\[eq:density\])–(\[eq:optical\_depth\]) and the opacity law in Table \[table:opacity\_law\], to give $$T^{4-b}=\frac{9 \dot{M}\kappa_i}{2^{a+7}\pi \sigma} \Omega^{2} H^{-a}\Sigma^{a+1},
\label{eq:temperature_density_relation}$$ with $a$, $b$, and $\kappa_i$ specified for each opacity regime. This relationship allows us to describe the disk temperature and column density self consistently, when one or the other is specified.
At a given radius, we solve this equation within each opacity regime, and determine whether the resulting temperature and density fall within the limits of that regime. Solutions which do not fall within these limits are discarded. The solution is not necessarily unique, as the disk may satisfy the conditions of multiple opacity regimes (e.g., [@1994APJ...427..987B; @2007ApJ...669..483Z]).
Conservation of angular momentum provides the closing relation by specifying the accreting column needed to drive the inflow caused by turbulence, $\dot{M}=2\pi \nu\Sigma$ [^3] . A common approach to modelling the turbulent viscosity $\nu$ is to adopt the $\alpha$-viscosity prescription, in which uncertainties in the form of the viscosity are gathered into a single parameter $\alpha\lesssim1$ , $$\nu=\alpha c_s H.
\label{eq:alpha_prescription}$$ Observational estimates of $\alpha$, derived from the inferred mass accretion rates of T-Tauri stars, and the time dependent behaviour of FU Orionis outbursts, dwarf nova, and X-ray transients, indicate $\alpha\sim0.001-0.1$, while numerical magnetohydrodynamical shearing box simulations yield $\alpha\sim 0.01$–$10^{-3}$ \[see @2007MNRAS.376.1740K and references therein\]. This results in an accreting column $$\begin{aligned}
\Sigma&=&\frac{\dot{M}}{2\pi \alpha c_s H}\label{eq:mdot_alpha_relation}\\
&\approx&1.6\times10^2 \text{g cm}^{-2} \left(\frac{\dot{M}}{10^{-6} M_J/\text{year}}\right)\left(\frac{\alpha}{10^{-3}}\right)^{-1}\nonumber\\
&&\times\left(\frac{M}{M_J}\right)^{\frac{1}{2}}\left(\frac{T}{1000\,\text{K}}\right)^{-1}\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{2}}\end{aligned}$$ for negligible self-gravity.
Degree of ionisation {#sec:thermal_ionisation}
====================
In this section we calculate the level of ionisation at the midplane of the circumplanetary disk. The disk is too dense for the penetration of cosmic rays and X-rays down to the midplane, and so the primary sources of ionisation are thermal ionisation and decaying radionuclides. We also consider two further ionising mechanisms produced by the action of MRI turbulence - the transport of ionisation from MRI active surface layers to the midplane by eddies, and ionisation from electric fields generated by MRI turbulence.
Thermal ionisation {#sec:thermal_ionisation}
------------------
Ionisation leads to the production of electrons, ions (with atomic number $j$), and charged dust grains with associated number density $n_e$, $n_{i,j}$, $n_g$, mass $m_e$, $m_{i,j}$, $m_g$, and charge $-q$, $+q$, $Z_g q$ respectively. Here, the grain mass and charge represent the mean value.
From this we define the total ion number density $n_i\equiv\sum_j n_{i,j}$, and average ion mass $m_i\equiv \left(n_i^{-1}\sum_j n_{i,j} m_{i,j}^{-1/2}\right)^{-2}$, where the summation runs over each ion species.
To calculate the level of thermal ionisation we use the Saha equation $$\label{eq:saha}
\frac{n_e n_{i,j}}{n_{j}}= g_e \left(\frac{2\pi m_e k T}{h^3}\right)^{\frac{3}{2}} \exp\left(-\frac{\chi_j}{k T}\right),$$ where $n_{j}$ is the number density of neutrals with atomic number $j$, $\chi_j$ is the ionisation potential of the $j^{\text{th}}$ ion species, $g_e=2$ is the statistical weight of an electron, and $h$ is Planck’s constant. Table \[table:ion\_species\] gives the atomic weight and first ionisation energy of five key contributing elements: hydrogen, helium, sodium, magnesium, and potassium [@CRC].
The exponential factor in the Saha equation gives rise to switch on/off behaviour in thermal ionisation, such that the bulk of atoms are ionised in a narrow temperature band around their ionisation temperature. Potassium has the lowest ionisation energy and is first to be ionised with an ionisation temperature of $T\sim10^3$K.
--------------- --------- --------------- ----------------------- --------------------- ---------------------- -----------
Atomic number Element Atomic weight Logarithmic abundance Abundance Ionisation potential Depletion
(amu) (eV) (dex)
1 H 1.01 12.00 9.21$\times10^{-1}$ 13.60 0
2 He 4.00 10.93 $7.84\times10^{-2}$ 24.59 0
11 Na 22.98 6.24 1.60$\times10^{-6}$ 5.14 $\delta$
12 Mg 24.31 7.60 $3.67\times10^{-5}$ 7.65 $\delta$
19 K 39.10 5.03 9.87$\times10^{-8}$ 4.34 $\delta$
--------------- --------- --------------- ----------------------- --------------------- ---------------------- -----------
We use solar photospheric abundances to model the elemental composition of the disk, as given in Table \[table:ion\_species\] . However heavy elements are encorporated into grains, reducing their gas phase abundance. We allow for depletion onto grains through a depletion factor $\delta$ (c.f., [@2000ApJ...543..486S]). The degree of depletion varies greatly between elements, however we make the simplifications that the abundance of elements other than hydrogen and helium are reduced by a constant factor, $10^{\delta}$. Grain depletion in the Orion nebula has been determined by comparing the abundances in the HII region (gas only) with that of Orion O stars (gas+dust; @1998MNRAS.295..401E). Magnesium, a key grain constituent, is depleted at the level $\delta_{\text{Mg}}=-0.92$, which we adopt for all depleted elements.
The abundance of the $j^{\text{th}}$ element is related to its logarithmic form, accounting for depletion onto grains: $X_j=\log_{10}(n_{j}/n_H)+12-\delta$, where the logarithmic abundance of hydrogen is defined to be $X_H=12$. The abundance is then $x_j=10^{X_j}/(\sum_i 10^{X_i})$, for which we take the logarithmic abundances of the remaining elements from .
Dust grains also act to reduce the ionisation fraction by soaking up electrons, acquiring charge through the competitive sticking of electrons and ions to their surface. The net charge is found through the balance of preferential sticking of electrons due to their higher thermal velocity, with the subsequent Coulomb repulsion that develops. The average charge acquired by a dust grain is [@1987ApJ...320..803D] $$Z_g=\psi\tau-\frac{1}{1+\sqrt{\tau_0/\tau}}
\label{eq:grain_charge}$$ where $$\begin{aligned}
\tau=\frac{a_g k T}{q^2},\\
\tau_0 \equiv \frac{8m_e}{\pi\mu m_p},\\
\mu\equiv \left( \frac{n_e s_e}{n_i}\right)^2\left(\frac{m_i}{m_p}\right),\end{aligned}$$ where $s_e$ is the electron sticking coefficient, $a_g$ the grain radius, and $\psi$ is the solution to the transcendental equation [@1941ApJ....93..369S]: $$1-\psi=\left(\mu \frac{m_p}{m_e}\right)^{\frac{1}{2}} e^{\psi}.
\label{eq:trancendental}$$ We solve this using the second order approximation [@ArmstrongKulesza] $$\psi=1-\ln(1+y)+\frac{\ln(1+y)}{1+\ln(1+y)}\ln[(1+y^{-1}) \ln(1+y)]$$ with $y\equiv e \sqrt{\mu m_p/m_e}$.
Charge fluctuations are small, with most grains having charge within one unit about this mean [@1979ApJ...232..729E]. Measurements and analytical estimates of the electron sticking coefficient suggest $s_e$ is in the range $10^{-3}$–$1$ [@1980PASJ...32..405U; @2010PhRvB..82l5408H]. As an approximation, we maximise the impact of grain charge removal by adopting $s_e\sim1$.
We adopt a constant gas to dust mass ratio ratio $\rho_d/\rho\equiv f_{dg}=10^{-2}$, grain size $a_g=0.1\mu$m, and grain bulk density $\rho_b=3$g cm$^{-3}$ [@1994ApJ...421..615P]. This leads to a grain number density $$\begin{aligned}
n_g&=&\frac{ m_n f_{dg} n}{\frac{4}{3}\pi a_g^3 \rho_b}\nonumber\\
&\approx&3.1\times10^{3}\,\text{cm}^{-3}\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)\left(\frac{f_{dg}}{10^{-2}}\right)\nonumber\\
&&\times\left(\frac{a_g}{0.1\,\mu\text{m}}\right)^{-3}\left(\frac{\rho_b}{3\,\text{g cm}^{-3}}\right)^{-1}.\end{aligned}$$ Grain evaporation, which removes grain species, will cause spatial variation of these properties. For instance, very few grains would be present where the temperature exceeds the vaporisation temperature of iron ($T\sim1500$K at $\rho\sim10^{-7}\,$gcm$^{-3}$; [@1994ApJ...421..615P]). However, we find that removing grains in this region (i.e., $f_{dg}=0$ for $r<7\,R_J$), or indeed uniformly across the disk (i.e., $f_{dg}=0$ for all $r$), has no effect on the boundary of the magnetically-coupled region owing to the overwhelming effectiveness of thermal ionisation here.
The final condition needed to determine the ionisation level is charge neutrality, $$n_i-n_e +Z_g n_g=0
\label{eq:charge}.$$ To solve equations (\[eq:saha\])–(\[eq:charge\]), we use Powell’s Hybrid Method for root finding [@Powell1970], with the routine `fsolve` from the Python library `scipy.optimize` [@scipython]. This method is a modified form of Newton’s Method, which checks that the residual is improved before accepting a Newton step. This optimisation allows for convergence despite the steep gradients caused by the exponential factor in the Saha equation.
Ionisation by decaying radionuclides, cosmic rays and X-rays
------------------------------------------------------------
Cosmic rays and the decay of radionuclides are the primary sources of ionisation in the outer disk where it is too cool for thermal ionisation. The short-lived radioisotope $^{26}$Al is the main contributor to ionisation by decaying radionuclides, yielding an ionisation rate $\zeta_R=7.6\times10^{-19}$s$^{-1}$ [@2009ApJ...690...69U]. Cosmic ray ionisation occurs at a rate $\zeta_{\text{CR}}=10^{-17}\,\text{s}^{-1}\exp(-\Sigma/\Sigma_{\text{CR}})$, where $\Sigma_{\text{CR}}=96\,\text{g}\,\text{cm}^{-2}$ is the attenuation depth of cosmic rays.
X-rays from the young star will also ionise the surface layers \[with $\zeta_{\text{XR}}=9.6\times10^{-17}\,\text{s}^{-1}\exp(-\Sigma/\Sigma_{\text{XR}})$ at the orbital radius of Jupiter for a star with Solar luminosity [@1999ApJ...518..848I; @2008ApJ...679L.131T]\], however the X-ray attenuation depth is so small ($\Sigma_{\text{XR}}=8\,\text{g}\,\text{cm}^{-2}$) that X-rays do not reach the midplane and do not contribute to midplane ionisation or accretion \[in contrast with *surface* ionisation calculations by @2013arXiv1306.2276T\].
Calculating the ionisation resulting from radioactive decay involves solving the coupled set of reaction rate equations for electrons, metal ions (number density $n_i$ with metal abundance $x_m$), and grains subject to charge neutrality. Molecular ions are the first ions produced as part of the reaction scheme, however, charge transfer to metals is so rapid that metal ions are more abundant [@2011ApJ...743...53F]. We model the metals as a single species, adopting the mass, $m_i$, and abundance, $x_i$, of the most abundant metal - magnesium . Free electrons and ions are formed through ionisation, and are removed through recombination (rate coefficient $k_{ei}$) and capture by grains (rate coefficients $k_{eg}$, $k_{ig}$ for electrons and ions, respectively). These processes are described by the following rate equations: $$\begin{aligned}
\frac{dn_i}{dt} &=& \zeta n - k_{ei}n_{i}n_{e}
-k_{ig}n_{g} n_{i}, \label{eq:ion_radioactive_decay}\\
\frac{dn_{e}}{dt} &=& \zeta n - k_{ei}n_{i}n_{e}
- k_{e g }n_{g} n_{e}, \label{eq:electron_radioactive_decay}\\
\frac{d Z_g}{dt}&=&
k_{ig} n_{i}
-k_{e g} n_{e}, \label{eq:grain_radioactive_decay}\\
0&=&n_i-n_e+Z_g n_g, \label{eq:neutrality_radioactive_decay}\end{aligned}$$ for which we have neglected grain charge fluctuations (see for example, @1980PASJ...32..405U [@2011ApJ...743...53F]). Anticipating that the resulting ionisation fraction will be low, we make the following simplifications: (i) the average grain charge will be low and so we approximate $Z_g\approx0$ in calculating the rate coefficients $k_{ig}, k_{eg}$ and (ii) recombination is inefficient such that charge capture by grains dominates and we set $k_{ei}=0$. The charge capture rate co-efficients for neutral grains are $$\begin{aligned}
k_{ig}&=&\pi a_g^2 \sqrt{\frac{8k_b T}{\pi m_i}}\nonumber\\
&\approx & 3.0\times10^{-5}\,\text{cm}^{3}\,\text{s}^{-1}\,\left(\frac{T}{10^3\text{\,K}}\right)^{\frac{1}{2}}\left(\frac{a_g}{0.1\mu\,\text{m}}\right)^2\nonumber\\
&&\times\left(\frac{m_i}{24.3\,m_p}\right)^{-\frac{1}{2}},\\
\label{eq:kmg}
k_{eg}&=&\pi a_g^2 \sqrt{\frac{8k_b T}{\pi m_e}}\nonumber\\
&\approx& 6.2\times10^{-3}\,\text{cm}^{3}\,\text{s}^{-1}\,\left(\frac{T}{10^3\text{\,K}}\right)^{\frac{1}{2}}\left(\frac{a_g}{0.1\mu\,\text{m}}\right)^2.
\label{eq:keg}\end{aligned}$$
Under these conditions the equilibrium electron and ion number density fractions are $$\begin{aligned}
\frac{n_e}{n}&=&\frac{\zeta }{k_{eg} n_{g}},\nonumber\\
&\approx& 5.2\times10^{-20}\,\left(\frac{T}{10^3\text{\,K}}\right)^{-\frac{1}{2}}\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\nonumber\\
&&\times\left(\frac{\zeta}{10^{-18}\,\text{s}^{-1}}\right)\left(\frac{\rho_b}{3\,\text{g\,cm}^{-3}}\right)\left(\frac{f_{dg}}{10^{-2}}\right)^{-1}\nonumber\\
&&\times\left(\frac{a_g}{0.1\mu\,\text{m}}\right),\\\label{eq:ne_fraction}
\frac{n_i}{n}&=&\frac{k_{eg}}{k_{ig}}\frac{n_e}{n_n},\nonumber\\
&\approx&1.1\times10^{-17}\,\left(\frac{T}{10^3\text{\,K}}\right)^{-\frac{1}{2}}\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\nonumber\\
&&\times\left(\frac{\zeta}{10^{-18}\,\text{s}^{-1}}\right)\left(\frac{\rho_b}{3\,\text{g\,cm}^{-3}}\right)\left(\frac{f_{dg}}{10^{-2}}\right)^{-1}\nonumber\\
&&\times\left(\frac{a_g}{0.1\mu\,\text{m}}\right)\left(\frac{m_i}{24.3\,m_p}\right)^{\frac{1}{2}}.\label{eq:ni_fraction}\end{aligned}$$ We insert these values into equation (\[eq:neutrality\_radioactive\_decay\]) to calculate an improved estimate of the grain charge: $$\begin{aligned}
Z_g&=&-\frac{n_i}{n_g}\nonumber\\
&\approx&-3.5\times 10^{-6} \,\left(\frac{T}{10^3\text{\,K}}\right)^{-\frac{1}{2}}\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\nonumber\\
&&\times\left(\frac{\zeta}{10^{-18}\,\text{s}^{-1}}\right)\left(\frac{\rho_b}{3\,\text{g\,cm}^{-3}}\right)^2\left(\frac{f_{dg}}{10^{-2}}\right)^{-2}\nonumber\\
&&\times\left(\frac{a_g}{0.1\mu\,\text{m}}\right)^4\left(\frac{m_i}{24.3\,m_p}\right)^{\frac{1}{2}}.\label{eq:grain_charge_rd}\end{aligned}$$ Charge capture by grains has removed a large fraction of the free electrons and so the average grain charge is small (validating our initial estimate, $Z_g\approx0$), and simply traces the ion density:
To calculate the charge resulting from the combined efforts of thermal ionisation, decay of radionuclides, and external ionisation sources we add the contributions linearly. A complete treatment would address the non-linear effects associated with using the combined charge particle population, rather than treating the populations as independent. However, as the drop-off of the radial thermal ionisation profile is so steep, the contribution of decaying radionuclides and cosmic rays within $r\lesssim55\,R_J$ is insignificant when compared to thermal ionisation. Similarly, thermal ionisation is highly inefficient beyond this distance, and so charge production is by radioactive decay and cosmic rays.
Ionisation from MRI turbulence
------------------------------
The action of MRI turbulence in the disk offer two further ionising mechanisms, which we describe below. We do not calculate the level of ionisation produced by these mechanisms, but rather determine their effectiveness within the circumplanetary disk.
Eddies within MRI active surface layers caused by cosmic ray ionisation may penetrate into the underlying dead zone, transporting ionised material with them . Turbulent mixing may deliver enough ionisation into the dead zone for magnetic coupling and reactivation of the dead zone [@2007ApJ...659..729T]. The vertical mixing time-scale for diffusion through a scale height is $$\tau_D=\frac{H^2}{\nu}=(\alpha\Omega)^{-1},$$ which is $1000$ dynamical times for Shakura-Sunyaev viscosity parameter $\alpha=10^{-3}$. However, free charges are removed through recombination and grain charge capture which lowers the ionisation fraction. From equation (\[eq:electron\_radioactive\_decay\]), we find that charges are removed on a time-scale $$\tau_R=\left( k_{ei}\overline{n_i}+k_{eg}\overline{n_g}\right )^{-1},$$ where the ion and grain number densities are vertically averaged along the path. We calculate the grain charge capture rate $k_{eg}$ for neutral grains, and the ion number density using the height averaged cosmic ray and constant radioactive decay ionisation rates assuming that ion capture by grains is small. We use a vertically uniform temperature, however we find no qualitative difference in the results using midplane or surface temperatures. For turbulent mixing to be effective in delivering ionisation to the midplane, it must be at least as rapid as charge removal (i.e., $\tau_D\gtrsim\tau_R$). Thus, we determine the effectiveness of midplane ionisation from active surface layers by comparing the charge removal and vertical mixing time-scales in §\[sec:results\].
Ionisation is also produced through currents generated by the action of the MRI turbulent field [@2005ApJ...628L.155I]. The electric field, $E$, associated with the MRI may be able to accelerate electrons to high enough energies that they are able to ionise hydrogen in some regions. Such MRI ‘sustained’ regions occur within the minimum mass solar nebula, reducing the vertical extent of the dead zone away from the midplane [@2012ApJ...760...56M]. Here we determine if self-sustained MRI occurs in circumplanetary disks.
Joule heating is the primary mechanism for converting work done by shear \[work per unit volume $W_{\text{S}}=(3/2)\alpha\Omega p$\] into the electron kinetic energy. The work dissipated per unit volume by Joule heating of an equipartition current \[i.e., the current $J_{\text{eq}}=cB_{\text{eq}}/(4 \pi H)$ associated with an equipartition field over a length scale $H$\], is $W_J=f_{\text{fill}} f_{\text{sat}} J_{\text{eq}} E$. Here $c$ is the speed of light, $f_{\text{fill}}$ is the filling factor representing the fraction of the total volume contributing to Joule heating, and $f_{\text{sat}}$ is the ratio of the saturation current in MRI unstable regions to the equipartition current. @2012ApJ...760...56M performed three dimensional shearing box simulations to determine the time, space, and ensemble averaged filling factor and MRI saturation current, finding $f_{\text{fill}}=0.264$ and $f_{\text{sat}}=13.1$. The total energy available for ionisation through Joule heating is limited to the work done by shear (i.e., $W_J \le W_S$), and so the electric field strength cannot exceed [@2012ApJ...760...56M][^4] $$E=\frac{3\alpha c_s B_{\text{eq}}}{4 c f_{\text{fill}} f_{\text{sat}}} \left(\frac{2Q}{1+\sqrt{1+4Q^2}}\right).
\label{eq:selfsustain}$$ Given this restriction, we calculate the maximum electron kinetic energy, $\epsilon$, available from Joule heating [@2005ApJ...628L.155I], $$\epsilon=0.43 q E l \sqrt{m_n/m_e}$$ where $l=1/(n \langle\sigma_{en}\rangle)\approx1\,\text{cm}\,(10^{15}\,\text{cm}^{-3}/n)$ is the electron mean free path, and $\langle\sigma_{en}\rangle=10^{15}\,\text{cm}^2$ is the momentum transfer rate co-efficient between elections and neutrals. For ionisation to be effective, the electron energy, $\epsilon$ must exceed the ionisation threshold of neutral particles within the disk.
Magnetic field strength {#sec:B_field}
=======================
Further to a possible proto-planetary dynamo field (e.g., Jupiter’s present day surface field is $4.2$G; ), the disk may accrete its own field from the protoplanetary disk [@1998ApJ...508..707Q; @2013arXiv1306.2276T]. As both MRI and vertical fields have been modelled extensively in protoplanetary disks, we consider both field geometries in driving accretion in circumplanetary disks. We calculate the magnetic field strength, $B$, required to drive accretion at the inferred accretion rate, $\dot{M}=10^{-6} M_J/\text{year}$.
Three dimensional stratified and unstratified shearing box, and global MRI simulations with a net vertical flux indicate that during accretion the MRI magnetic field saturates with [@1995ApJ...440..742H; @2004ApJ...605..321S; @2011ApJ...730...94S; @2013ApJ...763...99P] $$\alpha\approx0.5\beta^{-1}=0.5\frac{B^2}{8\pi p},
\label{eq:alpha_B_relation}$$ where $\beta\equiv 8\pi p/B^2$ is the plasma beta, and $p=c_s^2\rho$ is the pressure. This leads to a magnetic field strength $$B_{\text{MRI}}= \sqrt{16\pi \alpha c_s^2\rho},
\label{eq:alpha_magnetic_field}$$ which can be directly determined by the inflow rate as $$\begin{aligned}
B_{\text{MRI}}&=&\left(\frac{\dot{M}\Omega^2}{c_s}\right)^{\frac{1}{2}}\left(\frac{1+\sqrt{1+4Q^2}}{Q}\right)\nonumber\\
&\approx&0.66\text{\,G}\left(\frac{\dot{M}}{10^{-6} M_J/\text{year}}\right)^{\frac{1}{2}}\left(\frac{M}{M_J}\right)^{1/2}\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{1}{4}}\nonumber\\
&&\times\left(\frac{r}{10^2\,R_J}\right)^{-\frac{3}{2}}\left(\frac{Q^{-1}+\sqrt{Q^{-2}+4}}{2}\right).
\label{eq:BMRI}\end{aligned}$$
The equipartition field, $B_{\text{eq}}=\sqrt{8\pi p}$, defines the maximum field that the disk can support before magnetic pressure dominates over thermal pressure. From equation (\[eq:alpha\_B\_relation\]) we see that the MRI field is sub-equipartition, satisfying $$\frac{B_{\text{MRI}}}{B_{\text{eq}}}= \frac{v_a}{\sqrt{2}c_s}=\sqrt{2\alpha}
\label{eq:equipartition_ratio}$$ which is constant for a given $\alpha$, and where the Alfvén speed is $$\begin{aligned}
v_a&=&\frac{B}{\sqrt{4\pi \rho}},\nonumber\\
&\approx&8.9\times10^{-2}\,\text{km\,s}^{-1}\,\left(\frac{B}{1\text{\,G}}\right)\left(\frac{\rho}{10^{-9}\text{g\,cm}^{-3}}\right)^{-\frac{1}{2}}. \end{aligned}$$
Large-scale fields acting through disk winds and jets may also drive angular momentum transport and have been studied in the context of protoplanetary disks (e.g., @1993ApJ...410..218W [@1994ApJ...429..781S; @2013arXiv1301.0318B]). Magnetically-driven outflows have also been proposed for circumplanetary disks . If a vertical field drives the inflow the field strength must be at least $$\begin{aligned}
B_{\text{V}}&=&\sqrt{\frac{\dot{M} \Omega}{2r}},\nonumber\\
&\approx&0.16\text{\,G}\left(\frac{\dot{M}}{10^{-6} M_J/\text{year}}\right)^{\frac{1}{2}}\left(\frac{M}{M_J}\right)^{\frac{1}{4}}\left(\frac{r}{10^2\,R_J}\right)^{-5/4}.\end{aligned}$$
Magnetic coupling {#sec:magnetic_diffusivity}
=================
We are now in a position to calculate the level of magnetic diffusivity within the disk to identify which regions of the disk are subject to magnetically-driven transport. A minimum level of interaction between the disk and the magnetic field is needed for magnetically-controlled accretion.
Collisions disrupt the gyromotion of charged species around the magnetic field. Collisions between the electrons, ions, and neutrals occur at a rate $\nu_{ij}$ (for colliding species $i$ with $j$), with [@2008MNRAS.385.2269P] $$\begin{aligned}
\nu_{\text{ei}}&=&1.6\times10^{-2}\,\text{s}^{-1}\,\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{3}{2}}\left(\frac{n_e}{10\,\text{cm}^{-3}}\right)\nonumber \\
&&\times\left(\frac{n_n}{10^{15}\,\text{cm}^{-3}}\right),\\
\nu_{\text{en}}&=&6.7\times10^{6}\,\text{s}^{-1}\,\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{1}{2}}\left(\frac{\rho_n}{10^{-9}\,\text{g\,cm}^{-3}}\right),\nonumber\\\\
\nu_{\text{in}}&=&3.4\times10^{5}\,\text{s}^{-1}\,\left(\frac{\rho_n}{10^{-9}\,\text{g\,cm}^{-3}}\right),\end{aligned}$$ where $\rho_n=\rho-(\rho_i+\rho_e)$, and $n_n=\rho_n/m_n$ are the mass and number density of neutral particles, respectively. Electron–ions collisions are the dominant source of drag in the highly ionised inner region, however neutral drag dominates across the remainder of the disk. The Hall parameter for a species $j$, $\beta_j$, quantifies the relative strength of magnetic forces and neutral drag. It is the ratio of the gyrofrequency to the neutral collision frequency , $$\beta_j=\frac{|Z_j|eB}{m_j c} \frac{1}{\nu_{jn}}.$$ The Hall parameter is large, $\beta_j\gg1$, when magnetic forces dominate the equation of motion, and small, $\beta_j\ll1$, when neutral drag decouples the motion from the field.
The Hall parameters for ions, electrons, and grains are $$\begin{aligned}
\beta_i&\approx&4.6\times10^{-3}\left(\frac{B}{1\,G}\right)\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1},\\
\beta_e&\approx&1.1 \left(\frac{B}{1\,G}\right)\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\left(\frac{T}{10^3\,K}\right)^{-\frac{1}{2}}, \\
\beta_g&\approx&3.1\times10^{-3}\,Z_g\left(\frac{B}{1\,\text{G}}\right)\left(\frac{n}{10^{15}\,\text{cm}^{-3}}\right)^{-1}\left(\frac{T}{10^3\,K}\right)^{-\frac{1}{2}}\nonumber\\
&&\times\left(\frac{a_g}{0.1\,\mu\text{m}}\right)^{\frac{1}{2}}\left(\frac{\rho_b}{3\,\text{g\,cm}^{-3}}\right)^{\frac{1}{2}}.\end{aligned}$$ Ions and grains, being the more massive particles, have a lower gyrofrequency, and hence a lower Hall parameter. Thus, neutral collision are more effective at decoupling ions and grains than electrons. This leads to three regimes, according to the neutral density: (a) Ohmic regime, high density: electron–ion or neutral collisions are so frequent as to decouple both electrons and ions (i.e., $\beta_i\ll\beta_e\ll1$). (b) Hall regime, intermediate density: neutral collisions decouple ions, but the electrons remain tied to the field (i.e., $\beta_i \ll 1 \ll \beta_e$). (c) Ambipolar regime, low density: both the ions and electrons are coupled to the magnetic field, and drift through the neutrals. (i.e., $1\ll\beta_i\ll\beta_e$).
In each regime collisions produce magnetic diffusivity which affects the evolution of the magnetic field through the induction equation: $$\begin{aligned}
\frac{\partial \bold{B}}{\partial t}&=&\nabla(\bold{v}\times \bold{B})-\nabla\times[\eta_O(\nabla\times \bold{B})+\eta_H(\nabla\times \bold{B})\times \hat{\bold{B}}]\nonumber\\
&&-\nabla\times[\eta_A(\nabla\times \bold{B})_{\perp}],
\label{eq:induction}\end{aligned}$$ where $\bold{v}$ is the fluid velocity. The Ohmic ($\eta_O$), Hall ($\eta_H$), and Ambipolar diffusivities ($\eta_A$) are \[[@2008MNRAS.385.2269P], Wardle & Pandey (in preparation)\] $$\begin{aligned}
\label{eq:ohmic}
\eta_O&=&\frac{m_e c^2}{4\pi e^2n_e} (\nu_{\text{en}}+\nu_{\text{ei}})\nonumber\\
&\approx& 1.9\times10^{17}\text{cm}^{2}\, \text{s}^{-1}\,\left[\left(\frac{T}{10^3\,\text{K}}\right)^{\frac{1}{2}}\left(\frac{n_e}{10\,\text{cm}^{-3}}\right)^{-1} \right .\nonumber\\
&&\left .\times \left(\frac{10^{-9}\,\text{g\,cm}^{-3}}{\rho}\right)+ 2.4\times10^{-9}\,\left(\frac{T}{10^3\,\text{K}}\right)^{-\frac{3}{2}} \right ]\text{,}\end{aligned}$$ $$\begin{aligned}
\eta_H&=&\frac{cB}{4\pi e n_e}\left(\frac{1+\beta_g^2-\beta_i^2P}{1+\left(\beta_g+\beta_i P\right)^2}\right)\nonumber\\
&\approx&5.0\times10^{17}\,\text{cm}^{2}\,\text{s}^{-1}\,\left(\frac{B}{1\,\text{G}}\right)\left(\frac{n_e}{10\,\text{cm}^{-3}}\right)^{-1}\nonumber\\
&&\times\left(\frac{1+\beta_g^2-\beta_i^2P}{1+\left(\beta_g+\beta_i P\right)^2}\right),
\label{eq:hall}\end{aligned}$$ $$\begin{aligned}
\eta_A&=&\left(\frac{B^2}{4\pi\rho_i\nu_\text{ni}}\right)\left(\frac{\rho_n}{\rho}\right)^2\left(\frac{1+\beta_g^2+\left(1+\beta_i\beta_g\right)P}{1+\left(\beta_g+\beta_iP\right)^2}\right)\nonumber\\
&\approx&6.0\times10^{16}\,\text{cm}^{2}\,\text{s}^{-1}\left(\frac{B}{1\text{\,G}}\right)^{2}\left(\frac{\rho_n}{\rho}\right)^2\left(\frac{n_i}{10\,\text{cm}^{-3}}\right)^{-1}\nonumber\\
&&\times\left(\frac{\rho}{10^{-9}\,\text{g\,cm}^{-3}}\right)^{-1}\left(\frac{1+\beta_g^2+\left(1+\beta_i\beta_g\right)P}{1+\left(\beta_g+\beta_iP\right)^2}\right),
\label{eq:ambipolar}\end{aligned}$$ where $P=n_g\,\vline Z_g\vline/n_e$ is the Havnes parameter.
The magnetic field couples to the motion of the disk in regions of low magnetic diffusivity \[i.e., where $|\nabla\times(\bold{v}\times \bold{B})| \gg |\nabla\times[\eta (\nabla\times\bold{B})]|$, for each diffusivity, $\eta$\]. For MRI fields we require that the turbulent magnetic field grows faster than dissipation can destroy it such that [@2002ApJ...577..534S; @2013ApJ...764...65M] $$\begin{aligned}
\eta &<& v_{a,z}^2/\Omega\nonumber\\
&\approx&1.3\times10^{14}\,\text{cm}^{2}\,\text{s}^{-1}\,\left(\frac{B}{1\,\text{G}}\right)^2\left(\frac{\rho}{10^{-9}\text{\,g\,cm}^{-3}}\right)^{-1}\nonumber\\
&&\times\left(\frac{r}{10^2\,R_J}\right)^{\frac{3}{2}}\left(\frac{M}{M_J}\right)^{-\frac{1}{2}}
\label{eq:MRI_criterion}\end{aligned}$$ for each diffusivity $\eta=\eta_O$, $\eta_H$, and $\eta_A$. This condition is equivalent to the condition $\Lambda>1$, where $\Lambda=v_{a,z}^2/(\eta \Omega)$ is the Elsasser number. The coupling condition uses the Alfvén speed for the vertical component of the magnetic field. We calculate the vertical field component as $B_z\sim B_{\text{MRI}}/\sqrt{28}$, using results from @2004ApJ...605..321S.
If, instead, a vertical (rather than turbulent) field is responsible for angular momentum transport (e.g., through the action of a disk wind or jet), the condition is more relaxed as we only require that the magnetic field couples to the shear, with $$\begin{aligned}
\eta &<& c_s^2/\Omega\nonumber\\
&\approx &6.1\times10^{16}\,\text{cm}^{2}\,\text{s}^{-1}\,\left(\frac{T}{10^3\,\text{K}}\right)\left(\frac{r}{10^2\,R_J}\right)^{\frac{3}{2}}\left(\frac{M}{M_J}\right)^{-\frac{1}{2}}
\label{eq:LS_criterion}\end{aligned}$$ for each diffusivity.
Magnetic interaction still occurs for diffusivity at, or above the coupling threshold, however coupling is weak in these conditions and the connection between the dynamics of the disk and field is diminished.
Disk models {#sec:disk_models}
===========
We consider four circumplanetary disk models in this paper. We present two Shakura-Sunyaev $\alpha$ disks developed for this work: (i) a constant-$\alpha$ model in which the viscosity parameter is radially uniform (§\[sec:const\_alpha\_model\]), and (ii) a self-consistent accretion model in which the level of angular momentum transport is consistent with the strength of magnetic coupling or gravitational instability at all radii (§\[sec:thermally\_ionised\_model\]). For comparison we also describe two key circumplanetary disk models in the literature: (iii) the Minimum Mass Jovian Nebula (§\[sec:MMJN\]), and (iv) the Canup & Ward $\alpha$ disk (§\[sec:canup\_ward\]).
Constant-$\alpha$ model {#sec:const_alpha_model}
-----------------------
Here we take the traditional approach, adopting the $\alpha$-viscosity prescription with a radially-uniform $\alpha$. This allows for direct comparison with existing steady state circumplanetary disk models which adopt a constant $\alpha$. We take $\alpha=10^{-3}$ in keeping with the results of simulations (with net zero magnetic flux). However, the disk may accrete a net field which enhances transport, and so we also consider $\alpha=10^{-2}$.
To obtain the radial temperature profile for this model we insert equations (\[eq:H\_noselfgravity\]) and (\[eq:mdot\_alpha\_relation\]) into equation (\[eq:temperature\_density\_relation\]), yielding [@1997ApJ...486..372B] $$T^{\frac{3}{2}a-b+5}=\frac{9\kappa_i}{2^{2a+8}\sigma } \left(\frac{\mu m_p}{k}\right)^{\frac{3}{2}a+1} \alpha^{-(a+1)}\left(\frac{\dot{M}}{\pi}\right)^{a+2} \left(\frac{GM}{r^3}\right)^{a+\frac{3}{2}}.
\label{eq:const_alpha_temp_density_relationship}$$
We calculate all other properties, such as column density, by inserting this temperature profile into the relations given in §\[sec:disk\_structure\].
Self-consistent accretion model {#sec:thermally_ionised_model}
-------------------------------
The constant-$\alpha$ model implicitly assumes that the angular momentum transport mechanism operates at all radii, and to the right degree. Ionisation by cosmic rays and decaying radionuclides is insufficient to couple the disk and magnetic field [@2011ApJ...743...53F], and thermal ionisation is only active in the inner disk where $T\gtrsim10^3\,$K. Without gravitoturbulence from gravitational instability, or magnetically driven transport, which relies on magnetic coupling, little if any viscosity is produced throughout the bulk of the disk (i.e., $\alpha\approx0$). Thus, equation (\[eq:mdot\_alpha\_relation\]) is invalid across the majority of the disk.
Motivated by the inconsistency of the constant-$\alpha$ disk, we present an enhanced steady-state $\alpha$ disk in which the level of angular momentum transport (i.e., $\alpha$) driven by magnetic fields or gravitoturbulence is consistent with the level of magnetic coupling and strength of gravitational instability at all radii. To achieve this we divide the disk into three regions according to the mode of transport:
1. Saturated magnetic transport - the inner disk is hot enough for significant thermal ionisation allowing for strong magnetic coupling (i.e., $\eta_O, \eta_H, \eta_A$ are well below than the coupling threshold) and Toomre’s $Q\gg1$. Magnetically-driven angular momentum transport is maximally efficient and $\alpha$ saturates at its maximum value, which we take as $\alpha_{\text{sat}}=10^{-3}$. In this region the disk is identical to the constant-$\alpha$ disk.
2. Marginally coupled magnetic transport - in the majority of the disk, magnetic diffusivity exceeds the coupling threshold while self-gravity is still too weak for gravitoturbulence (i.e., Toomre’s $Q>1$). In this intermediate region accretion is magnetically driven, although at a reduced efficiency. @2002ApJ...577..534S determined the saturation level for MRI turbulence, and hence $\alpha$, for Ohmic and Ohmic+Hall MHD simulations in the non-linear regime (i.e., $\eta\Omega/v_{a,z}^2<1$; see their Fig. 20). They find that $\alpha$ is proportional to the ratio of the coupling threshold, $v_{a,z}^2/\Omega$, to Ohmic diffusivity. By extension we also assume that the effective $\alpha$ for non-turbulent accretion (i.e., for a vertical field) also adjusts according to the level of resistivity, using the analogous coupling threshold, $c_s^2/\Omega$. Thus, in this regime for the two modes of magnetic transport, we take $\alpha$ to be $$\label{eq:alphaSS02}
\alpha = \left\{
\begin{array}{lr}
\alpha_{\text{sat}} v_{a,z}^2/\left(\eta_O\Omega\right) & \text{for an MRI field,}\\
\alpha_{\text{sat}} c_s^2/\left(\eta_O\Omega\right) & \text{for a vertical field,}
\end{array}
\right.$$ which is at most $\alpha_{\text{sat}}$ [@2002ApJ...577..534S].
3. Gravoturbulent transport - in the outer disk magnetic coupling at the level required by equation (\[eq:alphaSS02\]) would result in a gravitationally unstable disk with Toomre’s $Q<1$, and so self-gravitational forces dominate. The cooling timescale determines whether the disk fragments or enters a gravoturbulent state. We find that the cooling time-scale is much longer than the dynamical time-scale, $\Omega^{-1}$, [@2007ApJ...662..642R; @2009ApJ...695L..53B] with $$\begin{aligned}
\Omega t_{\text{cool}}&=& \frac{\Sigma c_s^2 \Omega}{\sigma T_s^4}\nonumber\\
&=&\frac{8 c_s^3}{3G \dot {M}Q}\nonumber\\
&\sim& 1.9\times10^5\, \left(\frac{T}{120\,\text{K}}\right)^{\frac{3}{2}}\left(\frac{\dot{M}}{10^{-6}\,M_J\text{/year}}\right)^{-1}Q^{-1},\label{eq:t_cool}\end{aligned}$$ \[using equations (\[eq:Q\]) and (\[eq:surface\_temp\]), for a minimum midplane temperature $T=120\,$K set by the temperature of the Solar Nebula at the present day orbital radius of Jupiter according to the Minimum Mass Solar Nebula [@1981PThPS..70...35H]\] and so gravitoturbulence rather than fragmentation occurs [@2012MNRAS.427.2022M]. Either by the slow build up of surface density from inflow onto the disk coupled with heating by dissipation of turbulence [@2001ApJ...553..174G] or by time dependent evolution of gravitationally-unstable disks [@2011MNRAS.410..994F; @2013ApJ...767...63S], the disk likely evolves towards a state with $Q\sim1$. Thus, in this region we take $Q=1$.
We solve for the disk profile by inserting equation (\[eq:h\_self\_gravity\]), the scale height with self-gravity, into equation (\[eq:temperature\_density\_relation\]) requiring one final relation to close the set of equations. Each region has its own closing equation to account for the differences in the mode of transport :
1. In the saturated magnetic transport region, we use equation (\[eq:mdot\_alpha\_relation\]) with constant $\alpha=\alpha_{\text{sat}}$, inverted to give the surface density as a function of temperature.
2. In the marginally coupled magnetic transport region we solve for the midplane temperature numerically using `fsolve` from the Python library `scipy.optimize` [@scipython]. The solution is determined so that $\alpha$ calculated by inverting equation (\[eq:mdot\_alpha\_relation\]) is consistent with that from equation (\[eq:alphaSS02\]). To achieve this, at each iteration of the temperature solver we calculate the surface density, scale height and Q through equations (\[eq:Q\]), (\[eq:h\_self\_gravity\]) and (\[eq:temperature\_density\_relation\]) numerically using `fsolve`. These allow us to determine $\alpha$ from equation (\[eq:mdot\_alpha\_relation\]), and to also calculate the resulting ionisation fraction, magnetic field, and diffusivity (according to §\[sec:thermal\_ionisation\], §\[sec:B\_field\], and §\[sec:magnetic\_diffusivity\] respectively) for determining $\alpha$ from equation (\[eq:alphaSS02\]). Necessarily, $\alpha$ varies radially \[i.e., $\alpha\rightarrow\alpha(r)$\].
3. In the Gravoturbulent region, we set $Q=1$ and invert equation (\[eq:Q\]) to give the surface density as a function of temperature. We post-calculate $\alpha(r)$ using equation (\[eq:mdot\_alpha\_relation\]).
We solve the complete set of equations using the routine `fsolve` from the Python library `scipy.optimize` [@scipython].
Minimum Mass Jovian Nebula {#sec:MMJN}
--------------------------
The Minimum Mass Jovian Nebula (MMJN) is an adaptation of the Minimum Mass Solar Nebula used for modelling the Solar nebula [@1977ApSS..51..153W; @1981PThPS..70...35H]. The MMJN is produced by smearing out the solid mass of the satellites to form a disk, and augmenting it with enough gas to bring the composition up to solar (i.e., $f_{dg}\sim10^{-2}$).
We use the surface density for the MMJN given in @2003Icar..163..198M which follows a $\Sigma\propto r^{-1}$ profile, except in a transition region ($20R_J<r<26R_J)$ where the profile steepens, $$\Sigma=\left\{
\begin{array}{lr}
5.1\times10^5 \text{\,g\,cm}^{-2}\left(\frac{r}{14\,R_J}\right)^{-1} &r<20\,R_J,\\
3.6\times10^{5}\text{\,g\,cm}^{-2}\left(\frac{r}{20\,R_J}\right)^{-13.5} & 20\,R_J<r<26\,R_J,\\
3.1\times10^3 \text{\,g\,cm}^{-2}\left(\frac{r}{87\,R_J}\right)^{-1} &26\,R_J<r<150\,R_J.
\end{array}
\right.$$
We use the opacity ($\kappa=10^{-4}\,\text{cm}^{2}\,\text{g}^{-1}$; appropriate for absorption by hydrogen molecules) and temperature profile given by @1982Icar...52...14L, $$T = \left(240\,\text{K}\left(\frac{r}{15\,R_J}\right)^{-1}+(130\,\text{K})^{4}\right)^{1/4}.$$ The temperature profile follows $T\propto r^{-1}$ in the optically-thick inner regions, and is matched to the temperature of the ambient nebula ($T_{\text{neb}}=130\,$K) at the outer edge of the disk.
Canup & Ward $\alpha$ disk {#sec:canup_ward}
--------------------------
Canup & Ward (2002, 2006) model the circumplanetary disk as a steady-state, thin, axisymmetric, constant–$\alpha$ disk. They adopt the @1974MNRAS.168..603L surface density model, and use the plane-parallel stellar atmosphere model to calculate the midplane temperature. Heating sources are viscous dissipation, the ambient stellar nebula ($T_{\text{neb}}=150\,$K), and the hot young planet. The midplane temperature and density are solved self-consistently for a uniform opacity, however a range of opacities ($\kappa=10^{-4}$–1 cm$^{2}$ g$^{-1}$) are considered to account for uncertainty in the population of sub-micron grains. A range of inflow rates ($\dot{M}=10^{-8}$–$10^{-4} M_J/$year), and viscosity parameters ($\alpha=10^{-4}$–$10^{-2}$), are considered to model the disk at both early and late times. However, a low inflow rate ($\dot{M}= 2\times10^{-7}M_J/$year) is needed to match the ice line with the present day location of Ganymede and to ensure solid accretion is slow enough to account for Callisto’s partially differentiation. This indicates that the disk must be ‘gas-starved’ as compared with the MMJN. We calculate this disk model using the method given in [@2002AJ....124.3404C][^5], with parameters taken from @2006Natur.441..834C (i.e., $\alpha=6.5\times10^{-3}$, $\dot{M}=10^{-6} M_J/$year, and $\kappa=0.1$ cm$^{2}$g$^{-1}$).
Results {#sec:results}
=======
We are now in a position to apply the tools developed in §\[sec:disk\_structure\]–§\[sec:magnetic\_diffusivity\] to the models described in §\[sec:disk\_models\]. All figures are shown for a protoplanet in orbit around a solar-mass star at the current orbital distance of Jupiter (i.e., $M_*=1M_{\odot}$, and $d=5.2$au), calculated with the standard parameter set $\alpha=10^{-3}$, $\dot{M}=10^{-6}M_J/$year, and $M=M_J$, unless otherwise stated.
Disk structure {#disk-structure}
--------------
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Fig. \[fig:models\] shows the radial disk structure for each model. The constant-$\alpha$ disk, MMJN, and Canup & Ward disks are shown as the solid, long-dashed, and dot-dashed curves, respectively. The self-consistent accretion disk is shown for both an MRI (dotted curve) and vertical field (short-dashed curve). The curves are labelled $\alpha$, MMJN, CW, MRI, and V respectively.
The temperature profiles are shown in the top-left panel. The temperature profile for the constant-$\alpha$ and self-consistent accretion disks follow a power law with index changes at the transitions between opacity regimes. The self-consistent accretion disk profiles follow the constant-$\alpha$ profile out to $\sim30\,R_J$ where the temperature, and thermal ionisation level is high enough for good magnetic coupling. The stronger coupling requirement for an MRI field makes for a slightly hotter and more dense disk than for accretion driven by a vertical field, and so the disk is gravoturbulent beyond $200\,R_J$, where the temperature profile steepens. There is no corresponding gravoturbulent region for the self-consistent accretion disk with vertical field. Nevertheless, the self-consistent accretion disk is remarkably similar when either the MRI or verticals used for drive accretion. The profile for constant-$\alpha$ disk follows $T\propto r^{-1.1}$ in the outer regions where the opacity is primary from grains \[i.e., $a=0, b=0.74$; see equation (\[eq:const\_alpha\_temp\_density\_relationship\])\]. Of the parameter set $\alpha$, $\dot{M}$ and $M$, the temperature profile is most sensitive to changes in the inflow rate. An order of magnitude change in $\dot{M}$ only corresponds to a factor $\sim3$ change in the temperature across most of the disk, with little effect beyond $\sim40\,R_J$.The profiles are multivalued in the region $r\sim2$–$5\,R_J$, with a characteristic ‘S-shape’. Here the disk satisfies conditions for multiple opacity regimes, with the radially increasing, unstable branch corresponding to the H-scattering opacity regime. The viscous-thermal instability associated with this feature has been used to model outbursts in circumstellar disks surrounding T-Tauri stars - most notably FU Orionis outbursts by @1997ApJ...486..372B.
The constant-$\alpha$ and self-consistent accretion disks are hotter than the Canup & Ward and MMJN disks, which aim to model a later phase of the disk when the opacity is from ice grains (and necessarily lower; see the opacity profile in bottom-right panel of Fig. \[fig:models\]), and the disk is cool enough to form icy satellites. As inflow from the protoplanetary disk tapers, the disk cools, consistent with the evolution to an icy state recorded by the Solar System giant-planet satellite systems. For example, reducing the inflow rate by a factor of ten lowers the temperature to only $370\,$K at the disk outer edge.
The column density profile is shown in the top-right panel. The profile for the constant-$\alpha$ disk is generally shallow, decreasing by only a factor of $\sim12$ between the inner and outer edge. Like the @2002AJ....124.3404C disk, the column density is low compared with the MMJN, and so the disk is ‘gas starved’. Consequently, the disk mass is also low, with $M_{\text{disk}}=1.6\times10^{-3}M_J$, validating our neglect of self gravity. On the other hand, the column density in the self-consistent accretion disks increase beyond $\sim30\,R_J$ reaching $\Sigma=9.6\times10^4$gcm$^{-2}$ for a vertical field, and $\Sigma=2.5\times10^5$gcm$^{-2}$ for an MRI field. Consequently, the disk masses are large, with $M_{\text{disk}}=0.5\,M_J$ for the vertical field, and $M_{\text{disk}}=0.64\,M_J$ for the MRI field. The disk mass increases as the inflow rate from the protoplanetary disks tapers, such that a factor 10 reduction in the inflow rate leads to an inward extension of the gravoturbulent region, and a disk mass $M_{\text{disk}}=0.42\,M_J$, independent of the field geometry.
The centre-left panel of Fig. \[fig:models\] shows the aspect ratio for each model. The aspect ratio for the constant-$\alpha$ model ranges between $H/r=0.14$–$0.34$, with pressure dominating the scale height. Self-gravity is too weak to counteract the strong thermal pressure in the outer regions of the self-consistent accretion disks and so the disks are very thick, with the aspect ratio reaching a maximum of $H/r=0.63$, and $0.71$ for a vertical and MRI field, respectively. Our results agree with [@2013ApJ...767...63S] in that circumplanetary disks may be more aptly described as ‘slim’ (i.e., $H/r\lesssim1$) rather than ‘thin’.
The centre-right panel of Fig. \[fig:models\] shows the radial profile for Toomre’s $Q$. Toomre’s $Q$ is large for the low mass constant-$\alpha$ disk, however, despite the high temperatures the self-consistent accretion disks reach $Q\sim1$ at the outer edge where the column density is highest. We fix $Q=1$ in the gravoturbulent region in the self-consistent accretion disk with MRI field.
The bottom-left panel of Fig. \[fig:models\] shows the radial profile of the viscosity parameter, $\alpha$. The viscosity parameter is constant across the Canup & Ward and constant-$\alpha$ disks, and in the inner regions of the self-consistent accretion disks where magnetic coupling is good and $\alpha$ is saturates at its maximum value. Once the temperature drops below $\sim1000\,K$ thermal ionisation drops and with it the strength of magnetic coupling. Magnetic transport is less efficient with high diffusivity and so $\alpha$ is reduced, as per equation (\[eq:alphaSS02\]), reaching a minimum of $1.9\times10^{-7}$ for an MRI field, and $4.8\times10^{-7}$ for a vertical field. In the outer $\sim60\,R_J$ of the self-consistent MRI accretion disk, $\alpha$ increases radially to compensate for the decreasing column density. However, such a low required effective viscosity is potentially overwhelmed by other processes, such as stellar forcing or satellitesimal wakes which may contribute additional torque exceeding this level .
Note that a property of this model is that temperature increases with decreasing $\alpha$. This result is counter intuitive given that viscosity, and hence dissipation, are directly proportional to $\alpha$. However, for a fixed $\dot{M}$, increasing $\alpha$ enhances the effectiveness of the turbulence and so reduces the required active column density \[see equation (\[eq:mdot\_alpha\_relation\])\]. The associated reduction in optical depth lowers the midplane temperature relative to the surface temperature. Consequently, if we increase $\alpha_{\text{sat}}$ to $10^{-2}$ which is appropriate for MRI with net magnetic flux, we find that the midplane temperature reaches at most $2100\,$K. We also find that the saturated magnetic transport region (i.e. where the diffusivities are below the coupling threshold) only reaches out to $6\,R_J$, whereas the gravoturbulent region extends in as far as $120\,R_J$. However, we also note that increasing $\alpha_{\text{sat}}$ requires a further reduction of the minimum value of $\alpha$ to $2.2\times10^{-8}$ (at the boundary of the marginally coupled and gravoturbulent regions).
Opacity as a function of temperature is shown in the bottom–right panel of Fig. \[fig:models\], using the corresponding density profile \[i.e., $\kappa(\rho(r), T(r))$ vs $T(r)$\]. The opacity is complex and varies by four orders of magnitude throughout the disk. Despite differences in the temperature and density profiles, the opacity profile for the self-consistent accretion disks follow that of the constant-$\alpha$ disk. This is because the disks only deviate in the Grains opacity regime where the opacity is density independent (i.e., $a=0$).
Ionisation
----------
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Fig. \[fig:ionisation\_fraction\] shows the electron (solid curve), ion (dashed curve), and charge-weighted grain (dotted curve) number density fraction for the constant-$\alpha$ model (top-left panel), and self-consistent accretion disks with MRI field (centre-left panel) and vertical field (bottom-left panel).
In the constant-$\alpha$ disk, the ionisation fraction is high within the inner disk. Close to the planet the disk is almost fully ionised by thermal ionisation of hydrogen and helium, and thermal ionisation continues out to $\sim30\,R_J$ where the temperature exceeds $\sim1000$K and potassium is thermally ionised. In the abundance of free electrons grains acquire a large negative charge, $Z_g\sim-660$, but with little effect on the total electron density. Beyond this distance, the disk is not hot enough for significant thermal ionisation and so the ionisation fraction drops sharply. Ionisation is primarily by radioactive decay beyond $60\,R_J$, and the ionisation fraction is low (i.e., $n_e/n\sim10^{-19}$). In these conditions grains are mostly neutral, but still remove a large proportion of free electrons, reducing the electron density by a factor of $\sim190$ relative to the ions.
Thermal ionisation is strong over a larger portion of the self-consistent accretion disks, as the disk structure is reliant on a higher level of ionisation in the marginally magnetically coupled region. We rely on thermal ionisation to achieve magnetic coupling, as midplane ionisation from radioactive decay, cosmic rays and X-rays is too weak (see §\[sec:results\_coupling\]). Grain charging is important beyond $\sim40\,R_J$ for both field geometries, however it has a greater effect for the vertical field where the ionisation fraction is lower. All profiles are multivalued between $3\,R_J\le r\le5\,R_J$, in keeping with the temperature profiles.
Depletion onto grains removes heavy elements from the gas phase, and consequently reduces the ionisation fraction between $3\,R_J\lesssim r \lesssim 60\,R_J$ in the constant-$\alpha$ disk. There is no depletion close to the planet where ionisation is from the non-depleted hydrogen and helium, and in the outer disk ionisation by radioactive decay is so weak that neutral metals are abundant (i.e., $n_i/n_n\ll x_{\text{metals}})$ and the reaction rate is not limited by depletion. In the intermediate region depletion reduces the ionisation fraction by up to the depletion factor, $10^{-\delta}=0.12$. The lowered electron density leads to a slight increase (up to $10\%$) in grain charge. Depletion at this level has no appreciable effect on the structure of the self-consistent accretion disks.
Additional ionisation from MRI is ineffective for both the constant-$\alpha$ and fixed-temperature disks. Grain capture through vertical mixing rapidly removes ionisation in eddies produced in MRI active surface layers. If grains are absent, charges are removed by recombination quickly over a time-scale $\tau_R\approx4\Omega^{-1}$ at the outer edge. However, if grains are present, even at the level $f_{dg}\gtrsim10^{-11}$, grain charge capture is rapid. Thus, free charges are rapidly removed as they are mixed into the dead zone and so do not contribute to midplane ionisation.
For ionisation produced through acceleration by MRI electric fields, we find that the electron energy is at most $\epsilon\approx5\times10^{-3}$eV in the constant-$\alpha$ disk, and lower still in higher density self-consistent accretion disks. This energy is orders of magnitude too low to ionise any atomic species. Thus, there is no appreciable contribution from self-sustaining MRI ionisation in circumplanetary disks. Self-sustaining ionisation is more successful in protoplanetary disks where the density is lower such that electrons are able to be accelerated over a longer mean free path.
We have also calculated the charge number density fractions for the Canup & Ward $\alpha$ disk (top-right panel) and the MMJN (bottom-right panel) using the same method as given in §\[sec:thermal\_ionisation\]. In the Canup & Ward $\alpha$ disk thermal ionisation is high close to the planet with cosmic ray ionisation dominant beyond $20\,R_J$, similar to the constant-$\alpha$ disk. In the MMJN the ionisation fraction is very low ($n_e/n<10^{-16}$) due to both high surface density and low temperature.
Magnetic field strength {#magnetic-field-strength}
-----------------------
![Radial dependence of the magnetic field strength, B, for the $\alpha$ model (solid curve), and fixed temperature model with MRI field (dotted curve), and vertical field (short-dashed curve), Canup & Ward $\alpha$ disk (dot-dashed curve), and MMJN (long-dashed curve).[]{data-label="fig:magnetic"}](fig4){width="46.00000%"}
Fig. \[fig:magnetic\] shows the magnetic field strength for the constant-$\alpha$ model (solid curve), and self-consistent accretion disks with MRI field (dotted curve) and vertical field (dashed curve).
The MRI field strength for the constant-$\alpha$ disk varies between $B=0.28$–$250\,$G, and follows $B\propto r^{-1.1}$ across most of the disk. The field strength for the self-consistent accretion disk with MRI field is almost identical to that of the constant-$\alpha$ disk, except for a small deviation at the outer edge where the temperature profiles diverge. The vertical field required for self-consistent accretion has a similar dependency, with $B\propto r^{-5/4}$, but it is $\sim$5 times weaker and decreases monotonically. All disk model fields are sub-equipartition and are consistent with the with the estimate of $B=10$–$50$G at $10R_J$ by .
We have plotted the magnetic field strength required to drive accretion throughout the entire disk for the self-consistent accretion disk with MRI field, however beyond $200\,R_J$ accretion is powered by gravitoturbulence rather than magnetic fields. We have no information about the magnetic field in the gravoturbulent region.
For comparison we have calculated the MRI magnetic field strength for the Canup & Ward $\alpha$ disk and the MMJN, which we also show in Fig. \[fig:magnetic\]. We calculate the field strength the Canup & Ward disk using equation (\[eq:alpha\_magnetic\_field\]) for their $\alpha=6.5\times0^{-3}$, and for the MMJN using equation (\[eq:BMRI\]) assuming an accretion rate of $\dot{M}=10^{-6}\,M_J/$year.
Magnetic coupling {#sec:results_coupling}
-----------------
Fig. \[fig:diffusivity\] shows the Ohmic (solid curve), Hall (dashed curve), and Ambipolar (dotted curve) magnetic diffusivities scaled by the coupling threshold for the constant-$\alpha$ disk (top panel), and self-consistent accretion disk with MRI field (centre panel) and vertical field (bottom panel). The coupling threshold $\eta\Omega/v_a^2=1$ is used for the constant-$\alpha$ disk and self-consistent accretion disk with MRI field whereas $\eta\Omega/c_s^2=1$ is used for the self-consistent accretion disk with vertical field. The threshold is shown as a dotted horizontal line, with strong magnetic coupling in regions where each of the Ohmic, Hall and Ambipolar diffusivities are below the coupling threshold.
We find that all disks are dense enough that Ohmic diffusivity dominates over Hall and Ambipolar. The diffusivities follow the inverse of the ionisation fraction \[i.e., $\eta\propto n/n_e$, see equations (\[eq:ohmic\])–(\[eq:ambipolar\])\]. Within $30\,R_J$, the ionisation fraction is high and so the diffusivities are well below the coupling threshold, $\eta\Omega v_a^{-2}\ll1$ or $\eta\Omega c_s^{-2}\ll1$ . At $30\,R_J$ the diffusivities rise exponentially as thermal ionisation of potassium is suppressed by the low temperature. In the constant-$\alpha$ disk, ionisation from cosmic rays, X-rays and decaying radionuclides is too low for good magnetic coupling and so the majority of the disk, (i.e., $r>30\,R_J$), is uncoupled from the magnetic field. The magnetically coupled region is larger at higher inflow rates where the midplane temperature is higher (i.e., the disk is coupled within $90\,R_J$ for $\dot{M}=10^{-5}\,M_J/$year), however this also produces a higher disk scale height, (aspect ratio up to 0.79), violating the ‘thin-disk’ approximation. Diffusivity below the coupling threshold in the inner disk indicates that the evolution of the disk and magnetic field are locked together, however the bulk of the disk is uncoupled to the magnetic field and accretion cannot proceed in these regions.
The boundary of the magnetically coupled region is controlled by the exponential rise in the diffusivity at the ionisation temperature of potassium. For instance, if a vertical field is used instead of an MRI field, the scaled diffusivity is reduced by a factor $\left(v_a/c_s\right)^2=4\alpha$ \[using the MRI field to evaluate $v_a$; see equation (\[eq:equipartition\_ratio\])\], but the steepness of the diffusivity profile at the coupling boundary means that there is no change in the magnetically-coupled boundary. Similarly, depletion of heavy elements onto grains increases the diffusivity between $3\,R_J\le r \le 60\,R_J$, but does not change the radius of the magnetically-coupled region.
The diffusivity profile for the self-consistent accretion disk with MRI field follows the constant-$\alpha$ disk profiles out until $30\,R_J$, where Ohmic diffusivity reaches the coupling threshold. Here, the disk enters the marginally magnetic coupled region and the rise in the diffusivity is not as steep. Although magnetic coupling is only weak, as the diffusivities are above the coupling threshold, it is still enough to drive accretion at the level given by equation (\[eq:alphaSS02\]). This state of marginal coupling occurs out to $200\,R_J$, with Ohmic diffusivity up to $\sim10^4$ times greater than the coupling threshold. At the point where $Q=1$ gravitoturbulence becomes the dominant transport mechanism and the diffusivities resume their exponential rise.
The coupling criterion for a vertical field is less stringent, and so the diffusivities are lower relative to the coupling threshold within $r\sim30\,R_J$. As with the self-consistent accretion disk with MRI field, the sharp rise in the diffusivity is reduced once the diffusivities reach the coupling threshold as the disk transitions to marginal magnetic coupling. However, in contrast, the disk never reaches $Q=1$ and so there is no transition to the gravoturbulent region.
We have also calculated diffusivities for the Canup & Ward $\alpha$ disk (top-right panel) and the MMJN (bottom-right panel) for an MRI field. We show the absolute value of the Hall diffusivity for the Canup & Ward $\alpha$ disk as Hall diffusivity is negative beyond $r\sim70\,R_J$ (shown by a dotted curve when $\eta_H<0$). This occurs near the transition for ion re-coupling, and indicates that the Hall drift, between the field and neutrals, is in the opposite direction for a given field configuration. The diffusivities are above the coupling threshold for $r>10\,R_J$ for the Canup & Ward $\alpha$ disk and at all radii for the MMJN, preventing magnetically-driven accretion in these regions.
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Discussion {#sec:discussion}
==========
In this paper we modelled steady-state accretion within a giant planet circumplanetary disk, and determined the effectiveness of magnetic fields and gravitoturbulence in driving accretion. We modelled the disk as a thin Shakura-Sunyaev $\alpha$ disk, heated by viscous transport and solved for the opacity simultaneously with the disk midplane structure using the @2009ApJ...694.1045Z opacity law, including the effects of self-gravity. Thermal ionisation dominates within $r\lesssim30\,R_J$ where the disk reaches the ionisation temperature of potassium ($T\sim10^3\,$K), but drops rapidly in cooler regions where ionisation is primarily by radioactive decay. The midplane is too dense for penetration of cosmic rays or stellar X-rays. We considered both an MRI field and a vertical field in driving accretion, and found that a field of order $10^{-2}$–$10\,$G is needed to account for the inferred accretion rate onto the young Jupiter. To quantify the strength of interaction between the magnetic field and disk we calculated Ohmic, Hall, and Ambipolar diffusivities which cause slippage of the field lines relative to the bulk motion of the disk, decoupling their evolution.
In the standard constant-$\alpha$ disk, diffusivity is low enough for magnetic coupling in the inner region where potassium is thermally ionised. However, the remainder of the disk is too cool for thermal ionisation and so strong diffusivity prohibits magnetically-driven accretion throughout the bulk of the disk. The disk is gravitationally stable, with Toomre’s $Q\gg1$, and so there is no transport from gravitoturbulence either.
This is inconsistent with the assumption of a constant-$\alpha$, and so we presented an alternate model in which $\alpha$ varies radially, ensuring that the accretion rate (taken to be uniform through the disk) is consistent with the level of magnetic coupling and gravitational instability. We achieved this by dividing the disk into three regions according to the mode of accretion: (i) the inner disk is hot enough for strong magnetic coupling through thermal ionisation and inflow is magnetically driven with $\alpha$ saturated at its maximum value; (ii) Beyond $30\,R_J$ the disk is too cool for sufficient thermal ionisation of potassium and diffusivity exceeds the coupling threshold. Accretion is still magnetically driven, however as the magnetic coupling is weak, it occurs at a reduced efficiency with $\alpha$ inversely proportional to the level of magnetic coupling [@2002ApJ...577..534S]; (iii) The disk is gravitationally unstable in the outer regions where $Q\sim1$, and so gravitoturbulence is produced and drives accretion. Accretion is self-regulated so that the disk maintains marginal stability with $Q=1$. We calculated the disk structure for accretion driven by either MRI or vertical fields, finding very similar disk structures. With $Q\sim1$ at the outer edge, the disks are massive with $M_{\text{disk}}=0.5\,M_J$.
MHD analysis by @2011ApJ...743...53F and @2013arXiv1306.2276T argue against magnetically driven accretion through the midplane where the cosmic-ray and X-ray fluxes are too low. However, we find that midplane magnetic coupling relies primarily on thermal ionisation and so the disk temperature is crucial. @2011ApJ...743...53F use the surface temperature which is necessarily cooler than the midplane temperature, and so no thermal ionisation is expected. @2013arXiv1306.2276T considers both MMJN models and actively supplied accretion disk models, appropriate for a later, and so cooler, phase than we consider here. MMJN models are necessarily cold to match conditions recorded by the final, surviving generation of Jovian moons, however these are likely formed late after a succession of earlier generations were accreted by the planet [@2006Natur.441..834C]. Temperatures in actively accreting disks are controlled by the inflow rate which likely decreases as inflow from the protoplanetary disk tapers. @2013arXiv1306.2276T consider inflow rates that are lower than ours by a factor of 5–70, so these disks model a cooler stage and consequently thermal ionisation is limited to the inner $4\,R_J$ of their highest inflow disk. Additionally, we also consider accretion in regions which are only marginally coupled to the magnetic field. We find that while saturated magnetic transport (i.e. with strong magnetic coupling) is limited to the inner $30\,R_J$, magnetically driven accretion with marginal coupling can potentially occur across the entire disk.
We have modelled steady-state accretion within the disk, with the assumption that the disk evolves toward or through this state during the proto-planet accretion phase. Numerical simulations indicate that accretion disks, including circumplanetary disks, rapidly evolve away from a self-gravitating state toward a quasi-steady state [@2011MNRAS.410..994F; @2013ApJ...767...63S], however there may be other time-dependent processes, such as short time-scale variability of inflow from the protoplanetary disk. Observations of accretion onto giant planets are needed to determine the accretion timescales, and how rapidly the accretion rate can change.
The temperature profiles are multivalued in some regions of the disk, making the disks susceptible to viscous-thermal instability. This may lead to outbursts, undermining our steady-state assumption. This feature is only present when the inflow rate exceeds $\dot{M}=2\times10^{-8}\,M_J/$year, and so outbursting from the viscous-thermal instability will not occur at later time when the inflow rate has tapered off to below this value. While there is certainly the potential for outbursting at earlier times, our analysis centres on whether inflow driven by magnetic fields is plausible, rather than advocating a steady state solution.
There may also be additional torques on the circumplanetary disk, from stellar forcing or spiral waves generated by satellitesimals , which we have not included. It is not clear what level of transport these processes produce during this phase of giant planet accretion and whether they can be incorporated as additional sources within the Shakura-Sunyaev $\alpha$ formalism. We can model minor variations on the inflow parameters, such as a reduction in the accretion rate which reproduces the necessary cooling and disk mass lowering as inflow from the protoplanetary disk tapers. However these results are uncertain as they require yet lower values of $\alpha$ in the self-consistent accretion disk which are likely overwhelmed by the additional torques mentioned above.
Strong magnetic coupling near the surface of the planet will affect accretion onto the planet surface. The planetary magnetic field may channel the accretion flow onto the planet surface [@2011AJ....141...51L], effecting the spin evolution of the planet [@1996Icar..123..404T; @2011AJ....141...51L], and temperature of the planet. However magnetospheric accretion requires diffusivity in order for the inflow to transfer onto the planetary magnetic field from the disk field. Loading onto the planetary field lines is only expected to occur close to the surface, if at all (at $r\sim1$–$3\,R_J$; see ). However we find the diffusivity is very low at this distance, making loading of the gas onto the proto-planetary field lines from the disk field difficult. Magnetospheric accretion would require an additional source of diffusivity, such as electron momentum exchange with ion acoustic waves (e.g., see [@2006JGRA..111.1205P]), however it is not known how strong this effect is.
Finally, the circumplanetary disk is the formation site for satellites. The composition of the present day satellite systems around Jupiter and Saturn record conditions in their circumplanetary disks at the time of their formation. In particular, the rock/ice compositional gradients through the satellite systems set the disk ice line ($T\approx250\,$K) at the the location of Ganymede, $r=15R_J$, and Rhea, $r=8.7\,R_S$, in the Jovian and Saturnian systems, respectively [@2003Icar..163..198M]. The location of the ice line is often incorporated or used as a measure of success in circumplanetary disk models (e.g., @1982Icar...52...14L [@2003Icar..163..198M; @2002AJ....124.3404C]), however no moons have been discovered beyond the Solar System and so it is not clear how typical the Jovian system is, nor to what degree these systems can vary [@2013arXiv1306.1530K]. It is not our aim to reproduce the conditions for moon formation, but rather we are focussed on modelling the early phase of the disk, in which the disk is hot and there is the significant inflow onto Jupiter. Consequently, the ice line in our constant-$\alpha$ disk is at $r=139\,R_J$, and the self-consistent accretion disks are too hot for ice. Several generations of satellites may have formed in these conditions, but the present day satellites likely form at a later stage when the disk has cooled as inflow into the circumplanetary disk tapers with the dispersal of the protoplanetary disk [@1989oeps.book..723C; @2006Natur.441..834C; @2010ApJ...714.1052S]. Our results support the two stage circumplanetary disk evolution proposed by @1989oeps.book..723C in which the disk is initially hot and turbulent, but evolves to the cool quiescent disk as recorded by the giant planet satellite systems.
In summary, we have found that during the final gas accretion phase of a giant planet the circumplanetary disk is hot and steady-state accretion may be driven by a combination of magnetic fields and gravitoturbulence. Accretion maintains the disk at a high temperature so that there is thermal ionisation through most of the disk.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Yuri I. Fujii and Philippa K. Browning for valuable discussion and comments on the manuscript. We thank the anonymous referee for helpful comments which improved this manuscript. This work was supported in part by the Australian Research Council grant DP120101792. S.K. further acknowledges the support of an Australian Postgraduate Award and funding from the Macquarie University Postgraduate Research Fund scheme. This research has made use of NASA’s Astrophysical Data System.
\[lastpage\]
[^1]: E-mail: [email protected]; [email protected]
[^2]: We have used the mean particle mass of molecular H/He gas in the conversion from pressure to density even though it is not strictly valid where hydrogen is ionised. Hydrogen is only ionised within the inner $5\,R_J$, at temperatures above $3000\,$K, and we find that correcting the mean particle mass (to $\mu=1.24$) leads to at most a 15% change in the temperature in this region.
[^3]: is released in a boundary layer (thickness $\ll R_J$) above the planet surface where the disk angular velocity profile transitions sharply between keplerian and the planetary rotation rate [@1977MNRAS.178..195P]. This contributes an additional factor $\left(1-\sqrt{R_J/r}\right)$ to the right hand side to this viscosity-inflow relation. However, we find that this factor is only significant within $r<2\,R_J$, i.e., within the boundary layer.
[^4]: For consistency we insert our equation (\[eq:alpha\_B\_relation\]) into equation (32) of @2012ApJ...760...56M, and account for self-gravity which leads to stricter criterion, independent of plasma $\beta$: $f_{\text{whb}}=5.4\times10^{-2}$ for $Q=0$ \[c.f., their equation (36)\].
[^5]: The profiles shown in Fig. \[fig:models\] are calculated using the full expression $\chi=1+\frac{3}{2}[r_c/r-\frac{1}{5}]^{-1}$ (given below equation 20 in @2002AJ....124.3404C), however we found $\chi=1$ was needed to reproduce the profiles in @2002AJ....124.3404C. For the parameter set used here, we find that the approximation leads to at most a $37\%$ increase in the surface density, and $27\%$ reduction in the temperature profile. The difference is greatest at $r=60R_J$, but decreases toward the inner and outer boundaries.
| ArXiv |
---
abstract: 'We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of prime numbers. The proof uses the probabilistic method. Using the same techniques we improve the bounds obtained by He for gaps in geometric-progression-free sets.'
address: 'Department of Mathematics, Towson University, 8000 York Road, Towson, MD 21252'
author:
- Nathan McNew
bibliography:
- 'bibliography.bib'
title: 'Primitive and Geometric-Progression-Free Sets without large gaps'
---
Introduction
============
Despite the rich history of research on the gaps in the sequence of prime numbers, including many recent breakthroughs, the magnitudes of the largest gaps in this sequence are still poorly understood. Denoting by $p_1, p_2, \ldots$ the sequence of prime numbers, it has been known since 2001, due to Baker, Harman, and Pintz [@bhp], that $$p_n -p_{n-1} \ll p_n^{0.525}.$$ Assuming the Riemann Hypothesis gives a small improvement. Cramér [@cra] shows $$p_n -p_{n-1} \ll \sqrt{p_n}\log p_n.$$ Cramér [@Cra2] conjectures, however, that the bound $p_n -p_{n-1} \ll \log^2 p_n$ gives the true order of magnitude of the largest gaps. As for lower bounds, it follows immediately from the prime number theorem that there must exist gaps where $p_n -p_{n-1} \geq \log p_n$. This can be improved upon slightly. It has recently been shown by Ford, Green, Konyagin, Maynard and Tao [@fgkmt] that, for some positive constant $c$, the innequality $$p_n-p_{n-1} > \frac{c\log p_n \log \log p_n \log_4p_n}{\log_3 p_n}$$ holds infinitely often, improving on the previous result of Rankin [@rankinprimes] which included an additional triple $\log $ factor in the denominator. Here, and throughout the paper, $\log_i x$ will be used to denote the $i$-fold iterated logarithm when $i\geq 3$. Since $\log \log x$ is commonly used it will be used for readability when $i=2$.
Generalizing from the set of primes, one can consider any primitive set of integers. We say a set is primitive if no integer in the set divides another integer in the set. The study of primitive sets also has a rich history. For example, it is known that primitive sets can have counting function substantially larger than the prime numbers. Ahlswede, Khachatrian, and Sárközy [@AKS] showed there exists a primitive sequence $s_1<s_2<\cdots$ with $$n \asymp \frac{s_n}{\log \log s_n (\log_3 s_n)^{1+\epsilon}}$$ for sufficiently large $n$. Martin and Pomerance [@mp] show that this can be improved slightly, in fact there exists such a sequence where $$n \asymp \frac{s_n}{\log \log s_n \log_3 s_n \cdots \log_k s_n (\log_{k+1} s_n)^{1+\epsilon}}$$ for sufficiently large $n$ and any $k\geq 2$. This is, in a sense, best possible, as Erdős [@erdosprimitive] shows that any primitive sequence $s_1, s_2, \ldots$ must satisfy $$\sum_{n=1}^\infty \frac{1}{s_n \log s_n} < \infty.$$ Compared to the sequence of prime numbers, where the average gap grows like $\log x$, we see from these results that primitive sets can have substantially smaller gaps on average, on the order of $\log \log x \log_3 x \cdots \log_k x (\log_{k+1} x)^{1+\epsilon}$ for any $k\geq 2$. Nevertheless, it has not yet been possible to show that the largest gaps among these sequences is any smaller than what is known for the prime numbers.
We show here that there exist primitive sequences in which the gap between consecutive terms is substantially smaller than has been previously shown for the primes or any other primitive sequence. In particular, we get the following upper bound.
\[thm:primitive\] For any $\epsilon>0$ there exists a primitive sequence $q_1< q_2 < \cdots$ of integers in which the gap between any two consecutive terms is bounded above by $$q_n-q_{n-1} \leq \exp \left(\sqrt{2\log q_n \log \log q_n + (2+\epsilon)\log q_n \log_3 q_n}\right). \label{primitive bound}$$
The proof utilizes the probabilistic method, and so it is not constructive. It generalizes, however, to the related problem of geometric-progression-free sets, where the analogous problem has recently attracted attention.
If $r>1$ is rational (sometimes we insist it be integral), then a geometric progression of length $k$ with ratio $r$ is a progression of integers $(g_1,g_2,\ldots g_k$) in which $g_i=rg_{i-1}$. We say $S$ avoids geometric progressions of length $k$ if it is not possible to find $k$ integers from $S$ in a geometric progression. Note that primitive sets can be described as sets avoiding geometric progressions of length 2 in which we insist that the ratio $r$ must be an integer. For the remainder of the paper we will assume that our geometric progressions have length at least 3, and, unless otherwise stated, are allowed to have rational ratio.
In the case of geometric-progression-free sets, unlike primitive sets, there exist such sets with positive density. In particular, the squarefree numbers avoid geometric progressions and have density $\frac{6}{\pi^2}$, though this density isn’t best possible. (See [@mcnewgpf; @NO; @Rankin] for results on the maximum density of such a set.)
Because of this it is not clear, a priori, that there cannot exist such sets in which all of the gaps are bounded above by a fixed constant. In ergodic theory a set in which every gap is bounded by a constant is known as a *syndetic* set. Bieglböck, Bergelsen, Hindman and Strauss [@BBHS] first posed the question of whether there exists a syndetic set that is geometric-progression-free. This problem has become well-known as a good example of the difficulty inherent in studying problems that mix the additive and multiplicative structure of the integers, and remains open.
There has been partial progress toward this question for 2-syndetic sets (sets in which the difference between any two consecutive terms is at most two). He [@He] shows by a computer search that any subset of the range \[1,640\] containing at least one of any pair of consecutive numbers must contain three term geometric progressions. Recently Patil [@Patil] shows that any sequence of integers $s_1<s_2<\cdots$ with $s_n-s_{n-1} \leq 2$ must contain infinitely pairs $\{a,ar^2\}$ with $r$ an integer.
In general, one can avoid geometric progressions of length $k{+}1$ by taking the sequence of $k$-free numbers. Denoting by $s_1<s_2<\cdots$ the sequence of $k$-free numbers, the best known bound on the gaps, due to Trifonov [@trifonov] is that $$s_n-s_{n-1} \ll s_n^{\frac{1}{2k+1}} \log s_n.$$ Though this, again, is likely far greater than the truth.
He [@He] considers the existence of geometric-progression-free sets with gaps provably smaller than the bounds for $k$-free numbers. He shows the following.
For each $\epsilon>0$ there exists a sequence $b_1<b_2<\cdots$ avoiding 6-term geometric progressions satisfying $$b_n-b_{n-1} \ll_\epsilon \exp\left( \left(\frac{5\log 2}6 +\epsilon \right) \frac{\log b_n}{\log \log b_n}\right).$$ Furthermore, there exists a sequence $c_1<c_2<\cdots$ avoiding 5-term geometric progressions satisfying $$c_n-c_{n-1} \ll_\epsilon c_n^\epsilon$$ and a sequence $d_1<d_2<\cdots$ that avoids 3-term geometric progressions with integral ratio in which $$d_n-d_{n-1} \ll_\epsilon d_n^\epsilon.$$
The technique developed here allows us to treat 3-term geometric progressions with rational ratio and obtain a substantially smaller bound on the size of the gaps. In particular we prove the following in section \[sec:gpf\] .
\[thm:gpf\] For any $\epsilon>0$ there exists a sequence of integers $t_1< t_2< \cdots$ free of 3-term-geometric-progressions, such that $$t_n-t_{n-1} \leq \exp \left(2\sqrt{\log 2\log t_n + \tfrac{3+\epsilon}{2}\sqrt{\log 2\log t_n}\log \log t_n}\right). \label{gp bound}$$
Coprime subsets of intervals
============================
We first prove that in any short interval we can find a relatively large subset of integers that are pairwise coprime. Using the linear sieve of Rosser and Iwaniec (see for example Theorem 12.14 and Corollary 12.15 of [@cribro]) one can sieve an interval of length $y$ by primes up to nearly $\sqrt{y}$. The result can be stated as follows.
\[lem:sieve\] There exist positive constants $c_1$ and $c_2$ so that every interval of length $c_1y$ with $y\geq 2$ contains at least $\frac{y}{ \log^2 y}$ integers free of prime factors smaller than $\sqrt{y}$, and at most $\frac{c_2 y}{\log y}$ such integers.
Using this we can show that the short interval $[x-y,x]$ contains a reasonably large subset of pairwise coprime integers. Erdős and Selfridge [@erdself] (see also [@erdric]) prove that for sufficiently large $y$ and any $\epsilon>0$ any such interval has a pairwise coprime subset of size at least $y^{1/2-\epsilon}$, though their proof is not correct as written. We correct and refine the argument, using Lemma \[lem:sieve\] to show the following.
\[thm:coprimeset\] For sufficiently large $y$ and $x\geq y+1$, any interval $[x-y,x]$ contains a subset of pairwise coprime integers of size at least $ \frac{c_3 \sqrt{y}}{\log y}$ for some positive constant $c_3$.
Let $y'=y/c_1$, where $c_1$ is the constant from Lemma \[lem:sieve\]. That lemma then implies that the set $S \subset [x-y,x]$ consisting of integers in this interval free of prime factors smaller than $\sqrt{y'}$ contains at least $\frac{y'}{ \log^2 y'}$ integers.
Now, let $p \geq \sqrt{y'}$ be prime, and suppose $p|n$ for some $n \in S$. Then $n=pm$ where $m \in \left[\frac{x}{p} - \frac{y}{p},\frac{x}{p}\right]$ (an interval of length $\frac{c_1y'}{p}$). Since $n$ is free of prime factors smaller than $\sqrt{y'}$, $m$ will be free of such primes as well. While we can’t sieve this shorter interval of primes as large as $\sqrt{y'}$, we can use Lemma \[lem:sieve\] to sieve this interval of primes up to $\sqrt{\frac{y'}{p}}$, at least so long as $\frac{y'}{p}$ is at least two. Thus for each prime $\sqrt{y'}\leq p < \frac{y'}{2}$, the number of integers in $S$ divisible by the prime $p$ is at most $$\frac{c_2 y'}{p \log \frac{y'}{p}}.$$ For those primes $\frac{y'}{2}\leq p<y$, we can bound the number of integers in $S$ divisible by $p$ trivially by $\left \lceil \frac{y}{p}\right \rceil =O(1)$.
We now use Turan’s graph theorem to prove that a large subset of $S$ is pairwise coprime. Construct a graph in which the vertices are the elements of $S$ and the edges connect vertices corresponding to integers which share a prime factor. Adding together the total number of edges produced by each prime, we find that the total number of edges in the graph is at most $$\begin{aligned}
\frac{1}{2}\sum_{\sqrt{y'}\leq p < \frac{y'}{2}} \left(\frac{c_2 y'}{p \log \frac{y'}{ p}}\times \left(\frac{ c_2 y'}{p \log \frac{y'}{ p}}-1\right) \right) &+\frac{1}{2}\sum_{\frac{y'}{2}\leq p <y} \left \lceil \frac{y}{p}\right \rceil\left(\left \lceil \frac{y}{p}\right \rceil-1 \right) \\
&\leq \sum_{\sqrt{y'}\leq p < \frac{y'}{2}} \frac{c_2^2y'^2}{p^2 \log^2 \frac{y'}{p}} + \sum_{\frac{y'}{2} \leq p < y } O(1).\end{aligned}$$ By partial summation this expression is at most $\frac{c' y'^{3/2}}{\log^3 y'}$ for some constant $c'$.
Turan’s graph theorem states that any graph with $v$ vertices and $e$ edges has an independent set of vertices of size at least $\frac{v^2}{v+2e}$. Applying this to our graph we find there must be an independent set of vertices (corresponding to a set of pairwise coprime integers) of size at least $$\frac{\frac{y'^2}{\log^4 y'}}{\frac{y'}{\log^2 y'} + \frac{2c' y'^{3/2}}{\log^3 y'}} \gg \frac{\sqrt{y'}}{\log y'}$$ and the result follows.
\[rem:pfactors\] Note that in the construction above, the integers in the set were free of prime factors less than $\sqrt{y'}$, and thus have at most $$\frac{\log x }{\log \sqrt{y'}} = \frac{2 \log x}{\log y -\log c_1} = \frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)$$ prime factors.
Primitive sets without large gaps
=================================
Using these results we are now able to give a proof of Theorem \[thm:primitive\] using the probabilistic method.
We construct a primitive set according to the following probabilistic construction and then show that, with high probability, the set we constructed does not have any gaps greater than the bound .
Fix $\epsilon>0$. For each prime number $p_i$ we choose a corresponding positive-integer-valued random variable $X_i$ with distribution $$P(X_i = n) = \frac{C_{\epsilon}}{n\log^{1+\tfrac{\epsilon}{8}} (n+2)},$$ with $C_{\epsilon}$ chosen to normalize the distribution. (Note that the sum of these terms converges since the power on the logarithm is greater than 1. The purpose of adding two inside the logarithm is just to make the probability positive when $n$ is either 1 or 2.) We then construct the set of integers $Q = \{n \geq 2 : p_i|n \rightarrow \Omega(n) = X_i\}$, consisting of only those integers $n$ for which the total number of prime factors dividing $n$ agrees with the random variable $X_i$ corresponding to every single one of its prime divisors, $p_i$.
It is readily seen that this construction always produces a primitive set, since if $a \in Q$, and $a|b$ with $b>a$, then $\Omega(a) < \Omega(b)$, but any prime dividing $a$ also divides $b$, and so $b$ cannot be in $Q$.
We now show that we expect every interval of size to contain an element of this set. Let $$y = \exp \left(\sqrt{2\log x \log \log x + (2+\epsilon)\log x \log_3 x }\right), \label{primitivey}$$ and consider the interval $[x-y,x]$. Using Theorem \[thm:coprimeset\], along with the observation of Remark \[rem:pfactors\], there exists a subset $S$ of the integers in this interval containing at least $\frac{c_3 \sqrt{y}}{\log y}$ integers from this interval which are pairwise coprime. Furthermore, the integers in $S$ have no more than $\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)$ prime factors. Suppose $n \in S$, then the probability that $n \in Q$ is $$\begin{aligned}
P&(n \in Q) = \prod_{p_i|n} P(X_i = \Omega(n)) = \prod_{p_i|n} \frac{C_{\epsilon}}{\Omega(n)\log^{1+\tfrac{\epsilon}{8}} (\Omega(n)+2)} \\
&\geq \left(\frac{C_{\epsilon}}{\left(\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right)\log^{1+\tfrac{\epsilon}{8}}\left(\frac{2 \log x}{\log y} +O(1)\right)}\right)^{\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)} \notag \\
&= \exp\left(\left(-\frac{2 \log x}{\log y} {+} O\left(\frac{\log x}{\log^{2} y}\right)\right){\times} \left(\log\left(\frac{\log x}{\log y}\right){+}\left(1{+}\tfrac{\epsilon}{8}\right)\log_3 x {+}O_\epsilon\left(1 \right)\right)\right)\notag \\
&= \exp\left(-\frac{2 \log x}{ \log y}\left(\log\left(\frac{\log x}{\log y}\right) +\left(1+\frac{\epsilon}{8}\right) \log_3 x+ O_\epsilon(1)\right)\right). \notag\end{aligned}$$
Since the elements of $S$ are pairwise coprime, the probability that any one element of $S$ is included in $Q$ is independent of the probability of any other element is included. Thus the probability that no integer from the interval $[x~-~y,x]$ is included in $Q$ can be bounded as follows. $$\begin{aligned}
&P([x-y,y]\cap Q = \varnothing) \leq P(S\cap Q = \varnothing) = \prod_{n \in S} \left(1-P(n \in Q)\right) \\
& \leq \prod_{n \in S}\left(1{-} \exp\left(-\frac{2 \log x}{ \log y}\left(\log\left(\frac{\log x}{\log y}\right) +\left(1+\tfrac{\epsilon}{8}\right) \log_3 x+ O_\epsilon(1)\right)\right)\right) \\
&\leq \left(1- \exp\left(-\frac{2 \log x}{ \log y}\left(\log\left(\frac{\log x}{\log y}\right) +\left(1{+}\tfrac{\epsilon}{8}\right) \log_3 x+ O_\epsilon(1)\right)\right)\right)^{\frac{c_3 \sqrt{y}}{\log y}} \\
& \leq \exp\left( - \frac{c_3 \sqrt{y}}{\log y} {\times} \exp\left(-\frac{2 \log x}{ \log y}\left(\log\left(\frac{\log x}{\log y}\right) {+}\left(1{+}\tfrac{\epsilon}{8}\right) \log_3 x{+} O_\epsilon(1)\right)\right)\right) \\
& = \exp\left(\hspace{-.5mm} {-}\exp \left(\tfrac{1}{2}\log y {-}\log \log y {-} \frac{2\log x}{\log y}\left( \log \left(\frac{\log x}{\log y}\right)\hspace{-.5mm}{+}\left(1{+}\tfrac{\epsilon}{8}\right) \log_3 x {+} O_\epsilon\hspace{-.5mm}(1)\hspace{-.2mm}\right)\hspace{-.2mm}\right)\hspace{-.2mm}\right).\end{aligned}$$
Now, inserting our choice for the length $y$ of the interval, the innermost exponent above becomes $$\begin{aligned}
\tfrac{1}{2}&\sqrt{2\log x (\log \log x {+} \left(1{+}\tfrac{\epsilon}{2}\right) \log_3 x)} {-} \frac{2\log x\left(\log \left(\frac{\sqrt{\log x}}{\sqrt{\log \log x }}\right) \hspace{-0.5mm}{+}\left(1{+}\frac{\epsilon}{8}\right) \log_3 x{+}O_\epsilon(1)\right)}{\sqrt{2\log x (\log \log x + \left(1+\frac{\epsilon}{2}\right) \log_3 x)}} \\
&=\sqrt{\tfrac{1}{2}\log x}\left(\sqrt{\log \log x {+} \left(1{+}\tfrac{\epsilon}{2}\right) \log_3 x}- \frac{\log \log x {+} \left(1{+}\tfrac{\epsilon}{2}{-}\frac{\epsilon}{4}\right)\log_3 x+O_\epsilon(1)}{\sqrt{\log \log x + \left(1{+}\frac{\epsilon}{2}\right) \log_3 x}}\right) \\
&= \frac{\epsilon}{8}\frac{\sqrt{2\log x}}{\sqrt{\log \log x}}\left(\log_3 x+O_\epsilon(1)\right).\end{aligned}$$
Therefore the probability that none of the integers from the interval $[x-y,x]$ are included in $Q$, which is less than the probability that no integer in $S$ is included in $Q$ since $S \subset [x-y,x]$, is at most $\exp\left(-\exp\left(\frac{\epsilon}{8}\frac{\sqrt{2\log x}}{\sqrt{\log \log x}}\left(\log_3 x+O_\epsilon(1)\right)\right)\right)$. Using linearity of expectation, and by starting the sequence at a sufficiently high initial value $N$, we can ensure that the expected number of intervals of the form $[x-y,x]$ which do not contain an integer in $Q$ is at most $$\begin{aligned}
\sum_{x>N} P(&[x {-} y,x] \cap Q = \varnothing) \leq \sum_{x>N} \exp\left(-\exp\left(\frac{\epsilon\sqrt{2\log x}}{8\sqrt{\log \log x}}\left(\log_3 x{+}O_\epsilon(1)\right)\right)\right) <1\end{aligned}$$ since this series converges.
Because the expected number of intervals that do not contain an integer in $Q$ is less than 1, there must exist a sequence $Q$ which intersects every such interval, and thus satisfies the properties of the theorem.
Geometric-Progression-Free sets without large gaps {#sec:gpf}
==================================================
A very similar construction can be used to prove Theorem \[thm:gpf\], producing a set free of 3-term geometric progressions with gaps smaller than those obtained by He.
Following the method of proof of Theorem \[thm:primitive\], we construct a set similar to the squarefree numbers, in the sense that each prime number is only allowed to appear (if it appears at all) to one fixed power in any element of the set. As before, we construct this set probabilistically and then bound the probability that it omits any interval of the size given in .
For each prime $p_i$ choose a positive-integer-valued random variable $X_i$ with distribution $$P(X_i = n) = \frac{1}{2^n}.$$ Now construct the set of integers $T = \{n \geq 2 : p_i|n \rightarrow p_i^{X_i}||n\}$ consisting of those integers $n$ where the exponent on each of its prime divisors $p_i$ is equal to the random variable $X_i$. (If $p_i$ divides $n$ then $p_i^{X_i}$ is the largest power of $p_i$ that divides $n$.)
This set $T$ is free of 3-term geometric progressions of integers for essentially the same reason that the squarefree integers avoid such progressions. If $\{a,ar,ar^2\}$ is any geometric progression with $r\in \mathbb{Q}$, $r>1$ and $p$ divides the numerator of $r$ but not the denominator, then $p$ appears to different, positive, powers in $ar$ and $ar^2$, and hence both cannot be in $T$.
We now show that we expect every interval of size to contain an element of this set. Let $$y = \exp \left(2\sqrt{\log 2\log x +\tfrac{3+\epsilon}{2}\sqrt{\log 2 \log x}\log \log x }\right) \label{eq:gpfy}$$ and consider the interval $[x-y,x]$. We again use Theorem \[thm:coprimeset\] to obtain a pairwise coprime subset $S$ of this interval of size at least $\frac{c_3 \sqrt{y}}{\log y}$ consisting of integers having at most $\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)$ prime factors. The probability an integer $n$ from this set is contained in $T$ is $$\begin{aligned}
P(n \in T) = \prod_{p_i^\alpha||n} P(X_i = \alpha) &= \prod_{p_i^\alpha||n} \frac{1}{2^\alpha} = \left(\frac{1}{2}\right)^{\Omega(n)} \geq \left(\frac{1}{2}\right)^{\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)} \\
&= \exp\left(-\left(\frac{2 \log 2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right) \right).\end{aligned}$$
Exploiting the fact that elements of $S$ are pairwise coprime, the probability that none of the elements of $S$ are included in $T$ is $$\begin{aligned}
\prod_{n \in S} \left(1-P(n \in T)\right) & \leq \prod_{n \in S}\left(1-\exp\left(-\left(\frac{2 \log 2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right) \right)\right) \\
&\leq \left(1-\exp\left(-\left(\frac{2 \log 2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right) \right)\right)^{\frac{c_3 \sqrt{y}}{\log y}} \\
& \leq \exp\left( - \frac{c_3 \sqrt{y}}{\log y} \times \exp\left(-\left(\frac{2 \log 2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right) \right)\right) \\
& = \exp\left( -\exp \left(\tfrac{1}{2}\log y - \frac{2\log 2 \log x}{\log y} - \log \log y + O(1) \right)\right) .\end{aligned}$$
Inserting here for $y$ the innermost exponent above becomes $$\begin{aligned}
&\sqrt{\log 2\log x +\tfrac{3+\epsilon}2\sqrt{\log 2 \log x}\log \log x } - \frac{ \log 2 \log x }{\sqrt{ \log 2 \log x +\frac{3+\epsilon}{2}\sqrt{\log 2 \log x}\log \log x}} \\
&\hspace{3cm} - \frac{1}{2} \log \log x +O(1)\\
& =\sqrt{\log 2 \log x}\left(\sqrt{1 {+}\frac{(3{+}\epsilon)\log \log x}{2\sqrt{\log 2\log x}}} - \frac{1}{\sqrt{1 {+}\frac{(3+\epsilon)\log \log x}{2\sqrt{\log 2\log x}}}}\right) {-} \frac{1}{2} \log \log x +O(1)\\
&=\sqrt{\log 2 \log x}\left(\frac{(3+\epsilon)\log \log x}{2\sqrt{\log 2\log x}}+ O\left(\frac{(\log \log x)^2}{\log x}\right)\right) - \frac{1}{2} \log \log x +O(1)\\
&=\left(1+\frac{\epsilon}{2}\right) \log \log x +O\left(1\right).\end{aligned}$$ The fourth line above was obtained using the Taylor expansion $$\sqrt{1+x}-\frac{1}{\sqrt{1+x}} = x +O(x^2)$$ around $x=0$.
Therefore, the probability that no such integer is included in $T$ is at most $\exp\left(-\exp\left(\left(1+\frac{\epsilon}{2}\right) \log \log x +O\left(1\right)\right)\right)$. As before, by linearity of expectation, we may choose a sufficiently high initial value $N$ so that the expected number of intervals of the form $[x-y,x]$ which do not contain an integer in $T$ is at most $$\sum_{x>N} P([x - y,x] \cap T = \varnothing) \leq \sum_{x>N} \exp\left(-\exp\left(\left(1+\frac{\epsilon}{2}\right) \log \log x +O\left(1\right)\right)\right) <1$$ since this series is convergent. Thus there exists a geometric-progression-free sequence $T$ satisfying the properties of the theorem.
Final Remarks
=============
While we were able to show that there exist primitive sets in which the gap between consecutive terms was much smaller than what is known to be true, even conditionally for the primes, the method developed doesn’t seem to generalize to sequences of pairwise coprime integers.
Can one prove that there exists a sequence $v_1<v_2<\cdots$ of pairwise coprime integers in which the difference between consecutive terms $v_n{-}v_{n-1}$ is smaller than the best known upper bound for the gaps between primes?
aknowledgements {#aknowledgements .unnumbered}
===============
The author is grateful to Angel Kumchev, Greg Martin and Carl Pomerance for helpful discussions during the development of this paper, and to the anonymous referee for useful feedback.
| ArXiv |
---
abstract: 'We study an insulator-metal transition in a ternary chalcogenide glass (GeSe$_3$)$_{1-x}$Ag$_x$ for $x$=0.15 and 0.25. The conducting phase of the glass is obtained by using “Gap Sculpting" (Prasai et al, Sci. Rep. 5:15522 (2015)) and it is observed that the metallic and insulating phases have nearly identical DFT energies but have a conductivity contrast of $\sim 10^8$. The transition from insulator to metal involves growth of an Ag-rich phase accompanied by a depletion of tetrahedrally bonded 2 in the host network. The relative fraction of the amorphous Ag$_2$Se phase and GeSe$_2$ phase is shown to be a critical determinant of DC conductivity.'
author:
- Kiran Prasai
- Gang Chen
- David Drabold
title: 'Amorphous to amorphous insulator-metal transition in GeSe$_3$:Ag glasses'
---
Metal-Insulator transitions (MIT) and their associated science are among the cornerstones of condensed matter physics [@mott2012]. In this Letter, we describe the atomistics of a technically important but poorly understood MIT in GeSe:Ag glasses, a prime workhorse of conducting bridge memory (CBRAM) devices [@patent1; @valov2011]. By [*design*]{}, we construct a stable conducting model from a slightly favored insulating phase. Predictions are made for structural, electronic and transport properties. We demonstrate the utility of our “Gap sculpting" method [@prasai2015] as a tool of Materials Design.
We report metallic phases of amorphous (GeSe$_3$)$_{1-x}$Ag$_{x}$ at $x=0.15$ and $0.25$. These are canonical examples of Ag-doped chalcogenide glasses, which are studied in relation to their photo-response and diverse opto-electronic applications [@kolobov2006; @inbook2]. Ag is remarkably mobile making the material a solid electrolyte and is known to act as “network-modifier" in these glasses and alter the connectivity of network. Experiments have shown Se rich ternaries ((Ge$_y$Se$_{1-y}$)$_{1-x}$Ag$_x$ with y $< 1/3$) to be phase-separated into Ag-rich Ag$_2$Se phase and residual Ge$_t$Se$_{1-t}$ phase [@mitkova1999].
Using first-principles calculations, we show that stable amorphous phases with at least $\sim 10^8$ times higher electronic conductivity exist with only small ($\approx 0.04$ eV/atom) difference in total energy. These conducting states present the same basic structural order in the glass, but have a higher relative fraction of an [*a-*]{}Ag$_2$Se phase compared to the insulating states. It is known that amorphous materials are characterized by large numbers of degenerate conformations that are mutually accessible to each other at small energy cost, but those usually have identical macroscopic properties. The remarkable utility of these materials accrues from states with distinct properties, nevertheless readily accessible to each other.
We discover the conducting phase of GeSe$_3$Ag glass by [*designing*]{} atomistic models with a large density of states (DOS) near the Fermi energy [@prasai2015]. This is achieved by utilizing Hellmann-Feynman forces from the band edge states. These forces are used to bias the true forces in [*ab initio*]{} molecular dynamics (AIMD) simulations to form structures with a large DOS at the Fermi level. The biased force on atom $\alpha$, $F^{bias}_{\alpha}$, is obtained by suitably summing Hellmann Feynman forces for the band edge states (second term in Eq. \[eq\_a\]) with the total force from AIMD calculations, $F^{AIMD}_{\alpha}$. $$\label{eq_a}
{F}^{bias}_{\alpha} = {F}^{AIMD}_{\alpha}+\sum \limits_{i} \gamma_{i} \langle \psi_{i}| \frac{\partial H}{\partial R_{\alpha}}|\psi_{i} \rangle$$ Here, $\gamma$’s set the sign and magnitude of the HF forces from individual states [*i*]{}. To maximize the density of states near $\epsilon_F$, gap states closer to the valence edge will have $\gamma > 0$, whereas the states in the conduction edge will have $\gamma < 0$. The magnitude of $\gamma$ determines the size of biasing force (with $\gamma=0$ representing true AIMD forces). We have employed biased forces as an electronic constraint to model semiconductors and insulators in our recently published work [@prasai2016] where the biasing is done in just the opposite sense: to force to states out of the band gap region.
We start with conventional 240 atom models of (GeSe$_3$)$_{1-x}$Ag$_x$, $x$=0.15 and 0.25, at their experimental densities 5.03 and 5.31 gm/cm$^3$ [@piarristeguy2000] respectively. These were prepared using melt-quench MD simulations, followed by conjugate-gradient relaxation to a local energy minimum. The MD simulations are performed using the Vienna [*Ab initio*]{} Simulation Package (VASP) [@kresse1; @*kresse2]. Plane waves of up to 350 eV are used as basis and DFT exchange correlation functionals of Perdew-Burke-Ernzerhof [@perdew1996] were used. Brillouin zone (BZ) is represented by $\Gamma$-point for bulk of the calculations. For static calculations, BZ is sampled over 4 k-points. These models fit the experimental structure factor reasonably well (Figure \[fig1\]).
![The structure factor of (GeSe$_3$)$_{1-x}$Ag$_x$ models (solid red line) compared with experiment (black squares)[@piarristeguy2000][]{data-label="fig1"}](sq.eps){width="0.8\linewidth"}
We obtain conducting conformations by annealing the starting configurations using biased forces at 700 K for 18 ps. The electronic states in the energy range \[$\epsilon_{F}$–0.4 eV, $\epsilon_{F}$+0.4 eV\] are included in the computation of bias force and $\gamma = 3.0$ is used. The bias potential ($\Phi_{b}(R_{1},..,R_{3N})= \sum -\gamma_{i} \langle \psi_{i}|H(R_{1},..,R_{3N})|\psi_{i} \rangle$) shepherds the electronic states in the band edges into the band-gap region. Since we want any proposed metallic conformation to be a true minimum of the unbiased DFT energy functional, we relax instantaneous snapshots of biased dynamics (taken at the interval of 0.2 ps, leaving out the first 4 ps) to their nearest minima using conjugate gradient algorithm with true DFT-GGA forces. We study all relaxed snapshots by i) gauging the density and localization of states around Fermi energy and, ii) testing the stability of the configurations by annealing them at 300 K ([*n.b.*]{} glass transition temperatures ($T_g$) are 488 K and 496 K for compositions $x$=0.15 and 0.25 respectively [@arcondo2007]). At each composition, we selected five models that display a large density of extended states around Fermi energy and are stable against extended annealing at 300 K as the ‘metallized’ models. These metallized systems are, on average, 0.040$\pm$0.009 eV/atom above their insulating counterparts.
![The electronic density of states (DOS) of the insulating model (black curve) and the metallized model (red curve). Energy axis is shifted to have Fermi level at 0 eV (the broken vertical line)[]{data-label="fig2"}](f51_DOSall_Fermi_0.eps){width="\linewidth"}
![The (black curve) electronic density of states (DOS) and (orange drop lines) Inverse Participation Ratio (IPR) of the insulating model (a) and the metallized model (b). Energy axis for all datasets is shifted to have Fermi level at 0 eV (highlighted by the broken vertical line)[]{data-label="fig3"}](DOSnIPR25.eps){width="\linewidth"}
The metallized models, by construction, show a large density of states around Fermi energy (Fig. \[fig2\]) whereas the insulating models display small but well defined PBE gap of 0.41 eV and 0.54 eV for $x$=0.15 and 0.25 respectively. For disordered materials, a high DOS at $\epsilon_F$ [*alone*]{} may not produce conducting behaviour since these states can be localized (example: amorphous graphene, [@pablo]). We gauge the localization of these states by computing inverse participation ratio (IPR, [@ziman])(plotted for $x$=0.25 system in Figure \[fig3\]) and show that these states [*are*]{} indeed extended. We compute the electronic conductivity \[$\sigma(\omega)$\] using Kubo-Greenwood formula (KGF) in the following form: $$\label{eq_KGF}
\begin{aligned}
{\sigma}_{k}(\omega) = \frac{ 2 \pi e^{2} \hslash^{2}}{3 m^{2} \omega \Omega} \sum \limits_{j=1}^{N} \sum \limits_{i=1}^{N} \sum \limits_{\alpha=1}^{3}[F(\epsilon_{i},k)-F(\epsilon_{j},k)] \\
|\langle \psi_{j,k}|\bigtriangledown_{\alpha} |\psi_{i,k} \rangle|^{2} \delta(\epsilon_{j,k}-\epsilon_{i,k}-\hslash \omega)
\end{aligned}$$ It has been used with reasonable success to predict conductivity [@abtew2007; @*galli1990; @*allen1987]. Our calculations used 4 k-points to sample the Brillouin zone. To compensate for the sparseness in the DOS due to the size of the supercell, a Gaussian broadening () for the $\delta$-function is used. We note that the choice of between 0.01 eV and 0.1 eV does not significantly alter the computed values of DC conductivity \[$\sigma(\omega=0)$\] (Figure \[fig4\]). For the choice of =0.05 eV (which is small compared to the thermal fluctuation of Kohn-Sham states for disordered systems at room temperature. For a heuristic theory, see [@prasai2016eph]), the DC conductivity of metallic models are of the order of $10^{2}~\Omega^{-1}cm^{-1}$ at both concentrations. For the insulating model at $x$=0.15, this value is of order $10^{-6}~\Omega^{-1}cm^{-1}$ whereas for insulating model at $x$=0.25, this value is lower but can not be ascertained from our calculations. We find that the metallized models show, at least, $\sim 10^8$ times higher conductivity than the insulating models. The computed conductivity for metallic models are comparable to the DC conductivity values of liquid silicon ($\approx 10^4~\Omega^{-1}cm^{-1}$, [@glazov]).
![(a) Optical conductivity of insulating (black curve) and metallized (red curve) models for (GeSe$_3$)$_{0.75}$Ag$_{0.25}$ model computed using Kubo-Greenwood Formula (KGF). Brillouin zone sampling is done over 4 k-points. Average over 3 directions was taken to eliminate artificial anisotropy. (b) DC conductivity as a function of Gaussian approximant $\delta$E. black squares: insulating model at $x$=015, red triangles: metallic model at $x$=0.15, green diamonds: insulating model at $x$=0.25, blue circles: metallic model at $x$=0.25[]{data-label="fig4"}](KGF.eps){width="\linewidth"}
We track the atomic rearrangements associated with the metallization of network to identify the microscopic origin of metallicity. Recalling that these are inhomogenous glasses with phase separation into Ag-rich [*a-*]{}Ag$_2$Se phase and residual Ge-Se backbone, we note that the insulator-metal transition in these glasses can be viewed in terms of relative ratio of these two competing phases. In particular, we make the following three observations associated with the insulator-metal transition: i) Growth of Ag-Se phase, ii) Depletion of tetrahedral GeSe$_2$ phase, and iii) Growth of Ge-rich phase in host network. Below we briefly comment on these three observations, a more detailed account of structural rearrangements will be published later.
[*Growth of Ag-Se phase.*]{} We observe that the Ag-Se phase grows upon metallization. Se-Ag correlation ($r_{Ag-Se}=2.67$ Å) is found to increase from the insulating to metallic model (see Figure \[fig5\], also the increase in peak P2 in Figure \[fig6\]). For both Ag concentrations, Se-coordination around Ag is found to increase from insulating to metallic models. For $x$=0.15, Se-coordination around Ag increases from 3.47 to 3.72 (the later value is an average over 5 metallic models, see Figure \[fig5\]). For $x$=0.25, it increases from 3.23 to (on average) 3.53.
![The Ag-Se correlation in insulating (black) and metallized (red) models at two concentrations of silver (a) $x$=0.15 and (b) $x$=0.25. The histogram in inset shows the Se-coordination around Ag atoms (n$_{Ag}$(Se)) for insulating (black) and 5 metallic (red) confirmations at both values of x. The cutoff for computing coordination is 3.00 Å, highlighted by an arrow.[]{data-label="fig5"}](SeAgcorr_inset.eps){width="\linewidth"}
[*Depletion of tetrahedral GeSe$_2$.*]{} The network in the insulating phase is dominated by Se-rich tetrahedral 2, accompanied by a competing Ag-Se phase. The fraction of later phase is directly determined by Ag-concentration in the network. These two phases appear as two distinct peaks in total radial distribution function (RDF) (Figure \[fig6\]). Upon metallization, the growth of Ag-Se shifts the balance of stoichiometries in network and the host network becomes Se deficient. At composition $x$=0.25 (plotted in Fig. \[fig6\]), the network in metallic phase is dominated by the Ag-Se subnetwork (peak P2). The corresponding Ge-Se coordination number in metallic model is 3.22, slightly lower than 3.40 in insulating model. These values are 3.28 and 3.43 respectively for $x$=0.15. The correlation cutoff of 2.70 Å is taken to determine the coordinations.
[*Response of host network*]{}. The host network of Ge-Se consists of Se-rich tetrahedral GeSe$_2$ and non-tetrahedral Ge-rich phases including the ethane-like Ge$_2$Se$_3$ units. These subnetworks were reported in GeSe$_2$ by Boolchand and coworkers [@boolchand2000] and in ternary chalcogenide glasses by Mitkova and coworkers [@mitkova2006]. We find that these Ge-Se stoichiometries have different bondlength distributions: Se rich phases ($n_{Ge-Se} \geq 4$) have bondlengths smaller than 2.55 Å whereas Ge-rich phases ($n_{Ge-Se} < 4$) have bondlengths longer than 2.55 Å. In an insulating conformation, the former phase dominates and registers an RDF peak at $\approx$ 2.40 Å (Fig. \[fig6\]). For metallic conformations, fewer Se atoms are available to Ge. This increases the fraction of Ge-rich phases and the Ge-Se bondlength distribution shifts to longer distances. This is represented by a shift in Ge-Se pair correlation function in Figure \[fig6\] (inset) and appearence of peak P3 in total RDF. Due to increase in fraction of Ge$_2$(Se$_{1/2}$)$_6$, Ge-Ge correlation peak appears around 3.5 Å in metallic models. We note that it is such a Ge-Ge signal in Raman scattering and $^{119}$Sn M$\ddot{o}$ssbauer spectroscopy that led to experimental discovery of Ge-rich Ge$_2$(Se$_{1/2}$)$_6$ phase in stoichiometric bulk Ge$_x$Se$_{100-x}$ glasses [@boolchand2000].
![The total radial distribution function (g(r)) of the insulating and metallized models (black and red curves respectively) at $x$=0.25. Note the bifurcated first peak originates from Ge-Se correlation (P1 at 2.40 Å) and Ag-Se correlation (P2 at 2.67 Å). For metallized model, peak P3 arises due to depletion of tetrahedral 2 and formation of Ge-rich Ge-Se phases.[]{data-label="fig6"}](corr.eps){width=".8\linewidth"}
Now we comment further on the role of Ag-Se phase in metallicity. It is well known that the states around Fermi energy are mainly Se p-orbitals ([@prasai2011; @tafen2005], In GeSAg: [@akola2015]). The electronic structure of metallic model projected onto its constituent subnetworks (Ag-Se and Ge-Se) shows different electronic activity of Se-atoms in the two subnetworks. We find that [*individual*]{} Se atoms in Ag$_2$Se nework have twice as much projection around the Fermi energy than the Se atoms in Ge-Se network (Figure \[fig7\]). This suggests that a more concentrated Ag-Se network will enhance the conduction. Experimentally, growth of Ag-rich nanocrystals in GeS$_2$ matrix has been shown to enhance the [*electronic*]{} conductivity [@wang2012; @*waser2009]. The Se-atoms in Ag-Se phase are found in atomic state ($q_{Se} \sim 0$) where as those in Ge-Se network are ionic with negative charge ($q_{Se} \sim -1$ or $-2$) (See inset in Figure \[fig7\]).
![The density of states of metallic model projected onto Se-atoms in the two subnetworks: Ag-Se subnetwork (black curve) and Ge-Se subnetwork (red curve). Since these two subnetworks contain different number of Se atoms (23 and 59 for this plot), an average was taken to enable comparision. Bridging Se-atoms are not included in the calculation. Energy axis was shifted to have Fermi energy ($\epsilon_{F}$) at 0 eV. The inset shows Bader charges (q$_{Se}$) for the same two groups of Se atoms. Black filled circles represent Se in Ag-Se network, Red filled squares represent Se in Ge-Se network.[]{data-label="fig7"}](proj.eps){width="0.8\linewidth"}
Altogether, we have presented a direct simulation of conducting phase of CBRAM material GeSe$_3$Ag and it provides evidence of the conduction through interconnected regions of Ag$_2$Se phase in the glassy matrix [@wang2012; @*waser2009]. This work does not attempt to describe the conduction through Ag-nanowires which may be the mechanism of conduction in two terminal metal-electrolyte-metal devices [@inbook1]. It demonstrates the existence of metastable amorphous forms (“poly-amorphism") of the glass with drastically different optical response. The observation that the DFT energies of these states are only 0.04 eV/atom higher than insulating state suggests that these states might to accessible. Furthermore, we have shown that “Gap Sculpting" can be used to purposefully [*design*]{} metallic conformation.
We thank M. Mitkova and P. Boolchand for stimulating discussions. This work is supported by National Science Foundation under grant no. DMR 1506836, no. DMR 1507670 and no. DMR 1507166. We are thankful to Ohio Supercomputer center for computing resources.
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| ArXiv |
---
abstract: 'We have theoretically studied the photon statistical properties in a nonlinear whispering-gallery-mode microresonator coupled with two nanoparticles. By tuning the relative position of two nanoparticles, the photon statistical features of the system can be modified remarkably. Interestingly, a controllable switching between unconventional and conventional photon blockade can be realized by manipulating the angular positions of two nanoparticles. We also investigate the influence of the Kerr effect on the second order correlation function and find that there is an optimal choice for the relative position of two nanoparticles and the strength of Kerr effect that can generate strong antibunching. Furthermore, under the strong driving, two photon blockade can be achieved when the system is close to an exceptional point. Our work may provide an effective way to control photon statistical characteristics and have potential applications in quantum information science.'
author:
- 'Wen-An Li'
title: 'Tunable photon blockade in a whispering-gallery-mode microresonator coupled with two nanoparticles'
---
introduction
============
A single photon source, an indispensable device for generating photons at the single-photon level, plays a central role in diverse areas such as quantum cryptography [@1], quantum information processing [@2], single-photon transistor [@3], quantum computation [@4] and so on. The photon blockade is one of most attractive mechanisms for constructing a single photon source. In close analog to Coulomb blockade for electrons [@5; @6], the photon blockade is a striking quantum phenomenon, where the excitation of a photon blocks the transport of subsequent photons for the nonlinear cavity so that they are emitted one by one. As a consequence, the cavity can only host one photon at a time, acting as a “photon turnstile” [@7; @8]. In 2005, the photon blockade was first demonstrated experimentally with a single atom trapped in an optical cavity [@9]. Subsequently, the strong antibunching behaviors were predicted in various experimental setups including a quantum dot in a photonic crystal [@10; @11] and circuit cavity quantum electrodynamics systems [@12; @13; @14]. In these works, the observation of antibunching requires large nonlinearities with respect to the decay rate of the system, so it is known as “conventional photon blockade”.
Apart from the single photon blockade, the multi-photon blockade has also attracted much interest due to its potential applications in multiphoton quantum-nonlinear optics like an $n$-photon source ($n>1$) [@15]. To date, the multi-photon blockade has been studied in various configurations [@16; @17; @18; @19; @20]. For instance, the two- and three-photon blockade can be observed in a system consisting of a cavity with Kerr nonlinearity driven by a weak classical field [@16]. The prerequisite for realizing multi-photon blockade in this system is the presence of strong nonlinearities. Another method to realize three-photon blockade is based on the collective decay of two atoms trapped in a single-mode cavity with different coupling strengths [@17]. In this scheme, the two-photon and three-photon blockades strongly depend on the location of two atoms in the strong-coupling regime. Recently, the two-photon blockade was first observed in an atom-driven cavity quantum electrodynamics system [@21]. Although many progress on the study of multi-photon blockade has been made, the accomplishment of the multi-photon blockade is still challenging in experiments.
In 2010, Liew and Savona found a new mechanism for the photon blockade, where strong photon antibunching can be obtained even with nonlinearities much smaller than the decay rates of the cavity modes [@22]. This mechanism is referred to as the “unconventional photon blockade”. Its feature can be understood as destructive quantum interference between different excitation pathways from the ground state to the two-photon states. Since then, a sequence of theoretical proposals based on this mechanism were suggested in many different systems including, for example, a bimodal optical cavity with a quantum dot [@23; @24; @25; @26; @27; @28; @29], symmetric and antisymmetric modes in weakly nonlinear photonic molecules [@30], coupled optomechanical systems [@31; @32]. More recently, the unconventional photon blockade was experimentally observed in two coupled superconducting circuit resonators [@33] and in a quantum dot embedded in a bimodal micropillar cavity [@34].
In parallel, the physical systems described by non-Hermitian Hamiltonians have also attracted much interest [@rev1; @rev2; @rev3; @rev4; @rev5], because such Hamiltonians exhibit special degeneracies known as exceptional points, at which two or more eigenvalues and the corresponding eigenvectors coalesce. In 2001, the physical existence of the exceptional point was experimentally demonstrated in microwave cavities [@35]. Subsequently, a variety of unconventional effects have been observed in experiments, such as loss-induced coherence [@36; @37], unidirectional lasing [@38], wireless power transfer [@39], and exotic topological states [@40; @41]. In recent experiments [@42; @43; @44], by coupling two nanoscale scatters (i.e. nanoparticles) to a whispering-gallery-mode (WGM) micro-toroid cavity, the system can be steered in a precise and controlled manner to the exceptional point. The presence of two nanoparticles within the mode volume of the cavity leads to the asymmetric backscattering of counter-propagating optical waves, which can be adjusted by manipulating the relative position of two nanoparticles. In the vicinity of the exceptional points, some counterintuitive effects have been shown including loss-induced revival of lasing [@37], ultra-sensitive sensor [@42], chiral lasing [@43] and optomechanically induced transparency [@46].
Motivated by above works [@42; @43; @44; @46], one question that arises naturally is whether the asymmetric coupling of two counter-propagating optical waves affects the photon statistical properties of cavity modes. In the previous works [@25; @26; @27; @30], studies on the unconventional photon blockade are based on the *symmetric* coupling of the optical modes. According to the optimal conditions, the required Kerr nonlinearity decreases with increasing coupling strength of the optical modes. It means that strong photon antibunching with weak Kerr nonlinearity requires large optical coupling between optical modes, which is not easy to realize in the experiments. Here, we consider the *asymmetric* coupling of two optical modes in one resonator and study the new possibility of controlling the photon blockade by tuning the relative angular position of two nanoparticles along the circumference of the nonlinear microresonator. In fact, adjusting the relative position of two nanoparticles corresponds to the change in the relative phase of the coupling coefficients without increasing the amplitudes of coupling constants. We find that the relative phase of the coupling coefficients plays a crucial role in modifying the photon statistical properties of the system. By tuning the relative position of two nanoparticles, the photon statistical properties can be well controlled and the switching between unconventional and conventional photon blockade can be realized. We also investigate the influence of the Kerr nonlinearity strength on the photon statistics properties. Furthermore, in the vicinity of an exceptional point, two-photon blockade effect can be achieved under the strong driving. Our work, with weak nonlinearity but without requiring strong coupling between optical modes, can be realized within current experimental techniques.
The remainder of the paper is organized as follows. In Sec. II, the theoretical model and Hamiltonian are described for the nonlinear WGM microresonator coupled with two nanoparticles. In Sec. III, the output power spectra of the WGM microresonator system are presented. Subsequently, in Sec. IV, the photon statistical properties of present system is analytically and numerically discussed. Finally, a summary of the main results is given in Sec. V.
![Schematic diagram of the WGM microresonator with Kerr medium coupled with two nanoparticles which is coherently driven by a pump field at frequency $\omega_L$ through an optical tapered fiber waveguide. The WGM microresonator supports two counterpropagating modes (clockwise mode $\hat{a}_{\mathrm{C}}$ and anti-clockwise mode $\hat{a}_{\mathrm{A}}$), which can be asymmetrically coupled through backscattering by two nanoparticles. $\beta$ is the relative angle between two particles.[]{data-label="fig1"}](fig1){width="40.00000%"}
theoretical model
=================
As shown in Fig.\[fig1\], we consider a WGM microresonator with Kerr medium coupled to an optical fiber waveguide for in- and out-coupling of light. With its circular geometry, the WGM cavity supports clockwise and anti-clockwise travelling modes with degenerate eigenfrequencies $\omega_c$ and the same decay rate $\gamma=\gamma_{\mathrm{ex}}+\gamma_{\mathrm{in}}$. $\gamma_{\mathrm{ex}}$ is the external decay rate (the outgoing coupling coefficient) from the WGM microresonator into the tapered fiber and $\gamma_{\mathrm{in}}$ is the intrinsic decay rate. Two nanoparticles are placed in the evanescent field of the resonator, which can tune the coherent backscattering of clockwise and anti-clockwise travelling modes inside the resonator. In the presence of the optical loss, the system considered here is an open system and the Hamiltonian is non-Hermitian.
\[sec:level2\]Review of the two-mode approximation model
--------------------------------------------------------
In order to give a full description of this open system, we first briefly review the two-mode approximation model and the eigenmode evolution in a WGM microresonator with nanoscatter-induced broken spatial symmetry. The two-mode approximation model was first phenomenologically introduced for deformed microdisk cavities [@tma1; @tma2] and was later rigorously derived for the microdisk with two scatterers [@47]. The key idea is to model the dynamics in the slowly-varying envelope approximation in the time domain with a Schrödinger-like equation $id\Psi/dt=H\Psi$. Here, $\Psi$ is the two-component column vector $(\Psi_A, \Psi_C)^T$, where the superscript $T$ indicates the matrix transpose. The complex-valued entry $\Psi_A$ ($\Psi_C$) stands for all the field amplitudes of the anti-clockwise (clockwise) propagating waves. Since the microcavity is an open system, the corresponding effective Hamiltonian, $$H=
\left(
\begin{matrix}
\Omega & J_1 \\
J_2 & \Omega
\end{matrix}
\right)$$ is a $2\times2$ matrix, which is in general non-Hermitian. The real parts of the diagonal elements $\Omega$ are the frequencies and the imaginary parts are the decay rates of the resonant traveling waves. The complex-valued off-diagonal elements $J_1$ and $J_2$ are the backscattering coefficients, which describe the scattering from the clockwise (anti-clockwise) to the anti-clockwise (clockwise) travelling wave. In general, in the open system the backscattering is asymmetric, $|J_1|\neq |J_2|$, which is allowed because of the non-Hermiticity of the Hamiltonian. A short calculation shows that the complex eigenvalues of $H$ are $\Omega_{\pm}=\Omega\pm\sqrt{J_1J_2}$ and the complex (not normalized) right eigenvectors are $$\Psi_{\pm}=
\left(
\begin{matrix}
\sqrt{J_1} \\
\pm\sqrt{J_2}
\end{matrix}
\right).$$ Clearly, in the case of asymmetric backscattering one component of a given eigenvector is larger than the other component. Physically, it means that the eigenvectors show an imbalance of clockwise and anti-clockwise components if the backscattering is asymmetric. For the particular case of the WGM microresonator perturbed by two scatterers the matrix elements of $H$ are determined as follows [@42; @43; @47],
$$\begin{aligned}
&\Omega=\omega_c-i\frac{\gamma}{2}+\sum_{j=1}^2\epsilon_j, \\
&J_1=\sum_{j=1}^2\epsilon_j e^{-i2m\beta_j}, \\
&J_2=\sum_{j=1}^2\epsilon_j e^{i2m\beta_j},\end{aligned}$$
where $m$ is the azimuthal mode number, $\beta_j$ is the angular position of scatterer $j$ and $2\epsilon_j$ is the complex frequency splitting that is introduced by scatterer $j$ alone. $\epsilon_j$ can be calculated for the single-particle-microdisk system either fully numerically (using, e.g., the finite element method (FEM) [@fem], the boundary element method (BEM) [@bem]), or analytically using the Green’s function approach [@green]. In recent experiments, $\epsilon_j$ can be adjusted by tuning the distance between the resonator and the particles. Here, we take the position of one of the nanoparticles as the reference position. For example, take the orange particle (in Fig.1) as the first particle and set its angular position to be $\beta_1=0$, then the angular position of the second particle is $\beta_2=\beta$, where $\beta$ represents the relative angular position of the two scatters. Therefore, the asymmetric backscattering coefficients of counterpropagating waves, induced by the nanoparticles, can be reduced to $$J_{1,2}=\epsilon_1+\epsilon_2 e^{\mp i 2m\beta}.$$ It is noted that the relative angular $\beta$ is of great importance, since it can modify the photon statistical properties of the system (see discussions below).
Although the two-mode approximation model was given for the isolated microdisk cavity perturbed by two particles, it is still valid in the waveguide-cavity systems by assuming that there is no backscattering of light between the microcavity and the waveguides. It can be justified when the distances between cavity and waveguides are sufficiently large. Note that, the extension of the two-mode model to waveguide-cavity systems has been introduced and tested in recent experiments [@42; @43].
\[sec:level2\]The Hamiltonian of our model
------------------------------------------
Based on above discussions, we will give a theoretical description of our model. To make our scheme work, a driving laser of frequency $\omega_L$ is applied to the system via the evanescent coupling of the optical fiber and the resonator, the field amplitudes are given by $F=\sqrt{\gamma_{\mathrm{ex}} P_L/\hbar \omega_L}$, where $P_L$ is the pump power. In the frame rotating with the input field frequency $\omega_L$, the Hamiltonian of this system is described by $$\begin{aligned}
\label{eq1}
\nonumber
\hat{H}_{\mathrm{sys}}=&\Delta(\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}+
\hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}})+U (\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}\hat{a}_{\mathrm{C}}\\ \nonumber
&+\hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}}\hat{a}_{\mathrm{A}})
+J_1\hat{a}_{\mathrm{C}} \hat{a}_{\mathrm{A}}^\dag+J_2\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{A}} \\
&+iF(\hat{a}_{\mathrm{C}}^\dag-\hat{a}_{\mathrm{C}})\end{aligned}$$ where $\Delta=\Delta_c+\mathrm{Re}(\epsilon_1+\epsilon_2)$, and $\Delta_c=\omega_c-\omega_L$. The nonlinear Kerr coefficient is given by $U=\hbar \omega_c^2 c n_2/n_0^2V_{\mathrm{eff}}$, where $c$ is the speed of light in vacuum, $n_0$ and $n_2$ are the linear and nonlinear refractive index of the material and $V_\mathrm{eff}$ is the effective mode volume. $\hat{a}_{\mathrm{C}}$ ($\hat{a}_{\mathrm{A}}$) and $\hat{a}_{\mathrm{C}}^\dag$ ($\hat{a}_{\mathrm{A}}^\dag$) are the photon annihilation and creation operators of the clockwise modes (anti-clockwise modes), satisfying the bosonic commutation relations $[\hat{a}_{\mathrm{C}},\hat{a}_{\mathrm{C}}^\dag]=1$ and $[\hat{a}_{\mathrm{A}},\hat{a}_{\mathrm{A}}^\dag]=1$.
In above Hamiltonian (\[eq1\]), the first term denotes the energy of the WGM microresonator in the rotating frame. The second term represents the Kerr nonlinear interaction. The third and fourth terms are the coherent coupling of the clockwise mode with anti-clockwise mode. In general, $J_1\neq J_2$, which can be tuned by the relative angular position of two nanoparticles and the distance between nanoparticles and the WGM microresonator. Due to this asymmetric coupling between two counterpropagating modes, some interesting, controllable photon statistical properties will be shown in our system. The last term describes the interaction between the cavity field and the input field.
output power spectra of the WGM microresonator
==============================================
Before discussing the photon statistical properties of our system, we first study the system output power spectra. As mentioned above, when two nanoparticles are placed along the circumference of the resonator, the system exhibits fully asymmetric internal backscattering. The position of each particle can be controlled by a nanopositioner, which tunes the relative position and effective size of the nanotip in the WGM fields [@42; @43]. By carefully tuning the relative positions of two particles, the system can be steered to an exceptional point. For the present system, the nanoparticles induced frequency splitting of the optical modes can be derived as $\Delta\omega=\pm\sqrt{J_1J_2}$, thus the corresponding critical value of $\beta$ can be obtained as [@46] $$\beta_c=\frac{l\pi}{2m}\mp\frac{\mathrm{arg}(\epsilon_1)-\mathrm{arg}(\epsilon_2)}{2m} \quad (l=\pm1,\pm3,...),$$ where $\mp$ corresponds to $J_1=0$ or $J_2=0$. Here, $\epsilon_j$ ($j=1,2$) are complex numbers, and we assume $|\epsilon_1|=|\epsilon_2|$. $\mathrm{arg}(\epsilon_j)$ denotes the argument of complex number $\epsilon_j$. In the vicinity of the exceptional points, some unconventional effects may occur. Thus, it is of great interest to investigate the output power spectra of such coupled system when the relative position of two particles varies.
According to the Hamiltonian (\[eq1\]) above, the dynamics of the coupled system can be described by the quantum Langevin equations
\[eq3\] $$\begin{aligned}
\frac{d}{dt}\hat{a}_{\mathrm{C}}=&\left(-\frac{\gamma_{\mathrm{opt}}}{2}-i\Delta-2iU\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}\right)\hat{a}_{\mathrm{C}}-iJ_2\hat{a}_{\mathrm{A}}+F+\hat{a}_{\mathrm{in}}^{\mathrm{C}},\\
\frac{d}{dt}\hat{a}_{\mathrm{A}}=&\left(-\frac{\gamma_{\mathrm{opt}}}{2}-i\Delta-2iU\hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}}\right)\hat{a}_{\mathrm{A}}-iJ_1\hat{a}_{\mathrm{C}}+\hat{a}_{\mathrm{in}}^{\mathrm{A}},\end{aligned}$$
where $\hat{a}_{\mathrm{in}}^{\mathrm{C}}$ and $\hat{a}_{\mathrm{in}}^{\mathrm{A}}$ are the input vacuum noises of the cavity modes, respectively. $\gamma_{\mathrm{opt}}=\gamma_{\mathrm{in}}-\mathrm{Im}(\epsilon_1+\epsilon_2)$ is the total optical loss. Under the mean-field approximation, we assume that the mean values of these noise operators are zero, i.e., $\langle \hat{a}_{\mathrm{in}}^{\mathrm{C}}\rangle=\langle \hat{a}_{\mathrm{in}}^{\mathrm{A}}\rangle=0$. Here, we are interested in the influence of relative position of two nanoparticles on the system output power spectra. Under weak Kerr nonlinearity $U\ll \gamma_{\mathrm{in}}$, we can easily omit the Kerr interaction terms in above Eq.(\[eq3\]). Furthermore, we assume that all of the time derivatives in the quantum Langevin equations are set to be zero. Thus, it is easy to obtain steady-state values of the dynamical variables as
$$\begin{aligned}
\langle \hat{a}_{\mathrm{C}}\rangle=\frac{F(\gamma_{\mathrm{opt}}/2+i\Delta)}{(\gamma_{\mathrm{opt}}/2+i\Delta)^2+J_1J_2},\\
\langle \hat{a}_{\mathrm{A}}\rangle=\frac{-iFJ_1}{(\gamma_{\mathrm{opt}}/2+i\Delta)^2+J_1J_2},\end{aligned}$$
Note that $J_1J_2=\epsilon_1^2+\epsilon_2^2+2\epsilon_1\epsilon_2\cos(2m\beta)$. We can see that the $\beta$-dependent optical coupling rate indeed affects the intracavity optical intensity. By using the standard input-output relations, i.e., $\langle \hat{a}_{\mathrm{out}}\rangle=\langle \hat{a}_{\mathrm{in}}\rangle-\sqrt{\gamma_{\mathrm{ex}}}\langle \hat{a}_{\mathrm{C}}\rangle$, we obtain the the normalized power forward transmission spectra $$T=\left|\frac{\langle \hat{a}_{\mathrm{out}}\rangle}{\langle \hat{a}_{\mathrm{in}}\rangle}\right|^2=\left|1-\frac{\gamma_{\mathrm{ex}}}{F}\langle \hat{a}_{\mathrm{C}}\rangle\right|^2.$$ When the Kerr terms are included, the exact expression of $\langle \hat{a}_{\mathrm{C}}\rangle$ ($\langle \hat{a}_{\mathrm{A}}\rangle$) can not be obtained generally. Therefore, we numerically calculate the solutions to equations (\[eq3\]) under the mean-field approximation and plot the transmission rate versus detuning under different relative angular position of two particles in Fig. \[fig2\](a). Here, we have selected the experimentally accessible values $\gamma_{\mathrm{ex}}/\gamma_{\mathrm{in}}=1$, $\epsilon_1/\gamma_{\mathrm{in}}=1.5-0.1i$, $\epsilon_2/\gamma_{\mathrm{in}}=1.4999-0.101489i$, $U/\gamma_{\mathrm{in}}=0.059$ and $m=4$ [@42; @43; @46]. With these parameters, the exceptional point corresponds to the angular position at $\beta_c\approx0.4$. Fig. \[fig2\](a) shows the $\beta$-dependent transmission rate with two nanoparticles, featuring a asymmetric spectrum around the resonance due to the asymmetric backscattering between the clockwise- and anticlockwise-travelling waves. More interestingly, when $\beta$ is set to be $\pi/8$ (in the vicinity of the exceptional points), the transmission spectra demonstrates only one local minimum at the resonance. For $\beta=\pi/16$, strong absorption is shown around $\Delta/\gamma_{\mathrm{in}}=2$. However, by tuning the system close to the exceptional point (namely, $\beta=\pi/8$), a transparency window emerges. Hence, an optical switching (at $\Delta/\gamma_{\mathrm{in}}=2$) can be achieved by adjusting the relative angular position of two particles. For completeness, we plot the transmission spectra versus driving field detuning and relative angle between two nanoparticles in Fig. \[fig2\](b).
photon statistical properties of the WGM microresonator with two nanoparticles
==============================================================================
\[sec:level2\]General formalism
-------------------------------
To correctly account for the driven-dissipative character of the system, we introduce the quantum master equation for the system density matrix, $$\begin{aligned}
\label{eq7}
\frac{d\hat{\rho}}{dt}=-i[\hat{H}_{\mathrm{sys}}, \hat{\rho}]+\gamma_1\mathcal{L}[\hat{a}_{\mathrm{C}}]\hat{\rho}+\gamma_2\mathcal{L}[\hat{a}_{\mathrm{A}}]\hat{\rho}\end{aligned}$$ where $\mathcal{L}[\hat{x}]\hat{\rho}=\hat{x}\hat{\rho} \hat{x}^\dag-\frac{1}{2}\hat{x}^\dag \hat{x}\hat{\rho}-\frac{1}{2}\hat{\rho} \hat{x}^\dag \hat{x}$ is the Lindblad superoperator term for the collapse operator $\hat{x}$ acting on the density matrix $\hat{\rho}$ to account for losses to the environment. $\gamma_1$ and $\gamma_2$ denote the damping constant of clockwise mode and anti-clockwise mode, respectively. Here, the decay rates of the resonator modes are assumed to be equal, i.e., $\gamma_1=\gamma_2=\gamma_{\mathrm{opt}}$. The steady-state solution $\rho_{ss}$ of the density matrix $\hat{\rho}$ can be obtained by setting $d\hat{\rho}/dt=0$ in Eq. (\[eq7\]). To observe the photon blockade, we focus on the statistic properties of clockwise mode photons, which are described by the zero-delay-time second order correlation function of the steady state, defined by $$\label{eq9}
g_{\mathrm{C}}^{(2)}(0)=\frac{\langle \hat{a}_{\mathrm{C}}^\dag\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}} \hat{a}_{\mathrm{C}}\rangle}{\langle \hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}\rangle^2}=\frac{\mathrm{Tr}\left(\rho_{ss}\hat{a}^\dag_{\mathrm{C}}\hat{a}^\dag_{\mathrm{C}}\hat{a}_{\mathrm{C}}\hat{a}_{\mathrm{C}}\right)}{[\mathrm{Tr}(\rho_{ss}\hat{a}^\dag_{\mathrm{C}}\hat{a}_{\mathrm{C}})]^2}.$$ This physical quantity emphasizes the joint probability of detecting two photons at the same time. The value of $g_{\mathrm{C}}^{(2)}(0)<1$ ($g_{\mathrm{C}}^{(2)}(0)>1$) corresponds to sub-Poisson (super-Poisson) statistics of the cavity field, which is a nonclassical (classical) effect. This effect of the sub-Poisson photon statistics is often referred to as photon antibunching.
\[sec:level2\]Weak driving limit
--------------------------------
If the driving field coupling $F$ is very weak, due to photon blockade, only lower energy levels of the cavity field are occupied (the total excitation number of the system doesn’t exceed 2). In this case, the truncated state of the system can be expanded as $$\begin{aligned}
\label{eq10}
\nonumber
|\psi\rangle=&C_{00}|00\rangle+C_{10}|10\rangle+C_{01}|01\rangle \\
&+C_{11}|11\rangle+C_{20}|20\rangle+C_{02}|02\rangle.\end{aligned}$$ Here $|mn\rangle$ represents the fock state basis of the system with the number $m$ denoting the photon number in clockwise cavity mode, $n$ denoting the photon number in anti-clockwise cavity mode. $C_{mn}$ stands for the probability amplitude and $|C_{mn}|^2$ denotes occupying probability in the state $|mn\rangle$. Using Eq.(\[eq9\]) and Eq.(\[eq10\]), the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ can be approximately given as $$\label{eq11}
g_{\mathrm{C}}^{(2)}(0) \simeq \frac{2|C_{20}|^2}{|C_{10}|^4}.$$ The result of Eq.(\[eq11\]) can be used to approximately describe the photon statistical properties in the weak driving limit. To obtain the coefficients $C_{mn}$ in Eq.(\[eq10\]), we can substitute the state $|\psi\rangle$ into the Schroödinger’s equation $i\frac{\partial}{\partial t}|\psi\rangle=\widetilde{H}|\psi\rangle$, where $\widetilde{H}=\hat{H}_{\mathrm{sys}}-i\frac{\gamma_{\mathrm{opt}}}{2}(\hat{a}_{\mathrm{C}}^\dag \hat{a}_{\mathrm{C}}+\hat{a}_{\mathrm{A}}^\dag \hat{a}_{\mathrm{A}})$. Then, we get a set of equations for the coefficients
$$\begin{aligned}
&i\frac{\partial}{\partial t}C_{00}\simeq 0,\\
&i\frac{\partial}{\partial t}C_{10}=iF C_{00}+\bar{\Delta}C_{10}+J_2C_{01}-i\sqrt{2}F C_{20},\\
&i\frac{\partial}{\partial t}C_{01}=J_1C_{10}+\bar{\Delta}C_{01}-iF C_{11},\\
&i\frac{\partial}{\partial t}C_{11}=iFC_{01}+2\bar{\Delta}C_{11}+\sqrt{2}J_1C_{20}+\sqrt{2}J_2C_{02},\\
&i\frac{\partial}{\partial t}C_{20}=i\sqrt{2}FC_{10}+\sqrt{2}J_2C_{11}+2(\bar{\Delta}+U_1)C_{20},\\
&i\frac{\partial}{\partial t}C_{02}=\sqrt{2}J_1C_{11}+2(\bar{\Delta}+U_2)C_{02},\end{aligned}$$
where $\bar{\Delta}=\Delta-i\frac{\gamma_{\mathrm{opt}}}{2}$.
Under the weak driving condition $F\ll\gamma_{\mathrm{in}}$, we have $|C_{00}|\gg|C_{10}|, |C_{01}|\gg|C_{20}|, |C_{11}|, |C_{02}|$, thus $C_{20}$ and $C_{11}$ can be removed in the Eq.(14b) and Eq.(14c). The vacuum state $C_{00}$ approximately has unity occupancy. Then, the steady-state solution can be found by solving the coupled equations for the coefficients. For simplicity of presentation, only $C_{10}$ and $C_{20}$ are given as below: $$\begin{aligned}
\label{eq13}
\nonumber
&C_{10}=\frac{-iF\bar{\Delta}}{\bar{\Delta}^2-J_1J_2}, \\
&C_{20}=\frac{1}{2\sqrt{2}}\frac{F^2[J_1J_2U+2\bar{\Delta}^2(\bar{\Delta}+U)]}{(\bar{\Delta}^2-J_1J_2)[J_1J_2(\bar{\Delta}+U)-\bar{\Delta}(\bar{\Delta}+U)^2]}.\end{aligned}$$ With Eq.(\[eq11\]) and Eq.(\[eq13\]), we can approximately obtain the analytical expression of the second order correlation function and the optimal condition for the photon blockade. However, the exact expressions for the condition $g_{\mathrm{C}}^{(2)}(0)\approx 0$ are too cumbersome to be presented here. Interestingly, from Eq.(\[eq13\]), it is obvious that the second order correlation function is closely related to the relative angular position of two particles. In particular, in the vicinity of exceptional points, i.e., $\beta=\beta_c\approx0.4$, $g_{\mathrm{C}}^{(2)}(0)\simeq |\bar{\Delta}|^2/|\bar{\Delta}+U|^2$. It means that the system shows stronger photon antibunching effect as the Kerr nonlinearity increases. When $\beta\neq\beta_c$, the case becomes different. The in-depth discussions and results of numerical calculation by the master equation approach for different parameter conditions are presented in the following subsections.
![The second order correlation function $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]$ as a function of the detuning $\Delta/\gamma_{\mathrm{in}}$ under various relative angular positions $\beta$ of two nanoparticles. The parameters have been selected the same as in Fig.\[fig2\].[]{data-label="fig3"}](fig3){width="45.00000%"}
\[sec:level2\]Single photon blockade
------------------------------------
In this subsection, we study the photon statistical properties of the nonlinear WGM microresonator system with two nanoparticles by numerically solving the master equation (\[eq7\]). Figure 3 displays the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ of the cavity mode $\hat{a}_{\mathrm{C}}$ in a logarithmic scale as a function of the detuning $\Delta/\gamma_{\mathrm{in}}$ under various relative angular positions $\beta$ of two nanoparticles. Here, we consider the weak Kerr nonlinearity $U/\gamma_{\mathrm{in}}=0.059$. We can see that the profile of the second-order correlation function in a logarithmic scale $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]$ varying with the detuning $\Delta/\gamma_{\mathrm{in}}$ exhibits a peak-dip structure. With the increasing of the detuning $\Delta$, the value of $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]$ first arrives at the maximum and then at the minimum. For $\beta=\pi/4$, the maximum value at the peak is about $0.5$ while the minimum value at the dip is about $-3.0$. Interestingly, the photon statistics can be changed dramatically by tuning the value of relative angular position $\beta$. We find that the value of the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ is about $0.92$ when $\Delta/\gamma_{\mathrm{in}}=0.3$ and $\beta=\pi/8$. However, by tuning the parameter $\beta$ to $\pi/4$ and keeping $\Delta/\gamma_{\mathrm{in}}=0.3$, the value of $g_{\mathrm{C}}^{(2)}(0)$ rapidly decreases to $0.002$, which indicates the strong antibunching effect.
![Energy-level and transition path diagram of the WGM system with two nanoparticles. The quantum interference between different paths leads to strong antibunching effect.[]{data-label="fig4"}](fig4){width="46.00000%"}
The physical grounds behind the photon antibunching under the weak Kerr effect can be explained by the effect of quantum interference between different pathways, as shown in Fig.\[fig4\]. There are two paths for the system to reach the two photon state of clockwise cavity mode :(i) the direct path, i.e., $|00\rangle\stackrel{F}{\longrightarrow}|10\rangle\stackrel{F}{\longrightarrow}|20\rangle$, and (ii) tunnel-coupling-mediated transition $|00\rangle\stackrel{F}{\longrightarrow}|10\rangle\stackrel{J_1}{\longrightarrow}|01\rangle\stackrel{F}{\longrightarrow}|11\rangle\stackrel{J_2}{\longrightarrow}|20\rangle$. With proper choice of parameters, the photons coming from the two pathways would destructively interfere. In other words, the destructive quantum interference between the direct path and the indirect path can reduce the probability in the two-photon excited state, this is known as unconventional photon blockade. In present model, strong antibunching effect can be achieved through adjusting the relative phase of coupling coefficients $J_1$ and $J_2$, instead of increasing the amplitudes of them. Thus, the relative phase $\beta$ plays a crucial role in the photon statistical properties of the system. In particular, when the system is steered close to an exceptional point (i.e., $\beta\approx0.4$), the indirect path $|00\rangle\stackrel{F}{\longrightarrow}|10\rangle\stackrel{J_1}{\longrightarrow}|01\rangle\stackrel{F}{\longrightarrow}|11\rangle\stackrel{J_2}{\longrightarrow}|20\rangle$ is blocked due to the fact that $J_1=0$ or $J_2=0$ in the vicinity of the exceptional point. Only the direct path to the two photon state is allowed and then strong antibunching requires large nonlinearities, which is just the feature of the conventional photon blockade. Accordingly, a controllable switching between the unconventional and conventional photon blockade can be realized by tuning the relative angle $\beta$.
In our model, another factor affecting the photon statistical properties is the Kerr nonlinearity. Figure \[fig5\] plots the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ in a logarithmic scale versus Kerr nonlinearity $U/\gamma_{\mathrm{in}}$ under various values of $\beta$ by fixing the value of detuning at $\Delta/\gamma_{\mathrm{in}}=0.4$. In contrast to the conventional photon blockade, the value of $g_{\mathrm{C}}^{(2)}(0)$ does not always monotonically decrease with the increase of the strength of Kerr nonlinearity. It’s worth noting that there exists a local minimum value of $g_{\mathrm{C}}^{(2)}(0)$, which can be adjusted by tuning the value of $\beta$. It suggests that the photon antibunching can be further enhanced with an optimal choice for the relative position $\beta$ and Kerr coefficient $U$. In previous works [@25; @26; @27; @30], achieving strong photon antibunching with weak Kerr effect requires a large coupling strength between cavity modes. Here, we only need to tune the relative angular position $\beta$ of two nanoparticles without requiring the large coupling strength $J_{1,2}$. Note that, in the vicinity of the exceptional points, i.e. $\beta=\pi/8$, the local minimum in the curve disappears. With increasing the strength of Kerr effect, the value of $g_{\mathrm{C}}^{(2)}(0)$ monotonically decreases. The physical reason is that, at the exceptional point, quantum interference between different pathways is suppressed and the second order correlation function $g_{\mathrm{C}}^{(2)}(0)$ shows the features of conventional photon blockade.
![The second order correlation function in a logarithmic scale $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]$ versus Kerr nonlinearity $U/\gamma_{\mathrm{in}}$ under various values of $\beta$ by fixing $\Delta/\gamma_{\mathrm{in}}=0.4$. All other parameters are given the same as in Fig.\[fig2\]. The black dotted line denotes the position where $\mathrm{log}_{10}[g_{\mathrm{C}}^{(2)}(0)]=0$.[]{data-label="fig5"}](fig5){width="42.00000%"}
![The second order (blue solid curve) and third order (red dot-dashed curve) correlation function as a function of the detuning $\Delta/\gamma_{\mathrm{in}}$ under various relative positions $\beta$ in (a)-(c); (d) The second and third order correlation function as a function of relative positions $\beta$ by fixing the detuning $\Delta/\gamma_{\mathrm{in}}=-2.8$. Here we have selected $F/\gamma_{\mathrm{in}}=2$. All other parameters are given the same as in Fig.\[fig2\]. The black dotted line denotes the position where $\mathrm{log}_{10}[g_{\mathrm{C}}^{(n)}(0)]=0$ ($n=2,3$). The two-photon bunching and three-photon antibunching can be achieved in the grey area.[]{data-label="fig6"}](fig6){width="49.00000%"}
\[sec:level2\]Two photon blockade
---------------------------------
Next, we consider the strong-pumping case (e.g., $F=2\gamma_{\mathrm{in}}$), where the photon excitation becomes much stronger. This allows for the implementation of two-photon blockade where the presence of two photons suppresses the addition of further photons. To demonstrate the two photon blockade effect, we plot the equal-time second order field correlation function ($g_{\mathrm{C}}^{(2)}(0)=\langle \hat{a}_{\mathrm{C}}^{\dag 2}\hat{a}_{\mathrm{C}}^2\rangle/\langle \hat{a}_{\mathrm{C}}^\dag\hat{a}_{\mathrm{C}}\rangle^2$) and third order field correlation function ($g_{\mathrm{C}}^{(3)}(0)=\langle \hat{a}_{\mathrm{C}}^{\dag 3}\hat{a}_{\mathrm{C}}^3\rangle/\langle \hat{a}_{\mathrm{C}}^\dag\hat{a}_{\mathrm{C}}\rangle^3$) in logarithmic units as a function of the normalized detuning $\Delta/\gamma_{\mathrm{in}}$ under various values of $\beta$ in Fig.\[fig6\](a)-(c). Here, the system parameters are chosen as the same as those used in Fig.\[fig2\]. It is noteworthy that, in the vicinity of the exceptional points (i.e. $\beta=\pi/8\approx \beta_c$), clear signatures of two photon blockade phenomena ($g_{\mathrm{C}}^{(2)}(0)>1$, and $g_{\mathrm{C}}^{(3)}(0)<1$) are shown in the grey area of Figure.\[fig6\](c). When $\beta\neq\beta_c$, the two photon blockade phenomena disappear and at the mean time single photon blockade appears. The physical reason is that when the system is not near the exceptional points destructive quantum interference between different pathways leads to strong photon antibunching, so the two photon bunching is greatly suppressed. On the contrary, at the exceptional points, the two-photon bunching and three-photon antibunching can be realized under the strong driving because of the uneven energy levels of the system. This feature leads to an optical switching from the single-photon blockade to the two-photon blockade by just tuning the relative angular position of two nanoparticles. To show this switching operation, we plot the second order (blue solid curve) and third order (red dot-dashed curve) field correlation functions as a function of the relative angular position $\beta/\pi$ in Fig.\[fig6\](d) by fixing the detuning $\Delta/\gamma_{\mathrm{in}}=-2.8$. Moreover, Kerr effect are also crucial for the degree of three photon antibunching. Figure \[fig7\] plots the second order and third order field correlation functions versus Kerr nonlinearity strength by fixing the relative angular position $\beta=\pi/8$ and detuning $\Delta/\gamma_{\mathrm{in}}=-2.8$. From fig.\[fig7\], we find that two photon blockade effect occurs in the grey region. Therefore, the choice of relative position $\beta$ and Kerr nonlinearity $U$ is very important for achieving the two photon blockade in the system discussed here.
![The second order (blue solid curve) and third order (red dot-dashed curve) correlation function as a function of the Kerr nonlinearity strength $U/\gamma_{\mathrm{in}}$ by setting $\Delta/\gamma_{\mathrm{in}}=-2.8$, $\beta=\pi/8$, $F/\gamma_{\mathrm{in}}=2$. All other system parameters used here are the same as in Fig.\[fig2\]. The black dotted line denotes the position where $\mathrm{log}_{10}[g_{\mathrm{C}}^{(n)}(0)]=0$ ($n=2,3$). The two-photon blockade can be achieved in the grey area.[]{data-label="fig7"}](fig7){width="40.00000%"}
conclusions
===========
In conclusion, we have studied the photon statistical properties in the nonlinear WGM microresonator coupled with two nanoparticles. By tuning the relative angular position $\beta$ of two nanoparticles, the photon statistical properties of the system can be well controlled and the switching between unconventional and conventional photon blockade can be achieved. We also investigate the influence of the Kerr effect on the second order correlation function and find that there is an optimal choice for relative position $\beta$ and Kerr coefficient $U$ to generate strong antibunching. Moreover, under the strong driving, two photon bunching and three photon antibunching can be achieved when the system is steered to the exceptional points. Compared with previous schemes (i.e. requiring large optical coupling between cavity modes), our scheme presented here is a good candidate for the realization of nonclassical light generation using small nonlinearities and weak driving fields.
We acknowledge the anonymous referees for their constructive comments to improve the paper. This work was supported by the National Natural Science Foundation of China (NSFC) (No. 61604045 and 11604059), the Natural Science Foundation of Guangdong Province, China (2017A030313020), and the Scientific Research Project of Guangzhou Municipal Colleges and Universities (1201630455).
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| ArXiv |
---
abstract: 'The Richardson-Lucy unfolding approach is reviewed. It is extremely simple and excellently performing. It efficiently suppresses artificial high frequency contributions and permits to introduce known features of the true distribution. An algorithm to optimize the number of iterations has been developed and tested with five different types of distributions. The corresponding unfolding results were very satisfactory independent of the number of events, the number of bins in the observed and the unfolded distribution, and the experimental resolution.'
address: 'Universität Siegen, D-57068 Siegen, Germany'
author:
- 'G. Zech'
title: 'Iterative unfolding with the Richardson-Lucy algorithm'
---
unfolding; Richardson-Lucy; iterative unfolding
Introduction
============
In many experiments the measurements are deformed by limited acceptance, sensitivity or resolution of the detectors. To be able to compare and combine results from different experiments and to compare the published data to a theory, the detector effects have to be unfolded. While acceptance losses can be corrected for, unfolding resolution effects is quite involved. Naive methods produce oscillations in the unfolded distribution that have to be suppressed by regularization schemes.
Various unfolding methods have been proposed in particle physics [@any91; @cernworkshop; @cowan]. The data are usually treated in form of histograms. This is also the case in the Richardson-Lucy (R-L) method [@rich72; @lucy74] which is especially simple, reliable, independent of the dimension of the histogram and independent of the underlying metric.
Iterative unfolding with the R-L algorithm has initially been used for picture restoration. Shepp and Vardi [@shepp82; @vardi85], and independently Kondor, [@kondor83] have introduced it into physics. It corresponds to a gradual unfolding. Starting with a first guess of the smooth true distribution, this distribution is modified in steps such that the difference between its smeared version and the observed distribution is reduced. With increasing number of steps, the iterative procedure develops oscillations. These are avoided by stopping the iterations as soon as the unfolded distribution, when folded again, is compatible with the observed data within the uncertainties. We will discuss the details below. The R-L algorithm originally was derived using Bayesian arguments [@rich72] but it can also be interpreted in a purely mathematical way [@muelthei86; @muelthei2005]. It became finally popular in particle physics after it had been promoted by D’Agostini [@dago] with the label Bayesian unfolding. In Ref. [@lindemann] it was adapted to unbinned unfolding. In Ref. [@na38] the R-L algorithm was applied to a 4-dimensional distribution.
The present situation in particle physics is unsatisfactory for two reasons: i) There is a lack of comparative systematic studies of the different unfolding methods and ii) the way to fix the degree of smoothing, the regularization strength, is usually only vaguely defined.
In the following section we introduce the notation and formulate the mathematical relations. In Section 3 we discuss regularization and the problem of assigning errors to the unfolded distribution. In Section 4 the R-L iterative approach is described. A criterion is developed to fix the number of iterations that have to be applied and which determine the degree of regularization. Section 5 contains examples. We conclude with a summary and recommendations.
Definitions and basic relations
===============================
An event sample with variables $\{x_{1},\ldots,x_{n}\}$, the *input sample* is produced according to a statistical distribution $f(x)$. It is observed in a detector. The *observed sample* $\{x_{1}^{\prime},\ldots,x_{n^{\prime}}^{\prime}\}$ is distorted due the finite resolution of the detector and reduced because of acceptance losses. We distinguish between four different histograms: The *true histogram* with content $\theta_{j}$, $j=1,\ldots,N$ of bin $j$. $\theta_{j}\propto
\int_{bin\text{ }j}f(x)dx$ corresponds to $f(x)$. The *input histogram* contains the input sample. The content of its bin $j$ is drawn from a Poisson distribution with mean value $\theta_{j}$. The *observed histogram* contains the observed sample with $d_{i}$ events in bin $i$, $i=1,\ldots,M$. The expected number of events $t_{i}$ in bin $i$ is given by $t_{i}\propto
\int_{bin\text{ }i}f^{\prime}(x^{\prime})dx^{\prime}$ where the functions $f^{\prime}$ and $f$ are related through $f^{\prime}(x^{\prime})=\int
g(x^{\prime},x)f(x)dx$ with the response function $g(x^{\prime},x)$. We choose $M>N$ to constrain the problem. The result of the unfolding procedure is again a histogram, the *unfolded histogram*, with bin content $\hat{\theta}_{j}$. We are confronted with a standard inference problem where the wanted parameters are the bin contents $\theta_{j}$ of the true histogram. It is to be solved by a least square (LS) or a maximum likelihood (ML) fit. We discuss only one-dimensional histograms but the corresponding array may represent a multi-dimensional histogram with arbitrarily numbered cells as well.
The numbers $t_{i}$ and $\theta_{j}$ are related by the linear relation $$t_{i}=\sum_{j=1}^{N}A_{ij}\theta_{j} \label{transfer}%$$ with the response matrix $A_{ij}$ $$A_{ij}=\frac{\int_{bin\text{ }i}f^{\prime}(x^{\prime})dx^{\prime}}%
{\int_{bin\text{ }j}f(x)dx}\;.\;$$
$A_{ij}$ is the probability to observe an event in bin $i$ that belongs to the true bin $j$. We calculate $A_{ij}$ by a Monte Carlo simulation, but as we do not know $f(x)$, we have to use a first guess of it. If the size of the bins is smaller than the experimental resolution, the elements of the response matrix show little dependence on the distribution that is used to generate the events.
We assume that the observed values $d_{i}$ fluctuate according to the Poisson distribution with the expectation $t_{i}$ and the variance $\delta_{i}%
^{2}=t_{i}$.
The representation of the unfolded distribution by a histogram is a first smoothing step. We call it *implicit regularization*. With wide enough bins, strong oscillations in the unfolded histogram are avoided. LS or ML fits will produce the parameter estimates $\hat{\theta}_{j}$ together with reliable error estimates. With the prediction $t_{i}$ for $d_{i}$ we can define $\chi^{2}$,
$$\chi^{2}=%
%TCIMACRO{\dsum \limits_{i=1}^{M}}%
%BeginExpansion
{\displaystyle\sum\limits_{i=1}^{M}}
%EndExpansion
\frac{\left[ d_{i}-t_{i}\right] ^{2}}{t_{i}}\;,$$
and the log-likelihood $\ln L$ derived from the Poisson distribution, $$\ln L=\sum_{i=1}^{M}\left[ d_{i}\ln t_{i}-t_{i}\right] \;.\label{likstat}%$$ Minimizing $\chi^{2}$ or maximizing $\ln L$ determines the estimates of the parameters $\hat{\theta}_{j}$. The ML fit is applicable also with small event numbers $d_{i}$ and suppresses negative estimates of the parameter values. Negative values can occur in rare cases.
The regularization and the error assignment
===========================================
In particle physics the data are often distorted by resolution effects. This means that without regularization the number of events in neighboring bins of the unfolded histogram are negatively correlated and as a consequence local fluctuations are observed. More precisely, the fitted parameters $\hat{\theta}_{j}%
,\hat{\theta}_{j^{\prime}}$ in two true bins $j,j^{\prime}$ are anti-correlated if their events have sizable probabilities $A_{ij}%
,A_{ij^{\prime}}$ to fall into the same observed bin $i$. These specific correlations are taken into account in most unfolding methods. An exception is entropy regularization [@nara86; @sch94; @maga98] which also penalizes fluctuations between distant bins.
The $\chi^{2}$ surface of the unregularized fit near its minimum $\chi_{0}%
^{2}$ is rather shallow and large correlated parameter changes produce only small changes $\Delta\chi^{2}$ of $\chi^{2}$ of the fit. The location of the true parameter point in the parameter space is badly known but the surfaces of $\chi_{0}^{2}+\Delta\chi^{2}$ for not too small values of $\Delta\chi^{2}$ are well defined and fix the error intervals which should not be affected by the regularization. We are allowed to move the point estimate but the error intervals should not be shifted. The regularization should lead only to a small increase of $\chi^{2}$. The increase $\Delta\chi^{2}=\chi^{2}-\chi_{0}^{2}$ defines an $N$ dimensional error interval around the fitted point in the parameter space. It can be converted to a $p$-value $$p=\int_{\Delta\chi^{2}}^{\infty}u_{N}(z)dz \label{pvalue}%$$ where $u_{N}$ is the $\chi^{2}$ distribution for $N$ degrees of freedom. Strictly speaking, $p$ is a proper $p$-value only in the limit where the test quantity $\chi^{2}$ is described by a $\chi^{2}$ distribution. Fixing $p$ fixes the regularization strength. A large value of $p$ corresponds to a weak regularization and means that the unfolding result is well inside the commonly used error interval of the likelihood fit. The optimal value of a cut in $p$ depends on the unfolding method. Remark that here the value of $\chi_{0}^{2}$ of the fit is irrelevant; what is relevant is its change due to the regularization. A large value $\chi_{0}^{2}$ could indicate that something is wrong with the model.
In most applications outside physics, like picture restoration, the uncertainties of the unfolded distribution are of minor concern. Of interest are mainly the point estimates which are obtained with a regularization that the user chooses according to his personal experience. In physics problems, the error bounds are as important as the point estimates. The manipulations related to the regularization in most methods constrain the fit and therefore reduce the errors of the unfolded histogram as provided by the unconstrained fit [@hoecker; @truee]. As a consequence, these errors depend on the regularization strength and do not cover the true distribution with a fixed probability. Distributions with narrow structures that are compatible with the data may be excluded. An example for such a behavior is presented in Appendix 1. It is not possible to associate classical confidence intervals to explicitly regularized solutions. As stated above, standard error intervals are provided by fits without regularization.
In the iterative method the errors could in principle be calculated by error propagation but these errors would not be constrained and therefore usually be large and strongly correlated. Furthermore their interpretation would be difficult. Therefore it does not make sense to include them in the graphical representation. A very qualitative way to indicate the errors is presented in Appendix 2.
To document quantitatively the precision of the data, a fit with a small number of bins and without explicit regularization of the unfolded histogram should be done, such that by a wide enough binning artificial oscillations are sufficiently suppressed. The result together with the corresponding error matrix[^1] estimate contain the information that is necessary for a comparison with theoretical predictions or other experiments. An example is given in Appendix 2. Alternatively, the data vector and the response matrix could be kept. These items, however, require some explanation to non-experts.
In case we have a theoretical prediction in analytic form, depending on unknown parameters, we should avoid unfolding and the regularization problem and estimate the parameters directly [@bohm]. A direct fit does not require the construction of a response matrix and is independent of assumptions about the shape of the distribution used to simulate the experiment, parameter inference is possible even with very low event numbers where unfolding is problematic, the results are unbiased and the full information contained in the experimental data can be explored.
The Richardson-Lucy iteration
=============================
The method
----------
Replacing the expected number $t_{i}$ in relation (1) by the observed number $d_{i}$, the corresponding matrix relation $d=A\hat{\theta}$ can be solved iteratively for the estimate $\hat{\theta}$. The idea behind the iteration algorithm is the following: Starting with a preliminary guess $\hat{\theta
}^{(0)}$of $\theta$, the corresponding prediction for the observed distribution $d^{(0)}$ is computed. It is compared to $d$ and for a bin $i$ the ratio $d_{i}/d_{i}^{(0)}$ is formed which ideally should be equal to one. To improve the agreement, all true components are scaled proportional to their contribution $A_{ij}\hat{\theta}_{j}^{(0)}$ to $d_{i}^{(0)}$. This procedure when iterated corresponds to the following steps:
The prediction $d^{(k)}$ of the iteration $k$ is obtained in a *folding step* from the true vector $\hat{\theta}^{(k)}$: $$d_{i}^{(k)}=\sum_{j=1}^{N}A_{ij}\hat{\theta}_{j}^{(k)}\;. \label{folding}%$$
In an *unfolding step*, the components $A_{ij}\hat{\theta}_{j}^{(k)}$ are scaled with $d_{i}/d_{i}^{(k)}$ and added up into the bin $j$ of the true distribution from which they originated: $$\hat{\theta}_{j}^{(k+1)}=\sum_{i=1}^{M}A_{ij}\hat{\theta}_{j}^{(k)}\frac
{d_{i}}{d_{i}^{(k)}}\left/ \alpha_{j}\right. \;. \label{unfolding}%$$
Dividing by the acceptance $\alpha_{j}=\sum_{i}A_{ij}$ corrects for acceptance losses.
The result of the iteration converges to the maximum likelihood solution as was proven by Vardi et al. [@vardi85] and Mülthei and Schorr [@muelthei86] for Poisson distributed bin entries. Since we start with a smooth initial distribution, the artifacts of the unregularized ML estimate (MLE) occur only after a certain number of iterations.
The regularization is performed simply by interrupting the iteration sequence. As explained above, the number of applied iterations should be based on a $p$-value criterion which measures the compatibility of the regularized unfolding solution with the MLE.
To this end, first the number of iterations is chosen large enough to approach the asymptotic limit with the ML solution which provides the best estimate of the true histogram if we put aside our prejudices about smoothness. Folding the result and comparing it to the observed histogram, we obtain $\chi_{0}^{2}$ of the fit.
\[ptb\]
[oscil.EPS]{}
Of course, the MLE does not depend on the starting distribution but the regularized solution obtained by stopping the iteration does. We may choose it according to our expectation. In most cases the detailed shape of it does not matter, and a uniform starting distribution will provide reasonable results.
As may be expected, the speed of convergence decreases with the spatial frequency of the true distribution if we consider a Gaussian type of smearing described by a point spread function. This is shown in Fig. \[oscil\]. Here the true distributions consisting of a superposition of a uniform distribution of $1000$ events and a squared sine/cosine distributions of $9000$ events with $1$ to $6$ oscillations is smeared and distributed into $40$ bins. The corresponding histogram is unfolded to a $20$ bin histogram starting with a uniform histogram. The statistic $\Delta\chi^{2}$ for $20$ degrees of freedom is plotted as a function of the number of iterations. The discrete points are connected by a line. The horizontal line corresponds to a $p$-value of $0.5$. As expected, the number of required iteration steps that are needed to reach the $p=0.5$ value increases with the frequency of the distribution. This means that high frequency contributions and artificial fluctuations of correlated bins are strongly suppressed in the R-L approach. The reason can be inferred from Relation (\[unfolding\]): The parameters $\theta_{j},\theta_{j^{\prime
}}$ of bins $j,j^{\prime}$ that are correlated in that they have similar values $A_{ij}$, $A_{ij^{\prime}}$ are scaled in a similar way and relative fluctuations develop only slowly with increasing number of iterations.
*Remark*: By construction, the R-L method is invariant against an arbitrary re-ordering of the bins. A multidimensional histogram can be transformed to a one-dimensional histogram. A rather general class of distortions can be treated. This is also true for entropy regularization and methods based on truncation of the eigenvalue sequence in singular value decomposition (SVD) [@hoecker] but not for local regularization schemes like curvature suppression [@tikhonov] which is difficult to apply in higher dimensions.
\[ptb\]
[iter2b2040.EPS]{}
The regularization strength
---------------------------
Without recipes how to fix the regularization strength, unfolding methods are incomplete and the results are to a certain extent arbitrary. In most of the proposed methods a recommendation is missing or rather vague. In the iterative method, we have to find a criterion, based on a $p$-value, when to stop the iteration process. The optimum way may depend on several parameters: the number of events, the number of bins, the resolution and the shape of the true distribution. Not all combinations of these parameters can be investigated in detail. We will study some specific Monte Carlo examples to derive a stopping criterion and then test it with further distributions. It will be shown that a general prescription works reasonably well for all studied examples.
The unfolded histogram is compared to the input histogram. In all examples we take care that the estimates of the elements of the response matrix have negligible statistical uncertainties. If not stated differently, the iteration starts with a uniform distribution as a first guess for the true distribution. The observed histogram has, with two exceptions, $40$ bins and the unfolded histogram usually comprises $20$ bins. With the standard settings the value of $\chi_{0}^{2}$ should be compatible with the $\chi^{2}$ distribution with $20$ degrees of freedom because we have $40$ measurements and $20$ estimated parameters.
### Example 1: Two peaks {#example-1-two-peaks .unnumbered}
We start with a two-peak distribution, a superposition of two normal distributions $N(x|0.3;0.10)$, $N(x|0.75;0.08)$ and a uniform distribution $U(x)$ in the interval $0<x<1$. Here $N(x|\mu;\sigma$) is the normal distribution of $x$ with the mean value $\mu$ and the standard deviation $\sigma$. The number of events attributed to the three distributions is $25,000$, $15,000$ and $10,000$, respectively. The experimental distribution is observed with a Gaussian resolution $\sigma=0.07.$ It is of the same order as the width of the peaks. Events are accepted in the interval $0<x,x^{\prime}<1$.
In Fig. \[iter2b2040\] unfolding results for different values of the number of iterations are shown. The shaded histograms (input histograms) correspond to the observation with an ideal detector and are close to the true histogram. The left top plot displays the observed histogram as squares. With increasing number of iterations the unfolded histogram (squares) quickly approaches the true histogram. The agreement is quite good in a wide range of the number of iterations. It deteriorates slowly when increasing the number of iterations beyond $32$. At $1000$ iterations oscillations are visible and after $100,000$ iterations the sequence has approached the maximum likelihood solution with strong fluctuations and no explicit regularization. We find $\chi_{0}%
^{2}=23.4$ for $20$ degrees of freedom.
\[ptb\]
[iterbumbprob.EPS]{}
The variation of $\chi^{2}$ as a function of the number of iterations is shown in Fig. \[bumbprob\] top, left hand scale. The corresponding $p$-value (right hand scale) jumps within a few iterations from a negligible value to a value close to one. To judge the quality of the unfolding, we compute the quantity $X^{2}=\Sigma_{i}(\hat{\theta}_{i}-\theta_{i})^{2}/\theta_{i}$ which is available in toy experiments. It is difficult to estimate the range of values of $X^{2}$ that correspond to acceptable solutions, but qualitatively the agreement of the unfolded histogram with the true histogram improves with decreasing $X^{2}$. The dependence of $X^{2}$ from the iteration number is displayed at the top center of the same figure. The minimum is reached at $14$ iterations with a $p$-value of $0.98$ but there is little change between $8$ and $16$ iterations. The corresponding unfolding result is shown on the right hand side. Repeating the same experiment with ten times less events, i.e. $5,000$, we obtain the results displayed at the bottom of Fig. \[bumbprob\]. Here the best agreement is reached after $9$ iterations.
\[ptb\]
[probcut.EPS]{}
The study is repeated for $5$ different samples. The $p$-values are shown as a function of the number of iterations in Fig. \[probcut\]. All curves start rising nearly at the same iteration, remain close to each other at the beginning but separate at large $p$-values. With $5,000$ events the lowest value of the test quantity $X^{2}$ is always obtained for $8$ or $9$ iterations, while the corresponding $p$-values vary because of the small slopes near $p$-values of one. Therefore, we should base the cut of the chosen number of iterations on a lower $p$-value. The following choice has proven to be quite stable and efficient: We stop the iteration at twice the value at which the $p$-value crosses the $0.5$ line. For the left hand plot with $5,000$ events the crossing is close $4.5$ and thus $9$ iterations should be performed. With $50,000$ events this criterion leads to a choice of $15$ iterations. Actually, from the $X^{2}$ variation, acceptable values are located between $11$ and $16$ iterations. In Table 1 the results for the same distribution but different number of bins of the observed and the unfolded histogram and for different resolutions $\sigma$ are summarized. From left to right the columns contain the number of generated events, the number of bins in the observed and the true histograms, the standard deviation of the Gaussian response function, the number of applied iterations as based on the stopping criterion, $X^{2}$, the corresponding $p$-value, the number of iterations that minimizes $X^{2}$ and the minimal value of $X^{2}$. In each case two independent toy experiments have been performed. The results from the second one are given in parentheses. They are close to those of the first one. In all cases the recipe for the choice of the number of iterations leads in most cases to very sensible results. The $p$-values are close to $1$ in most cases and always above $0.95$.
For the resolution $0.1$ the optimal number of iterations and also the $X^{2}$ values differ considerably from the those found by the stopping criterion. The visual inspection shows however that the unfolded distributions that correspond to the stopping prescription agree qualitatively well with the true distributions. For comparison, the example with $50,000$ events and resolution $0.1$ has also been repeated with a likelihood fit and entropy penalty regularization. The regularization constant was varied until the minimum of $X^{2}$ was obtained. The results was $X^{2}=159$ significantly larger than the value $91$ obtained with iterative unfolding. With the prescription $\Delta\chi^2=1$ [@sch94], $X^{2}=873$ was obtained. Regularization with a curvature penalty is not suited for this example. Here the best value of $X^2$ is $700$.
\[c\][|l|l|l|l|l|l|l|l|]{}events & bins & $\sigma$ & $\#$ & $X^{2}$ & $p$-value & $\#_{best}$ & $X_{best}^{2}$\
50000 & 40/20 & 0.07 & 15 (15) & 33 (40) & 0.989 (0.986) & 15 (14) & 33 (40)\
5000 & 40/20 & 0.07 & 9 (8) & 25 (39) & 0.958 (0.980) & 9 (9) & 25 (39)\
50000 & 40/14 & 0.07 & 18 (16) & 25(32) & 0.978 (0.989) & 16 (17) & 25 (32)\
5000 & 40/14 & 0.07 & 9 (10) & 27 (40) & 0.997 (0.971) & 10 (8) & 26 (38)\
50000 & 40/30 & 0.07 & 13 (13) & 44 (45) & 1.000 (1.000) & 14 (15) & 44 (44)\
5000 & 40/30 & 0.07 & 7 (7) & 28 (39) & 0.997 (1.000) & 8 (8) & 27 (39)\
50000 & 40/20 & 0.05 & 8 (8) & 31 (21) & 1.000 (1.000) & 7 (11) & 31 (21)\
5000 & 40/20 & 0.05 & 5 (6) & 9 (22) & 0.997 (0.971) & 6 (5) & 9 (20)\
50000 & 40/20 & 0.10 & 33 (33) & 143 (148) & 1.000 (1.000) & 205 (176) & 91 (108)\
5000 & 40/20 & 0.10 & 15 (18) & 100 (57) & 1.000 (0.985) & 23 (23) & 77 (52)\
50000 & 80/20 & 0.7 & 15 (15) & 32 (37) & 0.991 (0.985) & 14 (15) & 32 (37)\
5000 & 80/20 & 0.7 & 8 (8) & 26 (36) & 0.970 (0.999) & 7 (8) &26 (36)\
### Interpolation for fast converging iterations
In situations where the response function is narrow, usually the iteration sequence converges quickly to a reasonable unfolded histogram, sometimes after a single iteration. Then one might want to stop the sequence somewhere between two iterations. This is possible with a modified unfolding function. We just have to introduce a parameter $\beta>0$ into (\[unfolding\]): $$\hat{\theta}_{j}^{(k+1)}=\left[ \sum_{i=1}^{M}A_{ij}\hat{\theta}_{j}%
^{(k)}\frac{\hat{d}_{i}}{d_{i}^{(k)}}\left/ \alpha_{j}\right. +\beta
\hat{\theta}_{j}^{(k)}\right] \;\left/ (1+\beta)\right. .
\label{unfoldingsmooth}%$$
The value $\beta=0$ produces the original sequence (\[unfolding\]), with $\beta=1$ the convergence is slowed down by about a factor of two and in the limit where $\beta$ approaches infinity, there is no change. It is proposed to choose $\beta$ such that at least $5$ iteration steps are performed.
\[ptb\]
[iterpub.EPS]{}
Subjective elements
-------------------
Unfolding is not an entirely objective procedure. The choice of the method and the kind of regularization depend at least partially on personal taste. For a given value of $\chi^{2}$ there exist an infinite number of unfolded histograms. There is no objective criterion which would allow us to choose the best solution. Given the R-L iterative unfolding, with the stopping criterion as defined above and a uniform starting distribution all parameters are fixed, but in some rare situations it may make sense to modify the standard method.
### Choice of the starting distribution
Instead of a uniform histogram we may choose a different starting histogram. As long as the corresponding distribution shows little structure, the unfolding result will not be affected very much. If we start in our Example 1 ($50,000$ events) with an exponential distribution $f(x)=e^{-x}$ the unfolded histogram is hardly distinguishable from that with a uniform starting distribution. The difference is less than $1\%$ in all bins except for the two border bins with only about $60$ entries where it amounts to $2\%$. In both cases $15$ iterations are required.
For an input distribution that is close to the true distribution, the results are in most cases again very similar to those of the uniform input distribution, but of course the number of required iterations is reduced to one ore two. The situation is different for distribution with sharp structures, for instance, if there is a narrow peak with a small smooth background. Starting with a uniform distribution a large number of iterations is required which may lead to oscillations in the background region. This unpleasant effect is avoided if we start with a distribution that includes a peak structure and where only few iterations are necessary.
We have to be careful when choosing a starting distribution different from a monotone function. Only statistically well established structures should be modeled in the starting distribution.
The starting distribution can be obtained by fitting a polynomial, spline functions or another sensible parametrization to the data with the method described in Ref. [@bohm].
### Manual smoothing
In the specific example with a narrow peak which we discuss below, starting with a uniform distribution we can also avoid the oscillations if we replace the oscillating part in the true input histogram by a smooth distribution before the last iteration step[^2].
Examples with various distributions
===================================
We test the R-L unfolding and the stopping criterion with four different distributions, a single peak distribution, an exponential distribution, a step distribution and a uniform distribution. The results are displayed in Fig. \[iterpub\]. The number of events and the number of iterations are indicated in the plots. The starting true function is uniform, except for the last column where a rough guess of the true distribution is used. The input histogram is shaded, the unfolded histogram is indicated by squares and the observed histogram is plotted as circles in the left hand graphs.
### Example 2: Single narrow peak {#example-2-single-narrow-peak .unnumbered}
We turn now to a more difficult problem and consider a distribution of $40,000$ events distributed according to $N(x|0.6;0.05)$ and $10,000$ events distributed uniformly. The Gaussian response function with $\sigma=0.07$ is wider than the peak. There is a problem because for the flat region we would be satisfied with few iterations while the peak region requires many iterations. Here $\operatorname{about}$ $60$ iterations are needed because relatively high frequencies are required to model the narrow peak. We get $\chi^{2}=27$ while the value of $\chi_{0}^{2}$ after $100,000$ iterations is $20.6$. The unfolded histogram is shown in Fig. \[iterpub\] top left together with the smeared histogram and the true histogram. The peak is well reproduced. The corresponding results for $5,000$ events is shown at the center of the first row. The right hand plot is obtained with a modified input distribution for the last iteration. The unfolding result after $18$ iterations is used as input, but the flat region is replaced by a uniform distribution and one additional iteration is applied. In this way the artificial oscillations in the background region are reduced.
To test the effect of an improved starting distribution, a superposition of a quadratic basic spline function (b-spline) and a uniform distribution was fitted to the data. Four parameters were adjusted, two normalization parameters, the location and the width of the b-spline bump. With this starting distribution, after a single iteration the input distribution is almost perfectly reproduced. The test quantity $X^{2}$ is $47$ compared to $216$ with a uniform starting distribution.
### Example 3: Exponential distribution {#example-3-exponential-distribution .unnumbered}
$50,000$ events are generated in the interval $1<x<5$ according to an exponential distribution $f(x)=e^{-x}$ and $\sqrt{x}$ is smeared with a Gaussian resolution of $\sigma=0.1$ which means that the smearing of $x$ increases proportional to $\sqrt{x}$. The events are observed in the interval $0.5<x^{\prime}<5$ and distributed into $40$ bins. The convergence is rather fast because the distribution is smooth even though we start with a uniform true distribution. We stop after $5$ iterations and get $\chi^{2}=31.5$ which corresponds to a $p$-value of $0.996$. The results are shown in the second row of Fig. \[iterpub\]. In fact the agreement of the unfolded distribution improves slightly with additional iterations and is optimum after $7$ iterations. With $5,000$ events the convergence is faster and a reasonable agreement is obtained after $3$ iterations. Starting with a first guess of an exponential distribution the result slightly improves (right hand plot).
### Example 4: Step function {#example-4-step-function .unnumbered}
A step function is rather exotic. The sharp edge is not easy to reconstruct. We locate the edge at the center of the interval and superpose two uniform distributions containing $40,000$ events in the interval $0<x<0.5$ and $10,000$ events in the interval $0.5<x<1$ with the resolution $\sigma=0.05$. The unfolding results shown in the third row of Fig. \[iterpub\] are disappointing. The $p$-value of $\ 0.99$ is reached after $25$ iterations with $\chi^{2}=20.63$ ($\chi_{0}^{2}=12.42$). A problem is that to model the sharp edge, many iterations are required while for the flat regions oscillations start after a few iterations. However if we replace the uniform starting distribution by the result displayed in the left hand plot replacing the $16$ bins of the flat region by uniform distributions the result (right hand plot) near the edge is not improved
### Example 5: Uniform distribution {#example-5-uniform-distribution .unnumbered}
A uniform distribution is easy to unfold. $50,000$ events generated in the interval $0<x<1$ with a Gaussian resolution of $\sigma=0.1$ and observed in the same interval are unfolded. As the iteration starts with a uniform distribution, no iteration is necessary and the result is optimal with a $p$-value close to one. The initial value of $\chi^{2}$ is $26.4$ and the minimum value is $19.3$ corresponding to the strongly oscillating ML solution. In the case of $5,000$ events $1$ iteration is applied.
\[c\][|l|rrr|rrr|rrr|rrr|rrr|]{}case & & & & &\
& $\chi^{2}$ & $X^{2}$ & \# & $\chi^{2}$ & $X^{2}$ & \# & $\chi^{2}$ & $X^{2}$ & \# & $\chi^{2}$ & $X^{2}$ & \# & $\chi^{2}$ & $X^{2}$ & \#\
50,000 & 27 & 216 & 60 & 31 & 33 & 15 & 29 & 20 & 10 & 24 & 600 & 14 & 26 & 3 & 0\
50,000 best & 29 & 209 & 51 & 31 & 33 & 15 & 30 & 19 & 7 & 18 & 488 & 48 & 26 & 3 & 0\
5,000 & 32 & 167 & 18 & 37 & 25 & 9 & 43 & 7 & 2 & 37 & 104 & 3 & 45 & 6 & 1\
5,000 best & 29 & 71 & 70 & 37 & 25 & 9 & 43 & 7 & 2 & 33 & 96 & 6 & 45 & 6 & 1\
### Test of the stopping criterion
In Table 2 we compare the result obtained with the stopping criterion to the result obtained with the optimal number of iterations (denoted by *best* in the table). In all cases the iteration starts with a uniform distribution. The agreement with the observed distribution, indicated by $\chi^{2}$, the compatibility of the unfolded distribution with the input distribution, measured with $X^{2}$ and the number of applied iterations are given. The stopping criterion produces very satisfactory results in all cases. With the exception of the single peak distribution with $5,000$ events, it is close to the optimum. Here the observed discrepancy between the number of iterations from the stopping criterion and the number derived from the minimum of $X^{2}$ is due to the fact that the distribution consists of a flat region where few iteration are needed and the peak region which requires many iteration to converge to an optimal result. Nevertheless also the solution with $18$ iteration is satisfactory.
Summary, conclusions and recommendations
========================================
Iterative unfolding with the R-L approach is extremely simple, independent of the number of dimensions, efficiently damps oscillations of correlated histogram bins and needs little computing time. A general stopping criterion has been introduced that fixes the number of iterations, e.g. the regularization strength, that should be applied. It has a simple statistical interpretation. Its stability has been demonstrated for five different distributions, two different event numbers, two different experimental resolutions and three binnings. The results are very satisfactory. The present study should be extended to more distributions with varying statistics and binning and also be applied to higher dimensions.
In most problems a uniform distribution should be used as starting distribution, but the dependence on its shape is negligible as long as this distribution does not contain pronounced structures. In cases where the observed distribution indicates that there are sharp structures in the true distribution, the iterative method permits to implement these in the input distribution. In this way the number of iterations is reduced and oscillations are avoided.
Standard errors, as we associate them commonly in particle physics to measurements, cannot be attributed to explicitly regularized unfolded histograms. We propose to indicate the precision of the graphical representation of the result qualitatively in a way that is independent of the regularization strength. For a quantitative documentation, the unfolding results without explicit regularization should be published together with an error matrix or its inverse. The widths of the bins of the corresponding histogram have to be large enough to suppress excessive fluctuations.
A quantitative comparison of the R-L unfolding with other unfolding methods is difficult, because in most of them a clear prescriptions for the choice of the regularization strength is missing or doubtful. A sensible comparison requires similar binning and regularization strengths in all methods. The latter could be measured with the $p$-value. Independent of the unfolding method that is used, in publications the values of $\chi^{2}$ obtained with and without regularization should be given to indicate the regularization strength and the reliability of the unfolded distribution.
Whenever it is possible to parametrize the true distribution, the parameters should be fitted directly.
Acknowledgment {#acknowledgment .unnumbered}
==============
I thank Gerhard Bohm for many valuable comments.
Appendix 1: The problem of the error assignment {#appendix-1-the-problem-of-the-error-assignment .unnumbered}
===============================================
\[ptb\]
[svderror.EPS]{}
In most unfolding schemes the oscillations are suppressed, either by introduction of a penalty term in the fit, or by reduction of the effective number of parameters [@blobel]. Both approaches constrain the fit and thus reduce the errors. As a consequence the assigned uncertainties do not necessarily cover the true distribution. An example is shown in Fig. \[svderror\] right hand side. The parameters of the LS fit have been orthogonalized with a singular value decomposition (SVD) [@hoecker]. The left hand plot shows the significance of the parameters which is defined as the ratio of the parameter and its error as assigned by the fit. The $20$ parameters are ordered with decreasing eigenvalues. A smooth cut is applied at parameter $\varepsilon_{c}=11$. Contributions are then weighted by $\phi(\varepsilon)=\varepsilon/(\varepsilon+\varepsilon_{c)}$. In this way oscillations are suppressed that might be caused by an abrupt cut, similar to Gibbs oscillations as observed with Fourier approximations [@hoecker; @blobel]. Obviously the number of $11$ effectively used parameters is insufficient to reproduce the peak and the true distribution is excluded. With the addition of further parameters oscillations start to develop. The problem is especially severe with low event numbers. With $10$ times more events the discrepancy between the true distribution and the unfolded one is considerably reduced.
Regularization with a curvature penalty reduces the statistical errors even in the limit where the resolution is perfect. The errors presented by an experiment that suffers from a limited resolution may be smaller than those of a corresponding experiment with an ideal detector where unfolding is not required.
Appendix 2: The documentation of the results {#appendix-2-the-documentation-of-the-results .unnumbered}
============================================
\[ptb\]
[ilustrate.EPS]{}
In the following we present a possible way to document unfolding results such that they can be compared to theoretical predictions and to other experiments.
The left hand plot of Fig. \[ilustrate\] shows the result of a ML fit of the content of the $10$ bins of a histogram without explicit regularization for Example 1 with $5,000$ events. The errors are indicated. They are large due to the strong negative correlation between adjacent bins which amounts to $80 \%$. The fitted values together with the error matrix can be used for a quantitative comparison with predictions. Instead of the error matrix its inverse could be presented. The inverse is in fact the item that is required for parameter fitting. Even more information is contained in the combination of the data vector and the response matrix. These items, however, require some explanation to non-experts.
The right hand side of Fig. \[ilustrate\] shows a possibility to indicate the precision of an explicitly regularized unfolded histogram. The plot is based on the same data as in the left hand plot. The vertical error bar corresponds to the uncertainty of the bin content neglecting correlations and the horizontal bars indicate the uncertainty in the location of the events. In the absence of acceptance corrections the vertical error of bin $i$ is simply equal to the square root of the bin content, $\sqrt{\theta_i}$. If the average acceptance of the events in the bin is $\alpha_i$, the error is $\theta_i /\sqrt{\alpha_i\theta_i}$. The horizontal bar indicates the experimental resolution. Such a graph is intended to show the likely shape of the distribution but is not to be used for a quantitative comparison with other results or predictions. It usually overestimates the uncertainties but for an experienced scientist it indicates quite well the precision of a result.
[99]{}
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[^1]: Instead of the error matrix its inverse could be published. The inverse is needed if data are combined or if parameters are estimated.
[^2]: A similar but more drastic proposal has been made in Ref. [@dago1].
| ArXiv |
---
abstract: 'Statistical techniques are used in all branches of science to determine the feasibility of quantitative hypotheses. One of the most basic applications of statistical techniques in comparative analysis is the test of equality of two population means, generally performed under the assumption of normality. In medical studies, for example, we often need to compare the effects of two different drugs, treatments or preconditions on the resulting outcome. The most commonly used test in this connection is the two sample $t$-test for the equality of means, performed under the assumption of equality of variances. It is a very useful tool, which is widely used by practitioners of all disciplines and has many optimality properties under the model. However, the test has one major drawback; it is highly sensitive to deviations from the ideal conditions, and may perform miserably under model misspecification and the presence of outliers. In this paper we present a robust test for the two sample hypothesis based on the density power divergence measure [@MR1665873], and show that it can be a great alternative to the ordinary two sample $t$-test. The asymptotic properties of the proposed tests are rigorously established in the paper, and their performances are explored through simulations and real data analysis.'
author:
- 'A. Basu'
- 'A. Mandal'
- 'N. Martin'
- 'L. Pardo'
bibliography:
- 'reference.bib'
date: 'September 28, 2014'
title: '**Robust Tests for the Equality of Two Normal Means based on the Density Power Divergence** '
---
**:** 62F35, 62F03.
: Robustness, Density Power Divergence, Hypothesis Testing.
Introduction: Motivation and Background
=======================================
In many scientific studies, often the main problem of interest is to compare different population groups. In medical studies, for example, the primary research problem could be to test for the difference between the location parameters of two different populations receiving two different drugs, treatments or therapy, or having two different preconditions. The normal distribution often provides the basic setup for statistical analyses in medical studies (as well as in other disciplines). Inference procedures based on the sample mean, the standard deviation and the one and two-sample $t$-tests are often the default techniques for the scenarios where they are applicable. In particular, the two sample $t$-test is the most popular technique in testing for the equality of two means, performed under the assumption of equality of variances. Its applicability in real life situations is, however, tempered by the known lack of robustness of this test against model perturbations. Even a small deviation from the ideal conditions can make the test completely meaningless and lead to nonsensical results. This problem is caused by the fact that the $t$-test is based on the classical estimates of the location and scale parameters (the sample mean and the sample standard deviation). Large outliers tend to distort the mean and inflate the standard deviation. This may lead to false results of both types, i.e. detecting a difference when there isn’t one, and failing to detect a true significance.
In this paper we are going to develop a class of robust tests for the two sample problem which evolves from an appropriate minimum distance technique in a natural way. This class of tests is indexed by two real parameters $%
\beta $ and $\gamma $, and we will constrain each of these parameters to lie within the $[0,1]$ interval. Our general minimum distance approach will allow us to study the likelihood ratio test in an asymptotic sense, as the likelihood ratio test is asymptotically equivalent to the test generated by the parameters $%
\beta =\gamma =0$. Normally we will work with the one parameter family of test statistics corresponding to $%
\beta =\gamma $; the outlier stability of the proposed tests increase with the tuning parameter $\gamma $.
Let $X$ and $Y$ be independent random variables whose distributions are modeled as normals having unknown means $\mu_1$ and $\mu_2$, respectively, with an unknown but common variance $\sigma^2$. We are interested in testing the null hypothesis $$H_{0}:\mu_1=\mu_2\text{ against }H_{1}:\mu_1\neq \mu_2,
\label{EQ:0}$$ under the above set up. It is well known that the exact two sample $t$-test (which is equivalent to the likelihood ratio test) rejects the null hypothesis in (\[EQ:0\]) if and only if $$t=\frac{\left\vert \bar{X}-\bar{Y}\right\vert }{S_{p}\sqrt{\frac{1%
}{n_1}+\frac{1}{n_2}}}>t_{\frac{\alpha }{2}}(n_1+n_2-2),$$ where $\bar{X}$ and $\bar{Y}$ are the sample means corresponding to the random samples $X_{1},X_{2},\ldots ,X_{n_1}$ and $%
Y_{1},Y_{2},\ldots ,Y_{n_2}$ obtained from the two distributions, $$S_{p}^{2}=\frac{(n_1-1)S_{1}^{2}+(n_2-1)S_{2}^{2}}{n_1+n_2-2},$$ $$S_{1}^{2}=\frac{1}{n_1-1}\sum_{i=1}^{n_1}\left( X_i-\bar{X}%
\right) ^{2},\quad S_{2}^{2}=\frac{1}{n_2-1}\sum_{i=1}^{n_2}\left( Y_{i}-%
\bar{Y}\right) ^{2},$$and $t_{\frac{\alpha }{2}}(n_1+n_2-2)$ is the $100(1-\frac{\alpha }{2})$-th quantile of the $t$-distribution with $n_1+n_2-2$ degrees of freedom. The $t$-test is the uniformly most powerful unbiased and invariant test for this hypothesis. Testing the equality of means of independent normal populations with unknown variances which are not necessarily equal, is referred to as the Behrens-Fisher problem.
In this paper we will use the density power divergence (DPD) measure [@MR1665873], which provides a natural robustness option for many standard inference problems. The density power divergence and its variants have been successfully used by many authors in a variety of inference problems; see, eg. [@MR1859416], [@MR2299175; @MR2466551], [@MR3011625; @basu2013], [@MR3117102]. However, the two sample problem requires a non-trivial extension of the currently existing techniques. Our purpose in this paper is to derive the asymptotic properties of the class of two sample tests based on the density power divergence and demonstrate their robust behavior in practical situations.
**Example 1 (Cloth Manufacturing data)**: In order to emphasize the need for applications early, we now present a motivational example. This example illustrates the use of quality control methods practiced in a clothing manufacturing plant. Levi-Strauss manufactures clothing from cloth supplied by several mills. The data used in this example (see Table [TAB:Staudte\_Sheather]{}) are for two of these mills and were obtained from the quality control department of the Levi plant in Albuquerque, New Mexico ([@lambert1987introduction], p. 86). In order to maintain the anonymity of these two mills we have coded them $A$ and $B$. A measure of wastage due to defects in cloth and so on is called *run-up*. It is quoted as percentage of wastage per week and is measured relative to computerized layouts of patterns on the cloth. Since the people working in the plant can often beat the computer in reducing wastage by laying out the patterns by hand, it is possible for run-up to be negative. From the viewpoint of quality control, it is desirable not only that the run-up be small but that the quality from week to week be fairly consistent. There are 22 measurements on run-up for each of the two mills and they are presented in Table \[TAB:Staudte\_Sheather\]. The $t$-test for the equality of the two means against the two-sided alternative has a $p$-value of 0.3428 and fails to reject the null hypothesis; however, when the presumed outliers (presented in bold fonts in Table \[TAB:Staudte\_Sheather\]) are removed from the dataset, the same two-sample $t$-test produces a $p$-value of 0.0308, leading to clear rejection. Choosing $\beta = \gamma$ to be the only parameter, the $p$-values of the DPD tests (to be developed in the next section) for testing the same hypotheses are presented in Figure [fig:Staudte\_Sheather\_book\_p\_val]{} as a function of $\gamma$. It is observed that the $p$-values of the tests with the full data and those with the outlier deleted data are practically identical for $\gamma = 0.2$ or larger, and lead to solid rejection. Thus, while the outliers mask the significance in case of the two sample $t$-test, the more robust DPD tests are able to capture the same.
-------- -------- --------- --------- ------------- --------- --------- ------------- ------------- --------- --------- --------
Mill A $0.12$ $1.01$ $-0.20$ $0.15$ $-0.30$ $-0.07$ $0.32$ $% $-0.32$ $-0.17$ $0.24$
0.27$
$0.03$ $0.35$ $-0.08$ $\bf{2.94}$ $0.28$ $1.30$ $\bf{4.27}$ $0.14$ $% $0.24$ $0.13$
0.30$
Mill B $1.64$ $-0.60$ $-1.16$ $-0.13$ $0.40$ $1.70$ $0.38$ $% $1.04$ $0.42$ $0.85$
0.43$
$0.63$ $0.90$ $0.71$ $0.43$ $1.97$ $0.30$ $0.76$ $\bf{7.02}$ $% $0.60$ $0.29$
0.85$
-------- -------- --------- --------- ------------- --------- --------- ------------- ------------- --------- --------- --------
: Cloth Manufacturing data.[]{data-label="TAB:Staudte_Sheather"}
Our primary motivation for studying the alternatives of the two sample $t$-test has been the need for developing such a test in the context of examples relating to medical data. However, examples abound in practically all scientific disciplines showing that this is a real necessity which is certainly not restricted to the medical field. The example considered above is one such, where the context does not have anything directly to do with a medical problem, but the importance of the problem and the need for a robust solution can immediately be appreciated.
The rest of the paper is organized as follows: In Section \[SEC:MDPDE\] the asymptotic distribution of the minimum DPD estimators in the two sample situation is described. In Section \[SEC:Test\] we introduce our robust two sample test statistic and develop the necessary theory. A large number of real data examples and extensive simulation results are presented in Section \[SEC:numerical\]. Finally Section \[SEC:concluding\] has some concluding remarks.
The Minimum DPD Estimator: Asymptotic Distribution {#SEC:MDPDE}
==================================================
For any two probability density functions $f$ and $g$, the density power divergence measure is defined, as the function of a single tuning parameter $\beta \geq 0$, as $$d_{\beta}(g,f)=\left\{
\begin{array}
[c]{ll}%
\int\left\{ f^{1+\beta}(x)-\left( 1+\frac{1}{\beta}\right) f^{\beta
}(x)g(x)+\frac{1}{\beta}g^{1+\beta}(x)\right\} dx, & \text{for}%
\mathrm{~}\beta>0,\\[2ex]%
\int g(x)\log\left( \displaystyle\frac{g(x)}{f(x)}\right) dx, &
\text{for}\mathrm{~}\beta=0.
\end{array}
\right. \label{EQ:definition_DPD}%$$ Let $X_{1},X_{2},\ldots ,X_n$ be a random sample of size $n$ from a $\mathcal{N}(\mu,\sigma^2)$ distribution, where both parameters are unknown. Let $f_{\mu,\sigma }(x)$ represent the density function of a $\mathcal{N}(\mu,\sigma^2)$ variable. For a given $\beta $, we get the minimum density power divergence estimators (MDPDEs) $\widehat{\mu }%
_{\beta }$ and $\widehat{\sigma }_{\beta}$ of $\mu$ and $\sigma$ by minimizing the following function over $\mu$ and $\sigma$ $$\int_{\mathbb{R}}f_{\mu,\sigma }^{1+\beta }(x)dx-\left( 1+\frac{1}{%
\beta }\right) \frac{1}{n}\sum_{i=1}^{n}f_{\mu,\sigma }^{\beta
}(X_i),\text{\qquad for }\beta >0, \label{1}$$ and $$-\frac{1}{n}\sum_{i=1}^{n}\log f_{\mu,\sigma }(X_i),\text{%
\qquad for }\beta =0. \label{EQ:1.0}$$ For $\beta =0$, the objective function in (\[EQ:1.0\]) is the negative of the usual log likelihood and has the classical maximum likelihood estimator as the minimizer. For a normal density the function in (\[1\]) simplifies to $$h_{n,\beta }(\mu,\sigma )=\frac{1}{\sigma ^{\beta }(2\pi )^{\frac{%
\beta }{2}}}\left\{ \frac{1}{\left( 1+\beta \right) ^{3/2}}-\frac{1}{%
n\beta }\sum_{i=1}^{n}\exp \left( -\frac{1}{2}\left( \frac{X_i-\mu}{{\sigma }}\right) ^{2}\beta \right) \right\} .$$ In order to get $\widehat{\mu }_{\beta }$ and $\widehat{\sigma}_\beta $, we have to solve the estimating equation $$\mathbf{h'}_{n,\beta }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta ) =%
\begin{pmatrix}
_{1}h'_{n,\beta }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta ) \\
_{2}h'_{n,\beta }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta )%
\end{pmatrix} = \boldsymbol{0}_2,
\label{h1}$$ where $$_{1}h_{n,\beta }^{\prime }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta ) =\left. \frac{%
\partial h_{n,\beta }(\mu,\widehat{\sigma}_\beta )}{\partial \mu}%
\right\vert _{\mu=\widehat{\mu }_{\beta }},\qquad _{2}h_{n,\beta }^{\prime }(\widehat{\mu }_{\beta },\widehat{\sigma}_\beta )
=\left. \frac{\partial h_{n,\beta }(\widehat{\mu }_{\beta },\sigma )}{%
\partial \sigma }\right\vert _{\sigma =\widehat{\sigma}_\beta },$$ and $\boldsymbol{0}_2$ represents a zero vector of length 2. We denote $$\mathbf{H}_{n,\beta }( \mu_0,\sigma_0 ) =\left(
\begin{array}{cc}
_{11}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right) &
_{12}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right)
\\
_{21}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right) &
_{22}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right)
\end{array}%
\right),$$where $$\begin{aligned}
_{11}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right) &
=\left. \dfrac{\partial ^{2}h_{n,\beta }\left( \mu,\sigma
_{0}\right) }{\partial \mu^{2}}\right\vert _{\mu =\mu_0},\qquad
_{12}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right)
=\left. \dfrac{\partial ^{2}h_{n,\beta }\left( \mu,\sigma \right) }{%
\partial \mu\partial \sigma }\right\vert _{\mu =\mu_0,\sigma
=\sigma_0}, \\
_{21}h_{n,\beta }^{\prime \prime }\left( \mu_0,\sigma_0\right) &
=\left. \dfrac{\partial ^{2}h_{n,\beta }\left( \mu,\sigma \right) }{%
\partial \sigma \partial \mu}\right\vert _{\mu =\mu_0,\sigma
=\sigma_0},\qquad _{22}h_{n,\beta }^{\prime \prime }\left( \mu
_{0},\sigma_0\right) =\left. \dfrac{\partial ^{2}h_{n,\beta }\left(
\mu_0,\sigma \right) }{\partial \sigma^2}\right\vert _{\sigma =\sigma
_{0}}.\end{aligned}$$Using a Taylor series expansion of the function in equation (\[h1\]), it is easy to show that $$\begin{aligned}
\sqrt{n}
\begin{pmatrix}
\widehat{\mu }_{\beta } - \mu_0\\
\widehat{\sigma}_\beta - \sigma_0
\end{pmatrix}
&=& \sqrt{n} \mathbf{H}_{n,\beta }^{-1}( \mu_0,\sigma_0 ) \boldsymbol{h}'_{n,\beta }(\mu_0,\sigma_0 ) + o_p(1) \nonumber\\
&=& \sqrt{n} \mathbf{J}_\beta^{-1}( \sigma_0 ) \boldsymbol{h}'_{n,\beta }(\mu_0,\sigma_0 ) + o_p(1),
\label{muSigma}\end{aligned}$$ where $$\boldsymbol{J}_{\beta }(\sigma_0) = \lim_{n \rightarrow \infty }%
\mathbf{H}_{n,\beta } ( \mu_0,\sigma_0 )
= \frac{1}{%
\sqrt{1+\beta }\left( 2\pi \right) ^{\beta /2}\sigma_0 ^{2+\beta }}\left(
\begin{array}{cc}
\frac{1}{1+\beta } & 0 \\
0 & \frac{\beta ^{2}+2}{\left( 1+\beta \right) ^{2}}%
\end{array}%
\right).$$ The joint distribution of $\widehat{\mu }_{\beta }$ and $\widehat{\sigma}_\beta$ then follows (see [@MR3011625]) from the result that $$\sqrt{n}\mathbf{h'}_{n,\beta}(\mu _0,\sigma_0)\underset{%
n\rightarrow \infty }{\overset{\mathcal{L}}{\longrightarrow }}\mathcal{N}%
\left( \boldsymbol{0}_{2},\boldsymbol{K}_{\beta }(\sigma_0)\right) ,
\label{1.1}$$ where $$\begin{aligned}
\boldsymbol{K}_{\beta }(\sigma_0) &=& \left( K_{ij,\beta
}(\sigma_0)\right) _{i,j=1,2} \nonumber\\
&=&
\frac{1}{\sigma_0 ^{2+2\beta
}\left( 2\pi \right) ^{\beta }}\left( \frac{1}{(1+2\beta )^{3/2}}\left(
\begin{array}{cc}
1 & 0 \\
0 & \frac{4\beta ^{2}+2}{1+2\beta }%
\end{array}%
\right) -\left(
\begin{array}{cc}
0 & 0 \\
0 & \frac{\beta ^{2}}{(1+\beta )^{3}}%
\end{array}%
\right) \right) .
\label{2}\end{aligned}$$ We will use the above results to obtain the MDPDEs of the parameters in the two sample setup mentioned below.
Suppose $X_{1},X_{2},\ldots ,X_{n_1}$ is a random sample of size $n_1$ from $X$ which has a $\mathcal{N}(\mu_1,\sigma^2)$ distribution, and $%
Y_{1},Y_{2},\ldots ,Y_{n_2}$ is a random sample of size $n_2$ from $Y$ which has a $\mathcal{N}(\mu_2,\sigma^2)$ distribution; all three parameters are unknown. Let $f_{\mu_1,\sigma }(x)$ and $f_{\mu
_{2},\sigma }(y)$ be the density functions of $X$ and $Y$ respectively. Let us denote the set of unknown parameters by $\boldsymbol{\eta }=(\mu_1,\mu
_{2},\sigma )^{T}$. The MDPDE of $\boldsymbol{\eta }$, denoted by $\widehat{\boldsymbol{\eta }}_{\beta }=(\widehat{\mu}_{1\beta },\widehat{\mu}_{2\beta
},\widehat{\sigma}_{\beta })^{T}$, is obtained by minimizing the following function $$h_{n_1,n_2,\beta }(\boldsymbol{\eta })=\frac{1}{n_1+n_2}%
\left( n_{1\text{ }}h_{n_1,\beta }(\mu_1,\sigma )+n_{2\text{ }%
}h_{n_2,\beta }\left( \mu_2,\sigma \right) \right) .
\label{hn12}$$ It may be noticed that $\hat{\mu}_{1\beta }$ is based only on the first term of the above function, and similarly $\hat{\mu}_{2\beta }$ depends only on the second term. Therefore, the estimating equations are given by $_{1}h_{n_{i},\beta }^{\prime }\left( \mu _{i},\sigma \right) =0$, $i=1,2$, and $_{2}h_{n_1,n_2,\beta }^{\prime }(\boldsymbol{\eta } )=0$, where $$_{2}h_{n_1,n_2,\beta }^{\prime }(\boldsymbol{\eta })=\frac{%
\partial h_{n_1,n_2,\beta }(\boldsymbol{\eta } )}{\partial \sigma }%
=\frac{1}{n_1+n_2}\left( n_1\,\allowbreak _{2}h_{n_1,\beta }^{\prime
}\left( \mu_1,\sigma \right) +n_2\,\allowbreak _{2}h_{n_2,\beta
}^{\prime }\left( \mu_2,\sigma \right) \right) .
\label{h2n12}$$ For $\beta =0$, the above equations can be explicitly solved to get the MDPDEs for this case. It is easily seen that $\widehat{\mu }_{10}=\bar{X%
}$ and $\widehat{\mu }_{20}=\bar{Y}$. Moreover, using equation (\[EQ:1.0\]) we get from (\[hn12\]) $$\begin{aligned}
& h_{n_1,n_2,\beta =0}(\widehat{\boldsymbol{\eta }}_0 ) \\
& =-\frac{1}{n_1+n_2}\left( n_1\frac{1}{n_1}\log
\prod_{i=1}^{n_1}f_{\widehat{\mu }_{10},\widehat{\sigma}_0 }(X_i)+n_2\frac{1}{n_2}\log
\prod_{i=1}^{n_2}f_{\widehat{\mu }_{20},\widehat{\sigma}_0 }(Y_{i})\right) \\
& =\frac{1}{n_1+n_2}\left( (n_1+n_2)\log \widehat{\sigma}_0 +\sum_{i=1}^{n_1}%
\frac{\left( X_i-\bar{X}\right) ^{2}}{2\widehat{\sigma}_0^2}+\sum_{i=1}^{n_2}%
\frac{\left( Y_{i}-\bar{Y}\right) ^{2}}{2\widehat{\sigma}_0^2}+(n_1+n_2)\log
\sqrt{2\pi }\right) .\end{aligned}$$ So, $$_{2}h_{n_1,n_2,\beta }^{\prime }(\widehat{\boldsymbol{\eta }}_0 )
=\frac{1}{\widehat{\sigma}_0 }-\frac{1}{%
\widehat{\sigma}_0 ^{3}(n_1+n_2)}\left\{ (n_1-1)S_{1}^{2} + (n_2-1)S_{2}^{2}\right\} ,$$which leads to the solution $$\widehat{\sigma }_{0}=\left( \frac{(n_1-1)S_{1}^{2}+(n_2-1)S_{2}^{2}}{%
n_1+n_2}\right) ^{\frac{1}{2}}. \label{sig0}$$ Therefore, for $\beta=0$ the MDPDEs turn out to be the MLEs of the corresponding parameters. The following theorem gives the asymptotic distribution of the MDPDE of $\boldsymbol{\eta }$ for a given $\beta$.
\[Th0\]We consider two normal populations with unknown means $\mu_1$ and $\mu_2$ and unknown but common variance $\sigma^2.$ Let $$w=\lim_{n_1,n_2\rightarrow \infty }\frac{n_1}{n_1+n_2}
\label{EQ:w}$$ be the limiting proportion of observations from the first population in the whole sample. We assume that $w \in (0,1)$. Then, the minimum density power divergence estimator $\widehat{\boldsymbol{\eta }}_{\beta }$ of $\boldsymbol{\eta}$ has the asymptotic distribution given by $$\sqrt{\frac{n_1n_2}{n_1+n_2}}(\widehat{\boldsymbol{\eta }}_{\beta }-%
\boldsymbol{\eta }_{0})\underset{n_1,n_2\rightarrow \infty }{\overset{%
\mathcal{L}}{\longrightarrow }}\mathcal{N}\left( \boldsymbol{0}_{3},%
\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)\right) , \label{eqTh0}$$where $\boldsymbol{\eta }_{0}=(\mu _{10},\mu _{20},\sigma_0)^{T}$ is the true value of $\boldsymbol{\eta }$, and $$\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)=\sigma_0^{2}\left(
\begin{array}{ccc}
\left( 1-w\right) \frac{\left( \beta +1\right) ^{3}}{\left( 2\beta +1\right)
^{\frac{3}{2}}} & 0 & 0 \\
0 & w\frac{\left( \beta +1\right) ^{3}}{\left( 2\beta +1\right) ^{\frac{3}{2}%
}} & 0 \\
0 & 0 & w\left( 1-w\right) \frac{\left( \beta +1\right) ^{5}}{\left( \beta
^{2}+2\right) ^{2}}\left( \frac{4\beta ^{2}+2}{(1+2\beta )^{5/2}}-\frac{%
\beta ^{2}}{(1+\beta )^{3}}\right)%
\end{array}%
\right) . \label{8}$$
See Appendix.
The Asymptotic Distribution of the DPD Test Statistic {#SEC:Test}
=====================================================
Let $f_{\mu_1,\sigma _{1}}(x)$ and $f_{\mu_2,\sigma _{2}}(y)$ be the density functions of $X\sim \mathcal{N}(\mu_1,\sigma _{1})$ and $Y\sim
\mathcal{N}(\mu_2,\sigma _{2})$ respectively. The density power divergence measure between the densities of $X$ and $Y$, for $\gamma >0$, is given by $$\begin{aligned}
d_{\gamma }(f_{\mu_1,\sigma _{1}},f_{\mu_2,\sigma _{2}}) =&\frac{1}{%
\sigma _{2}^{\gamma }\sqrt{1+\gamma }\left( 2\pi \right) ^{\gamma /2}}+%
\frac{1}{\gamma\sigma _{1}^{\gamma }\sqrt{1+\gamma }\left( 2\pi \right)
^{\gamma /2}} \\
& -\frac{\gamma +1}{\gamma \sigma _{2}^{\gamma -1}(\gamma \sigma
_{1}^{2}+\sigma _{2}^{2})^{1/2}\left( 2\pi \right) ^{\gamma/2 }} \\
& \times \exp \left\{ \frac{1}{2}\left[ -\left( \tfrac{\mu_2^{2}}{\left(
\frac{\sigma _{2}}{\sqrt{\gamma }}\right) ^{2}}+\tfrac{\mu_1^{2}}{\sigma
_{1}^{2}}\right) +\tfrac{\left( \sigma _{1}^{2}\mu_2+\mu_1\left( \frac{%
\sigma _{2}}{\sqrt{\gamma }}\right) ^{2}\right) ^{2}}{\left( \sigma
_{1}^{2}+\left( \frac{\sigma _{2}}{\sqrt{\gamma }}\right) ^{2}\right) \left(
\frac{\sigma _{2}}{\sqrt{\gamma }}\right) ^{2}\sigma _{1}^{2}}\right]
\right\} ,\end{aligned}$$ and for $\gamma =0$$$d_\gamma(f_{\mu_1,\sigma _{1}},f_{\mu_2,\sigma _{2}})=\log {\frac{\sigma
_{2}}{\sigma _{1}}}-\frac{1}{2}+\frac{\sigma _{1}^{2}}{2\sigma _{2}^{2}}+%
\frac{1}{2\sigma _{2}^{2}}(\mu_1-\mu_2)^{2}.$$To test the null hypothesis given in (\[EQ:0\]), under the assumption that $\sigma _{1}=\sigma _{2}=\sigma $, we will consider the divergence between the two normal populations with the estimated parameters; this yields $$d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })=\left\{
\begin{array}{ll}
\frac{\sqrt{1+\gamma }}{\gamma \left( \sqrt{2\pi }\widehat{\sigma }_{\beta
}\right) ^{\gamma }}\left[ 1-\exp \left\{ -\frac{\gamma }{2(\gamma
+1)}\left( \frac{\widehat{\mu }_{1\beta }-\widehat{\mu }_{2\beta }}{\widehat{%
\sigma }_{\beta }}\right) ^{2}\right\} \right] , & \text{ for }\gamma >0, \\
\frac{1}{2}\left( \frac{\widehat{\mu }_{1\beta }-\widehat{\mu }_{2\beta }}{%
\widehat{\sigma}_\beta }\right) ^{2}, & \text{ for }\gamma =0.%
\end{array}%
\right. \label{EQ:initial_statistic}$$Naturally, we will reject the null hypothesis for large values of $d_{\gamma }(f_{\widehat{\mu }%
_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }})$. To propose the test in a very general setup we have considered two possibly distinct tuning parameters $\gamma$ and $\beta$ in the above expression; the parameter $\gamma$ represents the tuning parameters of the divergence, and the parameter $\beta$ represents the tuning parameter of the MDPDEs. In order to determine the critical region of this test we will find (later in Theorem \[Th2\]) the asymptotic null distribution of the test statistic based on ([EQ:initial\_statistic]{}), standardized with a suitable scaling constant involving $n_1$ and $n_2$.
\[Th1\] For $\gamma >0$, let us define $\boldsymbol{t}_{\gamma }\left( \boldsymbol{\eta }\right) =(t_{\gamma, 1}(%
\boldsymbol{\eta }),t_{\gamma, 2}(\boldsymbol{\eta }),t_{\gamma, 3}(%
\boldsymbol{\eta }))^{T}$, with$$\begin{aligned}
t_{\gamma, 1}(\boldsymbol{\eta })& =\frac{\frac{\mu_1-\mu_2}{\sigma }}{%
\sqrt{1+\gamma }\left( \sqrt{2\pi }\right) ^{\gamma }\sigma ^{\gamma +1}}%
\exp \left\{ -\frac{1}{2}\tfrac{\gamma }{\gamma +1}\left( \tfrac{\mu
_{1}-\mu_2}{\sigma }\right) ^{2}\right\} , \label{t1} \\
%
t_{\gamma, 2}(\boldsymbol{\eta })& =-t_{1}(\boldsymbol{\eta }), \label{t2} \\
%
t_{\gamma, 3}(\boldsymbol{\eta })& = - \tfrac{\sqrt{1+\gamma }}{\left( \sqrt{%
2\pi }\right) ^{\gamma }\sigma ^{\gamma +1}}\left[ 1-\left( 1 - \tfrac{1}{%
1+\gamma }\left( \tfrac{\mu_1-\mu_2}{\sigma }\right) ^{2}\right) \exp
\left\{ -\tfrac{1}{2}\tfrac{\gamma }{\gamma +1}\left( \tfrac{\mu_1-\mu
_{2}}{\sigma }\right) ^{2}\right\} \right] . \label{t3}\end{aligned}$$Then, for $w \in (0,1)$ as defined in (\[EQ:w\]) we have $$\sqrt{\frac{n_1n_2}{n_1+n_2}}\left( d_{\gamma }(f_{\widehat{\mu }%
_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }})-d_{\gamma }(f_{\mu _{10},\sigma_0},f_{\mu
_{20},\sigma_0})\right) \underset{n_1,n_2\rightarrow \infty }{\overset%
{\mathcal{L}}{\longrightarrow }}\mathcal{N}\left( 0, \sigma_\gamma^2
\right) , \label{resTh1}$$ where $$\sigma_\gamma^2 = \boldsymbol{t}_{\gamma
}^{T}\left( \boldsymbol{\eta }_{0}\right) \boldsymbol{\Sigma }_{w,\beta
}(\sigma_0)\boldsymbol{t}_{\gamma }\left( \boldsymbol{\eta }_{0}\right),
\label{sigma_gamma}$$ and $\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)$ is given in (\[8\]).
See Appendix.
Notice that $\boldsymbol{t}_{\gamma }^{T}\left( \boldsymbol{\eta }%
_{0}\right) \boldsymbol{\Sigma }_{w,\beta }(\sigma_0)\boldsymbol{t}%
_{\gamma }\left( \boldsymbol{\eta }_{0}\right) \geq 0$. If $\mu _{10}\neq
\mu _{20}$, we observe that $\boldsymbol{t}_{\gamma }\left( \boldsymbol{%
\eta }_{0}\right) \neq \boldsymbol{0}_{3}$, and since $\boldsymbol{\Sigma }%
_{w,\beta }(\sigma_0)$ is positive definite matrix, we have $\boldsymbol{t}%
_{\gamma }^{T}\left( \boldsymbol{\eta }_{0}\right) \boldsymbol{\Sigma }%
_{w,\beta }(\sigma_0)\boldsymbol{t}_{\gamma }\left( \boldsymbol{\eta }%
_{0}\right) >0$. But for $\mu _{10}=\mu _{20}$, $\boldsymbol{t}%
_{\gamma }\left( \boldsymbol{\eta }_{0}\right) =\boldsymbol{0}_{3}$, and hence $\boldsymbol{t}_{\gamma }^{T}\left( \boldsymbol{\eta }_{0}\right)
\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)\boldsymbol{t}_{\gamma }\left(
\boldsymbol{\eta }_{0}\right) =0$. Therefore, to get the asymptotic distribution of the test statistic under the null hypothesis we need a higher order scaling involving $n_1$ and $n_2$ to the quantity given in (\[EQ:initial\_statistic\]).
\[Th2\]Let $w\in (0,1)$ as defined in (\[EQ:w\]) and $\gamma >0$. Then, under the null hypothesis, we have $$S_{\gamma }\left( \widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }\right) =\frac{2n_1n_2}{n_1+n_2}\frac{d_{\gamma
}(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }%
_{2\beta },\widehat{\sigma}_\beta })}{\lambda _{\beta ,\gamma
}\,\allowbreak (\widehat{\sigma}_\beta )}\underset{n_1,n_2\rightarrow
\infty }{\overset{\mathcal{L}}{\longrightarrow }}\chi ^{2}(1),
\label{EQ:test}$$where $$\lambda _{\beta ,\gamma }(\widehat{\sigma}_\beta )=\frac{\left( \beta
+1\right) ^{3}\left( 2\beta +1\right) ^{-\frac{3}{2}}}{\widehat{\sigma }%
_{\beta }^{\gamma }\left( 2\pi \right) ^{\frac{\gamma }{2}}\left( \gamma
+1\right) ^{\frac{1}{2}}}.
\label{lambda1}$$
See Appendix.
The above result indicates that the density power divergence test for the hypothesis in (\[EQ:0\]) can be based on the statistic $S_{\gamma }\left(
\widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta },\widehat{\sigma }_{\beta
}\right) $, where the critical region corresponding to significance level $%
\alpha $ is given by the set of points satisfying $$S_{\gamma }\left( \widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }\right) >\chi _{\alpha }^{2}(1).$$ Using the result of Theorem \[Th1\] we can get an approximation of the power function of the test statistic. We consider $\mu _{10}\neq \mu _{20}$. In the following we will let $\lambda$ denote the quantity defined in equation (\[lambda1\]) to keep the notation simple. The power function is then given by $$\begin{aligned}
\eta_{\gamma,\beta}(\mu _{10}, \mu _{20}, \sigma_0) &=& P\left( S_{\gamma }\left( \widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }\right) >\chi _{\alpha }^{2}(1) \right) \\
%
&=& P\left( \frac{2}{\lambda}\frac{n_1n_2}{n_1+n_2} d_{\gamma }(f_{\widehat{\mu }%
_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }}) >\chi _{\alpha }^{2}(1) \right) \\
%
&=& P\Bigg( \sqrt{\frac{n_1n_2}{n_1+n_2}}\left( d_{\gamma }(f_{\widehat{\mu }%
_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }_{2\beta },\widehat{%
\sigma }_{\beta }})-d_{\gamma }(f_{\mu _{10},\sigma_0},f_{\mu
_{20},\sigma_0})\right) \\
%
&& \ \ \ > \frac{\lambda}{2} \sqrt{\frac{n_1 + n_2}{n_1n_2}} \left(\chi _{\alpha }^{2}(1)
- \frac{2 n_1 n_2}{\lambda (n_1 + n_2)} d_{\gamma }(f_{\mu _{10},\sigma_0},f_{\mu
_{20},\sigma_0}) \right)\Bigg) \\
%
&= & 1 - \Phi_n\left( \frac{\lambda}{2\sigma_\gamma}\sqrt{\frac{n_1n_2}{n_1+n_2}} \left(\chi _{\alpha }^{2}(1)
- \frac{2 n_1n_2}{n_1+n_2} d_{\gamma }(f_{\mu _{10},\sigma_0},f_{\mu
_{20},\sigma_0}) \right)\right),\end{aligned}$$ where $\Phi _{n}$ is a sequence of distributions functions tending uniformly to the standard normal distribution function $\Phi$, and $\sigma_\gamma$ is defined in (\[sigma\_gamma\]). We observe that if $\mu _{10}\neq \mu _{20}$ $$\lim_{n_1,n_2\rightarrow \infty }\eta_{\gamma,\beta}(\mu _{10}, \mu _{20}, \sigma_0) =1.$$ Therefore, the test is consistent in the Frasar’s sense [@MR0093863].
\[Cor1\]Let $w\in (0,1)$ as defined in (\[EQ:w\]) and $\gamma =\beta
=0$. Then, under the null hypothesis defined in (\[EQ:0\]), we have $$S_{0}\left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{\sigma }%
_{0}\right) =\frac{n_1n_2}{n_1+n_2}\frac{\left( \bar{X}-%
\bar{Y}\right) ^{2}}{\widehat{\sigma }_{0}^{2}}\underset{%
n_1,n_2\rightarrow \infty }{\overset{\mathcal{L}}{\longrightarrow }}\chi
^{2}(1). \label{eqCor1}$$
The proof of the corollary is straightforward. The test statistic given in the above corollary is closely related to the likelihood ratio test. This correspondence is described in the next corollary.
\[Cor2\]For a given sample the value of the test statistic $S_{0}\left(
\widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{\sigma }_{0}\right) $, defined in (\[eqCor1\]), does not exactly match the value of the likelihood ratio test statistic$$-2\log \Lambda \left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{%
\sigma }_{0}\right) =(n_1+n_2)\log \left( 1+\frac{n_1n_2}{\left(
n_1+n_2\right) ^{2}}\frac{(\bar{X}-\bar{Y})^2}{%
\widehat{\sigma }_{0}^{2}}\right) ,$$where $\widehat{\sigma }_{0}^{2}$ is defined in (\[sig0\]). However, as $%
n_1,n_2\rightarrow \infty $, and $w \in (0,1)$ as defined in (\[EQ:w\]), both test statistics are asymptotically equivalent.
Let us denote $\Theta _{0}=\left\{ \left( \mu ,\mu ,\sigma \right)^T :\mu
\in \mathbb{R},\sigma \in \mathbb{R}^{+}\right\} ,$ $\Theta =\left\{ \left( \mu_1,\mu_2,\sigma \right)^T
:\mu_1,\mu_2\in \mathbb{R},\sigma \in \mathbb{R}^{+}\right\}$. The likelihood function is given by $$\mathcal{L}(\mu_1,\mu_2,\sigma
)=\prod_{i=1}^{n_1}\prod_{j=1}^{n_2}f_{\mu_1,\sigma }(X_i)f_{\mu
_{2},\sigma }(Y_{j}).$$It can be shown that $$\Lambda \left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{\sigma }%
_{0}\right) =\frac{\sup_{\mu_1,\mu_2,\sigma \in \Theta _{0}}\mathcal{L}%
(\mu_1,\mu_2,\sigma )}{\sup_{\mu_1,\mu_2,\sigma \in \Theta }%
\mathcal{L}(\mu_1,\mu_2,\sigma )}=\left( \frac{\sum_{i=1}^{n_1}%
\left( X_i-\widetilde{\mu }\right) ^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-%
\widetilde{\mu }\right) ^{2}}{\sum_{i=1}^{n_1}\left( X_i-\bar{X}%
\right) ^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\bar{Y}\right) ^{2}}\right)
^{-\frac{n_1+n_2}{2}},$$where $\widetilde{\mu }=\frac{n_1}{n_1+n_2}\bar{X}+\frac{n_2}{%
n_1+n_2}\bar{Y}$. Therefore, asymptotically, the likelihood ratio test rejects the null hypothesis $H_{0}$ if $$-2\log \Lambda \left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{%
\sigma }_{0}\right) =(n_1+n_2)\log \left( \frac{\sum_{i=1}^{n_1}\left(
X_i-\widetilde{\mu }\right) ^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\widetilde{%
\mu }\right) ^{2}}{\sum_{i=1}^{n_1}\left( X_i-\bar{X}\right)
^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\bar{Y}\right) ^{2}}\right) >\chi
^{2}(1).$$ Now $$\begin{aligned}
\sum_{i=1}^{n_1}\left( X_i-\widetilde{\mu }\right)
^{2}+\sum\limits_{i=1}^{n_2}\left( Y_{i}-\widetilde{\mu }\right) ^{2}&
=\sum\limits_{i=1}^{n_1}\left( X_i-\bar{X}\right) ^{2}+n_1\left(
\bar{X}-\widetilde{\mu }\right) ^{2}+\sum\limits_{i=1}^{n_2}\left(
Y_{i}-\bar{Y}\right) ^{2}+n_2\left( \bar{Y}-\widetilde{\mu }%
\right) ^{2} \\
& =\sum\limits_{i=1}^{n_1}\left( X_i-\bar{X}\right)
^{2}+\sum\limits_{i=1}^{n_2}\left( Y_{i}-\bar{Y}\right) ^{2}+\frac{%
n_1n_2}{n_1+n_2}(\bar{X}-\bar{Y})^{2}.\end{aligned}$$ So $$\begin{aligned}
-2\log \Lambda \left( \widehat{\mu }_{10},\widehat{\mu }_{20},\widehat{%
\sigma }_{0}\right) & = & (n_1+n_2)\log \left( \frac{\sum_{i=1}^{n_1}\left( X_i-\widetilde{\mu }\right)
^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\widetilde{\mu }\right) ^{2}}{%
\sum_{i=1}^{n_1}\left( X_i-\bar{X}\right)
^{2}+\sum_{i=1}^{n_2}\left( Y_{i}-\bar{Y}\right) ^{2}}\right)
\\
& =& (n_1+n_2)\log \left( 1+\frac{n_1n_2}{\left( n_1+n_2\right)
^{2}}\frac{(\bar{X}-\bar{Y})^{2}}{\widehat{\sigma }_{0}^{2}}%
\right)
\\
& = &\displaystyle \frac{n_1n_2}{\left( n_1+n_2\right)
}\frac{(\bar{X}-\bar{Y})^{2}}{\widehat{\sigma }_{0}^{2}} + R_{n_1,n_2},\end{aligned}$$where $R_{n_1,n_2} \rightarrow 0$ in probability as $n_1,n_2\rightarrow \infty$ and $w \in (0,1)$. Thus, the test statistics $-2\log \Lambda \left( \widehat{\mu }_{10},\widehat{\mu }%
_{20},\widehat{\sigma }_{0}\right)$ and $S_{0}\left( \widehat{\mu }_{10},%
\widehat{\mu }_{20},\widehat{\sigma }_{0}\right)$ are asymptotically equivalent.
Numerical Studies {#SEC:numerical}
=================
Simulation Study
----------------
In this section we study the performance of our proposed test statistics through simulated data. We have generated two random samples $%
X_{1},X_{2},\ldots ,X_{n_1}$ and $Y_{1},Y_{2},\ldots ,Y_{n_2}$ from $%
\mathcal{N}(\mu_1,\sigma^2)$ and $\mathcal{N}(\mu_2,\sigma^2)$ respectively; thus the total sample size is $n=n_1+n_2$. The value of $w$ in (\[EQ:w\]) is taken to be 0.6, and the sample size from the first population is $n_1=[wn]+1$, where $[x]$ denotes the integer part of $x$. Our aim is to test the null hypothesis given in (\[EQ:0\]). We have taken $%
\sigma^2=1$ in this study. We have compared the results of the ordinary two sample $t$-test and the density power divergence tests with four different values of the tuning parameter $\gamma =\beta =0,0.05, 0.1$ and 0.15; let DPD($\gamma $) represent the DPD test with tuning parameter $%
\gamma $. The nominal level of the tests are 0.05, and all tests are replicated 1,000 times.
In the first case we have taken $\mu_{1}=\mu_{2}=0$. Plot (a) in Figure \[fig:level\_power\] shows the observed levels of the five test statistics for different values of the sample size (obtained as the proportion of test statistics, in the $1000$ replications, that exceed the nominal $\chi ^{2}$ critical value at 5% level of significance). It is seen that the observed levels of the $t$-test are very close to the nominal level. On the other hand, the DPD tests are slightly liberal for very small sample sizes and lead to somewhat inflated observed levels. However, as the sample size increases the levels settle down rapidly around the nominal level.
Next, we have generated data with $\mu_1 = 0$ but $\mu_2 = 1$. The observed power of the tests are presented in plot (b) of Figure \[fig:level\_power\]. There is not much difference among the observed powers in this plot. The DPD tests have slightly higher power than the $t$-test in very small sample sizes. This, however, must be a consequence of the fact that the observed levels of these tests are higher than the nominal level (and higher than the observed level of the $t$-statistic) in small samples.
Now we check the performance of the tests under contaminated data. So, we have generated $n_2$ observations $Y_{1},Y_{2},\ldots ,Y_{n_2}$ from $0.95%
\mathcal{N}(\mu_2,1)+0.05\mathcal{N}(-10,1)$, whereas the $n_1$ observations representing the first population come from the pure $\mathcal{N%
}(\mu_1,1)$ distribution. To evaluate the stability of the level of the tests for testing the hypothesis in (\[EQ:0\]), we have taken $\mu_1=\mu
_{2}=0 $. Figure \[fig:level\_power\] (c) presents the levels for different values of the sample sizes. It may be observed that there is a drastic inflation in the levels for the $t$-test and DPD(0) test statistic, but the levels of the other DPD test statistics remain stable.
Figure \[fig:level\_power\] (d) shows the power of the tests under the contaminated setup considered in the previous paragraph, when $\mu_1 = 0$ and $\mu_2 = 1$. Here, the presence of the outliers lead to a sharp drop in power for the $t$-test and the DPD(0) test. On the other hand, the other tests are clearly more resistant, and hold their power much better as $\gamma$ increases.
On the whole, therefore, it appears that in comparison to the $t$-test, many of our DPD tests are quite competitive in performance when the data come from the pure model. Under contaminated data, however, the robustness properties of the DPD tests appear to be far superior, and they do much better at maintaining the stability of the level and the power in such cases.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(a) Simulated levels of the DPD tests for pure data; (b) simulated power of the DPD tests for pure data; (c) simulated levels of the DPD tests for contaminated data; (d) simulated power of the DPD tests for contaminated data.[]{data-label="fig:level_power"}](Two_sample_t_nonpara_mu1_0,mu2_0,sigma_1,w_0.6,n_20_1_100,rep_1000_v2.eps "fig:"){height="7.5cm" width="7.5cm"} ![(a) Simulated levels of the DPD tests for pure data; (b) simulated power of the DPD tests for pure data; (c) simulated levels of the DPD tests for contaminated data; (d) simulated power of the DPD tests for contaminated data.[]{data-label="fig:level_power"}](Two_sample_t_nonpara_mu1_0,mu2_1,sigma_1,w_0.6,n_20_1_100,rep_1000_v2.eps "fig:"){height="7.5cm" width="7.5cm"}
![(a) Simulated levels of the DPD tests for pure data; (b) simulated power of the DPD tests for pure data; (c) simulated levels of the DPD tests for contaminated data; (d) simulated power of the DPD tests for contaminated data.[]{data-label="fig:level_power"}](Two_sample_t_nonpara_mu1_0,mu2_0,sigma_1,w_0.6,p_0.05,muC_-10,n_20_1_100,rep_1000_v2.eps "fig:"){height="7.5cm" width="7.5cm"} ![(a) Simulated levels of the DPD tests for pure data; (b) simulated power of the DPD tests for pure data; (c) simulated levels of the DPD tests for contaminated data; (d) simulated power of the DPD tests for contaminated data.[]{data-label="fig:level_power"}](Two_sample_t_nonpara_mu1_0,mu2_1,sigma_1,w_0.6,p_0.05,muC_-10,n_20_1_100,rep_1000_v2.eps "fig:"){height="7.5cm" width="7.5cm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Comparison with Other Robust Tests
----------------------------------
In this section we compare the DPD test with some other popular robust tests. For comparison we have used a parametric test – the two sample trimmed $t$-test proposed by [@yuen1973approximate], as well as two non-parametric tests – the Kolmogorov-Smirnov test (KS-test) and the Wilcoxon two-sample test (which is also known as the Mann-Whitney $U$-test). For the two sample trimmed $t$-test we have trimmed 20% extreme observations from each of the data sets of $X$ and $Y$. The set up, the parameters taken for the simulation and the level of contamination are exactly the same as in the previous section. For comparison we have used only one DPD test in this case, that corresponding to tuning parameter 0.1. To emphasize the robustness properties of these tests we have also included the two sample $t$-test in this investigation. The results are presented in Figure \[fig:level\_power\_v1\].
Figure \[fig:level\_power\_v1\] (a) shows that the observed levels of all the robust tests are very close to the nominal level of 0.05 for the pure normal data. The same result is observed in Figure \[fig:level\_power\_v1\] (c) for the contaminated data. On the other hand, if we consider the observed power of the tests the DPD test is much more powerful than the other tests. Specifically, for the contaminated data, the DPD test does significantly better than the others in holding on to its power. Therefore, on the whole, the DPD tests are not only superior to the two sample $t$-test under contamination, but they also appear to be competitive or better than the other popular robust tests as far as this simulation study is concerned.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(a) Simulated levels of different tests for pure data; (b) simulated power of different tests for pure data; (c) simulated levels of different tests for contaminated data; (d) simulated power of different tests for contaminated data.[]{data-label="fig:level_power_v1"}](Two_sample_t_nonpara_mu1_0,mu2_0,sigma_1,w_0.6,n_20_1_100,rep_1000_v1.eps "fig:"){height="7.5cm" width="7.5cm"} ![(a) Simulated levels of different tests for pure data; (b) simulated power of different tests for pure data; (c) simulated levels of different tests for contaminated data; (d) simulated power of different tests for contaminated data.[]{data-label="fig:level_power_v1"}](Two_sample_t_nonpara_mu1_0,mu2_1,sigma_1,w_0.6,n_20_1_100,rep_1000_v1.eps "fig:"){height="7.5cm" width="7.5cm"}
![(a) Simulated levels of different tests for pure data; (b) simulated power of different tests for pure data; (c) simulated levels of different tests for contaminated data; (d) simulated power of different tests for contaminated data.[]{data-label="fig:level_power_v1"}](Two_sample_t_nonpara_mu1_0,mu2_0,sigma_1,w_0.6,p_0.05,muC_-10,n_20_1_100,rep_1000_v1.eps "fig:"){height="7.5cm" width="7.5cm"} ![(a) Simulated levels of different tests for pure data; (b) simulated power of different tests for pure data; (c) simulated levels of different tests for contaminated data; (d) simulated power of different tests for contaminated data.[]{data-label="fig:level_power_v1"}](Two_sample_t_nonpara_mu1_0,mu2_1,sigma_1,w_0.6,p_0.05,muC_-10,n_20_1_100,rep_1000_v1.eps "fig:"){height="7.5cm" width="7.5cm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Real Data Examples
------------------
**Example 2 (Lead Measurement data):** In Table \[TAB:Lake\] the lead measurement data ([@MR922042], p. 280) are presented. The numbers represent the values of $10(x-2)$, $x$ being the level of lead in the water samples from two lakes at randomly chosen locations. To test whether the average pollution levels of the two lakes are equal, we perform tests for equality of the means of the populations represented by the two different samples. The $p$-values of the DPD tests are plotted in Figure \[fig:Lake\_data\_p\_val\]; the solid line represents the $p$-values for the full data, while the dashed line represents for the $p$-values for the outlier deleted data. The less robust tests (corresponding to very small values of $\gamma$) register only borderline significance under full data, and for very small values of $\gamma$ the tests would fail to reject the equality hypothesis at the 1% significance level. However, for all value of $\gamma$, the tests would soundly reject the null hypothesis when the obvious outliers (displayed with bold fonts in Table \[TAB:Lake\]) are removed from the dataset. For higher values of $\gamma$ (0.2 or larger), the $p$-values with or without the outliers are practically identical, demonstrating that the outliers have little effect in such cases. The $p$-values for the two-sample $t$-test with and without the outliers are 0.02397 and 0.0004 respectively. As in Example 1, the presence of the outliers masks the significance of the two-sample $t$-test and the small $\gamma$ DPD tests, but the large $\gamma$ DPD tests successfully discount the effect of the outliers.
------------- --------- -------- --------- --------- --------- --------------- --------- --------------- --------- --------
First Lake $-1.48$ $1.25$ $-0.51$ $0.46$ $0.60$ ${\bf -4.27}$ $0.63$ $-0.14$ $-0.38$ $1.28$
$0.93$ $0.51$ $1.11$ $-0.17$ $-0.79$ $-1.02$ $-0.91$ $0.10$ $0.41$ $1.11$
Second Lake $1.32$ $1.81$ $-0.54$ $2.68$ $2.27$ $2.70$ $0.78$ ${\bf -4.62}$ $1.88$ $0.86$
$2.86$ $0.47$ $-0.42$ $0.16$ $0.69$ $0.78$ $1.72$ $1.57$ $% $1.62$
2.14$
------------- --------- -------- --------- --------- --------- --------------- --------- --------------- --------- --------
: Lead Measurement data.[]{data-label="TAB:Lake"}
**Example 3 (Ozone Control data):** [@MR0443210] report data from a study design to assess the effects of ozone on weight gain in rats. The experimental group consisted of 22 rats, each 70-day old kept in an ozone environment for 7 days. A control group of 23 rats, of the same age, were kept in an ozone-free environment. The weight gains, in grams, are listed in Table \[TAB:Ozone\]. We want to test for the equality of the means of the two groups. The $p$-values of the DPD tests are plotted in Figure \[fig:Ozone\_control\_data\_p\_val\]. The $p$-values of the two-sample $%
t$-test for the full data and the outlier deleted data are $0.0168$ and $%
3.4721\times 10^{-6}$ respectively. The conclusions of this example are similar to those of Examples 1 and 2.
----- ------------- ------------- -------- -------- ------------- --------- ------------- --------- ------------ -------------- -------- ---------
$X$ $\bf{41.0}$ $\bf{38.4}$ $24.4$ $25.9$ $21.9$ $18.3$ $13.1$ $27.3$ $28.5$ $\bf{-16.9}$ $26.0$ $17.4$
$21.8$ $15.4$ $27.4$ $19.2$ $22.4$ $17.7$ $26$ $29.4$ $% $26.6$ $22.7$
21.4 $
$Y$ $10.1$ $6.1$ $20.4$ $7.3$ $14.3$ $15.5$ $-9.9$ $6.8$ $% $17.9$ $-9.0$ $-12.9$
28.2$
$14.0$ $6.6$ $12.1$ $15.7$ $\bf{39.9}$ $-15.9$ $\bf{54.6}$ $-14.7$ $% $-9.0$
\bf{44.1}$
----- ------------- ------------- -------- -------- ------------- --------- ------------- --------- ------------ -------------- -------- ---------
: Ozone Control data[]{data-label="TAB:Ozone"}
**Example 4 (Newcomb’s Light Speed data)**: In 1882 Simon Newcomb, an astronomer and mathematician, measured the time required for a light signal to pass from his laboratory on the Potomac River to a mirror at the base of the Washington Monument and back. The total distance was $%
7443.73 $ meters. Table \[TAB:tNewcomb\] contains these measurements from three samples, as deviations from $24,800$ nanoseconds. For example, for the first observation, $28$, means that the time taken for the light to travel the required $7443.73$ meters is $24,828$ nanoseconds. The data comprises three samples, of sizes $20$, $20$ and $26$, respectively, corresponding to three different days. These data have been analyzed previously by a number of authors including [@MR0455205] and [@Voinov]. The $p$-values of the DPD statistics for the test of the equality of means between Day 1 and Day 2, and Day 1 and Day 3 are plotted in Figure \[fig:Newcomb\_data12\_p\_val\], and \[fig:Newcomb\_data13\_p\_val\] respectively.
The $p$-values for the two-sample $t$-tests for the (Day 1, Day 2) comparison are $0.1058$ for the full data case, and $0.3091$ for the outlier deleted case. The same for the (Day 1, Day 3) comparison are $0.0970$ and $%
0.2895$ respectively. However, for the large $\gamma $, the results from the DPD tests are clearly insignificant with or without the outliers. In this example, therefore, the outliers are forcing the outcome of the two-sample $t$-test (and the DPD tests for small $\gamma $) to the borderline of significance, but the robust tests give insignificant results with or without the outliers, preventing the false significance that is produced by the outliers in the $t$-test; this is unlike the previous three examples where the robust tests overcame a masking effect. These examples demonstrate that the robust DPD tests can give protection against spurious conclusions in both directions.
------- ------ ------ ------ ------ ------ ------------ ------ ------ ------ ----------- ------ ------ ------ ------ ------
day 1 $28$ $26$ $33$ $24$ $34$ $\bf{-44}$ $27$ $16$ $40$ $\bf{-2}$ $29$ $22$ $24$ $21$ $25$
$30$ $23$ $29$ $31$ $19$
day 2 $24$ $20$ $36$ $32$ $36$ $28$ $25$ $21$ $28$ $29$ $37$ $25$ $28$ $26$ $30$
$32$ $36$ $26$ $30$ $22$
day 3 $36$ $23$ $27$ $27$ $28$ $27$ $31$ $27$ $26$ $33$ $26$ $32$ $32$ $24$ $39$
$28$ $24$ $25$ $32$ $25$ $29$ $27$ $28$ $29$ $16$ $23$
------- ------ ------ ------ ------ ------ ------------ ------ ------ ------ ----------- ------ ------ ------ ------ ------
: Newcomb’s Light Speed data.[]{data-label="TAB:tNewcomb"}
**Example 5 (Na Intake data)**: Sodium chloride preference was determined in ten patients with essential hypertension and in 12 normal volunteers. All exhibited normal detection and recognition thresholds for the taste of sodium chloride. All were placed on a constant dry diet containing 9 mEq of Na+ and given, as their only source of fluids, a choice of drinking either distilled water or 0.15 M sodium chloride. Patients with essential hypertension consumed a markedly greater proportion of their total fluid intake as saline (38.2% vs 10.6%, average daily preference over one week) and also showed a greater total fluid intake (1,269 ml vs 668 ml, average daily intake over one week). The hypertensive patients consumed more than four times as much salt as did the normal volunteers. The data are given in Table \[TAB:Na\_intake\]. The $p$-values of the tests for the equality of means are plotted in Figure \[fig:Na\_intake\_data\_p\_val\]. The findings are similar to examples 1, 2 and 3.
----- --------- -------- -------- -------- ------------ -------- -------- -------- -------- ------- -----
$X$ $114.6$ $64.6$ $70.4$ $61.2$ $\bf{297}$ $60.9$ $73.7$ $15.7$ $53.3$
$Y$ $14.2$ $3.2$ $3.7$ $0.0$ $73.6$ $56.6$ $97.2$ $2.4$ $% $4.8$ $0$
0.0$
----- --------- -------- -------- -------- ------------ -------- -------- -------- -------- ------- -----
: Na Intake data.[]{data-label="TAB:Na_intake"}
[**Example 6 (Sri Lanka Zinc Content data)**]{}: The impact of a polluted environment on the health of the residents of an area is a common environmental concern. Large amounts of heavy metals in the body may signal a serious health threat to a community. One study, performed in Sri Lanka, sought to compare rural Sri Lankans with their urban counterparts in terms of the zinc content of their hair. A collection of individuals from rural Sri Lanka was recruited, samples of their hair were taken, and the zinc content in the hair was measured. An independent collection of students from an urban environment was studied, with the zinc content in samples of their hair being measured as well. The data are given in Table \[TAB:SriLanka\]. The $p$-values of the tests for the equality of the means are plotted in Figure \[fig:SriLanka\]. The results again indicate that the presence of outliers can mask the true significance in case of the two sample $t$-test and DPD tests for small values of $\gamma$, but for the large $\gamma$ DPD tests are much more stable in such situations.
------------- ------ ----- ---------------------------------------------------------------------------------------- -- -- -- -- -- -- -- --
Urban ($X$) 1120 230 **[4200]{} & 1200 & 1400 & 750 & 2101 & 430 & 690 & 600 & 834\
Rural ($Y$) & **[3619]{} & 1104 & 243 & 658 & 673 & 598 & 648 & 918 & 133 & 289 & 250\
& 304 & 555 & 640 & 933 & & & & & & &\
****
------------- ------ ----- ---------------------------------------------------------------------------------------- -- -- -- -- -- -- -- --
: Sri Lanka Zinc Content data.[]{data-label="TAB:SriLanka"}
Concluding Remarks {#SEC:concluding}
==================
Without any doubt, the two sample $t$-test is one of the most frequently used tools in the statistics literature. It allows the experimenter to perform tests of the comparative hypotheses, which are the default requirements to be passed before one may declare that a new drug or treatment is an improvement over an existing one. The two sample $t$-test is simple to implement and has several optimality properties. In spite of such desirable attributes, this test is deficient on one count, which is that it does not retain its desired properties under contamination and model misspecification. As few as one, single, large outlier can turn around the decision of the test, and can make the resulting inference meaningless. In this paper we have introduced a test based on the density power divergence; the theoretical properties of the test have been rigorously determined. More importantly, we have demonstrated, through several real data examples, that the DPD test is capable of uncovering both kinds of masking effects caused by outliers – blurring the true difference when one exists, and detecting a difference when there is actually none. The test is simple to use and easy to understand, and we trust that it has the potential to become a powerful tool for the applied statistician.
**Acknowledgments** This work was partially supported by Grants MTM-2012-33740 and ECO-2011-25706. The authors gratefully acknowledge the suggestions of two anonymous referees which led to an improved version of the paper.
Appendix {#appendix .unnumbered}
========
**Proof of Theorem \[Th0\]:** As $\widehat{\mu }_{i\beta }$ is the solution of the estimating equation $_{1}h_{n_{i},\beta }^{\prime }\left( \mu _{i},\sigma \right) =0$, we get from equation (\[muSigma\]) $$\sqrt{n_{i}}(\widehat{\mu }_{i\beta }-\mu _{i0})=\sqrt{n_{i}} \boldsymbol{J}_{11,\beta }^{-1}( \sigma
_{0})\,\allowbreak
_{1}h_{n_{i},\beta }^{\prime }\left( \mu _{i0},\sigma_0\right) +o_{p}(1) ,\quad i=1,2.$$ Hence, using (\[1.1\]) we get $$\sqrt{n_{i}}(\widehat{\mu }_{i\beta }-\mu _{i0})\underset{n_{i}\rightarrow
\infty }{\overset{\mathcal{L}}{\longrightarrow }}\mathcal{N}\left(
0,K_{11,\beta }\boldsymbol{(}\sigma_0)J_{11,\beta }^{-2}( \sigma_0) \right) ,\quad i=1,2,
\label{distmu}$$ where $$K_{11,\beta }\boldsymbol{(}\sigma_0)J_{11,\beta }^{-2}(\sigma
_{0})=\sigma_0^{2}\left( \beta +1\right) ^{3}\left( 2\beta +1\right) ^{-%
\frac{3}{2}}. \label{KJ1}$$ It is clear that $\widehat{\mu }_{1\beta }$ and $\widehat{\mu }_{2\beta }$ are based on two independent set of observations, hence, $Cov(\widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta })=0$. As $_{2}h_{n_1,n_2,\beta }^{\prime }(\widehat{\boldsymbol{\eta }}_\beta )=0$, taking a Taylor series expansion around $\boldsymbol{\eta }_0$ we get $$\begin{aligned}
_{2}h_{n_1,n_2,\beta }^{\prime }(\widehat{\boldsymbol{\eta }}_\beta ) =& \ %\allowbreak
_{2}h_{n_1,n_2,\beta }^{\prime }( \boldsymbol{\eta }_0) + \left. \frac{\partial }{\partial \mu_1}\,\allowbreak
_{2}h_{n_1,n_2,\beta }^{\prime }( \boldsymbol{\eta })\right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} (\widehat{\mu }_{1\beta }-\mu _{10}) \nonumber\\
& + \left. \frac{\partial }{\partial \mu_2}\,\allowbreak _{2} h_{n_1,n_2,\beta
}^{\prime \prime }( \boldsymbol{\eta }) \right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} (\widehat{%
\mu }_{2\beta }-\mu _{20}) \nonumber\\
& + \left. \frac{\partial }{\partial \sigma }\,\allowbreak _{2}h_{n_1,n_2,\beta
}^{\prime \prime }( \boldsymbol{\eta })\right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} \left(
\widehat{\sigma}_\beta -\sigma_0\right) +o_{p}\left(
(n_1+n_2)^{-1/2}\right) \nonumber\\
=& \ 0.
\label{2h'}\end{aligned}$$Notice that $$\begin{aligned}
\lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial }{\partial \mu_1}\,\allowbreak _{2}h_{n_1,n_2,\beta
}^{\prime }( \boldsymbol{\eta }) \right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} & = \lim_{n_1,n_2\rightarrow \infty } \frac{\partial }{%
\partial \mu_1}\left( \frac{n_1}{n_1+n_2}\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }\left( \mu _{10},\sigma_0\right) +\frac{%
n_2}{n_1+n_2}\,\allowbreak _{2}h_{n_2}^{\prime }(\mu _{10},\sigma
_{0})\right) \nonumber\\
& =\lim_{n_1,n_2\rightarrow \infty } \frac{n_1}{n_1+n_2} \lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial }{\partial \mu_1}\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }\left( \mu _{1},\sigma_0\right) \right\vert_{\mu_1 = \mu_{10} }\nonumber\\
&= w \boldsymbol{J}_{12,\beta }\left( \sigma_0\right) = 0.
\label{hmu2}\end{aligned}$$ Similarly we get $$\lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial }{\partial \mu_2}\,\allowbreak _{2}h_{n_1,n_2,\beta
}^{\prime }( \boldsymbol{\eta })\right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} =0.
\label{hmu1}$$ Moreover, $$\begin{aligned}
\lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial }{\partial \sigma }%
\allowbreak _{2}h_{n_1,n_2,\beta }^{\prime }\left( \boldsymbol{\eta } \right)\right\vert_{\boldsymbol{\eta }=\boldsymbol{\eta }_0} &=&\lim_{n_1,n_2\rightarrow \infty }\tfrac{%
n_1}{n_1+n_2}\,\allowbreak _{22}h_{n_1,\beta }^{\prime \prime }(\mu
_{10},\sigma_0)+\lim_{n_1,n_2\rightarrow \infty }\tfrac{n_2}{%
n_1+n_2}\,\allowbreak _{22}h_{n_2,\beta }^{\prime \prime }\left( \mu
_{20},\sigma_0\right) \nonumber\\
&=&w\boldsymbol{J}_{22,\beta }\left( \sigma_0\right) +(1-w)\boldsymbol{J}_{22,\beta }( \sigma_0) = \boldsymbol{J}_{22,\beta } ( \sigma_0).
\label{hsigma}\end{aligned}$$ Therefore, using equations (\[hmu2\]), (\[hmu1\]) and (\[hsigma\]) we get from equation (\[2h’\]) $$\sqrt{n_1+n_2}\left( \widehat{\sigma}_\beta -\sigma_0\right)
=-\boldsymbol{J}_{22,\beta }^{-1}\left( \sigma_0\right) \sqrt{n_1+n_2}%
\,\allowbreak _{2}h_{n_1,n_2,\beta }^{\prime }( \boldsymbol{\eta }_0) +o_{p}(1).
\label{sigmaL}$$ Applying (\[1.1\]) and (\[EQ:w\]) we get $$\begin{aligned}
& \lim_{n_1,n_2\rightarrow \infty } \text{$E$}\left[ \sqrt{n_1+n_2}\,\allowbreak _{2}h_{n_1,n_2,\beta
}^{\prime }(\boldsymbol{\eta }_0) \right] \\
& = \lim_{n_1,n_2\rightarrow \infty } \frac{\sqrt{n_1+n_2}}{n_1+n_2}E\left[ n_1\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }\left( \mu _{10},\sigma_0\right)
+n_2\,\allowbreak _{2}h_{n_2,\beta }^{\prime }\left( \mu _{20},\sigma
_{0}\right) \right] \\
& = \lim_{n_1,n_2\rightarrow \infty }\sqrt{\frac{n_1}{n_1+n_2}} \lim_{n_1,n_2\rightarrow \infty } E\left[ \sqrt{n_1}\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }(\mu _{10},\sigma_0)\right] \\
& \ \ \ + \lim_{n_1,n_2\rightarrow \infty } \sqrt{\frac{%
n_2}{n_1+n_2}} \lim_{n_1,n_2\rightarrow \infty } E\left[ \sqrt{n_2}\,\allowbreak _{2}h_{n_2,\beta
}^{\prime }\left( \mu _{20},\sigma_0\right) \right] \\
&= 0.\end{aligned}$$ Similarly we also have $$\begin{aligned}
& \lim_{n_1,n_2\rightarrow \infty } \text{$Var$}\left[ \sqrt{n_1+n_2}\,\allowbreak
_{2}h_{n_1,n_2,\beta }^{\prime }(\boldsymbol{\eta }_0) \right] \\
& =\lim_{n_1,n_2\rightarrow \infty } (n_1+n_2)\text{$Var$}\left[ \frac{1}{n_1+n_2}\left(
n_1\,\allowbreak _{2}h_{n_1,\beta }^{\prime }(\mu _{10},\sigma
_{0})+n_2\,\allowbreak _{2}h_{n_2,\beta }^{\prime }(\mu _{20},\sigma
_{0}\right) \right] \\
& = \lim_{n_1,n_2\rightarrow \infty } \frac{n_1}{n_1+n_2} \lim_{n_1,n_2\rightarrow \infty } \text{$Var$}\left[ \sqrt{n_1}\,\allowbreak
_{2}h_{n_1,\beta }^{\prime }(\mu _{10},\sigma_0)\right] \\
& \ \ \ + \lim_{n_1,n_2\rightarrow \infty } \frac{n_2}{%
n_1+n_2} \lim_{n_1,n_2\rightarrow \infty } \text{$Var$}\left[ \sqrt{n_2}\,\allowbreak _{2}h_{n_2,\beta
}^{\prime }(\mu _{20},\sigma_0)\right] \\
&= w \boldsymbol{K}_{22,\beta }(\sigma_0) + (1-w) \boldsymbol{K}_{22,\beta }(\sigma_0) \\
&= \boldsymbol{K}_{22,\beta }(\sigma_0) .\end{aligned}$$Hence, $$\sqrt{n_1+n_2}\,\allowbreak _{2}h_{n_1,n_2,\beta }^{\prime }\left(
\boldsymbol{\eta }_0\right) \underset{n_1,n_2\rightarrow
\infty }{\overset{\mathcal{L}}{\longrightarrow }}\mathcal{N}\left(
0,\boldsymbol{K}_{22,\beta }(\sigma_0)\right) .$$Now, from equation (\[sigmaL\]) we get $$\sqrt{n_1+n_2}\left( \widehat{\sigma}_\beta -\sigma_0\right)
\underset{n_1,n_2\rightarrow \infty }{\overset{\mathcal{L}}{%
\longrightarrow }}\mathcal{N}\left( 0,\boldsymbol{K}_{22,\beta }(\sigma_0)\boldsymbol{J}_{22,\beta
}^{-2}(\sigma_0)\right) ,
\label{sigma1}$$where $$\boldsymbol{K}_{22,\beta }(\sigma_0)\boldsymbol{J}_{22,\beta }^{-2}(\sigma_0)=\sigma_0^{2}%
\frac{\left( \beta +1\right) ^{5}}{\left( \beta ^{2}+2\right) ^{2}}\left(
\frac{4\beta ^{2}+2}{(1+2\beta )^{5/2}}-\frac{\beta ^{2}}{(1+\beta )^{3}}%
\right). \label{KJ2}$$As $\boldsymbol{J}_{12,\beta }(\sigma_0)=\boldsymbol{J}_{21,\beta }(\sigma_0)=0$, it is clear that $$\lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial^2 }{\partial \mu_1 \partial \sigma}\,\allowbreak h_{n_1,n_2,\beta
}( \boldsymbol{\eta })\right\vert_{\boldsymbol{\eta } = \boldsymbol{\eta }_0}
= \lim_{n_1,n_2\rightarrow \infty } \left. \frac{\partial^2 }{\partial \mu_2 \partial \sigma}\,\allowbreak h_{n_1,n_2,\beta
}( \boldsymbol{\eta }) \right\vert_{\boldsymbol{\eta } = \boldsymbol{\eta }_0}
=0.$$ Therefore, $Cov(\widehat{\mu }_{1\beta },\widehat{\sigma}_{\beta })=Cov(\widehat{\mu }_{2\beta },\widehat{\sigma}_{\beta })=0$. Moreover, $Cov(\widehat{\mu }_{1\beta },\widehat{\mu }_{2\beta })=0$. Combining the results in (\[distmu\]) and (\[sigma1\]) we get the variance-covariance matrix of $\sqrt{\frac{n_1n_2}{n_1+n_2}}\widehat{\boldsymbol{\eta }}_{\beta }$ as follows $$\boldsymbol{\Sigma }_{w,\beta }(\sigma_0)=\left(
\begin{array}{ccc}
\left( 1-w\right) \boldsymbol{K}_{11,\beta }\boldsymbol{(}\sigma_0) \boldsymbol{J}_{11,\beta
}^{-2}\left( \sigma_0\right) & 0 & 0 \\
0 & w \boldsymbol{K}_{11,\beta }\boldsymbol{(}\sigma_0) \boldsymbol{J}_{11,\beta }^{-2}(\sigma_0)
& 0 \\
0 & 0 & w\left( 1-w\right) \boldsymbol{K}_{22,\beta }(\sigma_0)\boldsymbol{J}_{22,\beta
}^{-2}\left( \sigma_0\right)
\end{array}%
\right) ,$$ where the values of the diagonal elements are given in (\[KJ1\]) and (\[KJ2\]). Hence, the theorem is proved. ${\blacksquare }$
**Proof of Theorem \[Th1\]**: A Taylor expansion of $%
d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })$ around $\boldsymbol{\eta }_{0}$ gives$$d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })=d_{\gamma }(f_{\mu
_{10},\sigma_0},f_{\mu _{20},\sigma_0})+\boldsymbol{t}_{\gamma
}^{T}\left( \boldsymbol{\eta }_{0}\right) (\widehat{\boldsymbol{\eta }}%
_{\beta }-\boldsymbol{\eta }_{0})+o_{p}\left( \left\Vert \widehat{%
\boldsymbol{\eta }}_{\beta }-\boldsymbol{\eta }_{0}\right\Vert \right),$$ where $\boldsymbol{t}_{\gamma }\left( \boldsymbol{\eta }_{0}\right) =\frac{%
\partial }{\partial \boldsymbol{\eta }}\left. d_{\gamma }(f_{\mu_1,\sigma
},f_{\mu_2,\sigma })\right\vert _{\boldsymbol{\eta }=\boldsymbol{\eta }%
_{0}}$; the expressions of the components $t_{\gamma, i}\left( \boldsymbol{\eta }%
_{0}\right) $, $i=1,2,3$, are given in (\[t1\])-(\[t3\]). Hence, the result directly follows from Theorem \[Th0\]. ${\blacksquare }$
**Proof of Theorem \[Th2\]**: If $\mu _{10}=\mu _{20}$, it is obvious that $d_{\gamma }(f_{\mu_{10},\sigma_0},f_{\mu _{20},\sigma_0})=0$, and $\boldsymbol{t}_{\gamma } ( \boldsymbol{\eta }_{0})=0$. Hence, a second order Taylor expansion of $d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })$ around $\boldsymbol{\eta }_{0}$ gives $$2d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })=(\widehat{\boldsymbol{%
\eta }}_{\beta }-\boldsymbol{\eta }_{0})^{T}\boldsymbol{A}_{\gamma }\left(
\sigma_0\right) (\widehat{\boldsymbol{\eta }}_{\beta }-\boldsymbol{\eta }%
_{0})+o_p(\left\Vert \widehat{\boldsymbol{\eta }}_{\beta }-\boldsymbol{\eta }%
_{0}\right\Vert ^{2}),
\label{gam}$$where $\boldsymbol{A}_{\gamma }(\sigma_0)$ is the matrix containing the second derivatives of $d_{\gamma }(f_{\mu_1,\sigma },f_{\mu_2,\sigma })\ $evaluated at $\mu_{10}=\mu_{20}$. It can be shown that $$\boldsymbol{A}_{\gamma }\left( \sigma_0\right) \boldsymbol{=}\ell
_{\gamma }(\sigma_0)\left(
\begin{array}{ccc}
1 & -1 & 0 \\
-1 & 1 & 0 \\
0 & 0 & 0%
\end{array}%
\right) ,$$ where $$\ell _{\gamma }(\sigma_0)=\sigma ^{-(\gamma +2)}\left( 2\pi \right) ^{-%
\frac{\gamma }{2}}\left( \gamma +1\right) ^{-\frac{1}{2}}.$$ Therefore, equation (\[gam\]) simplifies to $$2d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{%
\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })=\left(
\begin{pmatrix}
\widehat{\mu }_{1\beta } \\
\widehat{\mu }_{2\beta }%
\end{pmatrix}%
-%
\begin{pmatrix}
\mu _{10} \\
\mu _{20}%
\end{pmatrix}%
\right) ^{T}\boldsymbol{A}_{\gamma }^{\ast }\left( \sigma_0\right) \left(
\begin{pmatrix}
\widehat{\mu }_{1\beta } \\
\widehat{\mu }_{2\beta }%
\end{pmatrix}%
-%
\begin{pmatrix}
\mu _{10} \\
\mu _{20}%
\end{pmatrix}%
\right) +o_{p}\left( \left\Vert \widehat{\boldsymbol{\eta }}_{\beta }-%
\boldsymbol{\eta }_{0}\right\Vert ^{2}\right) ,$$ where $$\boldsymbol{A}_{\gamma }^{\ast }\left( \sigma_0\right) =\ell _{\gamma
}(\sigma_0)\left(
\begin{array}{cc}
1 & -1 \\
-1 & 1%
\end{array}%
\right) .$$ From Theorem \[Th0\] we know that $$\sqrt{\frac{n_1n_2}{n_1+n_2}}\left(
\begin{pmatrix}
\widehat{\mu }_{1\beta } \\
\widehat{\mu }_{2\beta }%
\end{pmatrix}%
-%
\begin{pmatrix}
\mu _{10} \\
\mu _{20}%
\end{pmatrix}%
\right) ^{T}\underset{}{\overset{\mathcal{L}}{\longrightarrow }}\mathcal{N}%
\left( \boldsymbol{0}_{2},\boldsymbol{\Sigma }_{w,\beta }^{\ast }(\sigma
_{0})\right),$$ where $$\boldsymbol{\Sigma }_{w,\beta }^{\ast }(\sigma_0)=K_{11,\beta }%
\boldsymbol{(}\sigma_0)J_{11,\beta }^{-2}\left( \sigma_0\right) \left(
\begin{array}{cc}
1-w & 0 \\
0 & w%
\end{array}
\right).$$ Therefore, $\frac{2 n_1n_2}{n_1+n_2} d_{\gamma }(f_{\widehat{\mu }_{1\beta },\widehat{\sigma }_{\beta
}},f_{\widehat{\mu }_{2\beta },\widehat{\sigma}_\beta })$ has the same asymptotic distribution (see [@MR801686]) as the random variable $$\sum\limits_{i=1}^{2}\lambda _{i,\beta ,\gamma }(\sigma_0)Z_{i}^{2},$$ where $Z_{1}$ and $Z_{2}$ are independent standard normal variables, and $$\lambda _{1,\beta ,\gamma }(\sigma_0)=0\text{, and }\lambda _{2,\beta
,\gamma }(\sigma_0)=K_{11,\beta }\boldsymbol{(}\sigma_0)J_{11,\beta
}^{-2}\left( \sigma_0\right) \ell _{\gamma }(\sigma_0)=\lambda _{\beta
,\gamma }(\sigma_0)$$are the eigenvalues of the matrix $\boldsymbol{\Sigma }_{w,\beta }^{\ast
}(\sigma_0)\boldsymbol{A}_{\gamma }^{\ast }\left( \sigma_0\right) $. Hence, $$\frac{2n_1n_2}{n_1+n_2}\frac{d_{\gamma
}(f_{\widehat{\mu }_{1\beta },\widehat{\sigma}_\beta },f_{\widehat{\mu }%
_{2\beta },\widehat{\sigma}_\beta })}{\lambda _{\beta ,\gamma
}\,\allowbreak (\sigma_{0} )} \underset{%
n_1,n_2\rightarrow \infty }{\overset{\mathcal{L}}{\longrightarrow }}\chi
^{2}(1).$$ Finally, since $\widehat{\sigma}_\beta $ is a consistent estimator of $%
\sigma $, replacing $\lambda _{\beta ,\gamma }(\sigma_0)$ by $\lambda
_{\beta ,\gamma }(\widehat{\sigma}_\beta )$ and by following Slutsky’s theorem we obtain the desired result. ${\blacksquare }$
| ArXiv |
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abstract: |
The muon anomalous magnetic moment is one of the most precisely measured quantities in particle physics. In a recent experiment at Brookhaven it has been measured with a remarkable 14-fold improvement of the previous CERN experiment reaching a precision of 0.54ppm. Since the first results were published, a persisting “discrepancy” between theory and experiment of about 3 standard deviations is observed. It is the largest “established” deviation from the Standard Model seen in a “clean” electroweak observable and thus could be a hint for New Physics to be around the corner. This deviation triggered numerous speculations about the possible origin of the “missing piece” and the increased experimental precision animated a multitude of new theoretical efforts which lead to a substantial improvement of the prediction of the muon anomaly $a_\mu=(g_\mu-2)/2$. The dominating uncertainty of the prediction, caused by strong interaction effects, could be reduced substantially, due to new hadronic cross section measurements in electron-positron annihilation at low energies. Also the recent electron $g-2$ measurement at Harvard contributes substantially to the progress in this field, as it allows for a much more precise determination of the fine structure constant $\alpha$ as well as a cross check of the status of our theoretical understanding.
In this report we review the theory of the anomalous magnetic moments of the electron and the muon. After an introduction and a brief description of the principle of the muon $g-2$ experiment, we present a review of the status of the theoretical prediction and in particular discuss the role of the hadronic vacuum polarization effects and the hadronic light–by–light scattering correction, including a new evaluation of the dominant pion-exchange contribution. In the end, we find a 3.2 standard deviation discrepancy between experiment and Standard Model prediction. We also present a number of examples of how extensions of the electroweak Standard Model would change the theoretical prediction of the muon anomaly $a_\mu$. Perspectives for future developments in experiment and theory are briefly discussed and critically assessed. The muon $g-2$ will remain one of the hot topics for further investigations.
address:
- 'Humboldt-Universität zu Berlin, Institut für Physik, Newtonstrasse 15, D-12489 Berlin, Germany'
- 'Institute of Physics, University of Silesia, ul. Uniwersytecka 4, PL-40007 Katowice, Poland'
- |
Regional Centre for Accelerator-based Particle Physics, Harish-Chandra Research Institute,\
Chhatnag Road, Jhusi, Allahabad - 211 019, India
author:
- Fred Jegerlehner
- Andreas Nyffeler
title: 'The Muon g-2'
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HU-EP-09/07, HRI-P-09-02-001, RECAPP-HRI-2009-003
---------------------------------------------------
,
muon, anomalous magnetic moment, precision tests 14.60.Ef,13.40.Em
| ArXiv |
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abstract: 'We have used an extension of our slow light technique to provide a method for inducing small density defects in a Bose-Einstein condensate. These sub-resolution, micron-sized defects evolve into large amplitude sound waves. We present an experimental observation and theoretical investigation of the resulting breakdown of superfluidity. We observe directly the decay of the narrow density defects into solitons, the onset of the ‘snake’ instability, and the subsequent nucleation of vortices.'
author:
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title: 'Observation of Quantum Shock Waves Created with Ultra Compressed Slow Light Pulses in a Bose-Einstein Condensate'
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Superfluidity in Bose condensed systems represents conditions where frictionless flow occurs because it is energetically impossible to create excitations. When these conditions are not satisfied, various excitations develop, and experiments on superfluid helium, for example, have provided evidence that the nucleation of vortex rings occurs when ions move through the fluid faster than a critical speed ([*1,2*]{}). Under similar conditions, shock waves would occur in a normal fluid ([*3*]{}). Such discontinuities are not allowed in a superfluid and instead topological defects, such as quantized vortices and solitons, are nucleated when the spatial scale of density variations becomes on the order of the healing length. This is the length scale at which the kinetic energy, associated with spatial variations of the macroscopic condensate wave function, becomes on the order of the atom-atom interaction energy ([*2,4*]{}). It is therefore also the minimum length over which the density of a condensate can adjust.
Bose-Einstein condensates (BECs) of alkali atoms ([*5*]{}) have provided a system for the study of superfluidity, which is theoretically more tractable than liquid helium and allows greater experimental control. An exciting recent development is the production of solitons and vortices. Experiments have employed techniques that manipulated the phase of the BEC ([*6-9*]{}), or provided the system with a high angular momentum which makes vortex formation energetically favorable ([*10,11*]{}). However, a direct observation of the formation of vortices via the breakdown of superfluidity has been lacking. Rather, rapid heating from ’stirring’ with a focused laser beam above a critical velocity was observed as indirect evidence of this process ([*12,13*]{}).
In this context, it is natural to ask what would happen if one were to impose a sharp density feature in a BEC with a spatial scale on the order of the healing length. Optical resolution limits have prevented direct creation of this kind of excitation. In this paper we present an experimental demonstration of creation of such defects in sodium Bose-Einstein condensates, using a novel extension of the method of ultra slow light pulse propagation ([*14*]{}) via electromagnetically induced transparency (EIT) ([*15,16*]{}).
Our slow-light setup is described in ([*14*]{}). By illuminating a BEC with a ‘coupling’ laser, we create transparency and low group velocity for a ‘probe’ laser pulse subsequently sent into the cloud. In a geometry where the coupling and probe laser beams propagate at right angles, we can control the propagation of the probe pulse from the side. By introducing a spatial variation of the coupling field, along the pulse propagation direction, we vary the group velocity of the probe pulse across the cloud. Here we accomplish this by blocking half of the coupling beam so it illuminates only the left hand side ($z<0$) of the condensate, setting up a light ‘roadblock’. In this way, we compress the probe pulse to a small spatial region at the center of the BEC, while bypassing the usual bandwidth requirements for slow light ([*17*]{}). The technique produces a short wavelength excitation by suddenly removing a narrow disk of the condensate, with the width of the disk determined by the width of the compressed probe pulse.
We find that this excitation results in short wavelength, large amplitude sound waves that shed off ‘gray’ solitons ([*18-20*]{}), and we make the connection to the formation of shock waves in classical fluids. The ‘snake’ (Kadomtsev-Petviashvili) instability is predicted to cause solitons to decay into vortices ([*21-24*]{}). This has been observed with optical solitons ([*25*]{}) and recently the JILA group predicted and observed that solitons in a BEC decay into vortex rings ([*9*]{}). Here we present a direct observation of the dynamics of the snake instability in a BEC and the subsequent nucleation of vortices. The images of the evolution are compared to numerical propagation of the Gross-Pitaevskii equation in two dimensions.
Details of our Bose-Einstein condensation apparatus can be found in ([*26*]{}). We use condensates with about 1.5 million sodium atoms in the state $|1 \rangle \equiv |3S_{1/2},F=1,M_F=-1
\rangle$ and trapped in a 4-Dee magnet. For the experiments presented here, the magnetic trap has an oscillator frequency $\omega_z=(2\pi) 21$ Hz along its symmetry direction and a frequency $\omega_x = \omega_y = 3.8 \omega_z$ in the transverse directions. The peak density of the condensates is then $9.1
\times 10^{13}~\mathrm{cm}^{-3}$. The temperature is $T \sim
0.5\,T_c$, where $T_c=300$ nK is the critical temperature for condensation, and so the vast majority ($\sim 90 \%$) of the atoms occupy the ground state.
To create slow and spatially localized light pulses, the coupling beam propagates along the $x$ axis (Fig. 2B), is resonant with the $D_1$ transition from the unoccupied hyperfine state $|2 \rangle
\equiv |3S_{1/2},F=2,M_F=-2 \rangle$ to the excited level $ |3
\rangle \equiv |3P_{1/2}, F=2, M_F=-2 \rangle$, and has a Rabi frequency $\Omega_c = (2\pi) 15$ MHz ([*27*]{}). We inject probe pulses along the $z$ axis, resonant with the transition and with peak Rabi frequency $\Omega_p = (2\pi) 2.5$ MHz. The pulses are Gaussian shaped with a $1/e$ half-width of $\tau = 1.3\;\mu$s. With the entire BEC illuminated by the coupling beam, we observe probe pulse delays of $4\;\mu$s for propagation through the condensates, corresponding to group velocities of 18 m/s at the center of the clouds. A pulse with a temporal half-width $\tau$ is spatially compressed from a length $2 \, c \tau$ in vacuum to ([*14,17,28,29*]{}) $$L=2 \tau V_g = 2 \tau \frac{|\Omega_c|^2}{\Gamma f_{13} \sigma_0
n_c}$$
inside the cloud, where $\Gamma = (2 \pi) 10$ MHz is the decay rate of state $|3 \rangle$, $n_c$ is the cloud density, $\sigma_0 = 1.65 \times 10^{-9}\;\mathrm{cm}^2$ is the absorption cross-section for light resonant with a two-level atom, and $f_{13}=1/2$ is the oscillator strength of the $|1 \rangle
\rightarrow |3 \rangle$ transition. The atoms are constantly being driven by the light fields into a dark state, a coherent superposition of the two hyperfine states $|1 \rangle$ and $ |2
\rangle$ ([*15*]{}). In the dark state, the ratio of the two population amplitudes is varying in space and time with the electric field amplitude of the probe pulse as
$$\psi_2 = - \frac{\Omega_p}{\Omega_c}\,\psi_1,$$
where $\psi_1,\;\psi_2$ are the macroscopic condensate wave functions associated with the two states $|1 \rangle$ and $
|2 \rangle$.
For the parameters listed above, the probe pulse is spatially compressed from 0.8 km in free space to only 50 $\mu$m at the center of the cloud, at which point it is completely contained within the atomic medium. The corresponding peak density of atoms in $|2 \rangle$, proportional to $|\psi_2|^2$, is $1/34$ of the total atom density. The $|1 \rangle$ atoms have a corresponding density depression.
From Eqs. 1 and 2, it is clear that to minimize the spatial scale of the density defect, we need to use short pulse widths and low coupling intensities. However, for all the frequency components of the probe pulse to be contained within the transmission window for propagation through the BEC ([*17*]{}), we need a pulse with a temporal width $\tau$ of at least $2 \sqrt{D} \Gamma /
{\Omega_c}^2 \approx 0.3~\mu$s (here $D \approx 520$ is the optical density of a condensate for on-resonance two level transitions) to avoid severe attenuation and distortion. Furthermore, we see from Eq. 2 that to maximize the amplitude of the density depression would favor use of a peak Rabi frequency for the probe of $\Omega_p \sim \Omega_c$. This also severely distorts the pulse.
Both of these distortion effects accumulate as the pulse propagates through large optical densities. This motivated us to introduce a roadblock in the condensate for a light pulse approaching from the left hand ($z<0$) side. By imaging a razor blade onto the right half of the condensate, we ramp the coupling beam from full to zero intensity over the course of a $12\;\mu$m region in the middle of the condensate, determined by the optical resolution of the imaging system. In the illuminated region ($z<0$), our bandwidth and weak-probe requirements are well satisfied and we get undistorted, unattenuated propagation through the first half of the cloud to the high-density, central region of the condensate. As the pulse enters the roadblock region of low coupling intensity, it is slowed down and spatially compressed. The exact shape and size of the defects which are created with this method are dependent on when absorption effects become important.
To accurately model the pulse compression and defect formation, we account for the dynamics of the slow light pulses, the coupling field, and the atoms self-consistently. At sufficiently low temperatures, the dynamics of the two-component condensate can be modelled with coupled Gross-Pitaevskii (GP) equations ([*4,5*]{}). Here we include terms to account for the resonant two-photon light coupling between the two components:
$$\begin{aligned}
i \hbar \frac{\partial}{\partial t}\psi_1 & = &
\left(-\frac{\hbar^2 \nabla^2}{2m} + V_1(\mathbf{r}) + U_{11}
|\psi_1|^2 + U_{12}
|\psi_2|^2 \right) \psi_1 \nonumber \\
& & \hspace{2 cm} - i \frac{\mid{\Omega_p}\mid^2}{2 \Gamma} \psi_1
- i \frac{{\Omega_p}^\ast \Omega_c}{2 \Gamma} \psi_2 \nonumber
- i N_c \sigma_e \hbar \frac{k_{2 \gamma}}{2 m} |\psi_2|^2 \psi_1, \\
i \hbar \frac{\partial}{\partial t}\psi_2 & = &
\left(-\frac{\hbar^2 (\nabla^2 + i \mathbf{k_{2 \gamma}} \cdot
\nabla )}{2m} + V_2(\mathbf{r}) + U_{22} |\psi_2|^2 + U_{12}
|\psi_1|^2
\right) \psi_2 \nonumber \\
& & \hspace{2 cm}- i \frac{\mid{\Omega_c}\mid^2}{2 \Gamma} \psi_2
- i \frac{\Omega_p {\Omega_c}^\ast}{2 \Gamma} \psi_1 - i N_c
\sigma_e \hbar \frac{k_{2 \gamma}}{2 m} |\psi_1|^2 \psi_2.\end{aligned}$$
Here $V_1(\mathbf{r}) = \frac{1}{2} m {\omega_z}^2(
\lambda^2 (x^2+y^2) + z^2)$, where $m$ is the mass of the sodium atoms, and $\lambda=3.8$. Due to the magnetic moment of atoms in state $|2 \rangle$, $V_2(\mathbf{r})= - 2 V_1(\mathbf{r})$, and atoms in this state are repelled from the trap. The EIT process involves absorption of a probe photon and stimulated emission of a coupling photon, leading to a $4.1$ cm/s recoil velocity. This is described by a term in the second equation, containing $\mathbf{k_{2 \gamma}} = \mathbf{k_p - k_c}$, the difference in wave vectors between the two laser beams. (Here we use a gauge where the recoil momentum is transformed away.) Atom-atom interactions are characterized by the scattering lengths, $a_{ij}$, via $U_{ij} = 4 \pi N_c \hbar^2 a_{ij}/m$, where $a_{11}=2.75\;\mathrm{nm}$, $a_{12}=a_{22} = 1.20\,a_{11}$ ([*30*]{}), and $N_c$ is the total number of condensate atoms. To obtain the light coupling terms in Eq. 3, we have adiabatically eliminated the excited state amplitude $\psi_3$ ([*31*]{}), as the relaxation from spontaneous emission occurs much faster than the light coupling and external atomic dynamics driving $\psi_3$. In our model, atoms in $|3 \rangle$ that spontaneously emit are assumed to be lost from the condensate, which is why the light coupling terms are non-Hermitian. Finally, the last term in each equation accounts for losses due to elastic collisions between high momentum $|2\rangle$ atoms and the nearly stationary $|1\rangle$ atoms ($\sigma_e = 8 \pi {a_{12}}^2$) ([*32*]{}).
The spatial dynamics of the light fields are described classically with Maxwell’s equations in a slowly varying envelope approximation, again using adiabatic elimination of $\psi_3$ :
$$\begin{aligned}
\frac{\partial}{\partial z}\Omega_p & = & - \frac{1}{2}f_{13}
\sigma_0 N_c(\Omega_p |\psi_1|^2 +
\Omega_c {\psi_1}^\ast \psi_2), \nonumber \\
\frac{\partial}{\partial x}\Omega_c & = & - \frac{1}{2}f_{23}
\sigma_0 N_c(\Omega_c |\psi_2 \mid^2 + \Omega_p \psi_1
{\psi_2}^\ast).\end{aligned}$$
In the region where the coupling beam is illuminating the BEC ($z<0$), the light coupling terms dominate the atomic dynamics and solving Eqs. 3 and 4 reduces to Eqs. 1 and 2 above.
We have performed numerical simulations in two dimensions ($x$ and $z$) to track the behavior of the light fields and the atoms. The probe and coupling fields were propagated according to Eq. 4 with a second order Runge-Kutta algorithm ([*33*]{}) and the atomic mean fields were propagated according to Eq. 3, with an Alternating-Direction Implicit variation of the Crank-Nicolson algorithm ([*33,34*]{}). In this way, Eqs. 3 and 4 were solved self-consistently ([*35*]{}). Profiles of the probe pulse intensity along $z$, through $x=0$, are shown in Fig. 1A. As the pulse runs into the roadblock, a dramatic compression of the probe pulse’s spatial length occurs. When the probe pulse enters the low coupling region, the Rabi frequency $|\Omega_p|$ becomes on the order of $|\Omega_c|$. So the density of state $|2 \rangle$ atoms, $N_c |\psi_2|^2$, increases in a narrow region, which is accompanied by a decrease in $N_c |\psi_1|^2$ (Fig. 1B). The half-width of the defect is $2\;\mu$m. As the compression develops, absorption/spontaneous emission events eventually start to remove atoms from the condensate and reduce the probe intensity.
Experimental results are shown in Fig. 2. Fig. 2A is an in-trap image of the original condensate of $|1 \rangle$ atoms, Fig. 2B diagrams the beam geometry, and Fig. 2C shows a series of images of state $|2 \rangle$ atoms as the pulse propagates into the roadblock. The corresponding optical density (OD) profiles along $z$ through $x=0$ are also shown. The OD is defined to be $-
\mathrm{ln}(I/I_0)$, where $(I/I_0)$ is the transmission coefficient. All imaging is done with near resonant laser beams propagating along the $y$-axis, and with a duration of 10 $\mu$s. There is clearly a build-up of a dense, narrow sample of $|2
\rangle$ atoms at the center of the BEC as the pulse propagates to the right. Note that the pulse reaches the roadblock at the top and bottom edges of the cloud before the roadblock is reached at the center, which is a consequence of the transverse variation in the density of the BEC, with the largest density along the center line. After the pulse compression is achieved, we shut off the coupling beam to avoid heating and phase shifts of the atom cloud due to extended exposure to the coupling laser, and the subsequent dynamics of the condensate are observed. (We observed that exposure to the coupling laser alone, for the exposure times used to create defects, causes no excitations of the condensates).
In considering the dynamics resulting from this excitation of a condensate, it should be noted that the roadblock ‘instantaneously’ removes a spatially selected part of $\psi_1$. The entire light compression happens in approximately $15\;\mu\mathrm{s}$. After the pulse is stopped and the coupling laser turned off, the $|2 \rangle$ atoms remaining in the condensate $\psi_2$ have a $4.1$ cm/s recoil and atoms which have undergone absorption and spontaneous emission events have a similarly sized but randomly directed recoil. So the $\psi_2$ component and the other recoiling atoms interact with $\psi_1$ for less than $0.5$ ms before leaving the region. Both of these time scales are short compared with the several millisecond timescale over which most of the subsequent dynamics of $\psi_1$ occur, as discussed in the following.
We first considered the one-dimensional dynamics along the $z$ axis. Snapshots of both condensate components, obtained from numerical propagation in 1D according to Eqs. 3 and 4, are shown at various times after the pulse is stopped at the roadblock (and the coupling laser turned off) (Fig. 3A). In the linear hydrodynamic regime, where the density defect has a relative amplitude $A \ll 1$ and a half-width $\delta \gg \xi$ (here $\xi=
1/\sqrt{8 \pi N_c |\psi_{1}|^2
a_{11}}$ is the local healing length which is $0.4\;\mu\mathrm{m}$ at the center of the ground state condensate in our experiment), one expects to see two density waves propagating in opposite directions at the local sound speed, $c_s = \sqrt{U_{11} |\psi_1
\mid^2/m}$, as seen experimentally in ([*36*]{}). However, for the parameters used in our experiment, the sound waves are seen to shed off sharp features propagating at lower velocities. Examination of the width, speed, and the phase jump across these features shows that they are gray solitons. Describing the slowly varying background wave function of the condensate with $\psi^{(0)}_{1}\!,$ the wave function in the vicinity of a gray soliton centered at $z_0$ is ([*18-20*]{}):
$$\psi_1(z,t) = \psi^{(0)}_{1}(z,t) \left( i \sqrt{1-\beta^2} +
\beta \; \mathrm{tanh}\left(\frac{\beta}{\sqrt{2}~\xi}
(z-z_0)\right) \right).$$
The dimensionless constant $\beta$ characterizes the ‘grayness’, with $\beta = 1$ corresponding to a stationary soliton with a $100\%$ density depletion. With $\beta \ne 1$, the solitons travel at a fraction of the local sound velocity, $c_s
\sqrt{1-\beta^2}$. As seen in the figure, after a shedding event, the remaining part of the sound wave continues to propagate at a reduced amplitude. Our numerical simulations show that the solitons eventually reach a point where their central density is zero and then oscillate back to the other side, in agreement with the discussions in ([*18,20*]{}).
In Fig. 3A, each of the two sound waves shed off two solitons. By considering the available free energy created by a defect, one finds that, when the defect size is somewhat larger than the healing length and the defect amplitude, $A$, is on the order of unity, the number of solitons that can be created is approximately $\sqrt{A} \delta /(2 \xi)$.
One obtains a simple physical estimate of the conditions necessary for soliton shedding by calculating the difference in sound speed associated with the difference in atom density between the center and back edge of the sound wave. As confirmed by our numerical simulations, this difference leads to development of a steeper back edge and an increasingly sharp jump in the phase of the wave function. This is the analog of shock wave formation from large amplitude sound waves in a classical fluid ([*3*]{}). When the spatial width of the back edge has decreased to the width of a soliton with amplitude $\beta = \sqrt{A/2}$ (according to Eq. 5), such a soliton is shed off the back. It’s subsonic speed causes it to separate from the remaining sound wave. Furthermore, by creating defects with sizes on the order of the healing length, we excite collective modes of the condensate, with wave vectors on the order of the inverse healing length. In this regime, the Bogoliubov dispersion relation is not linear ([*4,5*]{}), and accordingly some of the sound wave will disperse into smaller ripples, as seen in Fig. 3A.
Considering the evolution of a defect of relative density amplitude $A$ and half-width $\delta$ in an otherwise homogenous medium, we estimate that solitons of amplitude $\beta =
\sqrt{A/2}$ will be created after the two resulting sound waves have propagated a distance
$$z_{sol} = \frac{2 \delta}{A}\left( \frac{1 - \frac{\xi}{2
\delta}}{1 - \frac{{\pi}^2 \xi^2}{\delta^2 A}}\right).$$
This is in agreement with our numerical calculations. We conclude that the minimum soliton formation length is obtained for large amplitude defects with a width just a few times the healing length. This dictates the defect width picked in the experiments presented here. Narrower defects disperse, whereas larger defects, comparable to the cloud size, couple to collective, nonlocalized excitations of the condensate.
We explored the soliton formation experimentally by creating defects in a BEC with the light roadblock. We controlled the size of the defects by varying the intensity of the probe pulses, which had a width $\tau = 1.3~\mu$s. OD images of state $|1 \rangle$ condensates are shown (Fig. 4) in one particular case. Immediately after the defect is created, the trap is turned off, and the cloud evolves and expands for 1 ms and 10 ms, respectively. As seen from the 1 ms picture, a single deep defect is formed initially, which results in creation of 5 solitons after 10 ms of condensate dynamics and expansion. The initial defect created in the trap could not be resolved with our imaging system, which has a resolution of 5 $\mu$m. By varying the probe intensity, we find that the number of solitons formed scales linearly with the probe pulse energy, as expected.
To study the stability of the solitons, we first performed 2D numerical simulations of Eqs. 3 and 4. Fig. 3B shows density profiles, $N_c |\psi_1|^2$, obtained for the same parameters as used in Fig. 1B. Again, the profiles are shown at various times after the pulse is compressed and stopped. The deepest soliton (the one closest to the center) is observed to quickly curl and eventually collapse into a vortex pair. The wave function develops a $2 \pi$ phase shift in a small circle around the vortex cores, which shows that they are singly quantized vortices. Also, the core radius is approximately the healing length. (Upon collapse, a small sound wave between the two vortices carries away some of the remaining soliton energy.) This decay can be understood as resulting from variation in propagation speed along the transverse soliton front. As discussed in ([*22-25*]{}), a small deviation will be enhanced by the nonlinearity in the Gross-Pitaevskii equation, and thus, the soliton collapses about the deepest (and therefore slowest) point.
Fig. 5 shows experimental images of state $|1 \rangle$ condensates. After the defect is created, the condensate of $|1
\rangle$ atoms is left in the trap for a varying amount of time (as indicated on the figures). The trap was then abruptly turned off and, $15$ ms after release, we imaged a selected slice of the expanded condensate, with a thickness of 30 $\mu$m along $y$ ([*37*]{}). The release time of $15$ ms is picked large enough that the condensate structures are resolvable with our imaging system ([*38*]{}). The slice was optically pumped from state $|1 \rangle$ to the $|3S_{1/2},\,F=2 \rangle$ manifold for $10\;\mu\mathrm{s}$ before it was imaged with absorption imaging by a laser beam nearly resonant with the transition from $|3S_{1/2},\,F~=~2
\rangle$ to $|3P_{3/2},\,F~=~3 \rangle$. The total pump and imaging time was small enough that no significant motion due to photon recoils occurred during the exposure. The slice was selected at the center of the condensate by placing a slit in the path of the pump beam.
For the data in Fig. 5A, it is seen that the deepest soliton curls as it leaves certain sections behind, and at 1.2 ms it has nucleated vortices. This is a direct observation of the snake instability. In Fig. 5B, at 0.5 ms, the snake instability has caused a complicated curving structure in one of the solitons, and vortices are observed after 2.5 ms. The vortices are seen to persist for many milliseconds and slowly drift towards the condensate edge. We observed them even after $30$ ms of trap dynamics, long enough to study the interaction of vortices with sound waves reflected off the condensate boundaries. Preliminary results, obtained by varying the $y$ position of the imaged condensate slices, indicate a complicated 3D structure of the vortices. In addition, the defect has induced a collective motion of the condensate whereby atoms, originally in the sides of the condensate, attempt to fill in the defect. This leads to a narrow and dense central region, which then slowly relaxes (Fig. 5B).
We performed the experiment with a variety of Rabi frequencies for the probe pulses, and saw nucleation of vortices only for the peak $\Omega_p > (2 \pi) 1.4$ MHz. The free energy of a vortex is substantially smaller near the border of the condensate where the density is smaller, so smaller (and thus lower energy) defects will form vortices very near the condensate edges, seen as ‘notches’ in Figs. 3B and 5.
In conclusion, we have studied and observed how small wavelength excitations cause a breakdown of superfluidity in a BEC. Our results show how localized defects in a superfluid will quite generally either disperse into high frequency ripples or end up in the form of topological defects such as solitons and vortices, and we have obtained an analytic expression for the transition between the two regimes. By varying our experimental parameters, we can create differently sized and shaped defects, and also control the number of defects created, allowing studies of a myriad of effects. Among them are soliton-soliton collisions, more extensive studies of soliton stability, soliton-sound wave collisions, vortex-soliton interactions, vortex dynamics, interaction between vortices, and the interaction between the BEC collective motion and vortices.
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41. This work was supported by the Rowland Institute for Science, the Defense Advanced Research Projects Agency, the U.S. Airforce Office of Scientific Research, the U.S. Army Research Office OSD Multidisciplinary University Research Initiative Program, the Harvard Materials Research Science and Engineering Center sponsored by the National Science Foundation, and by the Carlsberg Foundation, Denmark. C.S. is supported by a National Defense Science and Engineering Grant sponsored by the Department of Defense.
Figure Captions {#figure-captions .unnumbered}
===============
**Fig. 1.** **(A)** Compression of a probe pulse at the light ’roadblock’, according to 2D numerical simulations of Eqs. 3 and 4. The solid curves indicate probe intensity profiles along $z$ (at $x=0$), normalized to the peak input intensity. The snapshots are taken at a sequence of times, indicated in the figure, where $t=0$ is defined as the time when the center of the probe pulse enters the BEC from the left. For reference, the atomic density profile of the original condensate is plotted (in arbitrary units) as a dashed curve. The gray shading indicates the relative coupling input intensity as a function of $z$, with white corresponding to full intensity, and the darkest shade of gray corresponding to zero. The spatial turn off of the coupling field is centered at $z=0$ and occurs over $12\;\mu$m, as in the experiment. The number of condensate atoms is $1.2 \times 10^6$ atoms, the peak density is $6.9 \times
10^{13}\; \mathrm{cm}^{-3}$, and the coupling Rabi frequency is $\Omega_c=(2 \pi) 8.0$ MHz. The probe pulse has a peak Rabi frequency of $\Omega_p=(2 \pi)2.5$ MHz and a $1/e$ half-width of $\tau = 1.3 \;\mu$s. **(B)** Creation of a narrow density defect in a BEC. Density profiles of the two condensate components, $N_c|\psi_1|^2$ (dashed) and $N_c|\psi_2|^2$ (solid), along $z$ at $x=0$ for a sequence of times. Note that the $z$ range of the plot is restricted to a narrow region around the roadblock at the cloud center. The densities are normalized relative to the peak density of the original condensate indicated by the red dashed curve. The other curves correspond to times $1
\;\mu$s (green), $4 \;\mu$s (blue), and $14\; \mu$s (black). The width of the probe pulse is $\tau = 4 \;\mu$s and the other parameters are the same as in (A). An animated version is provided in the supplemental material ([*40*]{}) as Animation 1.\
\
**Fig. 2.** **(A)** Experimental in-trap OD image of a typical BEC before illumination by the probe pulse and coupling field. The condensate contains $1.5 \times 10^6$ atoms. The imaging beam was -30 MHz detuned from the $|1 \rangle
\rightarrow |3P_{1/2}, F=2, M_F = -1 \rangle$ transition. **(B)** Top view of the beam configuration used to create and study localized defects in a BEC as discussed in the text. **(C)** Build up of state $|2 \rangle$ atoms at the road block. In-trap OD images (left) show the transfer of atoms from $|1\rangle$ to $|2 \rangle$ as the probe pulse propagates through the condensate and runs into the roadblock. The atoms in $|2\rangle$ were imaged with a laser beam -13 MHz detuned from the $| 2\rangle \rightarrow |3P_{3/2}, F=3, M_F = -2\rangle$ transition. To allow imaging, the probe pulse propagation was stopped at various times, indicated in the figure, by switching the coupling beam off ([*29*]{}). The figures on the right show the corresponding line cuts along the probe propagation direction through the center of the BEC. The probe pulse had a Rabi frequency $\Omega_p=(2 \pi) 2.4$ MHz and the coupling Rabi frequency was $\Omega_c= (2 \pi) 14.6$ MHz.\
\
**Fig. 3.** **(A)** Formation of solitons from a density defect. The plots show results of a 1D numerical simulation of Eqs. 3 and 4. The light roadblock forms a defect and the subsequent formation of solitons is seen. The defect is set up with the same parameters for the light fields as in Fig. 1B. The number of condensate atoms is $N_c=1.2 \times 10^6$ and the peak density is $7.5 \times 10^{13}\;\mathrm{cm}^{-3}$. The times in the plots indicate the evolution time relative to the time when the probe pulse stops at the roadblock and the coupling beam is switched off (at $t=8\;\mu\mathrm{s}$ with $t=0$ as defined in Fig. 1A). The solid and dashed curves show the densities of $|1\rangle$ atoms ($N_c | \psi_1|^2$) and $|2\rangle$ atoms ($N_c |\psi_2|^2$). The phase of $\psi_1$ is shown in each case with a dotted curve (with an arbitrary constant added for graphical clarity). In the first two frames, the $|2\rangle$ atoms quickly leave due to the momentum recoil, leaving a large-amplitude, narrow defect in $\psi_1$ (Because this is a 1D simulation, the momentum kick in the $x$ direction is ignored). **(B)** The snake instability and the nucleation of vortices. The plots show the density $N_c |\psi_1|^2$ from a numerical simulation in 2D, with white corresponding to zero density and black to the peak density ($6.9 \times
10^{13}\;\mathrm{cm}^{-3}$). The parameters are the same as in Fig. 1B and the times indicated are relative to the coupling beam turn-off at $t=21\;\mu\mathrm{s}$. The solitons curl about their deepest point, eventually breaking and forming vortex pairs of opposite circulation (seen first at 3.5 ms). Several vortices are formed and the last frame shows the vortices slowly moving towards the edge of the condensate. At later times, they interact with sound waves which have reflected off the condensate boundaries. Animated versions (Animations 2 and 3) are provided in the supplemental material ([*40*]{}).\
\
**Fig. 4.** Experimental OD images and line cuts (at $x=0$) of a localized defect (top) and the resulting formation of solitons (bottom) in a condensate of $|1 \rangle$ atoms. The imaging beam was detuned $-30$ MHz and $-20$ MHz, respectively, from the $|\,3S_{1/2},\;F=2\rangle \rightarrow |\,3P_{3/2},
F=3\rangle$ transition. Prior to imaging, the atoms were optically pumped to $|\,3S_{1/2}, F=2\rangle$ for $10\;\mu\mathrm{s}$. The probe pulse had a peak Rabi frequency $\Omega_p = (2 \pi) 2.4$ MHz. The coupling laser had a Rabi frequency of $\Omega_c = (2
\pi) 14.9$ MHz, was turned on $6\;\mu\mathrm{s}$ before the probe pulse maximum, and had a duration of $18\;\mu\mathrm{s}$.\
\
**Fig. 5.** Experimental OD images of a $|1\rangle$ condensate, showing development of the snake instability and the nucleation of vortices. In each case, the BEC was allowed to evolve in the trap for a variable amount of time after defect creation. **(A)** The deepest soliton (nearest the condensate center) is observed to curl due to the snake instability and eventually break, nucleating vortices at 1.2 ms. Defects were produced in BECs with $1.9\times 10^6$ atoms by probe pulses with a peak $\Omega_p = (2 \pi) 2.4$ MHz, and a coupling laser with $\Omega_c = (2 \pi)14.6$ MHz. The imaging beam was $-5$ MHz detuned from the $|\,3S_{1/2},\,F=2\rangle \rightarrow
|\,3P_{3/2},\,F=3\rangle$ transition. **(B)** The snake instability and behavior of vortices at later times. The parameters in this series are the same as in (A), except that the peak $\Omega_p = (2\pi) 2.0$ MHz, the number of atoms in the BECs was $1.4 \times 10^6\!,$ and the pictures were taken with the imaging beam on resonance.
Animation Captions {#animation-captions .unnumbered}
==================
**Animation 1.** An animation, based on 2D numerical calculations, showing creation of a narrow density defect in a BEC by the light roadblock. The parameters and conventions are the same as in Fig. 1B. Successive frames are spaced by $1~\mu$s. The solid curve shows the build-up of $|2 \rangle$ atoms as the probe pulse runs into the roadblock, and the dashed curve shows the corresponding depletion of the density of the condensate of $|1
\rangle$ atoms.
**Animation 2.** Animation of 40 ms of BEC dynamics based on 1D numerical simulations. Parameters and conventions are the same as in Fig. 3A, but the phase is not plotted here. Successive frames are spaced by 0.25 ms. The animation shows a narrow density defect in the $|1 \rangle$ condensate decaying into four solitons due to the steepening of the back edge of the sound waves. The high frequency ripples are due to the nonlinear part of the Bogoliubov dispersion curve. When the solitons reach a point where their amplitude, $\beta$, equals unity they turn around. Upon reaching the center of the condensate, they pass through each other unaffected.
**Animation 3.** Animation showing 30 ms of dynamics of the state $|1 \rangle$ condensate, based on 2D numerical simulations. Parameters and conventions are the same as in Fig. 3B. (The plot range of each frame is $96.8~ \mu$m$~\times~
31.2~\mu$m). Successive frames are spaced by 0.4 ms. The narrow density defect in the condensate decays into several solitons and the deepest solitons decay, via the snake instability, into vortices and release their remaining energy as sound waves. The vortices drift slowly, while some of the sound waves reflect off the condensate boundaries and subsequently interact with the vortices.
| ArXiv |
---
abstract: 'We consider the $\Lambda N\to NN$ weak transition, responsible for a large fraction of the non-mesonic weak decay of hypernuclei. We follow on the previously derived effective field theory and compute the next-to-leading one-loop corrections. Explicit expressions for all diagrams are provided, which result in contributions to all relevant partial waves.'
author:
- 'A. Pérez-Obiol'
- 'D. R. Entem'
- 'B. Juliá-Díaz'
- 'A. Parreño'
title: 'One-loop contributions in the EFT for the $\Lambda N \to NN$ transition'
---
Introduction
============
One of the major challenges in nuclear physics is to understand the interactions among hadrons from first principles. For more than twenty years, many research groups have directed their efforts to develop Effective Field Theories (EFT), working with the idea of separating the nuclear force in long-range and short-range components. The underlying premise was that low-energy processes, as the ones encountered in nuclear physics, should not be affected by the specific details of the high-energy physics.
The typical energies associated to nuclear phenomena suggest that the appropriate degrees of freedom are nucleons and pions (or the ground state baryon and pseudo scalar octets for processes involving strangeness), interacting derivatively as it is dictated by the effective chiral Lagrangian. The nuclear interaction is characterized by the presence of very different scales, going from the values of the masses of the light pseudo-scalar bosons to the ones of the ground-state octet baryons. The EFT formalism makes use of this separation of scales to construct an expansion of the Lagrangian in terms of a parameter built up from ratios of these scales. For example, in the study of the low-energy nucleon-nucleon interaction, a clear separation of scales is seen between the external momentum of the interacting nucleons, a soft scale which typically takes values up to the pion mass, and a hard scale corresponding to the nucleon mass. While the long-range part of this interaction is governed by the light scale through the pion-exchange mechanism, short-range forces are accounted for by zero-range contact operators, organized according to an increasing number of derivatives. These contact terms, which respect chiral symmetry, have values which are not constrained by the chiral Lagrangian, and therefore, their relative strength (encapsulated in the size of the low-energy coefficients, LECs) has to be obtained from a fit to nuclear observables. The large amount of experimental data for the interaction among pions and nucleons has made possible to perform successful EFT calculations of the strong nucleon-nucleon interaction up to fourth order in the momentum expansion (${\cal
O}(p^4)$), at next-to-next-to-next-to-leading order (N$^3$LO) in the heavy-baryon formalism [@Epelbaum:2012vx; @entem]. In the weak sector, the study of nucleon-nucleon Parity Violation (PV) with an Effective Field Theory at leading order has been undertaken in Ref. [@nucPV05], where the authors discuss existing and possible few-body measurements that can help in constraining the relevant (five) low-energy constants at order $p$ in the momentum expansion and the ones associated with dynamical pions.
In the strange sector, the experimental situation is less favorable due to the short life-time of hyperons, unstable against the weak interaction. This fact complicates the extraction of information regarding the strong interaction among baryons in free space away from the nucleonic sector. Nevertheless, SU(3) extensions of the EFT for nucleons and pions have been developed at leading order (LO) [@SW96; @KDT01; @H02; @BBPS05] and next-to-leading (NLO) order [@PHM06]. In the present work we consider the weak four-body $\Lambda N \to NN$ interaction, which is accessible experimentally by looking at the decay of $\Lambda-$hypernuclei, bound systems composed by nucleons and one $\Lambda$ hyperon. These aggregates decay weakly through mesonic ($\Lambda \to N \pi$) and non-mesonic ($\Lambda N \to NN$) modes, the former being suppressed for mass numbers of the order or larger than 5, due to the Pauli blocking effect acting on the outgoing nucleon. In contrast to the weak NN PV interaction, which is masked by the much stronger Parity Conserving (PC) strong NN signal, the weak $|\Delta S|=1 \, \Lambda N$ interaction has the advantage of presenting a change of flavor as a signature, favoring its detection in the presence of the strong interaction.
The first studies of the weak $\Lambda N$ interaction using a lowest order effective theory were presented in Refs. [@Jun; @PBH05; @PPJ11] . These works included the exchange of the lighter pseudoscalar mesons while parametrizing the short-range part of the interaction with contact terms at order ${\cal O}(q^0)$, where $q$ denotes the momentum exchanged between the interacting baryons. While the results of Ref. [@PPJ11] show that it is possible to reproduce the hypernuclear decay data with the lowest order effective Lagrangian, the stability of the momentum expansion has to be checked by including the next order in the EFT. If an effective field theory can be built for the weak $\Lambda N \to NN$ transition, the values for the LECs of the theory, which encode the high-energy components of the interaction, should vary within a reasonable and natural range when one includes higher orders in the calculation. Compared to the LO calculation, which involves two LECs, the unknown baryon-baryon-kaon vertices and the pseudoscalar cut-off parameter in the form-factor, the NLO calculation introduces additional unknowns. Namely, the parameters associated to the new contact terms (three when one neglects the small value of the momentum of the initial particles, a nucleon and a hyperon bound in the hypernucleus, in front of the momentum of the two outgoing nucleons) and the couplings appearing in the two-pion exchange diagrams. Therefore, in order to constrain the EFT at NLO, one needs to collect enough data, either through the accurate measure of hypernuclear decay observables, or through the measure of the inverse reaction in free space, $n p \to
\Lambda p$. Unfortunately, the small values of the cross-sections for the weak strangeness production mechanism, of the order of $10^{-12}$ mb [@Haidenbauer1995; @Parreno1998; @Inoue2001], has prevented, for the time being, its consideration as part of the experimental data set, despite the effort invested in extracting different polarization observables for this process [@Kishimoto2000; @Ajimura2001]. At present, quantitative experimental information on the $|\Delta
S|=1$ weak interaction in the baryonic sector comes from the measure of the total and partial decay rates of hypernuclei, and an asymmetry in the number of protons detected parallel and antiparallel to the polarization axis, which comes from the interference between the PC and PV weak amplitudes. Since observables from one hypernucleus to another can be related through hypernuclear structure coefficients, one has to be careful in selecting the data that can be used in the EFT calculation. For example, while one may indeed expect measurements from different p-shell hypernuclei, say, A=12 and 16, to provide with the same constraint, the situation is different when including data from s-shell hypernuclei like A=5. For the latter, the initial $\Lambda N$ pair can only be in a relative s-state, while for the former, relative p-states are allowed as well. In this paper we present the analytic expressions to be included at next-to-leading order in the effective theory for the weak $\Lambda N$ interaction. These expressions have been derived by considering four-fermion contact terms with a derivative operator insertion together with the two-pion exchange mechanism.
The paper is organized as follows. In Section II we introduce the Lagrangians and the power counting scheme we use to calculate the relevant Feynman diagrams. In Sections \[ss:loc\] and \[ss:nloc\] we present the LO and NLO potentials for the $\Lambda N\rightarrow NN$ transition, and a comparison between both contributions is performed in Section \[sec:bc\]. We conclude and summarize in Section \[sec:conclusions\].
Interaction Lagrangians and counting scheme {#sec2}
===========================================
The non-mesonic weak decay of the $\Lambda$ involves both the strong and electroweak interactions. The $\Lambda$ decay is mediated by the presence of a nucleon which in the simplest meson-exchange picture, exchanges a meson, e.g. $\pi$, $K$, with the $\Lambda$. Thus, computing the transition requires the knowledge of the strong and weak Lagrangians involving all the hadrons entering in the process. In this section we describe the strong and weak Lagrangians entering at leading order (LO) and next-to-leading order (NLO) in the $\Lambda N\to NN$ interaction.
![Weak vertices for the $\Lambda N\pi$, $\Sigma N\pi$ and $NNK$ stemming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nnpw "fig:") ![Weak vertices for the $\Lambda N\pi$, $\Sigma N\pi$ and $NNK$ stemming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nspw "fig:") ![Weak vertices for the $\Lambda N\pi$, $\Sigma N\pi$ and $NNK$ stemming from the Lagrangians in Eq. (\[eq:weakl\]). The weak vertex is represented by a solid black circle. \[vf2\]](nnkw "fig:")
The weak interaction between the $\Sigma$, $\Lambda$ and $N$ baryons and the pseudoscalar $\pi$ and $K$ mesons is described by the phenomenological Lagrangians:
$$\begin{aligned}
\label{eq:weakl}
{\mathcal{L}}_{\Lambda N\pi}^w=&-iG_Fm_\pi^2\overline{\Psi}_N(A+B\gamma^5)
{\vec{\tau}}\cdot\vec{\pi}\Psi_\Lambda
\\\nonumber
{\mathcal{L}}_{\Sigma N\pi}^w=&
-iG_Fm_\pi^2\overline{\Psi}_N(\vec{A}_{\Sigma_i}+\vec{B}_{\Sigma_i}\gamma^5)
\cdot\vec{\pi}\Psi_{\Sigma_i}\,,
\\\nonumber
{\cal L}^{w}_{NN K} =&
-iG_Fm_\pi^2 \, \left[ \, \overline{\psi}_{N} \left( ^0_1 \right)
\,\,( C_{K}^{PV} + C_{K}^{PC} \gamma_5) \,\,(\phi^{K})^\dagger
\psi_{N} \right.
\\ \nonumber
& \left. + \, \overline{\psi}_{N} \psi_{N}
\,\,( D_{K}^{PV}
+ D_{K}^{PC}
\gamma_5 ) \,\,(\phi^{K})^\dagger \,\,
\left( ^0_1 \right) \right] \ ,\end{aligned}$$
where $G_Fm_\pi^2=2.21\times10^{-7}$ is the weak Fermi coupling constant, $\gamma$ are the Dirac matrices and $\tau$ the Pauli matrices. The index $i$ appearing in the $\Sigma$ field refers to the different isospurion states for the $\Sigma$ hyperon: $$\Psi_{\Sigma\frac12}=
\left(\begin{array}{c}-\sqrt{\frac23}\Sigma_+\\\frac{1}{\sqrt3}\Sigma_0\end{array}\right)\,,
~~
\Psi_{\Sigma\frac32}=
\left(\begin{array}{c}0\\-\frac{1}{\sqrt3}\Sigma_+\\\sqrt{\frac23}\Sigma_0\\\Sigma_-\,
\end{array}\right)\,.$$ The PV and PC structures, $\vec{A}_{\Sigma_i}$ and $\vec{B}_{\Sigma_i}$ contain the corresponding weak coupling constants together with the isospin operators $\tau^a$ for $\frac12\to\frac12$ transitions and $T^a$ for $\frac12\to\frac32$ transitions. The weak couplings $A=1.05$, $B=-7.15$, $A_{\Sigma\frac12}=-0.59$, $A_{\Sigma\frac32}=2.00$, $B_{\Sigma\frac12}=-15.68$, and $B_{\Sigma\frac32}=-0.26$ [@DF96] are fixed to reproduce the experimental data of the corresponding hyperon decays, while the ones involving kaons, $C_K^{PC}=-18.9$, $D_K^{PC}=6.63$, $C_{K}^{PV}=0.76$ and $D_K^{PV}=2.09$, are derived using SU(3) symmetry.
![Weak vertices corresponding to the $\Lambda N\pi\pi$, and $\Lambda N$ interactions. The solid black circle represents the weak vertex. The corresponding Lagrangians are given in Eq. (\[eq:weakl2\]). \[vf3\]](nnppw "fig:") ![Weak vertices corresponding to the $\Lambda N\pi\pi$, and $\Lambda N$ interactions. The solid black circle represents the weak vertex. The corresponding Lagrangians are given in Eq. (\[eq:weakl2\]). \[vf3\]](nlw "fig:")
The other two weak vertices entering at the considered order (Fig. \[vf3\]) are obtained from the weak SU(3) chiral Lagrangian, $$\begin{aligned}
{\mathcal{L}}^w_{\Lambda N\pi\pi}=&
G_Fm_\pi^2\frac{h_{2\pi}}{f_\pi^2}(\vec{\pi}\cdot\vec{\pi})
\overline{\Psi}\Psi_\Lambda\,, \label{eq:weakl2}\\
{\mathcal{L}}^w_{\Lambda N} =& iG_Fm_\pi^2 h_{\Lambda N}
\overline{\Psi}\Psi_\Lambda \nonumber\,,\end{aligned}$$ with $
h_{2\pi}=(D+3F)/(8\sqrt6 G_Fm_\pi^2)=-10.13\text{ MeV}
$ and $
h_{\Lambda N}=-(D+3F)/(\sqrt6 G_F m_\pi^2)=81.02\text{ MeV} \,.
$ $D$ and $F$ are the couplings parametrizing the weak chiral SU(3) Lagrangian, and can be fitted through the pole model to the experimentally known hyperon decays. In that case, one finds that when s-wave amplitudes are correctly reproduced, p-wave amplitude predictions disagree with the experiment [@donoghue].
[cc]{} ![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](nnps "fig:")& ![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](nnpps "fig:")\
\
![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](lspw "fig:")& ![Strong vertices for the $NN\pi$, $NN\pi\pi$, $\Lambda \Sigma\pi$ and $\Lambda N K$ which arise from the Lagrangians in Eq. (\[eq:str\]). \[vf1\]](lnks "fig:")
The strong vertices for the interaction between our baryonic and mesonic degrees of freedom are obtained from the strong SU(3) chiral Lagrangian [@donoghue], $$\begin{aligned}
{\mathcal{L}}^s_{NN\pi}=&-\frac{g_A}{2f_\pi}\overline{\Psi}\gamma^\mu\gamma_5
{\vec{\tau}}\Psi\cdot\partial_\mu\vec{\pi}\,,\nonumber\\
{\mathcal{L}}^s_{NN\pi\pi} =&-\frac{1}{4f_\pi^2}\overline{\Psi}\gamma^\mu
{\vec{\tau}}\cdot(\vec{\pi}\times\partial_\mu\vec{\pi})\Psi\,,\nonumber\\
{\mathcal{L}}^s_{\Lambda\Sigma\pi}=&-\frac{D_s}{\sqrt{3}}\,\overline{\Psi}_\Lambda\gamma^\mu\gamma_5
\Psi_\Sigma\cdot\partial_\mu\vec{\pi} \,,\label{eq:str}
\\\nonumber
{\mathcal{L}}^{s}_{\Lambda N K} =&
\, \frac{D_s+3F_s}{2\sqrt3f_\pi} \, \overline{\Psi}_{N}
\gamma^\mu\gamma_5 \,\partial_\mu\phi_{K}
\Psi_\Lambda \,,\end{aligned}$$ where we have taken the convention which gives us $\Psi_\Sigma\cdot\vec{\pi}=\Psi_{\Sigma_+}\pi_-+\Psi_{\Sigma_-}\pi_++\Psi_{\Sigma_0}\pi_0$, and we consider, $g_A=1.290$, $f_\pi=92.4$ MeV, $D_s=0.822$, and $F_s=0.468$. These strong coupling constants are taken from $NN$ interaction models such as the Jülich [@JB] or Nijmegen [@nij99] potentials. The four interaction vertices corresponding to these Lagrangians are depicted in Fig. \[vf1\].
Once the interaction Lagrangians involving the relevant degrees of freedom have been presented, we need to define the power counting scheme which allows us to organize the different contributions to the full amplitude.
Power counting scheme {#ss:cs}
---------------------
The amplitude for the $\Lambda N\to NN$ transition is built as the sum of a medium and long-range one meson exchanges (i.e. $\pi$ and $K$), the contribution from the two-pion exchanges, and the contribution of the contact interactions up to ${\cal O} (q^2/M^2)$ as described below. The order at which the different terms enter in the perturbative expansion of the amplitudes is given by the so-called Weinberg power counting scheme [@W9091].
In our calculations we will employ the heavy baryon formalism [@jm]. This technique introduces a perturbative expansion in the baryon masses appearing in the Lagrangians, so that this new large scale does not disrupt the well-defined Weinberg power counting. It is worth noting that, in the heavy baryon formalism, terms of the type $\overline{\Psi}_B\gamma^5\Psi_B$ are subleading in front of terms like $\overline{\Psi}_B\Psi_B$, since they show up at one order higher in the heavy baryon expansion. In our calculation, we choose to keep both terms in our Lagrangians of Eqs. (\[eq:weakl\]) because the experimental values for the couplings $B_\Lambda$ and $B_\Sigma$ are much larger than $A_\Lambda$ and $A_\Sigma$. For example, $A_\Lambda=1.05$ and $B_\Lambda=-7.15$ [@donoghue].
Our calculation is characterized by the presence of different octet baryons in the relevant Feynman diagrams, contributing in both, the spinors and propagators.
The spinors for the incoming $\Lambda$ and $N$ with masses $M_\Lambda$ and $M_N$, energies $E_p^{\Lambda}$ and $E_p^N$, and momenta ${\vec{p}}$ and $-{\vec{p}}$ are $$\begin{aligned}
\nonumber
u_1(E_p^\Lambda,{\vec{p}}\,)=
\sqrt{\frac{E_p^\Lambda+M_\Lambda}{2M_\Lambda}}
\left(\begin{array}{c}
1\\\nonumber
\frac{{\vec{\sigma}_1}\cdot{\vec{p}}}{E_p^\Lambda+M_\Lambda}
\end{array}\right)\,,
\\\\\nonumber
u_2(E_p^N,-{\vec{p}}\,)=
\sqrt{\frac{E_p^N+M_N}{2M_N}}
\left(\begin{array}{c}
1\\
-\frac{{\vec{\sigma}_2}\cdot{\vec{p}}}{E_p^N+M_N}
\end{array}\right)\,,\end{aligned}$$ and for the outgoing nucleons with momenta ${\vec{p}\,'}$ and $-{\vec{p}\,'}$, and energy $E'\equiv\frac12\left(E_p^\Lambda+E_p^N\right)$, $$\begin{aligned}
\nonumber
\bar{u}_1(E',{\vec{p}\,'})=
\sqrt{\frac{E'+M_N}{2M_N}}
\left(\begin{array}{cc}
1&
-\frac{{\vec{\sigma}_1}\cdot{\vec{p}\,'}}{E'+M_N}
\end{array}\right)\,,
\\\\\nonumber
\bar{u}_2(E',-{\vec{p}\,'})=
\sqrt{\frac{E'+M_N}{2M_N}}
\left(\begin{array}{cc}
1&
ñ\frac{{\vec{\sigma}_2}\cdot{\vec{p}\,'}}{E'+M_N}
\end{array}\right)\,.\end{aligned}$$ The relativistic propagator of a baryon with mass $M_B$ and momentum $p$ reads $$\frac{i}{\cancel{p}-M_B+i\epsilon}
=\frac{i(\cancel{p}+M_B)}{p^2-M_B^2+i\epsilon} \,.$$ Making the heavy baryon expansion with these spinors and propagators introduces mass differences ($M_\Lambda-M_N$, $M_\Sigma-M_\Lambda$) in the baryonic propagators. A reasonable approach would be to consider these mass differences of order ${\cal O}\left({\vec{q}}^{\,2}/\Lambda^2\right)$ ($M_B={\overline{M}}+{\cal O}\left({\vec{q}}^{\,2}/\Lambda^2\right)$), and thus they would not enter in the loop diagrams. We have chosen to leave the physical masses in both the initial and final spinors and also in the intermediate propagators; i.e. we consider the mass differences as another scale in the heavy baryon expansion. The corresponding SU(3) symmetric limit is also given at the end of section \[ss:tped\], and can be easily obtained from our expressions by setting the mass differences, which we explicitly retain, to zero.
The procedure we follow to compute the different Feynman diagrams entering the transition amplitude is the following: first we write down the relativistic expressions for each diagram, and then afterwards, we perform the heavy baryon expansion.
In the next sections we will describe the LO and NLO contributions to the process $\Lambda N\to NN$, following the scheme presented here. The explicit expressions and details of the calculations are given in the Appendices.
Leading order Contributions {#ss:loc}
===========================
------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------
![One-pion and one-kaon exchange contributions to the transition.\[fig:loc\]](ope "fig:") ![One-pion and one-kaon exchange contributions to the transition.\[fig:loc\]](oke "fig:") ![One-pion and one-kaon exchange contributions to the transition.\[fig:loc\]](contactw0 "fig:")
------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------
For completeness, we rewrite here the LO EFT already presented in Ref. [@PPJ11], and then build the NLO contributions in the next section.
At tree level, the transition potential $\Lambda N\to NN$ involves the LO contact terms, and $\pi$ and $K$ exchanges, as depicted in Fig. \[fig:loc\]. First, the contact interaction can be written as the most general Lorentz invariant potential with no derivatives. The four-fermion (4P) interaction in momentum space at leading order (in units of $G_F$) is $$\begin{aligned}
V_{4P} ({\vec q} \, ) &=&
C_0^0 + C_0^1 \; {\vec \sigma}_1
{\vec \sigma}_2 \,,\label{eq:vlo}\end{aligned}$$ where $C_0^0$ and $C_0^1$ are low energy constants which need to be fitted by direct comparison to experimental data. In Ref. [@PPJ11] we presented several sets of values which were to a large extent compatible with the scarce data on hypernuclear decay.
The potentials for the one pion and one kaon exchanges, as functions of transferred momentum ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, read, respectively [@PRB97] $$\begin{aligned}
{V_{\pi}} ({\vec q}\,) =&
\nonumber
-\frac{G_F m_\pi^2g_{NN\pi}}{2 M_N}
\left(
A_\pi - \frac{B_\pi}{2 {\overline{M}}}{\vec \sigma}_1 \, {\vec q} \,
\right)
\frac{{\vec \sigma}_2 \, {\vec q}\,}{-q_0^2+{\vec q}^{\; 2}+m_\pi^2} \,
\\&\times {\vec{\tau}_1}\cdot{\vec{\tau}_2}{\rm ,}
\label{eq:pion}\\
{V_{K}} ({\vec q}\,) =&
-\frac{G_F m_\pi^2g_{\Lambda N K}}{2{\overline{M}}}
\left(
\hat{A} - \frac{\hat{B}}{2 M_N}{\vec \sigma}_1 \, {\vec q} \,
\right)
\nonumber\\&\times\frac{{\vec \sigma}_2 \, {\vec q}\,} {-q_0^2+{\vec q}^{\; 2}+m_K^2} \,
{\rm ,}
\label{eq:kaon}\end{aligned}$$ where $m_\pi=138$ MeV and $m_K=495$ MeV, $q_0\equiv
\frac12(M_\Lambda-M_N)$, $g_{NN\pi}\equiv \frac{g_A M_N}{f_\pi}$, $g_{\Lambda N
K}\equiv-\frac{D_s+3F_s}{2\sqrt3f_\pi}$, ${\overline{M}}\equiv\frac12(M_N+M_\Lambda)$, and $$\begin{aligned}
{\hat A} &=\left( \frac{ C^{PV}_{K}}{2} +
D^{PV}_{K} + \frac{
C^{PV}_{K}}{2} {\vec \tau}_1 {\vec \tau}_2
\,\right),
\\
{\hat B}&= \left( \frac{
C^{PC}_{K}}{2} +
D^{PC}_{K} + \frac{
C^{PC}_{K}}{2}
{\vec \tau}_1 \, {\vec \tau}_2 \right) \,.\end{aligned}$$
Next-to-leading order contributions {#ss:nloc}
===================================
Order Parity Structures
------- -------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
0 PC $1$, ${\vec{\sigma}_1}\cdot{\vec{\sigma}_2}$
${\vec{\sigma}_1}\cdot{\vec{q}}$, ${\vec{\sigma}_1}\cdot{\vec{p}}$, ${\vec{\sigma}_2}\cdot{\vec{q}}$,
${\vec{\sigma}_2}\cdot{\vec{p}}$, $({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}$, $({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{p}}$,
${\vec{q}}^2$, ${\vec{p}}^2$, $({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}){\vec{q}}^2$, $({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}){\vec{p}}^2$, $({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})$,
$({\vec{\sigma}_1}\cdot{\vec{p}})({\vec{\sigma}_2}\cdot{\vec{p}})$, $({\vec{\sigma}_1}+{\vec{\sigma}_2})\cdot({\vec{q}}\times{\vec{p}})$
${\vec{q}}\cdot{\vec{p}}$, $({\vec{\sigma}_1}\cdot{\vec{\sigma}_2}){\vec{q}}\cdot{\vec{p}}$, $({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}})$,
$({\vec{\sigma}_1}\cdot{\vec{p}})({\vec{\sigma}_2}\cdot{\vec{q}})$, $({\vec{\sigma}_1}-{\vec{\sigma}_2})\cdot({\vec{q}}\times{\vec{p}})$
: All possible PC and PV NLO operational structures connecting the initial and final spin and angular momentum states. There are a total of 18. \[tab:contacts\]
The NLO contribution to the weak decay process, $\Lambda N\to NN$, includes contact interactions with one and two derivative operators, caramel diagrams and two-pion-exchange diagrams.
NLO contact potential {#sec:nlocontact}
---------------------
In principle the NLO contact potential should include, in the center of mass, structures involving both the initial (${\vec{p}}\,$) and final (${\vec{p}\,'}$) momenta, or independent linear combinations, e.g. ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$ and ${\vec{p}}$. Table \[tab:contacts\] lists all these possible structures. At NLO there are 18 LECs —6 PV ones at order ${\cal O}\left(q/M\right)$, 7 PC ones at order ${\cal
O}\left(q^2/M^2\right)$ and 5 PV ones at order ${\cal
O}\left(q^2/M^2\right)$—, which must be fitted to experiment. This is not feasible with current experimental data on hypernuclear decay. A reasonable way to reduce the number of LECs and render the fitting procedure more tractable is to note that the pionless weak decay mechanism we are interested in takes place inside a bound hypernucleus. Thus, one can consider that in the $\Lambda N\rightarrow NN$ transition potential the initial baryons have a fairly small momentum. Moreover, the final nucleons gain an extra momentum from the surplus mass of the $\Lambda$ ($M_\Lambda-M_N=116$ MeV), which in most cases allow to consider, ${\vec{p}\,'}\gg{\vec{p}}$. In this case, one may approximate ${\vec{q}}\simeq{\vec{p}\,'}$ and ${\vec{p}}=0$. Within this approximation, the NLO part of the contact potential reads (in units of $G_F$): $$\begin{aligned}
V_{4P} ({\vec q} \, ) &=
C_1^0 \; \displaystyle\frac{{\vec \sigma}_1{\vec q}}{2 M_N}
\label{eq:vnlo}
+ \,C_1^1 \; \displaystyle\frac{{\vec \sigma}_2{\vec q}}{2 M_N}
+ {\im} \, C_1^2 \; \displaystyle
\frac{({\vec \sigma}_1 \times {\vec\sigma}_2)\;{\vec q}}{2 M_N}
\\
&+ C_2^0 \; \displaystyle\frac{{\vec \sigma}_1{\vec q}
\;{\vec\sigma}_2{\vec q}}{4 M_N^2}+
C_2^1 \; \displaystyle\frac{{\vec \sigma}_1
{\vec \sigma}_2 \; {\vec q}^{\; 2}}
{4 M_N^2}+
C_2^2 \; \displaystyle\frac{{\vec q}^{\,2}}{4 M_N^2} \,.
\nonumber\end{aligned}$$
----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- --
![ Caramel diagrams contributing to the process at NLO. The solid circle represents the weak vertex. \[fig:caramels\]](caramel1 "fig:") ![ Caramel diagrams contributing to the process at NLO. The solid circle represents the weak vertex. \[fig:caramels\]](caramel2 "fig:") ![ Caramel diagrams contributing to the process at NLO. The solid circle represents the weak vertex. \[fig:caramels\]](caramel3 "fig:")
(a) (b) (c)
----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------- --
Using strong and weak LO contact interactions and two baryonic propagators one can also build three diagrams that enter at NLO. These caramel-like diagrams are shown in Fig. \[fig:caramels\]. They only differ in the position of the strong and weak vertices and in the mass of upper-leg baryonic propagator. In order to write a general expression for the three caramel diagrams we label the mass of the upper-leg propagating baryon $M_\alpha$ ($M_a=M_N$, $M_b=M_\Lambda$ and $M_c=M_\Sigma$) and the corresponding strong and weak contact vertices $C_{S(s)}^\alpha+C_{T(s)}^\alpha{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}$ and $C_{S(w)}^\alpha+C_{T(w)}^\alpha{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}$, where $\alpha=a,b,c$ corresponds to the labels of Fig. \[fig:caramels\]. It is also convenient to define $M_\alpha=M_N+\Delta_\alpha$. In the heavy baryon formalism these diagrams only contribute with an imaginary part of the form $$\begin{aligned}
V_\alpha&=i\frac{G_Fm_\pi^2}{16\pi M_N}
(C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
\\&\times\nonumber
(C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
\\&\times\nonumber
\sqrt{(\Delta_b-\Delta_\alpha)(\frac12(\Delta_b+\Delta_\alpha)+M_N)+{\vec{p}}^2}\,.\end{aligned}$$ Few more details are given in App. \[sec:caramels\].
One pion corrections to the LO contact interactions, shown in Fig. \[fig:contact.corrections\], also enter at NLO. The net contribution of these diagrams is to shift the coefficients of the LO contact terms with functions dependent on $m_\pi$, $M_\Lambda-M_N$ and $M_\Sigma-M_N$.
![Corrections to the LO contact interactions. The contributions of all these diagrams can be accounted for by an adequate shift of the coefficients of the LO contact terms.[]{data-label="fig:contact.corrections"}](contacttots "fig:")\
![Corrections to the LO contact interactions. The contributions of all these diagrams can be accounted for by an adequate shift of the coefficients of the LO contact terms.[]{data-label="fig:contact.corrections"}](contacttots2 "fig:")
Two-pion-exchange diagrams {#ss:tped}
--------------------------
The two-pion-exchange contributions are organized according to the different topologies — balls, triangles, and boxes—, such that most of the integration techniques are shared by each class of diagrams. There are two types of ball diagrams, of which only one gives a non-zero contribution, depicted in Fig. \[fig:ball\]. In addition, there are four triangle diagrams, shown in Fig. \[fig:triangle\], and two box and crossed box diagrams, shown in Fig. \[fig:box\]. The topologies contain, respectively, zero, one, and two baryonic propagators, which may correspond to $N$ or $\Sigma$ baryons. All the diagrams contain two relativistic propagators from the 2$-\pi$ exchange.
--------------------------------------------------------------------------------------------------------------------------------
![The ball diagram contributing to the process at NLO. The solid circle represents the weak vertex.\[fig:ball\]](ball2 "fig:")
(a)
--------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------
![Triangle diagrams which contribute to the process at NLO. The solid circle represents the weak interaction vertex.\[fig:triangle\]](uptriangle1 "fig:") ![Triangle diagrams which contribute to the process at NLO. The solid circle represents the weak interaction vertex.\[fig:triangle\]](uptriangle2 "fig:") ![Triangle diagrams which contribute to the process at NLO. The solid circle represents the weak interaction vertex.\[fig:triangle\]](downtriangle1 "fig:") ![Triangle diagrams which contribute to the process at NLO. The solid circle represents the weak interaction vertex.\[fig:triangle\]](downtriangle2 "fig:")
(b) (c) (d) (e)
----------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------
![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box1 "fig:") ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box2 "fig:") ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box3 "fig:") ![Box diagrams which contribute to the process at NLO. The solid circle stands for the weak interaction vertex.\[fig:box\]](box4 "fig:")
(f) (g) (h) (i)
------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------
The technical details of the evaluation of the Feynman diagrams for the ball, triangle and box diagrams are given in the App. \[sec:balls\], \[sec:triangles\], and \[sec:boxs\] respectively. The main technique used is to introduce a number of master integrals, which appear in different diagrams, and which reduce the mathematical complexity of the problem (see App. \[sec:mi\]). Once they are defined, we derive a number of relations between the master integrals, which can in most cases be easily checked. Full details are provided to ensure the future use of these expressions.
Using the labels defined in Figs. \[fig:ball\], \[fig:triangle\] and \[fig:box\] we organize the contributions of all the $2-\pi$ exchange diagrams in Eq. (\[eq:tots\]). The corresponding coefficients in terms of the coupling constants, baryon and meson masses, and momenta can be read off from the full expressions given in the Appendices \[sec:balls\], \[sec:triangles\] and \[sec:boxs\]. $$\begin{aligned}
\label{eq:tots}
V_a=&c_{a1}\,{\vec{\tau}_1}\cdot{\vec{\tau}_2}\\\nonumber
V_b=&c_{b1}
\\\nonumber
V_c=&c_{c1}\,{\vec{\tau}_1}\cdot{\vec{\tau}_2}\\\nonumber
V_d=&
\left[c_{d1}+c_{d2}\,{\vec{\sigma}_1}\cdot{\vec{q}}+c_{d3}\,({\vec{q}}\cdot{\vec{p}})
+c_{d4}\,{\vec{\sigma}_1}\cdot({\vec{q}}\times{\vec{p}})\right]
({\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\\nonumber
V_e=&(c_{e1}+c_{e2}{\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\tau}_1}\cdot{\vec{\tau}_2})\end{aligned}$$ $$\begin{aligned}
\label{eq:tots2}
V_f=&
\Big[
c_{f1}
+c_{f2}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}+c_{f3}{\vec{\sigma}_1}\cdot{\vec{q}}+c_{f4}({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\nonumber\\
+&c_{f5}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})
+c_{f6}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}})
\\+&c_{f7}{\vec{\sigma}_1}\cdot({\vec{p}}\times{\vec{q}})\nonumber
+c_{f8}{\vec{\sigma}_2}\cdot({\vec{p}}\times{\vec{q}})\Big]
(c_{f1}'+c_{f2}'\,{\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\\nonumber
V_g=&
\Big[
c_{g1}
+c_{g2}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}+c_{g3}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})
\Big]
\\\times&\nonumber
(c_{g1}'+c_{g2}'\,{\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\+&\nonumber
\Big[
c_{g4}{\vec{\sigma}_1}\cdot{\vec{q}}+c_{g5}({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\Big]
(c_{g1}''+c_{g2}''\,{\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\\nonumber
V_h=&
\Big[
c_{h1}
+c_{h2}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}+c_{h3}{\vec{\sigma}_1}\cdot{\vec{q}}+c_{h4}({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\\+&c_{h5}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})\nonumber
+c_{h6}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}})
\\+&c_{h7}{\vec{\sigma}_1}\cdot({\vec{p}}\times{\vec{q}})\nonumber
+c_{h8}{\vec{\sigma}_2}\cdot({\vec{p}}\times{\vec{q}})\Big]
(c_{h1}'+c_{h2}'\,{\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\\nonumber
V_i=&
\Big[
c_{i1}
+c_{i2}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2}+c_{i3}({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})
\Big]
\\\times&\nonumber
(c_{i1}'+c_{i2}'\,{\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\+&\nonumber
\Big[
c_{i4}{\vec{\sigma}_1}\cdot{\vec{q}}+c_{i5}({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\Big]
(c_{i1}''+c_{i2}''\,{\vec{\tau}_1}\cdot{\vec{\tau}_2})\,.\end{aligned}$$ Considering the SU(3) limit where all the baryon masses are considered to take the same value ($q_0=q_0'=0$) the expressions above become much more simple. Defining $$\begin{aligned}
At(q)\equiv&\frac{1}{2q}\arctan\left(\frac{q}{2m_\pi}\right)
\\
L(q)\equiv&\frac{\sqrt{4m_\pi^2+q^2}}{q}\ln\left(\frac{\sqrt{4m_\pi^2+q^2}+q}{2m_\pi}\right),
\\q\equiv &\sqrt{{\vec{q}}^{\,2}},\end{aligned}$$ and extracting the baryonic poles and the polynomial terms, one obtains, $$\begin{aligned}
V_a=&\label{eq:va}
-\frac{h_{\Lambda N}}{192\pi^2f_\pi^4(M_\Lambda-M_N)}
(4m_\pi^2+q^2)L(q)({\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\
V_b=&
\frac{3g_A^2h_{2\pi}}{32\pi f_\pi^4}(2m_\pi^2+q^2)At(q)
\\\label{eq:vc}
V_c=&
-\frac{g_A^2h_{\Lambda N}}{384\pi^2
f_\pi^4(M_\Lambda-M_N)}(8m_\pi^2+5q^2)L(q)({\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\
V_d=&\nonumber
\frac{g_A}{64\pi^2 f_\pi^3 M_N}
L(q)({\vec{\tau}_1}\cdot{\vec{\tau}_2})
\left(
-2Bm_\pi^2-B{\vec{q}}^2+B({\vec{q}}\cdot{\vec{p}})
\right.\\&\left.+6A M_N ({\vec{\sigma}_1}\cdot{\vec{q}})-3iB {\vec{\sigma}_1}\cdot({\vec{q}}\times{\vec{p}})
\right)
\\
V_e=&
\frac{\sqrt{3}D_s}{384\pi^2 f_\pi^3 M_N}
L(q)
\left(
B_{\Sigma1}(4m_\pi^2+3{\vec{q}}^2)-4A_{\Sigma1} M_N({\vec{\sigma}_1}\cdot{\vec{q}})
\right)\,,\end{aligned}$$
$$\begin{aligned}
V_f=&\nonumber
\frac{g_A^3}{512\pi^2 f_\pi^3 M_N(4m_\pi^2+{\vec{q}}^2)}L(q)
(-3+2{\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\&\nonumber
\times\left[
\frac{1}{6}B(448m_\pi^4+4m_\pi^2(-24{\vec{q}}\cdot{\vec{p}}+47{\vec{q}}^2)+25{\vec{q}}^4
\right.\\&\left.\nonumber
-36{\vec{q}}^2({\vec{q}}\cdot{\vec{p}}))
+4iB(4m_\pi^2+{\vec{q}}^2){\vec{\sigma}_2}\cdot({\vec{q}}\times{\vec{p}})
\right.\\&\left.\nonumber
-4A M_N(8m_\pi^2+3{\vec{q}}^2){\vec{\sigma}_1}\cdot{\vec{q}}\right.\\&\left.\nonumber
+2iB(8m_\pi^2+3{\vec{q}}^2){\vec{\sigma}_1}\cdot({\vec{q}}\times{\vec{p}})
\right.\\&\left.\nonumber
+4B(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}})
\right.\\&\left.\nonumber
-4B(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})
\right.\\&\left.\nonumber
-4B(4m_\pi^2+{\vec{q}}^2)({\vec{q}}\cdot{\vec{p}}-{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{\sigma}_2})
\right.\\&\left.
-8iAM_N(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\right]
\\\nonumber\\
V_g=&\nonumber
\frac{D_s g_A^2}{256\sqrt{3}\pi^2 f_\pi^3 M_N(4m_\pi^2+{\vec{q}}^2)}L(q)
\\&\nonumber
\times\left[
-\frac{1}{6}B_{\Sigma2}(448m_\pi^4+188m_\pi^2{\vec{q}}^2+25{\vec{q}}^4)
\right.\\&\nonumber\left.
+4A_{\Sigma2}M_N(8m_\pi^2+3{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})
\right.\\&\nonumber\left.
+4B_{\Sigma2}(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})
\right.\\&\left.\nonumber
-4B_{\Sigma2}(4m_\pi^2+{\vec{q}}^2){\vec{q}}^2({\vec{\sigma}_1}\cdot{\vec{\sigma}_2})
\right.\\&\left.
-8iA_{\Sigma2}M_N(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\right]\,,\end{aligned}$$
$$\begin{aligned}
V_h=&\nonumber
\frac{g_A^3}{512\pi^2 f_\pi^3 M_N(4m_\pi^2+{\vec{q}}^2)}L(q)
(3+2{\vec{\tau}_1}\cdot{\vec{\tau}_2})
\\&\nonumber
\times\left[
\frac{1}{6}B(448m_\pi^4+4m_\pi^2(-24{\vec{q}}\cdot{\vec{p}}+47{\vec{q}}^2)+25{\vec{q}}^4
\right.\\&\nonumber\left.
-36{\vec{q}}^2({\vec{q}}\cdot{\vec{p}}))
-4iB(4m_\pi^2+{\vec{q}}^2){\vec{\sigma}_2}\cdot({\vec{q}}\times{\vec{p}})
\right.\\&\nonumber\left.
-4A M_N(8m_\pi^2+3{\vec{q}}^2){\vec{\sigma}_1}\cdot{\vec{q}}\right.\\&\nonumber\left.
-2iB(8m_\pi^2+3{\vec{q}}^2){\vec{\sigma}_1}\cdot({\vec{q}}\times{\vec{p}})
\right.\\&\nonumber\left.
+4B(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{p}})
\right.\\&\nonumber\left.
-4B(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})
\right.\\&\nonumber\left.
-4B(4m_\pi^2+{\vec{q}}^2)({\vec{q}}\cdot{\vec{p}}-{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{\sigma}_2})
\right.\\&\left.
+8iAM_N(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\right]
\\\nonumber\\
V_i=&\nonumber
\frac{D_s g_A^2}{256\sqrt{3}\pi^2 f_\pi^3 M_N(4m_\pi^2+{\vec{q}}^2)}L(q)
\\&\nonumber
\times\left[
\frac{1}{6}B_{\Sigma3}(448m_\pi^4+188m_\pi^2{\vec{q}}^2+25{\vec{q}}^4)
\right.\\&\nonumber\left.
+A_{\Sigma3}M_N(8m_\pi^2+3{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})
\right.\\&\nonumber\left.
+4B_{\Sigma3}(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})
\right.\\&\nonumber\left.
-4B_{\Sigma3}(4m_\pi^2+{\vec{q}}^2){\vec{q}}^2({\vec{\sigma}_1}\cdot{\vec{\sigma}_2})
\right.\\&\left.
+4iA_{\Sigma3}M_N(4m_\pi^2+{\vec{q}}^2)({\vec{\sigma}_1}\times{\vec{\sigma}_2})\cdot{\vec{q}}\right]\,.\end{aligned}$$
The isospin part for the potentials that contain $\Sigma$ propagators ($V_e$, $V_g$, $V_i$) is taken into account by making the replacements: $$\begin{aligned}
A_{\Sigma1}\to&\frac{2}{3}\left(\sqrt3 A_{\Sigma\frac12}+
A_{\Sigma\frac32}\right)
\vec{\tau_1}\cdot\vec{\tau_2}
\\
B_{\Sigma1}\to&
\frac{2}{3}\left(\sqrt3 B_{\Sigma\frac12}+
B_{\Sigma\frac32}\right)
\vec{\tau_1}\cdot\vec{\tau_2}\,,\end{aligned}$$
$$\begin{aligned}
A_{\Sigma2}\to&
-\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}
+\frac23(\sqrt3A_{\Sigma\frac12}+A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\
B_{\Sigma2}\to&
-\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}
+\frac23(\sqrt3B_{\Sigma\frac12}+B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$
$$\begin{aligned}
A_{\Sigma3}\to&
-\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}
-\frac23(\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\
B_{\Sigma3}\to&
-\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}
-\frac23(\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$
Note that Eqs. (\[eq:va\]) and (\[eq:vc\]) only have physical meaning away from the SU(3) limit.
Brief comparison of LO and NLO contributions {#sec:bc}
============================================
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(UP) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot\]](pottriangles "fig:")![(UP) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot\]](potboxes "fig:")
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(UP) Medium-Long range part of the potentials in the SU(3) limit for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials in the SU(3) limit for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot0\]](pottriangles0 "fig:")![(UP) Medium-Long range part of the potentials in the SU(3) limit for the one-pion-exchange, one-kaon-exchange, ball diagram and triangle diagrams. (DOWN) Medium-Long range part of the potentials in the SU(3) limit for the one-pion-exchange, one-kaon-exchange, box and crossed box diagrams. \[fig:pot0\]](potboxes0 "fig:")
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In Eqs. (\[eq:tots\]) and (\[eq:tots2\]) we provide the explicit momentum and spin structures arising from the different Feynman diagrams. Some features can be easily read off from the different terms. First, the ball (a) and first two triangle diagrams (b,c) only contribute to the parity conserving part of the transition potential. Most other diagrams have a non-trivial contribution, involving all allowed momenta and spin structures.
To provide a sample of the contribution of the different diagrams to the full amplitude, we consider one particular transition, $^3 S_1\rightarrow ^3S_1$. In particular, we compare the $\pi$ and $K$ exchanges with the ball, triangle and box diagrams for the $\Lambda n\rightarrow nn$ interaction. Since the transition is parity conserving, none of the parity violating structures of Table \[tab:contacts\] contribute. For structures of the type $({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})$ we have that $$\begin{aligned}
({\vec{\sigma}_1}\cdot{\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})=\frac{{\vec{q}}^{\,2}}{3}({\vec{\sigma}_1}\cdot{\vec{\sigma}_2})+\frac{{\vec{q}}^2}{3}\hat{S}_{12}(\hat{q}),\end{aligned}$$ where the tensor operator $\hat{S}_{12}(\hat{q})$ changes two units of angular momentum and does not contribute to this transition. The potential, therefore, depends only on the modulus of the momentum (or ${\vec{q}}^{\,2}$). To obtain the potential in position space we Fourier-transform the expressions for the one-meson-exchange contributions, Eqs. \[eq:pion\] and \[eq:kaon\], and the loop expressions in the appendices \[sec:balls\], \[sec:triangles\] and \[sec:boxs\]. More explicitly, $$\begin{aligned}
\tilde{V}(r)= {\cal F}\left[V({\vec{q}}^{\,2}) F({\vec{q}}^{\,2})\right]&\equiv
\int_{-\infty}^{\infty}\frac{d^3q}{(2\pi)^3}
e^{i{\vec{q}}\cdot{\vec{r}}}V({\vec{q}}^2)F({\vec{q}}^{\,2})
{}\end{aligned}$$ with $q\equiv|{\vec{q}}|$ and $r\equiv|{\vec{r}}|$ and where we have included a form factor in order to regularize the potential. Following the formalism developed in Ref. [@PRB97] we use a monopole form factor for the meson exchange contribution at each vertex, while the $2-\pi$ terms use a Gaussian form of the type $F({\vec{q}}^2)\equiv e^{-{\vec{q}}^{\,4}/\Lambda^4}$.
The expressions for each loop have been calculated using dimensional regularization and are shown in the appendices $B$, $C$ and $D$. They are written in terms of the couplings appearing in Sec. \[sec2\] and of the master integrals appearing in App. \[sec:mi\]. $\eta$ is the regularization parameter that appears when integrating in $D\equiv4-\eta$ dimensions. The modified minimal subtraction scheme ($\overline{MS}$) has been used—we have expanded in powers of $\eta$ the expressions for the different loop contributions and then subtracted the term $R\equiv-\frac{2}{\eta}+\gamma-1-\ln\left(4\pi\right)$—.
In Fig. \[fig:pot\], we show the respective contributions to the potential in position space. The contribution from the different $2-\pi$ exchange potentials are seen to be sizable at all distances. In particular, the box (f, g, h) and triangle (d) diagrams give larger contributions than the pion in the medium and long-range. The ball diagram (a) and the triangles (c), (e), (h) and (i) are attractive while all the others are repulsive. Notice that diagrams (d), (f) and (h) contribute with an imaginary part. This is characteristic of diagrams with a $\Lambda N\pi$ vertex, which may be on shell since $M_\Lambda>M_N+m_\pi$. This imaginary part is taking into account the amplitude for the possible $\Lambda
N\rightarrow NN\pi$ transition. We stress that the imaginary part of the box diagram (f) that comes from the baryonic pole has been extracted, so no iterated part is considered in Fig. \[fig:pot\].
Fig. \[fig:pot0\], shows the same potentials but taking $q_0=q_0'=0$. All diagrams seem to have a smaller contribution when the baryon mass differences are neglected. The attractive and repulsive character of the different potentials does not change except for the second box diagram and the second crossed box diagram, which turn to be attractive and repulsive, respectively, when taking the SU(3) limit.
Conclusions {#sec:conclusions}
===========
The weak decay of hypernuclei is dominated for large enough number of nucleons by the non-mesonic weak decay modes. In these modes, the bound $\Lambda$ particle decays in the presence of nucleons by means of a process which involves weak and strong interaction vertices describing the production and absorption of mesons. The relevant, experimentally known, partial and total decay rates of hypernuclei, are successfully described by meson-exchange models and also by a lowest-order effective field theory description of the weak $\Lambda N\to NN$ process, when appropriate nuclear wave functions are used for the initial and final nuclear systems. Nevertheless, the stability of the EFT approach which has to be tested by looking at higher orders in the theory, could not be analyzed yet, mainly because of the very scarce world-database for such observables, a situation which should be improved in the near future.
In this article we have presented the one-loop contribution to the previously obtained LO EFT for the weak $\Delta S=1$ $\Lambda N$ transition.
As expected, the structure of the transition amplitude is considerably more involved than the corresponding LO amplitude and contains more low-energy coefficients which ought to be fitted to data. In the present formal work we have solely presented the calculation of the amplitude terms and have not attempted to make any comparison to experimental data, therefore, no fit in order to extract the new unknowns has been performed. The different structures which appear in the obtained transition amplitude, involving spin, isospin and orbital degrees of freedom, produce sizable contributions to all relevant partial waves. To illustrate this fact, we have presented the potential in $r$ space corresponding to the different Feynman diagrams for the $^3S_1- ^3S_1$ partial wave. Box and cross-box diagrams are found to produce substantial contributions at distances of the order of 1 fm, larger than the ones corresponding to the one-pion-exchange and one-kaon-exchange mechanisms. In view of this result, it would be interesting to see if one-loop contributions play an equivalent role in other partial wave transitions, testing possible cancellations or enhancements that would leave the results for the decay rates either unchanged or modified. A complete analysis of the higher order terms would require a larger set of independent hypernuclear decay measurements and a more accurate measure of some observables, specially those related to the parity violating asymmetry for s-shell and p-shell hypernuclei. Moreover, it would be desirable to arrange for alternative experiments focused to obtain information on the weak $\Delta S=1$ interaction. A step in this direction was taken more than ten years ago by experimental groups at RCNP in Osaka (Japan) \[15,16\], by looking at the weak strangeness production reaction $np \to \Lambda
p$. Unfortunately, the small value for the cross-section for this process precluded the compilation of new data. We think that it is important to foster new experimental avenues of approaching the weak interaction among baryons in the strange sector, and even try to recover the Osaka experiment within the research plan of the new experimental facilities devoted to the study of strange systems.
To ease the use of the obtained EFT amplitudes, we have provided with the explicit analytic expressions for all diagrams which will in future work be implemented in the calculation of hypernuclear decay observables.
We thank J. Soto, J. Tarrús, J. Haidenbauer and A. Nogga for the helpful comments and discussions. This work is partly supported by grants FPA2010-21750-C02-02 and FIS2011-24154 from MICINN, 283286 from European Community-Research Infrastructure Integrating Activity ‘Study of Strongly Interacting Matter’, CSD2007-00042 from Spanish Ingenio-Consolider 2010 Program CPAN, and 2009SGR-1289 from Generalitat de Catalunya. A.P-O. acknowledges support by the APIF Ph.D. program of the University of Barcelona. B.J.D. is supported by the Ramon y Cajal program.
Caramel diagrams {#sec:caramels}
================
![First caramel-type Feynman diagram \[caramel1\]](caramel1g)
![Second caramel-type Feynman diagram \[caramel2\]](caramel2g)
![Third caramel-type Feynman diagram \[caramel3\]](caramel3g)
Using the same notation that is described in section \[sec:nlocontact\] we write a general expression for the three caramel diagrams that depends on the label $\alpha=a,b,c$, which corresponds, respectively, to the masses and vertices of Figs. \[caramel1\], \[caramel2\], and \[caramel3\]. The relativistic expression for our caramel diagrams is, $$\begin{aligned}
V_\alpha&=iG_Fm_\pi^2
(C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
(C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
\\&\times
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(E_p-l_0)^2-{\vec{l}}^2-M_N^2+i\epsilon}
\\&\times\frac{1}{(E_p^\Lambda+l_0)^2-{\vec{l}}^2-M_\alpha^2}\end{aligned}$$ In order to not miss the relativistic pole we must first integrate the temporal part ($l_0$) before heavy-baryon expand the expression. Proceeding in this manner one obtains a purely imaginary part (the real is suppressed in the heavy baryon expansion). $$\begin{aligned}
V_\alpha&=-\frac{G_Fm_\pi^2}{4M_N}
(C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
(C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
\\&\times
{\int\frac{d^3l}{(2\pi)^3}}\,
\frac{1}{(\Delta_b-\Delta_\alpha)(\frac12(\Delta_b+\Delta_\alpha)+M_N)+{\vec{p}}^2-{\vec{l}}^2}
\\&=i\frac{G_Fm_\pi^2}{16\pi M_N}
(C_{S(s)}^{\alpha}+C_{T(s)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
(C_{S(w)}^{\alpha}+C_{T(w)}^{\alpha}{\vec{\sigma}_1}\cdot{\vec{\sigma}_2})\,
\\&\times
\sqrt{
(\Delta_b-\Delta_\alpha)(\frac12(\Delta_b+\Delta_\alpha)+M_N)+{\vec{p}}^2}\end{aligned}$$
Ball diagrams {#sec:balls}
=============
In our calculation we have two different kind of ball diagrams depending on the position of the weak vertex, although only one of them actually contributes. Their contribution can be written in terms of the $B$ integrals defined in Appendix \[sec:mi\].
Here and in the following sections we first write the relativistic amplitude using $V=i \ M$ and then the corresponding heavy baryon expression.
![Kinematical variables of the first kind of ball-diagram.\[fball1\]](ball11)
For the first type of ball diagram, depicted in Fig. \[fball1\], we obtain the following contribution, $$\begin{aligned}
V_{\text{ball 1}}=& \frac{G_Fm_\pi^2 h_{2\pi}}{4f_\pi^4}
\delta_{ab}\ \epsilon^{abc}\tau^c
\nonumber\\
&\times {\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\frac{1}{(l-q)^2-m_\pi^2+i\epsilon}\nonumber
\\&\times
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})
u_1(E_p^\Lambda,{\vec{p}})\nonumber\\
&\times
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})
\gamma_\mu(q^\mu-2l^\mu)
u_2(E_p,-{\vec{p}})
\\=&0\,,\end{aligned}$$ which is shown to vanish due to the isospin factor, $\delta_{ab}\epsilon^{abc}\tau^c=0$.
![Kinematical variables of the second kind of ball-diagram.\[fball2g\]](ball2g)
The amplitude corresponding to the diagram in Fig. \[fball2g\] reads, $$\begin{aligned}
V_a&=&-i
\frac{G_Fm_\pi^2h_{\Lambda N}}{8f_\pi^4} ({\vec{\tau}_1}\cdot{\vec{\tau}_2}) \nonumber\\
&\times&{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\,\frac{1}{(l+q)^2-m_\pi^2+i\epsilon} \nonumber\\
&\times&
\frac{(2l^\mu+q^\mu)(q^\nu+2l^\nu)}{k_N^2-M_N^2+i\epsilon}
\nonumber\\
&\times&
{\overline{u}}_1({\overline{E}},{\vec{p}\,'}) \gamma_\mu ({\cancel{k}_N}+M_N)
u_1(E_p^\Lambda,{\vec{p}}) \nonumber\\
&\times&
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})
\gamma_\nu
u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion, $$\begin{aligned}
V_a
&=&\ \frac{G_Fm_\pi^2h_{\Lambda N}}{8\Delta Mf_\pi^4}
({\vec{\tau}_1}\cdot{\vec{\tau}_2})
(4 { B}_{20}
+4q_0 { B}_{10}
+q_0^2 { B}) \,,\nonumber\\\end{aligned}$$ where we have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$ and ${\vec{q}}={\vec{p}\,'}-{\vec{p}}$.
Triangle diagrams {#sec:triangles}
=================
Two up triangles and two down triangles contribute to the interaction. The final expressions are written in terms of the integrals $I$ defined in Appendix \[sec:mi\]. The amplitude for the first up triangle, depicted in Fig. \[uptri\], is
![Up triangle diagram contributing at NLO. \[uptri\]](uptriangle112)
$$\begin{aligned}
V_b=&-i\frac38\frac{G_F m_\pi^2h_{2\pi}g_A^2}{M_N f_\pi^4}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\,
\frac{(l^\mu+q^\mu)l^\nu}{k_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
\,{\overline{u}}_1({\overline{E}}_p,{\vec{p}\,'}){\overline{u}}_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu\gamma_5({\cancel{k}_N}+M_N)\gamma_\nu\gamma_5
u_2(E_p,-{\vec{p}})\,.\end{aligned}$$
Using heavy baryon expansion, $$\begin{aligned}
V_b
=\frac34\frac{G_F m_\pi^2h_{2\pi}g_A^2}{f_\pi^4}
\left[
(3-\eta)I_{22}+{\vec{q}}^2I_{23}+{\vec{q}}^2I_{11}
\right]\,,\end{aligned}$$ where, we have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=0$ and ${\vec{q}}={\vec{p}\,'}-{\vec{p}}$.
![Second up triangle contribution at NLO.[]{data-label="uptri2"}](uptriangle2g)
For the second up triangle, depicted in Fig. \[uptri2\], the relativistic amplitude is $$\begin{aligned}
V_c=&
-i\frac{G_Fm_\pi^2h_{\Lambda N} g_A^2}{8f_\pi^4(r_N^2-M_N^2)}
{\vec{\tau}_1}\cdot{\vec{\tau}_2}{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\,
\frac{(2l^\rho+q^\rho)(l^\mu+q^\mu)l^\nu}{k_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})\gamma_\rho({\cancel{k}_N}'+M_N)
u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu\gamma_5({\cancel{k}_N}+M_N)
\gamma_\nu\gamma_5 u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion, $$\begin{aligned}
V_c=&
\frac{G_Fm_\pi^2h_{\Lambda N} g_A^2}{8\Delta M f_\pi^4}
{\vec{\tau}_1}\cdot{\vec{\tau}_2}\left[
2(3-\eta)I_{32}+2{\vec{q}}^2I_{33}+2{\vec{q}}^2I_{21}
\nonumber\right.\\+&\left.
(3-\eta)q_0I_{22}+q_0{\vec{q}}^2I_{23}+q_0{\vec{q}}^2I_{11}
\right]\,,\end{aligned}$$ where, we have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=0$ and ${\vec{q}}={\vec{p}\,'}-{\vec{p}}$.
![“Down”-triangle contribution at NLO.\[downtri\]](downtriangle1g)
The amplitude for the first down triangle (Fig. \[downtri\]) is $$\begin{aligned}
V_d=&
i\frac{G_Fm_\pi^2g_A}{4 f_\pi^3}
(\vec{\tau}_1\cdot\vec{\tau}_2)
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\\\times&\nonumber\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}
\frac{(l^\nu+q^\nu)(2l^\mu+q^\mu)}{k_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})
\gamma_{\nu}\gamma_5
({\cancel{k}_N}+M_N)
(A+B\gamma_5)
u_1(E_p^\Lambda,{\vec{p}})
\\\times&\,\nonumber
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu u_2(E_p,-{\vec{p}})\,,\end{aligned}$$ with the heavy baryon expansion, it reduces to, $$\begin{aligned}
V_d=&
-\frac{G_Fm_\pi^2g_A}{8M_N f_\pi^3}
(\vec{\tau}_1\cdot\vec{\tau}_2)
\Big[
B(2I_{30}+7q_0I_{20}+7q_0^2I_{10}
\nonumber\\+&
2q_0^3I
-2(3-\eta)I_{32}-(3-\eta)q_0I_{22})
\\-&\nonumber
B(2I_{21}+q_0I_{11}+2I_{33}+q_0I_{23}){\vec{q}}^2
\\-&\nonumber
B(2I_{10}+2I_{21}+q_0 I+q_0I_{11})({\vec{q}}\cdot{\vec{p}})
\\+&\nonumber
2A\,M_N(2I_{21}
+q_0I_{11}-2I_{10}-q_0I){\vec{\sigma}_1}\cdot{\vec{q}}\\+&\nonumber
iB(-2I_{21}-q_0I_{11}+2I_{10}+q_0I){\vec{\sigma}_1}({\vec{q}}\times{\vec{p}})
\Big]\,.\end{aligned}$$ We have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=-M_\Lambda+M_N$ and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
![Second type of down-triangle involving the intermediate exchange of a $\Sigma$.[]{data-label="downtri2"}](downtriangle2g)
The second type of down-triangle diagram involves the intermediate exchange of the $\Sigma$ (Fig. \[downtri2\]). Its amplitude is $$\begin{aligned}
V_e=&
\frac{G_Fm_\pi^2D_s}{4\sqrt{3}f_\pi^3}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\nonumber\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\,
\frac{(2l^\mu+q^\mu)l^\nu}{k_N^2-M_\Sigma^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})(A_{\Sigma}+B_{\Sigma}\gamma_5)({\cancel{k}_N}+M_\Sigma)
\gamma_{\nu}\gamma_5 u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\mu u_2(E_p,-{\vec{p}}) \,.\end{aligned}$$ Using the heavy baryon expansion $$\begin{aligned}
V_e=&
-\frac{G_Fm_\pi^2D_s}{8\sqrt{3}M_N f_\pi^3}
\Big[
B_{\Sigma}\Big(-2I_{30}+(-5q_0-2\Delta M_\Sigma)I_{20}
\\+&\nonumber
2(3-\eta)I_{32}+2{\vec{q}}^2I_{33}+2{\vec{q}}^2I_{21}+{\vec{q}}^2I_{21}
\\+&\nonumber
q_0(-2q_0-\Delta M_\Sigma)I_{10}
+(3-\eta)q_0I_{22}+q_0{\vec{q}}^2I_{23}+q_0{\vec{q}}^2I_{11}\Big)
\\-&\nonumber
2A_{\Sigma} M_N(2I_{21}
+q_0I_{11})({\vec{\sigma}_1}\cdot{\vec{q}})
\Big]\,.\end{aligned}$$ The isospin is taken into account by replacing every $A_\Sigma$ and $B_\Sigma$ by $$\begin{aligned}
\frac{2}{3}\left(\sqrt3 A_{\Sigma\frac12}+
A_{\Sigma\frac32}\right)
\vec{\tau_1}\cdot\vec{\tau_2},
~~~
\frac{2}{3}\left(\sqrt3 B_{\Sigma\frac12}+
B_{\Sigma\frac32}\right)
\vec{\tau_1}\cdot\vec{\tau_2}\,,\end{aligned}$$ where, we have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=M_\Sigma-M_\Lambda$ and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
Box diagrams {#sec:boxs}
============
We have two kind of direct box diagrams and two cross-box ones. Direct box diagrams usually present a pinch singularity. This is because the poles appearing in the baryonic propagators get infinitesimally close to one another. In our integrals the denominators appearing in the baryonic propagators also contain terms proportional to $M_\Lambda-M_N$ and $M_\Sigma-M_\Lambda$, and this avoids the singularity.
The integrals entering in the expression of the amplitudes are the $J$ and $K$ defined in Appendix \[sec:mi\]. The amplitude for the first type of box diagram (Fig. \[box1\]) is
![Box diagram contributing at NLO.[]{data-label="box1"}](box1g)
$$\begin{aligned}
V_f=&
i\frac{G_Fm_\pi^2g_A^3}{8f_\pi^3}
(3-2{\vec{\tau}_1}\cdot{\vec{\tau}_2})
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\nonumber\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\,
\frac{1}{k_N^2-M_N^2+i\epsilon}a
\\\times&\nonumber\,
\frac{(l^\rho+q^\rho)(l^\nu+q^\nu)l^\mu}{r_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})
\gamma_\rho\gamma_5
({\cancel{k}_N}+M_N)
(A+B\gamma_5)
u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})
\gamma_\nu\gamma_5
({\cancel{r}_N}+M_N)
\gamma_\mu\gamma_5
u_2(E_p,-{\vec{p}}) \,.\end{aligned}$$
Using the heavy baryon expansion, $$\begin{aligned}
V_f&=-\frac{G_Fm_\pi^2g_A^3}{32M_N f_\pi^3}
(3-2{\vec{\tau}_1}\cdot{\vec{\tau}_2})\Bigg[
-4A M_N \left(4K_{22}
+ K_{11} {\vec{q}}^2
\nonumber\right.\\+&\left.\nonumber
2 K_{23} {\vec{q}}^2+K_{35} {\vec{q}}^2+(5-\eta)
K_{34}\right){\vec{\sigma}_1}\cdot {\vec{q}}\nonumber\\-&\nonumber
2B K_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{q}})
+
2B K_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{p}})
\nonumber\\-&\nonumber
4i A M_N K_{22} \left({\vec{\sigma}_1}\times{\vec{\sigma}_2}\right)\cdot{\vec{q}}-
2 B \left({\vec{p}}\cdot
{\vec{q}}-{\vec{q}}^2\right)
K_{22} {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\nonumber\\+&\nonumber
2 i B \left(K_{11} {\vec{q}}^2+2
K_{23} {\vec{q}}^2+K_{35} {\vec{q}}^2
\nonumber\right.\\+&\left.\nonumber
(4-\eta) K_{22}+(5-\eta) K_{34}\right)
{\vec{\sigma}_1}\cdot\left({\vec{p}}\times {\vec{q}}\right)
+2 i B K_{22}
{\vec{\sigma}_2}\cdot\left({\vec{p}}\times {\vec{q}}\right)
\nonumber\\-&\nonumber
2 B \Big(K_{11}{\vec{q}}^2
\left({\vec{p}}\cdot {\vec{q}}+2 q_0{}^2\right)+ K_{23}( 2{\vec{p}}\cdot
{\vec{q}}{\vec{q}}^2+2q_0^2{\vec{q}}^2+{\vec{q}}^4)
\nonumber\\+&\nonumber
K_{35}( {\vec{p}}\cdot {\vec{q}}{\vec{q}}^2+2{\vec{q}}^4)
+K_{22}((4-\eta) {\vec{p}}\cdot{\vec{q}}+{\vec{q}}^2+(6-2\eta)q_0^2)
\nonumber\\+&\nonumber
(5-\eta)K_{34}({\vec{p}}\cdot {\vec{q}}+2{\vec{q}}^2)
+K_{48} {\vec{q}}^4+K_{21}
{\vec{q}}^2 q_0+K_{33} {\vec{q}}^2 q_0
\nonumber\\-&\nonumber
K_{31} {\vec{q}}^2
-K_{43} {\vec{q}}^2
+2(5-\eta) K_{47} {\vec{q}}^2
+(3-\eta) K_{32}
q_0
\nonumber\\-&\nonumber
(3-\eta) K_{42}+(15-8\eta) K_{46}\Big)
\Bigg]\,,\end{aligned}$$ where we have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=M_N-M_\Lambda$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
![Second box-type Feynman diagram. \[box2\]](box3g)
The second box diagram (Fig. \[box2\]), which involves a $\Sigma$ propagator, contributes with $$\begin{aligned}
V_g=&
-i\frac{G_Fm_\pi^2g_A^2D_s}{4\sqrt{3} f_\pi^3}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m_\pi^2+i\epsilon}
\\\times&\nonumber\,
\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}\,
\frac{1}{k_N^2-M_\Sigma^2+i\epsilon}
\\\times&\nonumber\,
\frac{l^\rho(l^\nu+q^\nu)l^\mu}{r_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})(A_\Sigma+B_{\Sigma}\gamma_5)({\cancel{k}_N}+M_N)\gamma_\rho\gamma_5 u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5 u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using the heavy baryon expansion $$\begin{aligned}
V_g=&
\frac{G_Fm_\pi^2g_A^2D_s}{16\sqrt{3}M_N f_\pi^3}
\Big[
-2B_{\Sigma} K_{22} {\vec{q}}^2 {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\nonumber\\-&\nonumber\,
4A_{\Sigma} K_{22} M_N i\left({\vec{\sigma}_1}\times
{\vec{\sigma}_2}\right){\vec{q}}\\-&\nonumber\,
4 A_{\Sigma} M_N\left({\vec{q}}^2 K_{23}+5 K_{34}+{\vec{q}}^2
K_{35}+K_{22}\right){\vec{\sigma}_1}\cdot {\vec{q}}\\+&\nonumber\,
2B_{\Sigma} K_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{q}})
+
2 B_{\Sigma} \left({\vec{q}}^2 K_{22}+{\vec{q}}^4
K_{23}
\right.\\-&\nonumber\left.\,
{\vec{q}}^2 K_{31}+(3-\eta) (\Delta M-\Delta M_\Sigma) K_{32}
\right.\\+&\nonumber\left.\,
{\vec{q}}^2 (\Delta M -\Delta M_\Sigma)K_{33}+2(5-\eta) {\vec{q}}^2 K_{34}
+2 {\vec{q}}^4 K_{35}
\right.\\-&\nonumber\left.\,
(3-\eta)
K_{42}-{\vec{q}}^2 K_{43}+(15-8\eta) K_{46}
\right.\\+&\nonumber\left.\,
2(5-\eta) {\vec{q}}^2 K_{47}
+
{\vec{q}}^4K_{48}
+{\vec{q}}^2 K_{21} \left(\text{$\Delta $M}-\text{$\Delta
$M}_{\Sigma }\right) \right)
\Big]\,.\end{aligned}$$ To take into account the isospin we must replace every $A_\Sigma$ and $B_\Sigma$ by $$\begin{aligned}
A\to&
-\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}
+\frac23(\sqrt3A_{\Sigma\frac12}+A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\
B\to&
-\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}
+\frac23(\sqrt3B_{\Sigma\frac12}+B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$ We have used the master integrals with $q_0=-\frac{M_\Lambda-M_N}{2}$, $q_0'=M_\Sigma-M_\Lambda$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
![Crossed-box diagram contributing at NLO.[]{data-label="box2g"}](box2g)
The second crossed box diagram (Fig. \[box2g\]) includes a $\Sigma$-propagator and contributes to the potential with $$\begin{aligned}
V_h=&
i\frac{G_Fm_\pi^2g_A^3}{8f_\pi^3}
(3+2{\vec{\tau}_1}\cdot{\vec{\tau}_2})
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\nonumber\,
\frac{1}{l^2-m_\pi^2+i\epsilon}\,
\frac{1}{r_N^2-M_N^2+i\epsilon}\,
\frac{(l^\rho)(l^\nu+q^\nu)(l^\mu)}{k_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})
\gamma_\rho\gamma_5({\cancel{k}_N}+M_N)(A+B\gamma_5) u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5
u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion and the master integrals of Sec. \[sec:mi\], and redefining ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, $$\begin{aligned}
V_h=&
-\frac{G_Fm_\pi^2g_A^3}
{32M_N f_\pi^3}
(3+2{\vec{\tau}_1}\cdot{\vec{\tau}_2})\Big[
-2 iB J_{22}{\vec{\sigma}_2}\left({\vec{p}}\times {\vec{q}}\right)
\nonumber\\+&
2B J_{22} \left(-{\vec{p}}\cdot
{\vec{q}}+{\vec{q}}^2\right)
{\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\\+&\nonumber
2 i B\left(J_{22}+{\vec{q}}^2 J_{23}+(5+\eta) J_{34}+{\vec{q}}^2
J_{35}\right)
{\vec{\sigma}_1}\cdot\left({\vec{p}}\times {\vec{q}}\right)
\\+&\nonumber
4i A J_{22} M_N\left({\vec{\sigma}_1}\times {\vec{\sigma}_2}\right){\vec{q}}\\+&\nonumber
4 AM_N \left({\vec{q}}^2 J_{23}+5 J_{34}+{\vec{q}}^2
J_{35}+J_{22}\right){\vec{\sigma}_1}\cdot{\vec{q}}\\+&\nonumber
2BJ_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{p}})
-2BJ_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot{\vec{q}})
\\-&\nonumber
2B\left({\vec{q}}^2 q_0 J_{21}+\left(-{\vec{p}}\cdot {\vec{q}}+{\vec{q}}^2\right)
J_{22}+(-{\vec{p}}\cdot {\vec{q}}{\vec{q}}^2 +{\vec{q}}^4)
J_{23}
\right.\\-&\left.\nonumber
{\vec{q}}^2 J_{31}+(3-\eta) q_0 J_{32}+{\vec{q}}^2 q_0 J_{33}
\right.\\+&\left.\nonumber
(5-\eta)(-{\vec{p}}\cdot {\vec{q}}+2 {\vec{q}}^2 )J_{34}
+
(-{\vec{p}}\cdot {\vec{q}}{\vec{q}}^2 +2 {\vec{q}}^4 )J_{35}
\right.\\-&\left.\nonumber
(3-\eta) J_{42}
-{\vec{q}}^2 J_{43}+(15-8\eta)J_{46}
+
2(5-\eta) {\vec{q}}^2 J_{47}
\right.\\+&\left.\nonumber
{\vec{q}}^4 J_{48}\right)
\Big]\,.\end{aligned}$$ We have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=-\frac{M_\Lambda-M_N}{2}$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
![Second crossed-box-type Feynman diagram \[xbox2\]](box4g)
The amplitude for the crossed-box diagram with a $\Sigma$ propagator is $$\begin{aligned}
V_i=&
-i\frac{G_Fm_\pi^2g_A^2D_s}{16\sqrt{3}M_N^2f_\pi^3}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(l+q)^2-m_\pi^2+i\epsilon}
\nonumber\\\times&\,
\frac{1}{l^2-m_\pi^2+i\epsilon}\,
\frac{1}{r_N^2-M_N^2+i\epsilon}
\\\times&\nonumber\,
\frac{(l^\rho+{\vec{q}}^\rho)(l^\nu+q^\nu)(l^\mu)}{k_N^2-M_\Sigma^2+i\epsilon}
\\\times&\nonumber\,
{\overline{u}}_1({\overline{E}},{\vec{p}\,'})(A_\Sigma+B_\Sigma\gamma_5)({\cancel{k}_N}+M_\Sigma)\gamma_\rho\gamma_5
u_1(E_p^\Lambda,{\vec{p}})
\\\times&\nonumber\,
{\overline{u}}_2({\overline{E}}_p,-{\vec{p}\,'})\gamma_\nu\gamma_5({\cancel{r}_N}+M_N)\gamma_\mu\gamma_5
u_2(E_p,-{\vec{p}})\,.\end{aligned}$$ Using heavy baryon expansion and the master integrals of Sec. \[sec:mi\], and redefining ${\vec{q}}\equiv{\vec{p}\,'}-{\vec{p}}$, $$\begin{aligned}
V_i=&
\frac{G_Fm_\pi^2g_A^2D_s}{16\sqrt{3}M_N f_\pi^3}
\Big[
2B_\Sigma J_{22}{\vec{q}}^2 {\vec{\sigma}_1}\cdot {\vec{\sigma}_2}\\-&\nonumber
2iA_\Sigma J_{22} M_N
\left({\vec{\sigma}_1}\times {\vec{\sigma}_2}\text{)$\cdot$}{\vec{q}}\right.
\\-&\nonumber
A_\Sigma M_N \left({\vec{q}}^2 J_{11}+2 {\vec{q}}^2 J_{23}+5
J_{34}+{\vec{q}}^2 J_{35}+4J_{22}\right){\vec{\sigma}_1}\cdot{\vec{q}}\\-&\nonumber
2B_\Sigma J_{22} ({\vec{\sigma}_1}\cdot {\vec{q}})({\vec{\sigma}_2}\cdot {\vec{q}})
\\+&\nonumber
2B_\Sigma \left(
({\vec{q}}^2-(3-\eta)q_0(q_0+\Delta M_\Sigma) J_{22}
\right.\\+&\left.\nonumber
({\vec{q}}^4-{\vec{q}}^2q_0^2-{\vec{q}}^2q_0\Delta M_\Sigma)J_{23}
-{\vec{q}}^2 J_{31}
\right.\\-&\left.\nonumber
(3-\eta)(2q_0+\Delta M_\Sigma)J_{32}
-(2 {\vec{q}}^2 q_0 +{\vec{q}}^2\Delta M_\Sigma)J_{33}
\right.\\+&\left.\nonumber
2(5-\eta)
{\vec{q}}^2 J_{34}
+2 {\vec{q}}^4 J_{35}-(3-\eta) J_{42}-{\vec{q}}^2 J_{43}
\right.\\+&\left.\nonumber
(15-8\eta)J_{46}+
2(5-\eta) {\vec{q}}^2 J_{47}+{\vec{q}}^4 J_{48}
\right.\\-&\left.\nonumber
{\vec{q}}^2 q_0 J_{11}
\left(q_0+\text{$\Delta $M}_{\Sigma }\right)-{\vec{q}}^2 J_{21}
\left(2 q_0+\text{$\Delta $M}_{\Sigma }\right)\right)
\Big]\,.\end{aligned}$$ To take into account the isospin we must replace every $A_\Sigma$ and $B_\Sigma$ by $$\begin{aligned}
A_\Sigma\to&
-\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}
-\frac23(\sqrt3A_{\Sigma\frac12}+2A_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\\
B_\Sigma\to&
-\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}
-\frac23(\sqrt3B_{\Sigma\frac12}+2B_{\Sigma\frac32}){\vec{\tau}_1}\cdot{\vec{\tau}_2}\,.\end{aligned}$$ We have used the master integrals with $q_0=\frac{M_\Lambda-M_N}{2}$, $q_0'=M_\Sigma-M_\Lambda+\frac{M_\Lambda-M_N}{2}$, and ${\vec{q}}={\vec{p}}'-{\vec{p}}$.
Master integrals {#sec:mi}
================
Definitions
-----------
We need the following integrals in order to calculate the Feynman diagrams. The $B$’s, $I$’s, $J$’s and $K$’s appear, respectively, in the ball, triangle, box and crossed box diagrams: $$B_{;\mu;\mu\nu}
\equiv\,
\frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\frac{1 }{l^2-m^2+i\epsilon}\,
\frac{(1;l_\mu;l_\mu l_\nu)}{(l+q)^2-m^2+i\epsilon} \,,$$ $$\begin{aligned}
I_{;\mu;\mu\nu;\mu\nu\rho}
\equiv\,&
\frac{1}{i}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m^2+i\epsilon}\,
\frac{1}{(l+q)^2-m^2+i\epsilon}\,
\nonumber\\&\nonumber
\times\frac{1}{-l_0-q_0'+i\epsilon}
(1;{l_{\mu}};{l_{\mu}}{l_{\nu}};{l_{\mu}}{l_{\nu}}{l_{\rho}}) \,,\end{aligned}$$ $$\begin{aligned}
J_{;\mu;\mu\nu;\mu\nu\rho}
\equiv\,&
\frac1i{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m^2+i\epsilon}\,
\frac{1}{(l+q)^2-m^2+i\epsilon}\,
\\&\times
\frac{1}{-l_0-q_0'+i\epsilon}
\,\frac{(1;{l_{\mu}};{l_{\mu}}{l_{\nu}};{l_{\mu}}{l_{\nu}}{l_{\rho}})}{-l_0+i\epsilon} \,,\end{aligned}$$ $$\begin{aligned}
K_{;\mu;\mu\nu;\mu\nu\rho}
\equiv\,&
\frac1i{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m^2+i\epsilon}\,
\frac{1}{(l+q)^2-m^2+i\epsilon}\,
\\&\times
\frac{1}{-l_0-q_0'+i\epsilon}
\frac{(1;{l_{\mu}};{l_{\mu}}{l_{\nu}};{l_{\mu}}{l_{\nu}}{l_{\rho}})}{l_0+i\epsilon} \,.\end{aligned}$$ The strategy is to calculate explicitly the integrals with no subindex (no integrated momenta in the numerators), and then relate the others to simpler integrals. To do so we also need to explicitly calculate the following integrals: $$A(m)\equiv\,
\frac{1}{i}
{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{l^2-m^2+i\epsilon} \,,$$ $$\begin{aligned}
A_{;\mu;\mu\nu}(q,q')\equiv\,&
\frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\frac{1}{(l+q)^2-m^2+i\epsilon}
\\&\times
\frac{1}{-l_0-q_0'+i\epsilon}(1;l_\mu;l_\mu l_\nu) \,,\end{aligned}$$ $$\begin{aligned}
C_{;\mu;\mu\nu;\mu\nu\rho}(q_0,q_0')\equiv\,&
\frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\,
\frac{1}{(l+q)^2-m^2+i\epsilon}\,
\\&\times
\frac{1}{-l_0-q_0'+i\epsilon}
\frac{(1;l_\mu;l_\mu l_\nu;l_\mu l_\nu l_\rho)}{-l_0+i\epsilon} \,,\end{aligned}$$ $$\begin{aligned}
D_{;\mu;\mu\nu;\mu\nu\rho}(q_0,q_0')\equiv\,&
\frac{1}{i}{\int\frac{d^4l}{(2\pi)^4}}\,
\frac{1}{(l+q)^2-m^2+i\epsilon}
\\&\times
\frac{1}{-l_0-q_0'+i\epsilon}
\frac{(1;l_\mu;l_\mu l_\nu;l_\mu l_\nu l_\rho)}{l_0+i\epsilon} \,.\end{aligned}$$ The integrals can be divided depending on their subindexes being temporal or spatial. We show explicitly all the cases for the integrals $J$. The same definitions are used for all the other integrals. Therefore, to know any other integral one needs to replace in Eq. (\[eq:many\]) $J$ by $A$, $B$, $I$, etc. $$\begin{aligned}
J_\mu\equiv\,&
\delta_{\mu0}J_{10}+\delta_{\mu i}J_{11}{\vec{q}}_i \label{eq:many}\\\nonumber\\
J_{\mu\nu}\equiv\,&
\delta_{\mu0}\delta_{\nu0}J_{20}
+(\delta_{\mu0}\delta_{\nu i}
+\delta_{\mu i}\delta_{\nu 0})J_{21}{\vec{q}}_i
\nonumber\\&
+\delta_{\mu i}\delta_{\nu j}(J_{22}\delta_{ij}
+J_{23}{\vec{q}}_i{\vec{q}}_j)
\nonumber\\\nonumber\\
J_{\mu\nu\rho}\equiv\,&
\delta_{\mu0}\delta_{\nu0}\delta_{\rho0}J_{30}
+\delta\delta\delta_{\{\mu\nu\rho 00i\}}{\vec{q}}_iJ_{31}
\nonumber\\&
+\delta\delta\delta_{\{\mu\nu\rho 0ij\}}
(\delta_{ij}J_{32}+{\vec{q}}_i{\vec{q}}_jJ_{33})
\nonumber\\&
+\delta_{\mu i}\delta_{\nu j}\delta_{\rho k}
(\delta{\vec{q}}_{\{ijk\}}J_{34}+{\vec{q}}_i{\vec{q}}_j{\vec{q}}_kJ_{35})
\nonumber\\\nonumber\\
J_{\mu\nu\rho\sigma}\equiv\,&
\delta_{\mu0}\delta_{\nu0}\delta_{\rho0}\delta_{\sigma0}J_{40}
+\delta\delta\delta\delta_{\{\mu\nu\rho\sigma000i\}}{\vec{q}}_iJ_{41}
\nonumber\\&
+\delta\delta\delta\delta_{\{\mu\nu\rho\sigma00ij\}}
(\delta_{ij}J_{42}+{\vec{q}}_i{\vec{q}}_jJ_{43})
\nonumber\\&
+\delta\delta\delta\delta_{\{\mu\nu\rho\sigma0ijk\}}
(\delta{\vec{q}}_{\{ijk\}}J_{44}+{\vec{q}}_i{\vec{q}}_j{\vec{q}}_kJ_{45})
\nonumber\\&
+\delta_{\mu i}\delta_{\nu j}\delta_{\rho k}\delta_{\sigma l}
(\delta\delta_{\{ijkl\}}J_{46}
+\delta{\vec{q}}{\vec{q}}_{\{ijkl\}}J_{47}
\nonumber\\&
+{\vec{q}}_i{\vec{q}}_j{\vec{q}}_k{\vec{q}}_lJ_{48})\,. \nonumber\end{aligned}$$ All coefficients $J_{10}$, $J_{11}$, etc. have been written explicitly as functions of $I$, $J$, $K$, which can be integrated numerically, and the other simpler functions. The following definitions have been employed: $$\begin{aligned}
\delta{\vec{q}}_{\{ijk\}}=&\,
\delta_{ij}{\vec{q}}_k
+\delta_{ik}{\vec{q}}_j
+\delta_{jk}{\vec{q}}_i\,,
\\
\delta{\vec{q}}{\vec{q}}_{\{ijkl\}}=&\,
\delta_{ij}{\vec{q}}_k{\vec{q}}_l
+\delta_{ik}{\vec{q}}_j{\vec{q}}_l
+\delta_{il}{\vec{q}}_j{\vec{q}}_k \nonumber\\
&
+\delta_{jk}{\vec{q}}_i{\vec{q}}_l
+\delta_{jl}{\vec{q}}_i{\vec{q}}_k
+\delta_{kl}{\vec{q}}_i{\vec{q}}_j\,,
\\
\delta\delta_{\{ijkl\}}=&\,
\delta_{ij}\delta_{kl}
+\delta_{ik}\delta_{jl}
+\delta_{il}\delta_{jk}\,.\end{aligned}$$ The other quantities, $\delta\delta\delta_{\{\mu\nu\rho00i\}}$, $\delta\delta\delta_{\{\mu\nu\rho0ij\}}$, etc, are not meant to be contracted with the indexes $i$, $j$, and $k$ appearing in the rest of the expressions. They only indicate how many of the indexes $\mu$, $\nu$, $\rho$, and $\sigma$ must be temporal and how many spatial. It does not matter the order in which $0$, $i$, $j$, and $k$ are assigned to $\mu$, $\nu$, $\rho$, and $\sigma$, since all the integrals $J_{\mu\nu}$, $J_{\mu\nu\rho}$, etc, are symmetric with respect to these indexes. For example $$J_{00i}=J_{0i0}=J_{i00}={\vec{q}}_i J_{31} \,.$$
Results for the master integrals
--------------------------------
We have regularized the master integrals via dimensional regularization, where the integrals depend on the momentum dimension $D_\eta$, or more specifically, on the parameter $\eta$, defined through $D_\eta=4-\eta$, and on the renormalization scale $\mu$, for which we have taken $\mu=m_\pi$. In the following we use, $$\begin{aligned}
R=&-\frac{2}{\eta}-1+\gamma-\log(4\pi) \,,
\\
q_0''=&q_0'-q_0 \,.\end{aligned}$$ The integrals $A(m)$, $A(q_0,q_0')$ and $B(q_0,|{\vec{q}}|)$ appear, for example, in [@scherer02]. We have checked that both results coincide. It is important to maintain the $-i\epsilon$ prescription, otherwise the integrals may give a wrong result. We take it into account by replacing $q_0'\to q_0'-i\epsilon$ when evaluating the integrals.
### $A(m), A(q_0,q'_0)$ and $B(q_0,{\vec{q}})$
We have, $$A(m)=
-\frac{1}{8\pi^2}m^2\left(\frac12R+\log\left(\frac{m}{\mu}\right)\right) \,.$$ $$\begin{aligned}
A(q_0,q_0')\equiv
-\frac{q_0''}{8\pi^2}
\left[
\pi\frac{\sqrt{m^2-q_0''^2}}{q_0''}
+1-R-2\log\left(\frac{m}{\mu}\right)
\right. & \nonumber\\\left.
-\frac{2 \sqrt{{q_0''}^2(m^2-{q_0''}^2)} \tan
^{-1}\left(\frac{\sqrt{{q_0''}^2}}{\sqrt{m^2-{q_0''}^2
}}\right)}{{q_0''}^2}
\right] &\end{aligned}$$ $$\begin{aligned}
B(q_0,{\vec{q}})&=
-\frac{1}{16\pi^2}\left[-1+R+2\log\left(\frac{m}{\mu}\right)+2L(|q|)\right]\end{aligned}$$ with $$\begin{aligned}
L(|q|)\equiv &\frac{w}{|q|}\log\left(\frac{w+|q|}{2m}\right) \,,\end{aligned}$$ $w\equiv \sqrt{4m^2+|q|^2}$, $|q|\equiv \sqrt{{\vec{q}}^2-q_0^2}$, and $q^2\equiv q_0^2-{\vec{q}}^2\le0$.
### $C(q_0,q_0')$ and $D(q_0,q_0')$
$$\begin{aligned}
C(q_0,q_0')\equiv&
-\frac{1}{16\pi^2}{\int_0^1dx}{\int_{0}^1dy}\Bigg[
3y^{-\frac12}(1-y)
\\&
\left[
-\frac43
-\frac12(R-1+\log(4))
-\frac12\log\left(\frac{s_{xy}}{4\mu^2}\right)
\right]
\\&+y^{-\frac12}(1-y)(m^2+q_0''r_0')
s_{xy}^{-1}
\\&
-\pi(q_0''+r_0')s_x^{-\frac12}
\Bigg] \,,\end{aligned}$$ with $s_x=m^2-q_0^2+x(q_0^2-q_0''^2)$, $s_{xy}=m^2+(1-y)(-q_0^2+x(q_0^2-q_0''^2))$.
$$\begin{aligned}
D(q_0,q_0')=&
-C(q_0,q_0')+\frac{1}{q_0'}\frac{1}{4\pi}\sqrt{m^2-q_0^2} \,.\end{aligned}$$
### $I(q_0,|{\vec{q}}|,q_0')$
$$\begin{aligned}
I(q_0,q,q_0')&=
-\frac{1}{8\pi^2}\int_0^1dx\int_0^1dy
\left[
\frac{\pi}{2}\frac{1}{\sqrt{s_x}}
\right.\\&\left.
-\frac34y^{-\frac12}(1-y){C_q'}\frac{1}{s_{xy}}
+\frac12y^{\frac12}(1-y){C_q'}^3\frac{1}{s_{xy}^2}
\right]\end{aligned}$$ with $C_q'=-q_0(1-x)+q_0'$, $s_x\equiv -q^2x(1-x)-\left(q_0'-q_0+q_0x\right)^2+m_\pi^2$, and $s_{xy}\equiv -q^2x(1-x)-\left(q_0'-q_0+q_0x\right)^2(1-y)+m_\pi^2$.
### $J(q_0,|{\vec{q}}|,q_0')$ and $K(q_0,|{\vec{q}}|,q_0')$
$$\begin{aligned}
J{}=&
-\frac{1}{8\pi^2}\int_0^1dx{\int_{0}^1dy}\,y(1-y)
\Bigg\{
\left(-{C_q'}^3-{C_q'}^2{C_q}\right.\\&\left.
-{C_q'}{C_q}^2-{C_q}^3
+2 s_x ({C_q'}+{C_q})\right)\frac{3\pi}{8s_{xy}^{\frac52}}
\\&
+({C_q'}+{C_q})\frac{\pi}{8s_{xy}^{\frac32}}
+
\frac{105}{16}{\int_0^1dz}\,z^3\sqrt{1-z}
\Big[
-\frac{3}{s_{xyz}}
\\&
+\left({C_q'}^2-5 {C_q'}{C_q}+{C_q}^2-9s_x\right)\frac{2}{7s_{xyz}^2}
\\&+
\left(-9s_x^2+2s_x\left({C_q'}^2-5 {C_q'}{C_q}+{C_q}^2\right)+
\right.\\&\left.
3 {C_q'}^3{C_q}+{C_q'}^2{C_q}^2+3 {C_q'}{C_q}^3\right)\frac{8}{35s_{xyz}^3}
\\&+\left(-3s_x^3+s_x^2\left({C_q'}^2-5 {C_q'}{C_q}+{C_q}^2\right)
\right.\\&\left.
+s_x\left(3 {C_q'}^3{C_q}+{C_q'}^2{C_q}^2+3 {C_q'}{C_q}^3\right)
\right.\\&\left.
-{C_q'}^3 {C_q}^3\right)\frac{16}{35s_{xyz}^4}
\Big]
\Bigg\}\end{aligned}$$ with $C_{q}\equiv -q_0(1-x)$, $C_q'\equiv -q_0(1-x)+q_0'$, $s_x\equiv -q^2x(1-x)+m_\pi^2$, $s_{xy}\equiv s_x-{C_q}^2+y({C_q}^2-{C_q'}^2)$, and $
s_{xyz}\equiv s_x+z\cdot y({C_q}^2-{C_q'}^2)-z{C_q}^2$. $$\begin{aligned}
K&=
-J
+\frac{1}{8\pi q_0'}\int_0^1 dx
\frac{1}{\sqrt{m^2+(1-x)({\vec{q}}^2x-q_0^2)}} \,.\end{aligned}$$
Results for the master integrals with $q_0=q_0'=0$
--------------------------------------------------
$$\begin{aligned}
A(m)&=
-\frac{1}{8\pi^2}m^2\left(\frac12R+\log\left(\frac{m}{\mu}\right)\right)
\\
A(0,0)&=-\frac{m}{8\pi}
\\
B(0,{\vec{q}})&=
-\frac{1}{16\pi^2}\left[-1+R+2\log\left(\frac{m}{\mu}\right)+2L(q)\right]
\\
C(0,0)&=-\frac{1}{4\pi^2}
\left(
-\frac{R}{2}-\frac12
-\log(\frac{m}{\mu})\right)
\\
I(0,{\vec{q}},0)&=
-\frac{1}{4\pi}At(q)
\\
J(0,{\vec{q}},0)&=\frac{1}{2\pi^2{\vec{q}}^2}L(q),\end{aligned}$$
where $L(q)$ and $At(q)$ are defined with $$\begin{aligned}
At(q)\equiv&\frac{1}{2q}\arctan\left(\frac{q}{2m_\pi}\right)
\\
L(q)\equiv&\frac{\sqrt{4m_\pi^2+q^2}}{q}\log\left(\frac{\sqrt{4m_\pi^2+q^2}+q}{2m_\pi}\right)\,.\end{aligned}$$
Relations between master integrals
----------------------------------
### $A_\mu(q_0,q_0')$
$$\begin{aligned}
A_{10}&=-A(m)-q_0'A
\\
A_{11}&=-A\end{aligned}$$
### $A_{\mu\nu}(q,q')$
$$\begin{aligned}
A_{20}&=
\left[
(q_0+q_0')A(m)+
{q_0'}^2A\right]
\\A_{21}&=
A(m)+q_0'A
\\A_{22}&=
\frac{1}{D_\eta-1}\left[
q_0''A(m)+({q_0''}^2-m^2)A\right]
\\A_{23}&=A\end{aligned}$$
### $B_\mu(q)$
$$\begin{aligned}
B_{10}&=-\frac{q_0}{2}B
\\
B_{11}&=-\frac12B\end{aligned}$$
### $B_{\mu\nu}(q)$
$$\begin{aligned}
B_{20}&=
\frac{1}{2(D_\eta-1)q^2}\Bigg[
(q^2+q_0^2(D_\eta-2))A(m)
\\&
-\left(2{\vec{q}}^2m^2+\frac12q^2(q^2-D_\eta q_0^2)\right)B
\Bigg]
\\\\
B_{21}&=
\frac{q_0}{2(D_\eta -1)q^2}\left[
(D_\eta -2)A(m)+\left(\frac{ D_\eta}{2}q^2-2m^2\right)B\right]
\\\\
B_{22}&=
-\frac{1}{2(D_\eta -1)}\left[A(m)+\left(2m^2-\frac{q^2}{2}\right)B\right]
\\\\
B_{23}&=
\frac{1}{2(D_\eta -1)q^2}\left[
(D_\eta -2)A(m)+\left(\frac{D_\eta}{2}q^2-2m^2\right)B
\right]\end{aligned}$$
### $C_\mu(q_0,q_0')$
$$\begin{aligned}
C_{10}&=-A
\\
C_{11}&=-C\end{aligned}$$
### $C_{\mu\nu}(q_0,q_0')$
$$\begin{aligned}
C_{20}&=-A_{10}
\\
C_{21}&\equiv
-A_{11}
\\
C_{22}&=\frac{1}{D_\eta-1}(C_{20}+2q_0C_{10}+(q_0^2-m^2)C)
\\
C_{23}&=C\end{aligned}$$
### $C_{\mu\nu\rho}(q_0,q_0')$
$$\begin{aligned}
C_{30}&=-A_{20}
\\
C_{31}&=-A_{21}
\\
C_{32}&=-A_{22}
\\
C_{33}&=-A_{23}
\\
C_{34}&\equiv-C_{22}
\\
C_{35}&=-6C_{11}-3C_{23}-4C\end{aligned}$$
### $D_\mu(q_0,q_0')$
$$\begin{aligned}
D_{10}&=A
\\
D_{11}&=-D\end{aligned}$$
### $D_{\mu\nu}(q_0,q_0')$
$$\begin{aligned}
D_{20}&\equiv A_{10}
\\
D_{21}&\equiv A_{11}
\\
D_{22}&=\frac{1}{D_\eta-1}(D_{20}+2q_0D_{10}+(q_0^2-m^2)D)
\\
D_{23}&=D\end{aligned}$$
### $D_{\mu\nu\rho}(q_0,q_0')$
$$\begin{aligned}
D_{30}&\equiv A_{20}
\\
D_{31}&\equiv A_{21}
\\
D_{32}&\equiv A_{20}
\\
D_{33}&\equiv A_{21}
\\
D_{34}&\equiv -D_{22}
\\
D_{35}&\equiv-6D_{11}-3D_{23}-4D\end{aligned}$$
### $I_\mu$
$$\begin{aligned}
I_{10}&=
-B-q_0'I
\\
I_{11}&=
\frac{1}{2{\vec{q}}^2}
\left[-A(0,q_0',r_0)+A
-2q_0B+(q_0^2-{\vec{q}}^2-2q_0q_0')I
\right]\end{aligned}$$
### $I_{\mu\nu}$
$$\begin{aligned}
I_{20}&=-B_{10}-q_0'I_{10}
\\
I_{21}&=-B_{11}-q_0'I_{11}
\\
I_{22}&=\frac{1}{(D_\eta -2){\vec{q}}^{\,2}}
\left[-I_{({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2I_{({\vec{l}}^2)}\right]
\\
I_{23}&=\frac{1}{(D_\eta -2){\vec{q}}^{\,4}}
\left[(D_\eta -1)I_{({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^2I_{({\vec{l}}^2)}\right]\end{aligned}$$
$$\begin{aligned}
I_{({\vec{l}}^2)}&=
-A(q,q')-m^2I_0-B_{10}-q_0'I_{10}
\\
I_{({\vec{l}}\cdot{\vec{q}})^2}&=
\frac12{\vec{q}}^2
\left[
A_{11}(q,q')
-2q_0B_{11}+(q^2-2q_0q_0')I_{11}
\right]\end{aligned}$$
### $I_{\mu\nu\rho}$
$$\begin{aligned}
I_{30}&=-B_{20}-q_0'I_{20}
\\\\
I_{31}&=-B_{21}-q_0'I_{21}
\\\\
I_{32}&=
-B_{22}-q_0'I_{22}
\\\\
I_{33}&=
-B_{23}-q_0'I_{23}
\\\\
I_{34}&=
\frac{-I_{({\vec{l}}\cdot{\vec{q}})^3}+{\vec{q}}^2I_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}}
{{\vec{q}}^4(D_\eta -2)}
\\\\
I_{35}&=
\frac{(D_\eta +1)I_{({\vec{l}}\cdot{\vec{q}})^3}-3{\vec{q}}^2I_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}}
{{\vec{q}}^6(D_\eta -2)}\end{aligned}$$
$$\begin{aligned}
I_{({\vec{l}}\cdot{\vec{q}})^3}&=
\frac12{\vec{q}}^2
\left[-A_{22}(0,q_0')-{\vec{q}}^2A_{23}(0,q_0')-{\vec{q}}^2A(0,q_0')
\right.\\&-2{\vec{q}}^2 A_{11}(0,q_0')
A_{22}+{\vec{q}}^2A_{23}+q^2I_{22}+q^2{\vec{q}}^2I_{23}
\\&\left.
-2q_0B_{22}-2q_0{\vec{q}}^2B_{23}
-2q_0q_0'I_{22}
-2q_0q_0'{\vec{q}}^2I_{23}
\right]\end{aligned}$$
$$\begin{aligned}
I_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}&=
{\vec{q}}^2\left(
-A_{11}-m^2I_{11}-B_{21}-q_0'I_{21}
\right)\end{aligned}$$
### $J_\mu$
$$\begin{aligned}
J_{10}&\equiv-I
\\
J_{11}&\equiv \frac{1}{2{\vec{q}}^2}
\left[
-C(0,q_0')+C
-2q_0I+q^2J\right]\end{aligned}$$
### $J_{\mu\nu}$
$$\begin{aligned}
J_{20}&\equiv -I_{10}
\\
J_{21}&\equiv -I_{11}
\\
J_{22}&\equiv\frac{1}{(D_\eta -2){\vec{q}}^{\,2}}
\left[-J_{({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2J_{({\vec{l}}^2)}\right]
\\
J_{23}&\equiv\frac{1}{(D_\eta -2){\vec{q}}^{\,4}}
\left[(D_\eta -1)J_{({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^2J_{({\vec{l}}^2)}\right]\end{aligned}$$
$$\begin{aligned}
J_{({\vec{l}}^2)}&=
-C-m^2J-I_{10}
\\\\
J_{({\vec{l}}\cdot{\vec{q}})^2}&=
\frac12
\left[
C_{11}+q^2J_{11}
-2q_0I_{11}
\right]{\vec{q}}^2\end{aligned}$$
### $J_{\mu\nu\rho}$
$$\begin{aligned}
J_{30}&\equiv-I_{20}
\\
J_{31}&\equiv-I_{21}
\\
J_{32}&\equiv-I_{22}
\\
J_{33}&\equiv-I_{23}
\\
J_{34}&\equiv
\frac{-J_{({\vec{l}}\cdot{\vec{q}})^3}+{\vec{q}}^2J_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}}
{{\vec{q}}^4(D_\eta -2)}
\\
J_{35}&\equiv
\frac{(D_\eta +1)J_{({\vec{l}}\cdot{\vec{q}})^3}-3{\vec{q}}^2J_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}}
{{\vec{q}}^6(D_\eta -2)}\end{aligned}$$
$$\begin{aligned}
J_{({\vec{l}}\cdot{\vec{q}})^3}&=
\frac{{\vec{q}}^2}{2}\left[
-C_{20}(0,q_0')
-{\vec{q}}^2C_{21}(0,q_0')
-{\vec{q}}^2C(0,q_0')
\right.\\&\left.
-2{\vec{q}}^2C_{11}(0,q_0')
+C_{20}+C_{21}{\vec{q}}^2
\right.\\&\left.
+q^2(J_{22}+J_{23}{\vec{q}}^2)
-2q_0(I_{22}+I_{23}{\vec{q}}^2)
\right]
\\
J_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}&=
-{\vec{q}}^2\left[C_{11}
+m^2J_{11}
+I_{21}
\right]\end{aligned}$$
### $J_{\mu\nu\rho\sigma}$
$$\begin{aligned}
J_{40}&\equiv -I_{30}
\\
J_{41}&\equiv -I_{31}
\\
J_{42}&\equiv -I_{32}
\\
J_{43}&\equiv-I_{33}
\\
J_{44}&\equiv -I_{34}
\\
J_{45}&\equiv -I_{35}
\\
J_{46}&=
2\,\frac{-J_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2J_{{\vec{l}}^4}}
{{\vec{q}}^2(D-2)(2D+3)}
\\J_{47}&=
\frac{-(2D+3)J_{({\vec{l}}\cdot{\vec{q}})^4}+2(2+D){\vec{q}}^2J_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^4J_{{\vec{l}}^4}}
{{\vec{q}}^6(D-2)(2D+3)}
\\J_{48}&=
\frac{(D+4)J_{({\vec{l}}\cdot{\vec{q}})^4}-6{\vec{q}}^2J_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}}
{{\vec{q}}^8(D-2)}\end{aligned}$$
$$\begin{aligned}
J_{({\vec{l}}\cdot{\vec{q}})^4}&=
\frac{{\vec{q}}^4}{2}\left[
3C_{34}+{\vec{q}}^2C_{35}
\right.\\&\left.
+q^2(3J_{34}+{\vec{q}}^2J_{35})
\right.\\&\left.
-2q_0(3I_{34}+{\vec{q}}^2I_{35})
\right]
\\
J_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}&=
-{\vec{q}}^2\Big[C_{22}+{\vec{q}}^2C_{23}
+m^2(J_{22}+J_{23}{\vec{q}}^2)
+I_{32}
\\&
+{\vec{q}}^2I_{33}
\Big]
\\
J_{{\vec{l}}^4}&=
-(C_{22}(D_\eta -1)+C_{23}{\vec{q}}^2)
\\&
-m^2(J_{22}(D_\eta -1)+J_{23}{\vec{q}}^2)
-(I_{32}(D_\eta -1)+I_{33}{\vec{q}}^2)\end{aligned}$$
### $K_\mu$
$$\begin{aligned}
K_{10}&=I
\\
K_{11}
&\equiv
\frac{1}{2{\vec{q}}^2}
\left[
-D(0,q_0')+D+q^2K
+2q_0I
\right]\end{aligned}$$
### $K_{\mu\nu}$
For the first two cases we apply the following tricks, $$\begin{aligned}
K_{20}&\equiv I_{10}
\\
K_{21}&\equiv I_{11}
\\
K_{22}&\equiv\frac{1}{(D_\eta -2){\vec{q}}^{\,2}}
\left[-K_{({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2K_{({\vec{l}}^2)}\right]
\\
K_{23}&\equiv \frac{1}{(D_\eta -2){\vec{q}}^{\,4}}
\left[(D_\eta -1)K_{({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^2K_{({\vec{l}}^2)}\right]\end{aligned}$$
Giving the following results, $$\begin{aligned}
K_{({\vec{l}}^2)}&=-D-m^2K+I_{10}-r_0K_{10}
\\
K_{({\vec{l}}\cdot{\vec{q}})^2}&=
\frac12
\left[
D_{11}+q^2K_{11}
+2q_0I_{11}
\right]{\vec{q}}^2\end{aligned}$$
### $K_{\mu\nu\rho}$
$$\begin{aligned}
K_{30}&\equiv I_{20}
\\
K_{31}&\equiv I_{21}
\\
K_{32}&\equiv I_{22}
\\
K_{33}&\equiv I_{23}
\\
K_{34}&=
\frac{-K_{({\vec{l}}\cdot{\vec{q}})^3}+{\vec{q}}^2K_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}}
{{\vec{q}}^4(D_\eta -2)}
\\
K_{35}&\equiv
\frac{(D_\eta +1)K_{({\vec{l}}\cdot{\vec{q}})^3}-3{\vec{q}}^2K_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}}
{{\vec{q}}^6(D_\eta -2)}\end{aligned}$$
$$\begin{aligned}
K_{({\vec{l}}\cdot{\vec{q}})^3}&=
\frac{{\vec{q}}^2}{2}\left[
-D_{22}(0,q_0')
-{\vec{q}}^2D_{23}(0,q_0')
-{\vec{q}}^2D(0,q_0')
\right.\\&\left.
-2{\vec{q}}^2D_{11}(0,q_0')
+D_{22}
+{\vec{q}}^2D_{23}
\right.\\&\left.
+q^2(K_{22}+K_{23}{\vec{q}}^2)
+2q_0(I_{22}+I_{23}{\vec{q}}^2)
\right]
\\
K_{({\vec{l}}\cdot{\vec{q}}){\vec{l}}^2}&=
-{\vec{q}}^2\left[D_{11}
+m^2K_{11}
-I_{21}
+r_0K_{21}\right]\end{aligned}$$
### $K_{\mu\nu\rho\sigma}$
$$\begin{aligned}
K_{40}&\equiv I_{30}
\\
K_{41}&\equiv I_{31}
\\
K_{42}&\equiv I_{32}
\\
K_{43}&\equiv I_{33}
\\
K_{44}&\equiv I_{34}
\\
K_{45}&\equiv I_{35}
\\
k_{46}&=
2\,\frac{-K_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}+{\vec{q}}^2K_{{\vec{l}}^4}}
{{\vec{q}}^2(D-2)(2D+3)}
\\K_{47}&=
\frac{-(2D+3)K_{({\vec{l}}\cdot{\vec{q}})^4}+2(2+D){\vec{q}}^2K_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}-{\vec{q}}^4K_{{\vec{l}}^4}}
{{\vec{q}}^6(D-2)(2D+3)}
\\K_{48}&=
\frac{(D+4)K_{({\vec{l}}\cdot{\vec{q}})^4}-6{\vec{q}}^2K_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}}
{{\vec{q}}^8(D-2)}\end{aligned}$$
$$\begin{aligned}
K_{({\vec{l}}\cdot{\vec{q}})^4}&=
\frac12\left[
2D_{10}{\vec{q}}^4+{\vec{q}}^4D
\right.\\&\left.
+q^2(K_{22}{\vec{q}}^2+K_{23}{\vec{q}}^4)
\right.\\&\left.
+2q_0(I_{22}{\vec{q}}^2+
I_{23}{\vec{q}}^4)
\right]
\\
K_{{\vec{l}}^2({\vec{l}}\cdot{\vec{q}})^2}&=
-\Big[D_{22}+D_{23}{\vec{q}}^2
+m^2(K_{22}+K_{23}{\vec{q}}^2)
-I_{32}
\\&
-I_{33}{\vec{q}}^2
\Big]{\vec{q}}^2
\\
K_{{\vec{l}}^4}&=
-(D_{22}(D_\eta-1)+D_{23}{\vec{q}}^2)
\\&
-m^2(K_{22}(D_\eta-1)+K_{23}{\vec{q}}^2)
\\&
+(I_{32}(D_\eta-1)+I_{33}{\vec{q}}^2)\end{aligned}$$
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| ArXiv |
---
abstract: 'The effects of delayed feedback terms on nonlinear oscillators has been extensively studied, and have important applications in many areas of science and engineering. We study a particular class of second-order delay-differential equations near a point of triple-zero nilpotent bifurcation. Using center manifold and normal form reduction, we show that the three-dimensional nonlinear normal form for the triple-zero bifurcation can be fully realized at any given order for appropriate choices of nonlinearities in the original delay-differential equation.'
author:
- |
Victor G. LeBlanc\
Department of Mathematics and Statistics\
University of Ottawa\
Ottawa, ON K1N 6N5\
CANADA
title: 'Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators'
---
Introduction
============
Delay-differential equations are used as models in many areas of science, engineering, economics and beyond [@BBL; @HFEKGG; @Kuang; @LM; @SieberKrauskopf; @SC; @SS; @VTK; @WuWang; @ZW]. It is now well understood that retarded functional differential equations (RFDEs), a class which contains delay-differential equations, behave for the most part like ordinary differential equations on appropriate infinite-dimensional function spaces. As such, many of the techniques and theoretical results of finite-dimensional dynamical systems have counterparts in the theory of RFDEs. In particular, versions of the stable/unstable and center manifold theorems in neighborhoods of an equilibrium point exist for RFDEs [@HVL]. Also, techniques for simplifying vector fields via center manifold and normal form reductions have been adapted to the study of bifurcations in RFDEs [@FM1; @FM2].
One of the challenges of applying these finite-dimensional techniques to RFDEs lies in the so-called [*realizability problem*]{}. This problem stems from the fact that the procedure to reduce an RFDE to a center manifold often leads to algebraic restrictions on the nonlinear terms in the center manifold equations. Specifically, suppose $B$ is an arbitrary $m\times m$ matrix. For the sake of simplicity, suppose additionally that all eigenvalues of $B$ are simple. Let $C([-r,0],\mathbb{R})$ be the space of continuous functions from the interval $[-r,0]$ into $\mathbb{R}$, and for any continuous function $z$, define $z_t\in
C([-r,0],\mathbb{R})$ as $z_t(\theta)=z(t+\theta)$, $-r\leq\theta\leq 0$. It is then possible [@FM3] to construct a bounded linear operator $\mathcal{L}:C([-r,0],\mathbb{R})\longrightarrow\mathbb{R}$ such that the infinitesimal generator $A$ for the flow associated with the functional differential equation $$\dot{z}(t)=\mathcal{L}\,z_t
\label{linfde1}$$ has a spectrum which contains the eigenvalues of $B$ as a subset. Thus, there exists an $m$-dimensional subspace $P$ of $C([-r,0],\mathbb{R})$ which is invariant for the flow generated by $A$, and the flow on $P$ is given by the linear ordinary differential equation (ODE) $$\dot{x}=Bx.$$
Now, suppose (\[linfde1\]) is modified by the addition of a nonlinear delayed term $$\dot{z}(t)=\mathcal{L}\,z_t+az(t-\tau)^2,
\label{nonlinfde1}$$ where $a\in\mathbb{R}$ is some coefficient and $\tau\in [0,r]$ is the delay time. Then the center manifold theorem for RFDEs [@HVL] can be used to show that the flow for (\[nonlinfde1\]) admits an $m$-dimensional locally invariant center manifold on which the dynamics associated with (\[nonlinfde1\]) are given by a vector field which, to quadratic order, is of the form $$\dot{x}=Bx+ag(x),
\label{realizeode1}$$ where $g:\mathbb{R}^m\longrightarrow\mathbb{R}^m$ is a fixed homogeneous quadratic polynomial which is completely determined by $\mathcal{L}$ and $\tau$, and $a$ is the same coefficient which appears in (\[nonlinfde1\]). We immediately notice that for fixed $\mathcal{L}$ and $\tau$, (\[realizeode1\]) has at most one degree of freedom in the quadratic term, corresponding to the one degree of freedom in the quadratic term in (\[nonlinfde1\]). However, whereas one degree of freedom is sufficient to describe the general scalar quadratic term involving one delay in (\[nonlinfde1\]), it is largely insufficient (if $m>1$) to describe the general homogeneous quadratic polynomial $f:\mathbb{R}^m\longrightarrow\mathbb{R}^m$. Therefore, there exist $m$-dimensional vector fields $\dot{x}=Bx+f(x)$ (where $f$ is homogeneous quadratic) which can not be realized by center manifold reduction (\[realizeode1\]) of any RFDE of the form (\[nonlinfde1\]). The realizability problem has received considerable attention in the literature [@BuonoBelair; @ChoiLeBlanc1; @ChoiLeBlanc2; @FM3; @FM4].
In this paper, we will be interested in a realizability problem for a class of second-order scalar delay-differential equations of the form $$\ddot{x}(t)+b\dot{x}(t)+ax(t)-F(x(t),\dot{x}(t))=\alpha x(t-\tau)+\beta\dot{x}(t-\tau)+G(x(t-\tau),\dot{x}(t-\tau)),
\label{premaineq}$$ where $a$, $b$, $\alpha$ and $\beta$ are real parameters, $\tau>0$ is a delay term, and the nonlinear functions $F$ and $G$ are smooth and vanish at the origin, along with their first order partial derivatives. This class contains many interesting applications which have been studied in the literature, including Van der Pol’s oscillator with delayed feedback [@Atay; @deOlivera; @JiangYuan; @WeiJiang; @WuWang], as well as models for stabilization of an inverted pendulum via delayed feedback [@SieberKrauskopf].
Both the Van der Pol oscillator [@WuWang] and the inverted pendulum system [@SieberKrauskopf] have been shown to possess points in parameter space where a bifurcation via a non-semisimple triple-zero eigenvalue occurs. In [@SieberKrauskopf], this bifurcation is in fact characterized as the [*organizing center*]{} for their model, since it includes in its unfolding Bogdanov-Takens and steady-state/Hopf mode interactions and the associated complex dynamics of these codimension two singularities. As far as we are aware, a complete theoretical analysis and classification of all possible dynamics near the non-semisimple triple-zero bifurcation has yet to be done, although a rather thorough investigation was undertaken in [@DumortierIbanez]. Numerical tools are used in [@SieberKrauskopf] to illustrate the complexity of this singularity in their model, including many global bifurcations. It is stated in [@SieberKrauskopf] that because of the presence of invariant tori, a full versal unfolding of the triple-zero singularity must include terms other than those appearing at cubic order in their model, and conclude by wondering whether full realizability of the nonlinear normal form for the triple-zero bifurcation is possible for their delay-differential equation.
Other relevant work includes [@CampbellYuan], where the authors study a class of coupled first-order delay-differential equations which includes (\[premaineq\]) as a special case (if one writes (\[premaineq\]) as a first order system), and compute quadratic and cubic normal form coefficients in term of DDE coefficients for both non-semisimple double-zero and triple-zero bifurcations. Higer-order terms for these normal form are not considered.
In this paper, we will first show that the non-semisimple triple-zero singularity occurs generically in (\[premaineq\]), and then prove that the full nonlinear normal form for the non-semisimple triple-zero bifurcation, at any prescribed order, can be realized by center manifold normal form reduction of (\[premaineq\]) for appropriate choices of nonlinear functions $F$ and $G$. In section 2, we present the functional analytic framework in which we will study this problem. Section 3 gives a brief summary of the center manifold and normal form procedure for RFDEs which was developed by Faria and Magalh$\tilde{\mbox{\rm a}}$es [@FM1; @FM2]. Our main result is stated and proved in section 4. We end with some concluding remarks in section 5.
Functional analytic setup
=========================
As mentioned in the introduction, we consider a general class of second order nonlinear differential equations for the real-valued function $x(t)$ of the form (\[premaineq\]), which we rewrite as a first order system $$\begin{array}{rcl}
\dot{x}(t)&=&y(t)\\
&&\\
\dot{y}(t)&=&-a\,x(t)+\alpha\,x(t-\tau)-b\,y(t)+\beta\,y(t-\tau)+F(x(t),y(t))+G(x(t-\tau),y(t-\tau)).
\end{array}
\label{maineq}$$ where $a$, $b$, $\alpha$ and $\beta$ are real parameters, $\tau>0$ is a delay term, and the nonlinear functions $F$ and $G$ are smooth and vanish at the origin, along with their first order partial derivatives. In many applications, we have $a>0$, so we will assume this condition throughout (although other cases of $a$ can be treated in a similar manner).
The characteristic equation corresponding to (\[maineq\]) is $P(\lambda)=0$, where $$P(\lambda)= \lambda^2+b\lambda+a-(\alpha+\beta\lambda)e^{-\lambda\tau}.
\label{char}$$ A straightforward computation shows that when $$\begin{array}{c}
\alpha=a\\
\\
\tau=\tau_0={\displaystyle\frac{\beta}{a}+\frac{\sqrt{\beta^2+2a}}{a}}\\
\\
b=\beta-a\tau_0\\
\\
3\beta^2\neq 2a\,\,\,(\mbox{\rm or equivalently}\,\,\, a\tau_0\neq 3\beta)
\end{array}
\label{param}$$ then $P(0)=P'(0)=P''(0)=0$, $P'''(0)\neq 0$, and $P$ has no other roots on the imaginary axis. Therefore $0$ is a triple eigenvalue for the linearization of (\[maineq\]) at the origin, with geometric multiplicity one.
For the parameter values (\[param\]), we write (\[maineq\]) as $$\begin{array}{rcl}
\dot{x}(t)&=&y(t)\\
&&\\
\dot{y}(t)&=&-a\,(x(t)-x(t-\tau_0))+a\tau_o\,y(t)-\beta\,(y(t)-y(t-\tau_0))\\&&\\
&&+F(x(t),y(t))+G(x(t-\tau_0),y(t-\tau_0)).
\end{array}
\label{maineqsys}$$
Let $C=C\left( \left[ -\tau_0,0\right] ,\mathbb{R}^2\right)$ be the Banach space of continuous functions from $\left[ -\tau_0,0\right] $ into $\mathbb{R}^2$ with supremum norm. We define $z_{t}\in C$ as $$z_{t}\left( \theta\right)
=z\left( t+\theta\right) =\left(\begin{array}{c} x(t+\theta)\\y(t+\theta)\end{array}\right),-\tau_0\leq\theta\leq 0.$$ We view (\[maineqsys\]) as a retarded functional differential equation of the form $$\dot{z}\left( t\right) ={\cal L}\,z_{t}+{\cal F}\left( z_{t}\right) ,
\label{y1}$$ where ${\cal L}:C\rightarrow\mathbb{R}^2$ is the bounded linear operator $${\cal L}\,\phi=\int_{-\tau_0}^{0}\left[ d\eta\left(\theta\right) \right] \phi\left(
\theta\right)=\left( \begin{array}{cc}0&1\\-a&a\tau_0-\beta\end{array}\right)\phi(0)+\left(\begin{array}{cc}0&0\\ a&\beta\end{array}\right)\phi(-\tau_0)$$ and ${\cal F}$ is the smooth nonlinear function from $C$ into $\mathbb{R}^2$ $${\cal F}(\phi)=\left(\begin{array}{c}0\\F(\phi(0))+G(\phi(-\tau_0))\end{array}\right).$$
Let $A$ be the infinitesimal generator of the flow for the linear system $\dot{z}={\cal L}\,z_t$, with spectrum $\sigma(A)\supset\,\{0\}$, and $P$ be the three-dimensional invariant subspace for $A$ associated with the eigenvalue $0$. Then it follows that the columns of the matrix $$\Phi=\left(\begin{array}{ccc}1&\theta&\frac{1}{2}\theta^2\\0&1&\theta\end{array}\right)$$ form a basis for $P$.
In a similar manner, we can define an invariant space, $P^{\ast},$ to be the generalized eigenspace of the transposed system, $A^{T}$ associated with the triple nilpotency having as basis the rows of the matrix $\Psi=$$\left( \psi_{1},\ldots,\psi_{m}\right)$. Note that the transposed system, $A^{T}$ is defined over a dual space $C^{\ast}=C\left(
\left[ 0,\tau_0\right] ,\mathbb{R}^2\right),$ and each element of $\Psi$ is included in $C^{\ast}.$ The bilinear form between $C^{\ast}$ and $C$ is defined as $$\left( \psi,\phi\right) =\psi\left( 0\right) \phi\left( 0\right)
-\int\limits_{-r}^{0}\int\limits_{0}^{\theta}\psi\left( \zeta-\theta\right)
\text{ }\left[ d\eta\left( \theta\right) \right] \text{ }\phi\left(
\zeta\right) \text{ }d\zeta. \label{y11}$$ Note that $\Phi$ and $\Psi$ satisfy $\dot{\Phi}=B\Phi,$ $\dot{\Psi}=-\Psi B,$ where $B$ is the $3\times 3$ matrix $$B=\left(\begin{array}{ccc}0&1&0\\0&0&1\\0&0&0\end{array}\right).
\label{BBdef}$$
We can normalize $\Psi$ such that $\left( \Psi
,\Phi\right) =I$, and we can decompose the space $C$ using the splitting $C=P\oplus Q$, where the complementary space $Q$ is also invariant for $A$.
Faria and Magalh$\tilde{\mbox{\rm a}}$es [@FM1; @FM2] show that (\[y1\]) can be written as an infinite dimensional ordinary differential equation on the Banach space $BC$ of functions from $[-\tau_0,0]$ into ${\mathbb R}^2$ which are uniformly continuous on $[-\tau_0,0)$ and with a jump discontinuity at $0$, using a procedure that we will now outline. Define $X_0$ to be the function $$X_0(\theta)=\left\{\begin{array}{lc}
\left(\begin{array}{cc}1&0\\0&1\end{array}\right)&\theta=0\\[0.15in]
\left(\begin{array}{cc}0&0\\0&0\end{array}\right)&-\tau_0\leq\theta<0,
\end{array}\right.$$ then the elements of $BC$ can be written as $\xi=\varphi+X_0\lambda$, with $\varphi\in C$ and $\lambda\in {\mathbb R}^2$, so that $BC$ is identified with $C\times {\mathbb R}^2$.
Let $\pi:BC\longrightarrow P$ denote the projection $$\pi(\varphi+X_0\lambda)=\Phi [(\Psi,\varphi)+\Psi(0)\lambda],$$ where $\varphi\in C$ and $\lambda\in {\mathbb R}^2$. We now decompose $z_t$ in (\[y1\]) according to the splitting $$BC=P\oplus\mbox{\rm ker}\,\pi,$$ with the property that $Q\subsetneq\,\mbox{\rm ker}\,\pi$, and get the following infinite-dimensional ODE system which is equivalent to (\[y1\]): $$\begin{array}{rcl}
\dot{u}&=&Bu+\Psi(0)\,{\cal F}(\Phi\,u+v)\\[0.15in]
{\displaystyle\frac{d}{dt}\,v}&=&A_{Q^1}v+(I-\pi)X_0\,{\cal F}(\Phi\,u+v),
\end{array}
\label{projfdep}$$ where $u\in {\mathbb R}^3$, $v\in Q^1\equiv Q\cap C^1$, ($C^1$ is the subset of $C$ consisting of continuously differentiable functions), and $A_{Q^1}$ is the operator from $Q^1$ into $\mbox{\rm ker}\,\pi$ defined by $$A_{Q^1}\varphi=\dot{\varphi}+X_0\,[{\cal L}\,\varphi-\dot{\varphi}(0)].$$
Faria and Magalh$\tilde{\mbox{\bf a}}$es normal form
====================================================
Consider the formal Taylor expansion of the nonlinear terms $\mathcal{F}$ in (\[y1\]) $${\cal F}(\phi)=\sum_{j\geq 2}\,{\cal F}_j(\phi),\,\,\,\,\,\phi\in\,C,$$ where ${\cal F}_j(\phi)=V_j(\phi,\ldots,\phi)$, with $V_j$ belonging to the space of continuous multilinear symmetric maps from $C\times\cdots\times C$ ($j$ times) to $\mathbb{R}^2$. If we denote $f_j=(f_j^1,f_j^2)$, where $$\begin{array}{rcl}
f_{j}^1(u,v)&=&\Psi(0)\,{\cal F}_j(\Phi\,u+v)\\[0.15in]
f_j^2(u,v)&=&(I-\pi)\,X_0\,{\cal F}_j(\Phi\,u+v),
\end{array}$$ then (\[projfdep\]) can be written as $$\begin{array}{rcl}
\dot{u}&=&{\displaystyle Bu+\sum_{j\geq
2}\,f_j^1(u,v)}\\[0.15in]
{\displaystyle\frac{d}{dt}\,v}&=&{\displaystyle A_{Q^1}v+\sum_{j\geq
2}\,f_j^2(u,v)}.
\end{array}
\label{y4}$$
It is easy to see that the non-resonance condition of Faria and Magalh$\tilde{\mbox{\rm a}}$es (Definition (2.15) of [@FM2]) holds. Consequently, using successively at each order $j$ a near identity change of variables of the form $$(u,v)=(\hat{u},\hat{v})+U_j(\hat{u})\equiv (\hat{u},\hat{v})+
(U^1_j(\hat{u}),U^2_j(\hat{u})),
\label{nfcv}$$ (where $U^{1,2}_j$ are homogeneous degree $j$ polynomials in the indicated variable, with coefficients respectively in $\mathbb{R}^3$ and $Q^1$) system (\[y4\]) can be put into formal normal form $$\begin{array}{rcl}
\dot{u}&=&{\displaystyle Bu+\sum_{j\geq
2}\,g_j^1(u,v)}\\[0.15in]
{\displaystyle\frac{d}{dt}\,v}&=&{\displaystyle A_{Q^1}v+\sum_{j\geq
2}\,g_j^2(u,v)}
\end{array}
\label{y5}$$ such that the center manifold is locally given by $v=0$ and the local flow of (\[y1\]) on this center manifold is given by $$\dot{u}=Bu+\sum_{j\geq 2}\,g_j^1(u,0).
\label{y6}$$ The nonlinear terms in (\[y6\]) are in normal form in the classical sense with respect to the matrix $B$.
Realizability of the normal form for the triple-zero nilpotency
===============================================================
It was shown in [@DumortierIbanez; @YuYuan] that the classical normal form for the general nonlinear vector field $$\left(\begin{array}{c}\dot{u}_1\\\dot{u}_2\\\dot{u}_3\end{array}\right)=\left(\begin{array}{ccc}0&1&0\\0&0&1\\0&0&0\end{array}\right)\,\left(\begin{array}{c}u_1\\u_2\\u_3\end{array}\right)+\left(\begin{array}{c}r_1(u_1,u_2,u_3)\\r_2(u_1,u_2,u_3)\\r_3(u_1,u_2,u_3)\end{array}\right)$$ (where ${\displaystyle r_j(0,0,0)=0, \frac{\partial r_j}{\partial u_k}(0,0,0)=0, \,\,\mbox{\rm for}\,j,k=1,2,3}$) is $$\begin{array}{ccl}
\dot{u}_1&=&u_2\\
\dot{u}_2&=&u_3\\
\dot{u}_3&=&{\displaystyle \sum_{j\geq 2}\,\left(\,\sum_{i=0}^j\,a_{(j-i),i}\,u_1^{j-i}u_2^i\,+\,u_1^Iu_3\sum_{i=0}^{J}\,b_{N(J-i),i}\,u_1^{J-i}u_3^i\,\right)},
\end{array}$$ where $$\begin{array}{ll}
N=1,\,\,\,J=\frac{1}{2}(j-1),\,\,I=J,\,\,\,\,\,&\mbox{\rm when $j$ is odd},\\&\\
N=2,\,\,\,J=\frac{j}{2}-1,\,\,I=J+1,&\mbox{\rm when $j$ is even}.
\end{array}$$
Thus, if $B$ is the matrix (\[BBdef\]), and $H^j$ is the space of homogeneous polynomial mappings of degree $j\geq 2$ from $\mathbb{R}^3$ into $\mathbb{R}^3$, then the homological operator $L_B\equiv Dh(u)\cdot Bu-B\cdot h(u)$ acting on $H^j$ is such that $$H^j=L_B(H^j)\oplus W_j,
\label{homsplit}$$ where $W_j\subset H^j$ is the subspace of dimension $$\begin{array}{cl}
{\displaystyle\frac{3j+2}{2}}&\,\,\,\,\mbox{\rm if $j$ is even}\\&\\
{\displaystyle\frac{3j+3}{2}}&\mbox{\rm if $j$ is odd}
\end{array}$$ spanned by $$\left\{\left(\begin{array}{c}0\\0\\u_1^{j-i}u_2^i\end{array}\right)\,,\,i=0,\ldots,j\,\right\}\,\bigcup\,
\left\{\left(\begin{array}{c}0\\0\\u_1^{j-1-i}u_3^{i+1}\end{array}\right)\,,\,
\begin{array}{ll}
i=0,\ldots,\frac{1}{2}(j-2)\,\,\,\,&\mbox{\rm $j$ even}\\&\\
i=0,\ldots,\frac{1}{2}(j-1)&\mbox{\rm $j$ odd}.
\end{array}\,\right\}$$
Now, if $F(z_1,z_2)$ and $G(z_1,z_2)$ are real-valued functions such that $F$, $G$ and their first-order partial derivatives vanish at the origin, then we may write the Taylor series $$F(z_1,z_2)=\sum_{j\geq 2}\,\hat{F}_j(z_1,z_2),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
G(z_1,z_2)=\sum_{j\geq 2}\,\hat{G}_j(z_1,z_2),$$ where the $\hat{F}_j$ and $\hat{G}_j$ are homogeneous degree $j$ polynomials. The first equation in (\[y4\]) then reduces to $$\dot{u}=Bu+\sum_{j\geq 2}\left(\begin{array}{c}0\\
\\
\kappa_1\,(\hat{F}_j((u_1,u_2)+v(0))+\hat{G}_j((u_1-\tau_0u_2+\frac{1}{2}\tau_0^2u_3,u_2-\tau_0u_3)+v(-\tau_0)))\\
\\
\kappa_2\,(\hat{F}_j((u_1,u_2)+v(0))+\hat{G}_j((u_1-\tau_0u_2+\frac{1}{2}\tau_0^2u_3,u_2-\tau_0u_3)+v(-\tau_0)))\end{array}\right),
\label{prenf}$$ where $$\kappa_1=\frac{3(a\tau_0-4\beta)}{2\tau_0(a\tau_0-3\beta)^2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\kappa_2=\frac{6}{\tau_0^2(a\tau_0-3\beta)}\neq 0.$$ We note that when $v=0$ in (\[prenf\]), then $$\begin{array}{ccl}
\hat{F}_j(u_1,u_2)+\hat{G}_j(u_1-\tau_0u_2+\frac{1}{2}\tau_0^2u_3,u_2-\tau_0u_3)&=&
{\cal A}_j(u_1,u_2)+u_1u_3{\cal B}_{j-2}(u_1,u_3)+\\
&&\\
&&u_2u_3{\cal C}_{j-2}(u_1,u_2,u_3)+u_3^2{\cal D}_{j-2}(u_3),
\end{array}$$ where $$\begin{aligned}
{2}\label{Adef}
{\cal A}_j(u_1,u_2)=&\,\hat{F}_j(u_1,u_2)+\hat{G}_j(u_1-\tau_0u_2,u_2)\\\label{Bdef}
u_1u_3{\cal B}_{j-2}(u_1,u_3)=&\,\hat{G}_j(u_1+\frac{1}{2}\tau_0^2u_3,-\tau_0u_3)-\hat{G}_j(u_1,0)-\hat{G}_j(\frac{1}{2}\tau_0^2u_3,-\tau_0u_3)\\\notag
u_2u_3{\cal C}_{j-2}(u_1,u_2,u_3)=&\,\hat{G}_j(u_1-\tau_0u_2+\frac{1}{2}\tau_0^2u_3,u_2-\tau_0u_3)-\hat{G}_j(u_1-\tau_0u_2,u_2)-\\\notag
&\,\hat{G}_j(u_1+\frac{1}{2}\tau_0^2u_3,-\tau_0u_3)+\hat{G}_j(u_1,0)\\\notag
u_3^2{\cal D}_{j-2}(u_3)=&\,\hat{G}_j(\frac{1}{2}\tau_0^2u_3,-\tau_0u_3).\end{aligned}$$
For a given integer $j\geq 2$, let $\zeta(u_1,u_3)$ be a homogeneous degree $j$ polynomial such that $\zeta(0,u_3)=\zeta(u_1,0)=0$. Then there exists a homogeneous polynomial of degree $j$, $\xi(u_1,u_3)$ such that $$\zeta(u_1,u_3)=\xi(u_1+\frac{1}{2}\tau_0^2u_3,-\tau_0u_3)-\xi(u_1,0)-\xi(\frac{1}{2}\tau_0^2u_3,-\tau_0u_3).
\label{magic}$$
[**Proof of lemma:**]{} If we write $$\xi(u_1,u_3)=\sum_{i=0}^j\,\gamma_{j-i,i}\,u_1^{j-i}u_3^i$$ then a lengthy but straightforward computation shows that $$\begin{array}{l}
{\displaystyle \xi(u_1+\frac{1}{2}\tau_0^2u_3,-\tau_0u_3)-\xi(u_1,0)-\xi(\frac{1}{2}\tau_0^2u_3,-\tau_0u_3)=}\\
\\
{\displaystyle\gamma_{j,0}\left(\sum_{k=1}^{j-1}\left(\begin{array}{c}j\\k\end{array}\right)\left(\frac{1}{2}\tau_0^2\right)^k\,u_1^{j-k}u_3^k\right)+
\sum_{i=1}^{j-1}\,\gamma_{j-i,i}\left(
\sum_{k=0}^{j-i-1}\left(\begin{array}{c}j-i\\k\end{array}\right)\left(\frac{1}{2}\tau_0^2\right)^k(-\tau_0)^i\,u_1^{j-i-k}u_3^{i+k}\right)},
\end{array}$$ where $$\left(\begin{array}{c}j\\k\end{array}\right)=\frac{j!}{k!(j-k)!}.$$ Now, since $\zeta(0,u_3)=\zeta(u_1,0)=0$, we have that $$\zeta(u_1,u_3)=\sum_{i=1}^{j-1}\,\epsilon_{j-i,i}u_1^{j-i}u_3^i.$$ Thus, we see for example that we may solve (\[magic\]) by arbitrarily setting $\gamma_{j,0}=0$, $\gamma_{0,j}=0$, and choosing $\gamma_{j-i,i}$, $i=1,\ldots,j-1$ such that the following triangular linear algebraic system is satisfied $$\begin{array}{c}
{\displaystyle\left(\begin{array}{c}j-1\\0\end{array}\right)(-\tau_0)\,\gamma_{j-1,1} = \epsilon_{j-1,1}}\\
\\
{\displaystyle\left(\begin{array}{c}j-1\\1\end{array}\right)\left(\frac{1}{2}\tau_0^2\right)(-\tau_0)\,\gamma_{j-1,1}+
\left(\begin{array}{c}j-2\\0\end{array}\right)(-\tau_0)^2\,\gamma_{j-2,2}=\epsilon_{j-2,2}}\\
\\
\vdots\\
\\
{\displaystyle\left(\begin{array}{c}j-1\\j-2\end{array}\right)\left(\frac{1}{2}\tau_0^2\right)^{j-2}(-\tau_0)\,\gamma_{j-1,1}+\hdots+
\left(\begin{array}{c}1\\0\end{array}\right)(-\tau_0)^{j-1}\,\gamma_{1,j-1}=\epsilon_{1,j-1}}
\end{array}$$ This ends the proof of the lemma. [ ]{}
Now, recalling the splitting (\[homsplit\]), let $\Theta_j(u_1,u_2,u_3)$ be a homogeneous degree $j$ polynomial such that $\Theta_j\in W_j$. We may write $$\Theta_j(u)=\left(\begin{array}{c}0\\0\\q_j(u_1,u_2)\end{array}\right)+\left(\begin{array}{c}0\\0\\u_1u_3s_{j-2}(u_1,u_3)\end{array}\right),$$ where $q_j$ is a homogeneous polynomial of degree $j$, and $s_{j-2}$ is a homogeneous degree $j-2$ polynomial.
The degree $j$ term in (\[prenf\]) for $v=0$ can be written as $$\begin{array}{l}
\left(\begin{array}{c}0\\
\\
\kappa_1\,(\hat{F}_j(u_1,u_2)+\hat{G}_j(u_1-\tau_0u_2+\frac{1}{2}\tau_0^2u_3,u_2-\tau_0u_3))\\
\\
\kappa_2\,(\hat{F}_j(u_1,u_2)+\hat{G}_j(u_1-\tau_0u_2+\frac{1}{2}\tau_0^2u_3,u_2-\tau_0u_3))\end{array}\right)=\\
\\
R_j(u_1,u_2,u_3)+\left(\begin{array}{c}0\\0\\\kappa_2\,{\cal A}_j(u_1,u_2)\end{array}\right)+
\left(\begin{array}{c}0\\0\\\kappa_2\,u_1u_3{\cal B}_{j-2}(u_1,u_3)\end{array}\right)\end{array}$$ where $R_j$ is in the range of the homological operator, $R_j\subset L_B(H^j)$, and ${\cal A}_j$ and ${\cal B}_{j-2}$ are as in (\[Adef\]) and (\[Bdef\]). Using the previous lemma, we know that if we choose $\hat{G}_j$ such that $\kappa_2\,{\cal B}_{j-2}=s_{j-2}$, and then set $\kappa_2\,\hat{F}_j(u_1,u_2)=q_j(u_1,u_2)-\kappa_2\,\hat{G}_j(u_1-\tau_0u_2,u_2)$, then $$\Theta_j(u)=\left(\begin{array}{c}0\\0\\\kappa_2\,{\cal A}_j(u_1,u_2)\end{array}\right)+
\left(\begin{array}{c}0\\0\\\kappa_2\,u_1u_3{\cal B}_{j-2}(u_1,u_3)\end{array}\right).
\label{theteq}$$
We can now state and prove the following realizability theorem:
Given an integer $\ell\geq 2$ and a polynomial vector field on $\mathbb{R}^3$ of the form $$\dot{u}=Bu+\sum_{j=2}^{\ell}\,w_j(u),
\label{nf2}$$ where $B$ is the matrix (\[BBdef\]) and $w_j\in W_j$ as in (\[homsplit\]), there exist polynomial functions $F$ and $G$ in (\[maineqsys\]) such that the Faria and Magalh$\tilde{\mbox{\it a}}$es center manifold and normal form reduction (\[y6\]) of (\[prenf\]) up to order $\ell$ is (\[nf2\]). \[mainthm\]
[**Proof:**]{} Applying successively at each order $j$ (from $j=2$ to $j=\ell$) a near identity change of variables of the form (\[nfcv\]), and setting $v=0$, we transform (\[prenf\]) into $$\dot{u}=Bu+\sum_{j=2}^{\ell}\,\left[\left(\begin{array}{c}0\\0\\\kappa_2\,{\cal A}_j(u_1,u_2)\end{array}\right)
+\left(\begin{array}{c}0\\0\\\kappa_2\,u_1u_3{\cal B}_{j-2}(u_1,u_3)\end{array}\right)+\Lambda_j(u_1,u_2,u_3)\right]+O(|u|^{\ell+1}),
\label{postnf}$$ where $\Lambda_2(u_1,u_2,u_3)=0$ and for $j\geq 3$, $\Lambda_j\in W_j$ is an extra contribution to the terms of order $j$ coming from the transformation of the lower order $(<j)$ terms. Therefore, we set $$\Theta_j(u)=w_j(u)-\Lambda_j(u),\,\,\,\,\,j=2,\ldots,\ell,$$ and use $(\ref{theteq})$ to conclude that the truncation of (\[postnf\]) at order $\ell$ is (\[nf2\]). [ ]{}
Conclusion
==========
In this paper, we have solved the realizability problem for the normal form of the non-semisimple triple-zero singularity in a class of delay differential equations (\[premaineq\]) which includes delayed Van der Pol oscillators, as well as certain models for the control of an inverted pendulum as special cases. It is apparent from the complexity of the dynamics of (\[premaineq\]) near the triple-zero nilpotency reported in previous work [@CampbellYuan; @DumortierIbanez; @SieberKrauskopf] that high-order normal forms will be required for a complete classification of this singularity. Although such a complete classification of the dynamics near a triple-zero nilpotency is beyond the scope of this paper, our results allow us to conclude that the full range of complexity of this singularity is attainable within the class of delay-differential equations (\[premaineq\]).
Although we have not done so, we believe that the results in this paper could be suitably generalized to studying realizability of higher order nilpotencies in higher-order scalar delay-differential equations such as $$x^{(n)}(t)+\sum_{j=0}^{n-1}\,a_j\,x^{(j)}(t)-F(x(t),\ldots,x^{(n-1)}(t))=\sum_{j=0}^{n-1}\,\alpha_j\,x^{(j)}(t-\tau)+G(x(t-\tau),\ldots,x^{(n-1)}(t-\tau))$$ where $n\geq 3$ is an integer.
[**Acknowledgments**]{}
This research is partly supported by the Natural Sciences and Engineering Research Council of Canada in the form of a Discovery Grant.
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| ArXiv |
---
abstract: 'We investigate mode coupling in a two dimensional compressible disc with radial stratification and differential rotation. We employ the global radial scaling of linear perturbations and study the linear modes in the local shearing sheet approximation. We employ a three-mode formalism and study the vorticity (W), entropy (S) and compressional (P) modes and their coupling properties. The system exhibits asymmetric three-mode coupling: these include mutual coupling of S and P-modes, S and W-modes, and asymmetric coupling between the W and P-modes. P-mode perturbations are able to generate potential vorticity through indirect three-mode coupling. This process indicates that compressional perturbations can lead to the development of vortical structures and influence the dynamics of radially stratified hydrodynamic accretion and protoplanetary discs.'
author:
- |
A. G. Tevzadze$^1$, G. D. Chagelishvili$^{1,2}$, G. Bodo$^3$ and P. Rossi$^3$\
$^1$ Georgian National Astrophysical Observatory, Chavchavadze State University, Tbilisi, Georgia\
$^2$ Nodia Institute of Geophysics, Georgian Academy of Sciences, Tbilisi, Georgia\
$^3$ INAF – Osservatorio Astronomico di Torino, strada dell’Osservatorio 20, I-10025 Pino Torinese, Italy
title: Linear coupling of modes in 2D radially stratified astrophysical discs
---
accretion, accretion discs – hydrodynamics – instabilities
Introduction
============
The recent increased interest in the analysis of hydrodynamic disc flows is motivated, on one hand, by the study of turbulent processes, and, on the other, by the investigation of regular structure formation in protoplanetary discs. Indeed, many astrophysical discs are thought to be neutral or having ionization rates too low to effectively couple with magnetic field. Among these are cool and dense areas of protoplanetary discs, discs around young stars, X-ray transient and dwarf nova systems in quiescence (see e.g. Gammie and Menou 1998, Sano et al. 2000, Fromang, Terquem and Balbus 2002). Observational data shows that astrophysical discs often exhibit radial gradients of thermodynamic variables (see e.g. Sandin et al. 2008, Issela et al. 2007). To what extent these inhomogeneities affect the processes occurring in the disc is still subject open to investigations. It has been found that strong local entropy gradients in the radial direction may drive the Rossby wave instability (Lovelace et al. 1999, Li et al. 2000) that transfers thermal to kinetic energy and leads to vortex formation. However, in astrophysical discs, radial stratification is more likely weak. In this case, the radial entropy (temperature) variation on the global scale leads to the existence of baroclinic perturbations over the barotropic equilibrium state. This more appropriate situation has recently become a subject of extensive study.
Klahr and Bodenheimer (2003) pointed out that the radial stratification in the disc can lead to the global baroclinic instability. Numerical results show that the resulting state is highly chaotic and transports angular momentum outwards. Later Klahr (2004) performed a local 2D linear stability analysis of a radially stratified flow with constant surface density and showed that baroclinic perturbations can grow transiently during a limited time interval. Johnson and Gammie (2005) derived analytic solutions for 3D linear perturbations in a radially stratified discs in the Boussinesq approximation. They find that leading and trailing waves are characterized by positive and negative angular momentum flux, respectively. Later Johnson and Gammie (2006) performed numerical simulations, in the local shearing sheet model, to test the radial convective stability and the effects of baroclinic perturbations. They found no substantial instability due to the radial stratification. This result reveals a controversy over the issue of baroclinic instability. Presently, it seems that nonlinear baroclinic instability is an unlikely development in the local dynamics of sub-Keplerian discs with weak radial stratification.
Potential vorticity production, and the formation and development of vortices in radially stratified discs have been studied by Petersen et al. (2007a,b) by using pseudospectral simulations in the anelastic approximation. They show that the existence of thermal perturbations in the radially stratified disc flows leads to the formation of vortices. Moreover, stronger vortices appear in discs with higher temperature perturbations or in simulations with higher Reynolds numbers, and the transport of angular momentum may be both outward and inward.
Keplerian differential rotation in the disc is characterized by a strong velocity shear in the radial direction. It is known that shear flows are non-normal and exhibit a number of transient phenomena due to the non-orthogonal nature of the operators (see e.g. Trefethen et al. 1993). In fact, the studies described above did not take into account the possibility of mode coupling and energy transfer between different modes due to the shear flow induced mode conversion. Mode coupling is inherent to shear flows (cf. Chagelishvili et al. 1995) and often, in many respects, defines the role of perturbation modes in the system dynamics and the further development of nonlinear processes. Thus, a correct understanding of the energy exchange channels between different modes in the linear regime is vital for a correct understanding of the nonlinear phenomena.
Indications of the shear induced mode conversion can be found in a number of previous studies. Barranco and Marcus 2005 report that vortices are able to excite inertial gravity waves during 3D spectral simulations. Brandenburg and Dintrans (2006) have studied the linear dynamics of perturbation SFH to analyze nonaxisymetric stability in the shearing sheet approximation. Temporal evolution of the perturbation gain factors reveal a wave nature after the radial wavenumber changes sign. Compressible waves are present, along with vortical perturbations, in the simulation by Johnson & Gammie (2005b) but their origin is not particularly discussed.
In parallel, there are a number of papers that focus on the investigation of the shear induced mode coupling phenomena. The study of the linear coupling of modes in Keplerian flows has been conducted in the local shearing sheet approximation (Tevzadze et al. 2003,2008) as well as in 2D global numerical simulations (Bodo et al. 2005, hereafter B05). Tevzadze et al. (2003) studied the linear dynamics of three-dimensional small scale perturbations (with characteristic scales much less then the disc thickness) in vertically (stably) stratified Keplerian discs. They show, that vortex and internal gravity wave modes are coupled efficiently. B05 performed global numerical simulations of the linear dynamics of initially imposed two-dimensional pure vortex mode perturbations in compressible Keplerian discs with constant background pressure and density. The two modes possible in this system are effectively coupled: vortex mode perturbations are able to generate density-spiral waves. The coupling is, however, strongly asymmetric: the coupling is effective for wave generation by vortices, but not viceversa. The resulting dynamical picture points out the importance of mode coupling and the necessity of considering compressibility effects for processes with characteristic scales of the order or larger than the disc thickness. Bodo et al. (2007) extended this work to nonlinear amplitudes and found that mode coupling is an efficient channel for energy exchange and is not an artifact of the linear analysis. B05 is particularly relevant to the present study, since it studies the dynamics of mode coupling in 2D unstratified flows and is a good starting point for a further extension to radially stratified flows. Later, Heinemann & Papaloizou (2009a) derived WKBJ solutions of the generated waves and performed numerical simulations of the wave excitation by turbulent fluctuations (Heinemann & Papaloizou 2009b).
In the present paper we study the linear dynamics of perturbations and analyze shear flow induced mode coupling in the local shearing sheet approximation. We investigate the properties of mode coupling using qualitative analysis within the three-mode approximation. Within this approximation we tentatively distinguish vorticity, entropy and pressure modes. Quantitative results on mode conversion are derived numerically. It seems that a weak radial stratifications, while being a weak factor for the disc stability, still provides an additional degree of freedom (an active entropy mode), opening new options for velocity shear induced mode conversion, that may be important for the system behavior. One of the direct result of mode conversion is the possibility of linear generation of the vortex mode (i.e., potential vorticity) by compressible perturbations. We want to stress the possibility of the coupling between high and low frequency perturbations, considering that high frequency oscillations have been often neglected during previous investigations in particular for protoplanetary discs.
Conventionally there are two distinct viewpoints commonly employed during the investigation of hydrodynamic astrophysical discs. In one case (self gravitating galactic discs) the emphasis is placed on the investigation of the dynamics of spiral-density waves and vortices, although normally present in numerical simulations, are thought to play a minor role in the overall dynamics. In the other case (non-self-gravitating hydrodynamic discs) the focus is on the potential vorticity perturbations and density-spiral waves are often thought to play a minor role. Here, discussing the possible (multi) mode couplings, we want to draw attention to the possible flaws of these simplified views (see e.g. Mamatsashvili & Chagelishvili 2007). In many cases, mode coupling makes different perturbation to equally participate in the dynamical processes despite of a significant difference in their temporal scales.
In the next section we present mathematical formalism of our study. We describe three mode formalism and give schematic picture of the linear mode coupling in the radially sheared and stratified flow. Numerical analysis of the mode coupling is presented in Sec. 3. We evaluate mode coupling efficiencies at different radial stratification scales of the equilibrium pressure and entropy. The paper is summarized in Sec. 4.
Basic equations
===============
The governing ideal hydrodynamic equations of a two-dimensional, compressible disc flows in polar coordinates are: $${\partial \Sigma \over \partial t} + {1 \over r} {\partial \left( r
\Sigma V_r \right) \over
\partial r} + {1 \over r} {\partial \left( \Sigma V_\phi \right)\over
\partial \phi} = 0~,~~~~~~~~~~~~~~~~~~~~~~$$ $${\partial V_r \over \partial t} + V_r{\partial V_r \over
\partial r}+ {V_\phi \over r}{\partial V_r \over
\partial \phi} - {V_\phi^2 \over r} = -{1 \over \Sigma}
{\partial P \over \partial r} - {\partial \psi_g \over \partial r}
~,~~~~~~$$ $${\partial V_\phi \over \partial t} + V_r{\partial V_\phi \over
\partial r}+ {V_\phi \over r}{\partial V_\phi \over
\partial \phi} +
{V_r V_\phi \over r} = -{1 \over \Sigma r}{\partial P\over
\partial \phi}~,~~~~~~~~~$$ $${\partial P \over \partial t} + V_r{\partial P \over
\partial r}+ {V_\phi \over r}{\partial P \over
\partial \phi}
= - {\gamma P} \left( {1 \over r} {\partial (r V_r) \over
\partial r} + {1 \over r} {\partial V_\phi \over \partial
\phi} \right)~,$$ where $V_r$ and $V_\phi$ are the flow radial and azimuthal velocities respectively. $P(r,\phi)$, $\Sigma(r,\phi)$ and $\gamma~$ are respectively the pressure, the surface density and the adiabatic index. $\psi_g$ is the gravitational potential of the central mass, in the absence of self-gravitation $~(\psi_g \sim -{1 / r})$. This potential determines the Keplerian angular velocity: $${\partial \psi_g \over \partial r} = \Omega_{Kep}^2 r ~,~~~~
\Omega_{Kep} \sim r^{-3/2};$$
Equilibrium state
-----------------
We consider an axisymmetric $(\partial / \partial \phi \equiv 0),~$ azimuthal $(\bar {V}_{r} = 0)~$ and differentially rotating basic flow: $\bar {V}_{\phi}= \Omega(r)r$. In the 2D radially stratified equilibrium (see Klahr 2004), all variables are assumed to follow a simple power law behavior: $$\bar {\Sigma}(r) = \Sigma_0 \left( {r \over r_0}
\right)^{-\beta_\Sigma},~~~~\bar {P}(r) = P_0\left( {r \over r_0}
\right)^{-\beta_P} ~,$$ where overbars denote equilibrium and $\Sigma_0$ and $P_0$ are the values of the equilibrium surface density and pressure at some fiducial radius $r = r_0$. The entropy can be calculated as: $$\bar S = \bar P \bar \Sigma^{-\gamma} = - \left(r \over r_0
\right)^{-\beta_S} ~,$$ where $$\beta_S \equiv \beta_P - \gamma \beta_\Sigma ~.$$ $S$ is sometimes called potential temperature, while the physical entropy can be derived as $\bar S = C_V \log S + \bar S_0$.
This equilibrium shows a deviation from the Keplerian profile due to the radial stratification: $$\Delta \Omega^2(r) = \Omega^2(r) - \Omega^2_{Kep}= {1 \over r {\bar
{\Sigma}(r)}} {\partial {\bar{P}(r)}\over \partial r} =
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = - {P_0
\over \Sigma_0} {\beta_P \over r_0^2}\left( {r \over r_0}
\right)^{\beta_\Sigma-\beta_P-2} ~.$$ Hence, the described state is sub-Keplerian or super-Keplerian when the radial gradient of pressure is negative ($\beta_P>0$) or positive ($\beta_P<0$), respectively. Although these discs are non-Keplerian, they are still rotationally supported, since the deviation from the Keplerian profile is small: $~\Delta
\Omega^2(r)\ll \Omega^2_{Kep}$.
Linear perturbations
--------------------
We split the physical variables into mean and perturbed parts: $$\Sigma(r,\phi) = {\bar {\Sigma}(r)} + {{\Sigma}^\prime(r,\phi)} ~,$$ $$P(r,\phi) = {\bar{P}(r)} + P^\prime(r,\phi) ~,$$ $$V_r(r,\phi) = V_r^\prime(r,\phi) ~,$$ $$V_\phi(r,\phi) = \Omega(r) r + V_\phi^\prime(r,\phi) ~.$$ In order to remove background trends from the perturbations we employ the global radial power law scaling for perturbed quantities: $$\hat \Sigma(r) \equiv \left({r \over r_0}\right)^{-\delta_\Sigma}
\Sigma^\prime(r) ~,$$ $$\hat P(r) \equiv \left({r \over r_0}\right)^{-\delta_P} P^\prime(r)
~,$$ $$\hat {\bf V}(r) \equiv \left({r \over r_0}\right)^{-\delta_V} {\bf
V}^\prime(r) ~.$$
After the definitions one can get the following dynamical equations for the scaled perturbed variables: $$\left\{ {\partial \over \partial t} + \Omega(r) {\partial \over
\partial \phi} \right\} {\hat \Sigma \over \Sigma_0} +
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$\left( {r \over r_0} \right)^{-\beta_\Sigma-\delta_\Sigma+\delta_V}
\left[ {\partial \hat V_r \over \partial r} + {1 \over r} {\partial
\hat V_\phi \over
\partial \phi} + {1+\delta_V-\beta_\Sigma \over r} \hat V_r \right]
= 0 ~,$$ $$\left\{ {\partial \over \partial t} + \Omega(r) {\partial \over
\partial \phi} \right\} \hat V_r - 2\Omega(r) \hat V_\phi +$$ $${c_s^2 \over \gamma} \left({r \over r_0}
\right)^{\beta_\Sigma+\delta_P-\delta_V} {\partial \over \partial r}
{\hat P \over P_0} + c_s^2 {\delta_P \over \gamma r_0} \left( {r
\over r_0} \right)^{\beta_\Sigma+\delta_P-\delta_V-1} {\hat P \over
P_0} +$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ c_s^2 {\beta_P \over \gamma r_0}
\left( {r \over r_0}
\right)^{2\beta_\Sigma+\delta_\Sigma-\beta_P-\delta_V-1} {\hat
\Sigma \over \Sigma_0} = 0 ~,$$ $$\left\{ {\partial \over \partial t} + \Omega(r) {\partial \over
\partial \phi} \right\} \hat V_\phi + \left( 2 \Omega(r) +
r {\partial \Omega(r) \over \partial r} \right) \hat V_r +
~~~~~~~~~~~~~~~$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ {c_s^2 \over \gamma r_0}
\left( {r \over r_0} \right)^{\beta_\Sigma+\delta_P-\delta_V-1}
{\partial \over \partial \phi} {\hat P \over P_0} = 0 ~,$$ $$\left\{ {\partial \over \partial t} + \Omega(r) {\partial \over
\partial \phi} \right\} {\hat P \over P_0} +$$ $$\gamma \left( {r \over r_0} \right)^{-\beta_P+\delta_V-\delta_P}
\left[ {\partial \hat V_r \over
\partial r} + {1 \over r} {\partial \hat V_\phi \over \partial
\phi} + {1+\delta_V-\beta_P/\gamma \over r} \hat V_r \right] = 0 ~,$$ where $c_s^2 = \gamma P_0/\Sigma_0$ is the squared sound speed at $r=r_0$.
Local approximation
-------------------
The linear dynamics of perturbations in differentially rotating flows can be effectively analyzed in the local co-rotating shearing sheet frame (e. g., Goldreich & Lynden-Bell 1965; Goldreich & Tremaine 1978). This approximation simplifies the mathematical description of flows with inhomogeneous velocity. In the radially stratified flows the spatial inhomogeneity of the governing equations comes not only from the equilibrium velocity, but from the pressure, density and entropy profiles as well. In this case we first re-scale perturbations in global frame in order to remove background trends from linear perturbations, rather then use complete form of perturbations to the equilibrium (see Eqs. 14-16). Hence, using the re-scaled linear perturbation ($\hat P$, $\hat
\Sigma$, $\hat {\rm \bf V}$) we may simplify local shearing sheet description as follows. Introduction of a local Cartesian co-ordinate system: $$x \equiv r - r_0~,~~~~ y \equiv r_0 (\phi - \Omega_0 t)~,~~~~{x
\over r_0} ,~ {y \over r_0} \ll 1~,$$ $${\partial \over \partial x} = {\partial \over \partial r}~,~~~
{\partial \over \partial y} = {1 \over r_0}{\partial \over
\partial \phi}~,~~~ {\partial \over \partial t} =
{\partial \over \partial t} - r_0 \Omega_0 {\partial \over
\partial y},$$ where $\Omega_0$ is the local rotation angular velocity at $r=r_0$, transforms global differential rotation into a local radial shear flow and the two Oort constants define the local shear rate: $$A \equiv {1 \over 2} r_0 \left[ {\partial \Omega(r) \over \partial
r}\right]_{r=r_0}~,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$B \equiv - {1 \over 2} \left[ r{\partial \Omega(r)\over \partial r}
+ 2\Omega(r) \right]_{r=r_0}= -A - \Omega_0~.$$ Hence, the equations describing the linear dynamics of perturbations in local approximation read as follows: $$\left\{ {\partial \over \partial t} + 2Ax {\partial \over
\partial y} \right\} {\hat P \over \gamma P_0} +$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left[ {\partial \hat V_x \over
\partial x} + {\partial \hat V_y \over \partial y} +
{1+\delta_V-\beta_P/\gamma \over r_0} \hat V_x \right] = 0 ~,$$ $$\left\{ {\partial \over \partial t} + 2Ax {\partial \over
\partial y} \right\} {\hat V_x} - 2\Omega_0 \hat V_y +$$ $$~~~~~~~~~~~~~~~~~~~~~ c_s^2 \left[ {\partial \over \partial x} {\hat
P \over \gamma P_0} + {\delta_P + \beta_P/\gamma \over r_0} {\hat P
\over \gamma P_0} - {\beta_P \over \gamma r_0} {\hat S \over \gamma
P_0}\right] = 0 ~,$$ $$\left\{ {\partial \over \partial t} + 2Ax {\partial \over
\partial y} \right\} {\hat V_y} - 2B \hat V_y + c_s^2
{\partial \over \partial y} {\hat P \over \gamma P_0} =0 ~,$$ $$\left\{ {\partial \over \partial t} + 2Ax {\partial \over
\partial y} \right\} {\hat S \over \gamma P_0} - {\beta_S
\over \gamma r_0} \hat V_x = 0 ~,$$ where $\hat S $ is the entropy perturbation: $$\hat S \equiv \hat P - c_s^2 \hat \Sigma ~.$$ Now we may adjust the global scaling law of perturbations in order to simplify the local shearing sheet description (see Eqs. 25,26): $$1 + \delta_V - \beta_P/\gamma = 0 ~,$$ $$\delta_P + \beta_P/\gamma = 0 ~.$$
Let us introduce spatial Fourier harmonics (SFHs) of perturbations with time dependent phases: $$\left( \begin{array}{c} {\hat V}_x({\bf r},t) \\ {\hat V}_y({\bf r},t) \\
{{\hat P}({\bf r},t) / \gamma P_0} \\ {{\hat S}({\bf r},t) / \gamma
P_0} \end{array} \right) =
\left( \begin{array}{r} u_x({\bf k}(t),t) \\ u_y({\bf k}(t),t) \\
-{\rm i} p({\bf k}(t),t) \\ s({\bf k}(t),t)
\end{array} \right) \times$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\exp \left( {\rm i} k_x(t) x + {\rm i} k_y y \right) ~,$$ with $$k_x(t) = k_x(0) - 2Ak_y t~.$$
Using the above expansion and Eqs. (27-30), we obtain a compact ODE system that governs the local dynamics of SFHs of perturbations: $${{\rm d} \over {\rm d} t} p - k_x(t) u_x - k_y u_y = 0 ~,$$ $${{\rm d} \over {\rm d} t} u_x - 2 \Omega_0 u_y + c_s^2 k_x(t) p -
c_s^2 k_P s = 0 ~,$$ $${{\rm d} \over {\rm d} t} u_y - 2 B u_x + c_s^2 k_y p = 0 ~,$$ $${{\rm d} \over {\rm d} t} s - k_S u_x = 0 ~.$$ where $$k_P = {\beta_P \over \gamma r_0} ~~~ k_S = {\beta_S \over \gamma
r_0} ~.$$ The potential vorticity: $$W \equiv k_x(t)u_y - k_y u_x - 2B p ~,$$ is a conserved quantity in barotropic flows: $W = {\rm const.}$ when $k_P=0$.
Perturbations at rigid rotation
-------------------------------
The dispersion equation of our system can be obtained in the shearless limit ($A=0$, $B=-\Omega$). Hence, using Fourier expansion of perturbations in time $\propto \exp({\rm i} \omega t)$, in the shearless limit, we obtain: $$\omega^4 - \left( c_s^2 k^2 + 4 \Omega_0^2 - c_s^2 \eta \right)
\omega^2 - c_s^4 \eta k_y^2 = 0 ~,$$ where $$\eta \equiv k_P k_S = {\beta_P \beta_S \over \gamma^2 r_0^2} ~.$$ Solutions of the Eq. (40) describe a compressible density-spiral mode and a convective mode that involves perturbations of entropy and potential vorticity. For weakly stratified discs $(\eta \ll
k^2)$, we find the frequencies are: $$\bar \omega_{p}^2 = c_s^2 k^2 + 4 \Omega_0^2 ~,$$ $$\bar \omega_{c}^2 = - {c_s^4 \eta k_y^2 \over c_s^2 k^2 + 4
\Omega_0^2} ~.$$ High frequency solutions ($\bar \omega_{p}^2$) describe the density-spiral waves and will be referred later as the P-modes. Low frequency solutions ($\bar \omega_{c}^2$), instead, describe radial buoyancy mode due to the stratification. In barotropic flows ($\eta=0$) this mode is degenerated into stationary zero frequency vortical solution. Therefore, we may refer to it as a baroclinic mode. The mode describes instability when $\eta>0$. In this case the equilibrium pressure and entropy gradients point in the same direction. Klahr (2004) has anticipated such result, although worked in the constant surface density limit ($\beta_\Sigma=0$). The same behavior has been obtained for axisymmetric perturbations in Johnson and Gammie (2005). For comparison, in our model baroclinic perturbations are intrinsically non-axisymmetric. Hence, our result obtained in the rigidly rotating limit shows that the local exponential stability of the radial baroclinic mode is defined by the Schwarzschild-Ledoux criterion: $${{\rm d} \bar P \over {\rm d} r} {{\rm d} \bar S \over {\rm d} r}
> 0 ~.$$
The dynamics of linear modes can be described using the modal equations for the eigenfunctions: $$\left\{ {{\rm d}^2 \over {\rm d} t^2} + \bar \omega_{p,c}^2 \right\}
\Phi_{p,c}(t) = 0 ~,$$ where $\Phi_p(t)$ and $\Phi_c(t)$ are the eigenfunctions of the pressure and convective (baroclinic) modes, respectively. The form of these functions can be derived from Eqs. (34-41) in the shearless limit: $$\Phi_{p,c}(t) = (\bar \omega_{p,c}^2+c_s^2 \eta) p(t) - 2 \Omega_0
W(t) - c_s^2 k_P k_x s(t) ~.$$ All physical variables in our system ($p$, $u_x$, $u_y$, $s$) can be expressed by the two modal eigenfunctions and their first time derivatives ($\Phi_{p,c}$, $\Phi_{p,c}^\prime$). Hence, we can fully derive the perturbation field of a specific mode individually by setting the eigenfunction of the other mode equal to zero.
As we will see later, the Keplerian shear leads to the degeneracy of the convective buoyancy mode. In this case only the shear modified density-spiral wave mode eigenfunction can be employed in the analysis.
Perturbations in shear flow: mode coupling
------------------------------------------
It is well known that velocity shear introduces non-normality into the governing equations that significantly affects the dynamics of different perturbations. In this case we benefit from the shearing sheet transformation and seek the solutions in the form of the so-called Kelvin modes. These originate from the vortical solutions derived in seminal paper by Kelvin (1887). In fact, as it was argued lately (see e.g., Volponi and Yoshida 2002), the shearing sheet transformation leads to some sort of generalized modal approach. Shear modes arising in such description differ from linear modes with exponential time dependence in many respects. Primarily, phases of these continuous spectrum shear modes vary in time through shearing wavenumber; their amplitudes can be time dependent; and most importantly, they can couple in limited time intervals. On the other hand, shear modes can be well separated asymptotically, where analytic WKBJ solution for the each mode can be increasingly accurate. In the following, we will simply refer to these shearing sheet solution as “modes”.
The character of shear flow effects significantly depend on the value of velocity shear parameter. To estimate the time-scales of the processes we compare the characteristic frequencies of the linear modes $|\bar \omega_p|$, $|\bar \omega_c|$ and the velocity shear $|A|$. In order to speak about the modification of the linear mode by the velocity shear, the basic frequency of the mode should be higher than the one set by shear itself: $\omega^2 > A^2$. Otherwise the modal solution can not be used to calculate perturbation dynamics, since perturbations will obey the shear induced variations at shorter timescales.
In quasi-Keplerian differentially rotating discs with weak radial stratification: $$\bar \omega_p^2 \gg A^2 ~~~~ {\rm and} ~~~ \bar \omega_{c}^2 \ll
A^2 ~, {~~~ \rm when ~~~} {\beta_P \beta_S \over \gamma^2} \ll 1 ~.$$ In this case the convective mode diverges from its modal behavior and is strongly affected by the velocity shear: the thermal and kinematic parts obey shear driven dynamics individually. Therefore, we tentatively distinguish shear driven vorticity (W) and entropy (S) modes. On the contrary, the high frequency pressure mode is only modified by the action of the background shear. Hence, we assume the above described three mode (S, W and P) formalism as the framework for our further study.
For the description of the P mode in differential rotation we define the function: $$\Psi_p(t) = \omega_p^2(t) p(t) - 2\Omega_0 W(t) - c_s^2 k_P k_x(t)
s(t) ~,$$ where $$\omega_p^2(t) = c_s^2 k^2(t) - 4B \Omega_0 ~.$$ This can be considered as the generalization of the $\Phi_P(t)$ eigenfunction for the case of the shear flow, by accounting for the temporal variation of the radial wavenumber.
In order to analyze the mode coupling in the considered limit, we rewrite Eqs. (34-39) as follows: $$\left\{ {{\rm d}^2 \over {\rm d} t^2} + f_p {{\rm d} \over {\rm d}
t} + \omega_p^2 - \Delta \omega_p^2 \right\} \Psi_p = \chi_{pw} W +
\chi_{ps} s ~,$$ $$\left\{ {{\rm d} \over {\rm d} t} + f_s \right\} s = \chi_{s p 1}
{{\rm d} \Psi_p \over {\rm d} t} + \chi_{s p 2} \Psi_p + \chi_{s w}
W ~,$$ $${{\rm d} W \over {\rm d} t} = \chi_{ws} s ~,$$ where $f_p$ and $\Delta \omega_p^2$ describe the shear flow induced modification to the P-mode $$f_p = 4 A { k_x k_y \over k^2} - 2 {(\omega_p^2)^\prime \over
\omega_p^2 } ~,$$ $$\Delta \omega_p^2 = {(\omega_p^2)^{\prime \prime} \over \omega_p^2}
+ f_p {(\omega_p^2)^\prime \over \omega_p^2 } + 8AB{k_y^2 \over k^2}
~,$$ parameter $f_s$ describes the modification to the entropy mode $$f_s = c_s^2 \eta {k_x^2 (\omega_p^2)^\prime \over k^2 \omega_p^4} ~,$$ and $\chi$ parameters describe the coupling between the different modes: $$\chi_{pw} = 2 \Omega_0 \Delta \omega_p^2(t) + 4A {k_y^2 \over k^2}
\omega_p^2 ~,$$ $$\chi_{ps} = c_s^2 k_P k_x \left( \Delta \omega_p^2 + 4B {k_y \over
k_x} {(\omega_p^2)^\prime \over \omega_p^2} - 8AB {k_y^2 \over k^2}
\right) ~,$$ $$\chi_{s p 1} = {k_S k_x \over k^2 \omega_p^2 }~,$$ $$\chi_{s p 2} = -{k_S k_x \over k^2 \omega_p^2 } \left(
{(\omega_p^2)^\prime \over \omega_p^2} + 2B {k_y \over k_x} \right)
~,$$ $$\chi_{s w} = -{2 \Omega k_S k_x \over k^2 \omega_p^2} \left(
{(\omega_p^2)^\prime \over \omega_p^2} + 2B {k_y \over k_x} + {k_y
\omega_p^2 \over 2 \Omega k_x } \right) ~,$$ $$\chi_{w s} = -c_s^2 k_P k_y ~.$$ Here prime denotes temporal derivative.
Equations (50-52) describe the linear dynamics of modes and their coupling in the considered three mode model. In this limit, our interpretation is that the homogeneous parts of the equations describe the individual dynamics of modes, while the right hand side terms act as a source terms and describe the mode coupling. This tentative separation is already fruitful in a qualitative description of mode coupling.
Dynamics of the density-spiral wave mode in the differential rotation can be described by the homogeneous part of the Eq. (50). Homogeneous part of Eq. (51) describes the modifications to the entropy dynamics. Inhomogeneous parts of the Eqs. (50-52) reveal coupling terms between the three linear modes that originate due to the background velocity shear and radial stratification. We analyze the mode coupling dynamics numerically, but use the coupling $\chi$ coefficients for qualitative description.
The sketch illustration of the mode coupling in the above described three-mode approximation can be seen in Fig. \[coupling\]. The figure reveals a complex picture of the three mode coupling that originates by the combined action of velocity shear and radial stratification.
The temporal variation of the coupling coefficients during the swing of the perturbation SFHs from leading to trailing phases is shown in Fig. \[chi\]. The relative amplitudes of the $\chi_{pw}$ and $\chi_{ps}$ parameters reveal that potential vorticity is a somewhat more effective source of P mode perturbations when compared to the entropy mode. On the other hand, it seems that S mode excitation sources due to potential vorticity ($\chi_{sw}$) can be stronger when compared with the P-mode sources ($\chi_{sp1}$, $\chi_{sp2}$).
The effect of the stratification parameters on the mode coupling is somewhat more apparent. First, we may conclude that the excitation of the entropy mode, which depends on the parapeters $\chi_{sp1}$, $\chi_{sp2}$ and $\chi_{sw}$ is generally a stronger process for higher entropy stratification scales $k_S$ (see Eqs. 58-60). Second, we see that the generation of the potential vorticity depending on the $\chi_{ws}$ parameter proceeds more effectively at hight pressure stratification scale $k_P$. And third, we see profound asymmetry in the three-mode coupling: P-mode is not coupled with the W-mode [*directly*]{}.
A quantitative estimate of the mode excitation parameters can be done using a numerical analysis. In this case, the amplitudes of the generated W and S modes can be estimated through the values of potential vorticity or entropy outside the coupling area. In order to quantify the second order P mode dynamics we define its modal energy as follows: $$E_P(t) \equiv |\Psi_p(t)^\prime|^2 + \omega_p(t)^2 |\Psi_P(t)|^2 ~.$$ This quadratic form is a good approximation to the P mode energy in the areas where it obeys adiabatic dynamics: $k_x(t)/k_y \gg 1$.
The presented qualitative analysis suggests that perturbations of the density-spiral waves can generate entropy perturbations not only due to the flow viscosity (not included in our formalism), but also kinematically, due to the velocity shear induced mode coupling. The generated entropy perturbations should further excite potential vorticity through baroclinic coupling. Hence, it seems that in baroclinic flows, contrary to the barotropic case, P-mode perturbations are able to generate potential vorticity through a three-mode coupling mechanism: P $\to$ S $\to$ W. We believe that traces of the described mode coupling can be also seen in Klahr 2004, where the process has not been fully resolved due to the numerical filters used to remove higher frequency oscillations.
![ Mode coupling scheme. In the zero shear limit two second order modes P-mode and buoyancy mode with eigenfunctions $\Phi_p$ and $\Phi_c$ are uncoupled. In the shear flow, when the characteristic time of shearing is shorter then the buoyancy mode temporal variation scale ($A^2 > \bar \omega_c^2$), we use three mode formalism. In this limit we consider the coupling of the P, W, and S modes. $\chi$ parameters describe the strength of the coupling channel. Asymmetry of the mode coupling is revealed in the fact that compressible oscillations of the pressure mode are not able to directly generate potential vorticity, but still do so via interaction with S-mode and farther baroclinic ties with W-mode.[]{data-label="coupling"}](Coupling.eps){width="80mm"}
![The coupling $\chi$ parameters vs. the ratio of radial to azimuthal wavenumbers $k_x(t)/k_y$ when latter passes through zero value during the time interval $\Delta t = 4 \Omega_0^{-1}$. Here $k_y = H^{-1}$, $k_P = k_S = 0.5 H^{-1}$. []{data-label="chi"}](chi.eps){width="80mm"}
Numerical Results
=================
In order to study the mode coupling dynamics in more detail we employ numerical solutions of Eqs. (34-37). We impose initial conditions that correspond to the one of the three modes and use a standard Runge-Kutta scheme for numerical integration (MATLAB ode34 RK implementation). Perturbations corresponding to the individual modes at the initial point in time are derived in the Appendix A.
W-mode: direct coupling with S and P-modes
------------------------------------------
In this subsection we consider the dynamics of SFH when only the perturbations of potential vorticity are imposed initially. As it is known from previous studies (see Chagelishvili et al. 1997, Bodo et al. 2005) vorticity perturbations are able to excite acoustic modes nonadiabatically in the vicinity of the area where $k_x(t)=0$. Here we observe a similar, but more complex, behavior of mode coupling. The W-mode is able to generate P and S-modes simultaneously. Fig. \[SFH\_w1\] shows the evolution of the W-mode perturbations in a flow with growing baroclinic perturbations ($\eta>0$). The results show the excitation of both S and P-mode perturbations due to mode coupling that occurs in a short period of time in the vicinity of $t=10$. The following growth of the negative potential vorticity is due to the baroclinic coupling of entropy and potential vorticity perturbations.
Fig. \[SFH\_w2\] shows the evolution of potential vorticity SFH in flows with negative $\eta$. After the mode coupling and generation of P and S-modes, we observe a decrease of potential vorticity. This represents the well known fact that stable stratification (positive Richardson number) can play a role of “baroclinic viscosity” on the vorticity perturbations.
Numerical calculations show that the efficiency of the mode coupling generally decreases as we increase the azimuthal wavenumber $k_y$ corresponding to an increase of the density-spiral wave frequency: lower frequency waves couple more efficiently.
To test the effect of background stratification parameters on the mode coupling, we calculate the amplitude of the entropy and the energy of the P-mode perturbations generated in flows with different pressure and entropy stratification scales. The amplitudes are calculated after a $10 \Omega_0^{-1}$ time interval from the change in sign of the radial wave-number. In this case, modes are well isolated and the energy of the P mode can be well defined.
Fig. \[surf\_w\] shows the results of these calculations. It seems that the mode coupling efficiency is higher with stronger radial gradients. In particular, numerical results generally verify our qualitative results that the S-mode generation predominantly depends on the entropy stratification scale $k_S$. Therefore, P-mode excitation is stronger for higher values of $\eta$.
![Evolution of the W-mode SFH in the flow with $k_x(t)=-30H^{-1}$, $k_y=2H^{-1}$ and equilibrium with growing baroclinic perturbations $k_P=k_S=0.2H^{-1}$. Mode coupling occurs in the vicinity of $t=10 \Omega_0^{-1}$, where $k_x(t)=0$. Excitation of the P and S-modes are clearly seen in the panels for pressure ($P$) and entropy ($S$) perturbations. Perturbations of the potential vorticity start to grow due to the baroclinic coupling with entropy perturbations. []{data-label="SFH_w1"}](SFH_w1.eps){width="84mm"}
![Evolution of the W-mode SFH in the flow with $k_x(t)=-30H^{-1}$, $k_y=2H^{-1}$ and equilibrium with positive $\eta$: $k_P=-0.2H^{-1}$, $k_S=0.2H^{-1}$. Interestingly, SFH dynamics shows the decay of potential vorticity after the mode coupling and excitation of S- and W-modes at $t=10 \Omega_0^{-1}$. The latter fact is normally anticipated process in the flows that are baroclinically stable. []{data-label="SFH_w2"}](SFH_w2.eps){width="84mm"}
![Surface graph of the generated S and P-mode amplitudes at $ky=2H^{-1}$, $kx=-60H^{-1}$, and different values of $k_P$ and $k_S$. Initial perturbations are normalized to set E(0)=1. Excitation amplitudes of the entropy perturbations show stronger dependence of the $k_S$ (left panel), while both entropy and pressure scales are important (approximately $k_S k_P$ dependence) for the generation of P-modes (right panel). See electronic edition of the journal for color images.[]{data-label="surf_w"}](surf_w.eps){width="84mm"}
S-mode: direct coupling with W and P-modes
------------------------------------------
Fig. \[SFH\_s1\] shows the evolution of the S-mode SFH in a flow with growing baroclinic perturbations. Here we observe two shear flow phenomena: mode coupling and transient amplification. Entering the nonadiabatic area (around $t = 10$) the entropy SFH is able to generate the P-mode, while undergoing transient amplification itself. The transient growth of entropy is unsubstantial and the growth rate decreases with the growth of $k_y$. The W-mode is instead constantly coupled to entropy perturbations through baroclinic forces, although higher entropy perturbations at later times give an higher rate of growth of potential vorticity. The total energy of perturbations is however dominated, at the end, by the P mode.
Fig. \[surf\_s\] shows the dependence of the W and P-mode generation on the pressure and entropy stratification scales. As expected from qualitative estimates, P-mode excitation depends almost solely on the pressure stratification scale $k_P$, while the generation of potential vorticity generally grows with $\eta$.
![Evolution of the S-mode SFH in the flow with $k_x(t)=-30H^{-1}$, $k_y=2H^{-1}$ and equilibrium with growing baroclinic perturbations $k_P=k_S=0.2H^{-1}$. Perturbations of the potential vorticity are coupled grow from the beginning due to the baroclinic coupling with entropy perturbations. Excitation of the P-mode is clearly seen in the panel for pressure ($P$), while the panel for entropy perturbations ($S$) shows swing amplification in the nonadiabatic area around $k_x(t)=0$. Change of the amplitude of the entropy SFH affects the growth factor of potential vorticity SFH.[]{data-label="SFH_s1"}](SFH_s1.eps){width="84mm"}
![Surface graph of the generated W and P-mode amplitudes at $ky=2H^{-1}$, $kx=-60H^{-1}$, and different values of $k_P$ and $k_S$. Initial perturbations are normalized to set E(0)=1. Excitation amplitudes of the entropy perturbations show predominant dependence on the $k_S$ (left panel), while only pressure stratification scale $k_P$ is important for the generation of P-modes (right panel). See electronic edition of the journal for color images.[]{data-label="surf_s"}](surf_s.eps){width="84mm"}
P-mode: direct coupling with S-mode and indirect coupling with W-mode
---------------------------------------------------------------------
Fig. \[SFH\_p1\] shows the evolution of an initially imposed P-mode SFH in a flow with growing baroclinic perturbations. The [*oscillating*]{} behavior of the entropy perturbation for $t < 10$ is given by the P-mode. This oscillating component has a zero mean value when averaged over time-scales longer than the wave period. The existence of the [*aperiodic*]{} S-mode is instead characterized by a nonzero mean value. When the azimuthal wavenumber $k_y(t)$ changes sign at $t = 10$, we can observe the appearance of a nonzero mean value (marked on the plot by the horizontal dashed line), indicating that the high frequency oscillations of the P-mode are able to generate the aperiodic perturbations of the S-mode. The aperiodic part of the entropy perturbation is than able to generate potential vorticity perturbations. However, as we see from Eq. (54) and Fig. \[coupling\], there is no direct coupling between P and W-modes. Therefore, the P-mode generates the S-mode by shear flow induced mode conversion, while the W-mode is further generated because of its baroclinic ties with the entropy SFH. We describe this situation as the three-mode coupling or in other words, indirect coupling of the P to the W-mode. Note, that although the S and W-mode generation is apparent from the dynamics of entropy and potential vorticity SFH, energetically it plays a minor role as compared to the compressible energy carried by the P-mode.
Fig \[SFH\_p2\]. shows that P-mode generates potential vorticity with a positive sign. However, the sign of the generated potential vorticity depends on the initial phase of the P-mode. Hence, our numerical results show generation of the W-mode with both positive and negative signs.
It is interesting also to look at the P-mode dynamics in flows stable to baroclinic perturbations (see Fig. \[SFH\_p2\]). The initially imposed P-mode is able to generate the S-mode and consequently the W-mode, that gives a growth of the potential vorticity with time. Apart from the intrinsic limitations (the dependence of the sign of the generated potential vorticity on the initial phase of the P-mode and the low efficiency of the W-mode generation), this process demonstrates the fact that potential vorticity can be actually generated in flows with positive radial buoyancy ($\eta<0$) and Richardson number.
Fig. \[surf\_p\] shows the dependence of the S and W-mode generation on the pressure and entropy stratification scales. In good agreement with qualitative estimates, the S-mode excitation depends strongly on the entropy stratification scale $k_S$, while the generation of the potential vorticity generally grows with $\eta$.
![Evolution of the P-mode SFH in the flow with $k_x(t)=-30H^{-1}$, $k_y=2H^{-1}$ and equilibrium with growing baroclinic perturbations $k_P=k_S=0.2H^{-1}$. Mode coupling occurs in the vicinity of $t=10\Omega_0^{-1}$, where W and S-modes are excited. The amplitude of the generated aperiodic contribution to the entropy perturbation is marked by the red doted line. Farther, this component leads to the baroclinic production of potential vorticity with negative sign.[]{data-label="SFH_p1"}](SFH_p1.eps){width="84mm"}
![Same as in previous figure but for $k_P=-0.2H^{-1}$ and $
k_S=0.2H^{-1}$. Perturbations are stable to baroclinic forces. However, production of the potential vorticity with positive sign is still observed.[]{data-label="SFH_p2"}](SFH_p2.eps){width="84mm"}
![Surface graph of the generated S and W-mode amplitudes at $ky=2H^{-1}$, $kx=-60H^{-1}$, and different values of $k_P$ and $k_S$. Initial perturbations are normalized to set E(0)=1. Excitation amplitudes of the entropy perturbations mainly depend on the $k_S$ (left panel), while both pressure and entropy stratification scales are important for the generation of W-mode perturbations (right panel). See electronic edition of the journal for color images.[]{data-label="surf_p"}](surf_p.eps){width="84mm"}
Conclusion and Discussion
=========================
We have studied the dynamics of linear perturbations in a 2D, radially stratified, compressible, differentially rotating flow with different radial density, pressure and entropy gradients. We employed global radial scaling of linear perturbations and removed the algebraic modulation due to the background stratification. We derived a local dispersion equation for nonaxisymmetric perturbations and the corresponding eigenfunctions in the zero shear limit. We show that the local stability of baroclinic perturbations in the barotropic equilibrium state is defined by the Schwarzschild-Ledoux criterion.
We study the shear flow induced linear coupling and the related possibility of the energy transfer between the different modes of perturbations using qualitative and a more detailed numerical analysis. We employ a three-mode formalism and describe the behavior of S W and P-modes under the action of the baroclinic and velocity shear forces in local approximation.
We find that the system exhibits an asymmetric coupling pattern with five energy exchange channels between three different modes. The W-mode is coupled to S and P-modes: perturbations of the potential vorticity are able to excite entropy and compressible modes. The amplitude of the generated S-mode grows with the increase of entropy stratification scale of the background ($k_S$) while the amplitude of the generated P-mode perturbations grows with the increase of background baroclinic index ($\eta$). The S-mode is coupled to the W and P-modes: the amplitude of the generated P-mode perturbations grows with increase of the background pressure stratification scale ($k_P$), while the amplitude of the W-mode grows with the increase of baroclinic index. The P-mode is coupled to the S-mode: the amplitude of generated entropy perturbations grows with the increase of the background entropy stratification scale. On the other hand, there is no direct energy exchange channel from P to W mode and, therefore, no direct conversion is possible. Our results, however, show that the P-mode is still able to generate W-mode through indirect three mode P-S-W coupling scheme. This linear inviscid mechanism indicates that compressible perturbations are able to generate potential vorticity via aperiodic entropy perturbations.
The dynamics of radially stratified discs have been already studied by both, linear shearing sheet formalism and direct numerical simulations. However, previous studies focus on the baroclinic stability and vortex production by entropy perturbations, neglecting the coupling with higher frequency density waves.
The most vivid signature of density wave excitation in radially stratified disc flows can be seen in Klahr (2004). The numerical results presented on the linear dynamics of perturbation SFH show high frequency oscillations after the radial wavenumber changes sign. However, focusing on the energy dynamics, the author filters out high frequency oscillations from the analysis.
The purpose of numerical simulations by Johnson and Gammie (2006) was the investigation of the velocity shear effects on the radial convective stability and the possibility of the development of baroclinic instability. Therefore, no significant amount of compressible perturbations is present initially, and it is hard to judge if high frequency oscillations appear later in simulations. Petersen et al. (2007a), (2007b) employed the anelastic approximation that does not resolve the coupling of potential vorticity and entropy with density waves. Moreover, if produced, high frequency density waves soon develop into spiral shocks (see e.g., Bodo et al 2007). The anelastic gas approximation does intentionally neglect this complication and simplifies the description down to low frequency dynamics.
Numerical simulations of hydrodynamic turbulence in unstratified disc flows showed that the dominant part of turbulent energy is accumulated into the high frequency compressional waves (see, e.g., Shen et al. 2006). On the other hand, it is vortices that are thought to play a key role in hydrodynamic turbulence in accretion discs, as well as planet formation in protoplanetary disc dynamics. Therefore, any link and possible energy exchange between high frequency compressible oscillations and aperiodic vortices can be an important factor in the above described astrophysical situations.
Based on the present findings we speculate that density waves can participate in the process of the development of regular vortical structures in discs with negative radial entropy gradients. Numerical simulations have shown that thermal (entropy) perturbations can generate vortices in baroclinic disc flows (see e.g., Petersen et al. 2007a, 2007b). Hence, vortex development through this mechanism depends on the existence of initial regular entropy perturbations, i.e., thermal plumes, in differentially rotating baroclinic disc flows.
It seems that compressional waves with linear amplitudes can heat the flow through two different channels: viscous dissipation and shear flow induced mode conversion. However, there is a strict difference between the entropy production by the kinematic shear mechanism and viscous dissipation. In the latter case, compressional waves first need to be tightly stretched down to the dissipation length-scales by the background differential rotation to be subject of viscous dumping. As a result, the entropy produced by viscous dissipation of compressional waves takes a shape of narrow stretched lines. This thermal perturbations can baroclinically produce potential vorticity of similar configuration. However, this is clearly not an optimal form of potential vorticity that can lead to the development of the long-lived vortical structures. On the contrary, entropy perturbations produced through the mode conversion channel can have a form of a localized thermal plumes. These can be very similar to those used in numerical simulations by Petersen et al. (2007a), (2007b). In this case compressional waves can eventually lead to the development of persistent vortical structures of different polarity. Hence, high frequency oscillations of the P mode can participate in the generation of anticyclonic vortices that further accelerate dust trapping and planetesimal formation in protoplanetary discs with equilibrium entropy decreasing radially outwards.
Using the local linear approximation we have shown the possibility of the potential vorticity generation in flows with both, positive and negative radial entropy gradients (Richardson numbers). In fact, the standard alpha description of the accretion discs implies *positive* radial stratification of entropy and hence, weak baroclinic decay of existing vortices. In this case there will be a competition between the “baroclinic viscosity” and potential vorticity generation due to mode conversion. Hence, it is not strictly overruled that a significant amount of compressional perturbations can lead to the development of anticyclonic vortices even in flows with positive entropy gradients. In this case, radial stratification opens an additional degree of freedom for velocity shear induced mode conversion to operate. Although, the viability of this scenario needs further investigation.
This paper presents the results obtained within the linear shearing sheet approximation. At nonlinear amplitudes, the P mode leads to the development of shock waves. These shocks induce local heating in the flow. Therefore, a realistic picture of entropy production and vortex development in radially stratified discs with significant amount of compressible perturbations needs to be analyzed by direct numerical simulations.
Acknowledgments {#acknowledgments .unnumbered}
===============
A.G.T. was supported by GNSF/PRES-07/153. A.G.T. would like to acknowledge the hospitality of Osservatorio Astronomico di Torino. This work is supported in part by ISTC grant G-1217.
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Initial conditions
==================
Here we present the approximations used to derive the analytic form of the initial conditions corresponding to individual modes in radially stratified shear flows. These conditions are used to construct the initial values of perturbations in the numerical integration of the ODEs governing the linear dynamics of perturbations in these flows. We employ different methods for high and low frequency modes.
P-mode
------
P-mode perturbations are intrinsically high frequency and well separated from low frequency modes everywhere outside the coupling region $k_x/k_y < 1$. In order to construct P-mode perturbations we use convective eigenfunction derived in the shearless limit and account for shear flow effects only in the adiabatic limit: $$\Psi_{c}(t) = (\omega_{c}^2(t) + c_s^2 \eta) P(t) - 2 \Omega_0 W(t)
- c_s^2 k_P k_x(t) s(t) ~,$$ where $$\omega_c^2(t) = -{c_s^2 \eta k_y^2 \over c_s^2 k^2(t) - 4B\Omega_0}
~.$$ Although this form of the eigenfunction is not valid function for describing W and S modes individually in a sheared medium, it has proved to be a good tool for excluding both modes from the initial spectrum: $$\Psi_{c}(0) = 0 ~.$$ Assuming that we are looking for P-mode perturbations with wave-numbers satisfying the condition $ k_x(0)/k_y \gg 1 $ we may use the zero potential vorticity condition: $$W(0) = 0 ~.$$ Hence, Eqs. (A3,A4) yield the full set of initial conditions for the high frequency P-mode SFH of perturbations: $$p(0) = P_0 ~, ~~~~~ u_x(0) = U_0 ~,$$ $$u_y(0) = {1 \over k_x(0)} \left( k_y U_0 + 2BP_0 \right) ~,$$ $$s(0) = {\omega_c^2(0)+c_s^2 \eta \over c_s^2 k_p k_x(0) } P_0 ~,$$ where $P_0$ and $U_0$ are free parameters corresponding to the two P-modes in the system. Specific values of these two parameters define whether the potential or kinetic part of the wave harmonic is present initially.
Low frequency modes
-------------------
In order to derive the initial conditions for the S and W modes individually we employ the second order equation for radial velocity perturbation that can be derived from Eqs. (34-37): $$\left\{ {{\rm d}^2 \over {\rm d} t^2} + c_s^2 k^2 - 4B\Omega_0 -
c_s^2 \eta \right\} u_x = -c_s^2 k_y W + 4Ac_s^2 k_y p ~.$$ $$\left\{ {{\rm d}^2 \over {\rm d} t^2} + c_s^2 k^2 - 4B\Omega_0
\right\} u_y = c_s^2 k_x(t) W + 2 B c_s^2 k_P s ~,$$ For low frequency perturbations $${{\rm d}^2 \over {\rm d} t^2} \left( \begin{array}{c} u_x \\ u_y
\end{array} \right) \sim \omega^2_c \left( \begin{array}{c} u_x \\ u_y
\end{array} \right) ~.$$ Assuming that $\omega_c^2(0) \ll c_s^2 k^2(0)$ and neglecting the corresponding terms in Eqs. (A6-A7) leads to the following algebraic system: $$\left[c_s^2 k^2 - 4B\Omega_0 \right] u_x = -c_s^2 k_y W + 4Ac_s^2
k_y p ~.$$ $$\left[c_s^2 k^2 - 4B\Omega_0 \right] u_y = c_s^2 k_x(t) W + 2 B
c_s^2 k_P s ~.$$ Hence, we can derive the initial conditions for the low frequency modes as follows: $$p(0) = {B \over 2A c_s^2 k_y^2 + B\omega_p^2(0)} \left( 2 \Omega_0
W_0 + c_s^2 k_p k_x(0) S_0 \right) ~,$$ $$u_x(0) = {1 \over \omega_p^2(0)} \left( -c_s^2 k_y W_0 + 4A c_s^2
k_y p(0) \right) ~,$$ $$u_y(0) = {1 \over \omega_p^2(0)} \left( c_s^2 k_x(0) W_0 + 2B c_s^2
k_p S_0 \right) ~,$$ where $$\omega_p^2(0) = c_s^2 (k_x^2(0) + k_y^2) - 4B\Omega_0 ~.$$ Eqs. (A11-A14) give the initial values of perturbation SFHs for S-mode when $$W_0 = 0 ~,~~~ S_0 \not= 0 ~,$$ and W-mode when $$W_0 \not= 0 ~,~~~ S_0 = 0 ~.$$
| ArXiv |
---
abstract: 'We provide a quantitative version of the isoperimetric inequality for the fundamental tone of a biharmonic Neumann problem. Such an inequality has been recently established by Chasman adapting Weinberger’s argument for the corresponding second order problem. Following a scheme introduced by Brasco and Pratelli for the second order case, we prove that a similar quantitative inequality holds also for the biharmonic operator. We also prove the sharpness of both such an inequality and the corresponding one for the biharmonic Steklov problem.'
author:
- 'D. Buoso'
- 'L.M. Chasman'
- 'L. Provenzano'
bibliography:
- 'bibliography.bib'
title: On the stability of some isoperimetric inequalities for the fundamental tones of free plates
---
Introduction
============
The stability of isoperimetric inequalities is an important question that has gained significant interest in recent decades. For example, the celebrated Faber-Krahn inequality for the smallest eigenvalue of the Dirichlet Laplacian, $$\lambda_1(\Omega)\ge\lambda_1(\Omega^*),$$ can be improved in the following quantitative form: $$\lambda_1(\Omega)\ge\lambda_1(\Omega^*)(1+C\mathcal{A}(\Omega)^2),
\label{quantfk}$$ for some constant $C>0$. Here $\Omega\subset\mathbb{R}^N$ is a bounded open set, $N\geq2$, $\Omega^*$ is a ball such that $|\Omega|=|\Omega^*|$, and $\mathcal A(\Omega)$ is the so-called Fraenkel asymmetry of the domain $\Omega$ (see for the definition). Quantitative versions of the type have also been established for other isoperimetric inequalities involving eigenvalues of the Laplace operator, see, e.g., [@brascosteklov; @brasco2015; @brascopratelli].
Fewer isoperimetric inequalities have been established for eigenvalues of the biharmonic operator, namely for the first nontrivial eigenvalue of the Dirichlet (“clamped plate”) problem [@ashbaugh; @nadirashvili], of the Neumann (“free plate”) problem [@chasmanpreprint; @chasman], and of the Steklov problem introduced in [@buosoprovenzano] (see also [@buosoprovenzano0]). An isoperimetric inequality is still missing for another Steklov problem introduced in [@kuttler68], the conjectured optimizer being the regular pentagon (see, e.g., [@antunesgazzola; @bucurgazzola11] and the references therein).
Among these inequalities, the first one that has been given in quantitative form is the inequality for Steklov problem in [@buosoprovenzano], namely $$\lambda_2(\Omega)\le\lambda_2(\Omega^*)(1-C\mathcal{A}(\Omega)^2),
\label{quantitative_bp}$$ where $\lambda_2(\Omega)$ is the first nontrivial eigenvalue of the biharmonic Steklov problem $$\label{SteklovPDE}
\begin{cases}\Delta^2u-\tau\Delta u=0 &\text{in $\Omega$,}\\
\frac{\partial^2 u}{\partial \nu^2}= 0 &\text{on $\partial\Omega$,}\\
\tau\frac{\partial u}{\partial \nu}
-{\rm div}_{\partial\Omega}\Big(D^2u\cdot \nu\Big)-\frac{\partial\Delta u}{\partial \nu} = \lambda u &\text{on $\partial\Omega$,}
\end{cases}$$ where $\tau$ is a strictly positive constant.
In this paper we provide a quantitative form for the isoperimetric inequality for the first non-trivial eigenvalue of the following biharmonic Neumann problem: $$\label{NeumannPDE}
\begin{cases}\Delta^2u-\tau\Delta u=\lambda u &\text{in $\Omega$,}\\
\frac{\partial^2 u}{\partial \nu^2}= 0 &\text{on $\partial\Omega$,}\\
\tau\frac{\partial u}{\partial \nu}
-{\rm div}_{\partial\Omega}\Big(D^2u\cdot \nu\Big)-\frac{\partial\Delta u}{\partial \nu} = 0 &\text{on $\partial\Omega$.}
\end{cases}$$ We recall that for $N=2$, problem describes the transverse vibrations of an unconstrained thin elastic plate with shape $\Omega\subset \mathbb{R}^2$ when at rest. The constant $\tau$ represents the ratio of lateral tension to lateral rigidity and is taken to be non-negative.
When $\tau>0$ and $\Omega\subset\mathbb{R}^N$ is a smooth connected bounded open set, it is known that the spectrum of the Neumann biharmonic operator $\Delta^2-\tau\Delta$ consists entirely of non-negative eigenvalues of finite multiplicity, repeated according to their multiplicity: $$0=\lambda_1(\Omega)<\lambda_2(\Omega)\leq\cdots\leq\lambda_j(\Omega)\leq\cdots.$$ Note that since constant functions satisfy problem with eigenvalue $\lambda=0$, the first positive eigenvalue is $\lambda_2$, which is usually called the “fundamental tone” of the plate. In [@chasman], the author proved that $$\label{iso_neumann}
\lambda_2(\Omega)\leq \lambda_2(\Omega^*)$$ with equality if and only if $\Omega=\Omega^*$. The proof of inequality is based on Weinberger’s argument for the Neumann Laplacian, taking suitable extensions of the eigenfunctions of the ball as trial functions (see [@weinberger]). In [@brascopratelli], the authors carry out a more careful analysis of such an argument, improving Weinberger’s inequality to a quantitative form. In a similar way, we start from the proof of and improve the result to the quantitative inequality by means of this finer analysis.
The question of sharpness is another important issue that has to be addressed when dealing with quantitative isoperimetric inequalities. More precisely, given an inequality of the form $$%\label{quantitative_general}
\lambda_2(\Omega)\leq\lambda_2(\Omega^*)\left(1-\Phi({\rm dist}(\Omega,\mathcal B))\right),$$ where $\Phi$ is some modulus of continuity, ${\rm dist}(\cdot,\cdot)$ is a suitable distance between open sets and $\mathcal B$ is the family of all balls in $\mathbb R^N$, we say that it is sharp if there exists a family $\lbrace\Omega_{\varepsilon}\rbrace_{\varepsilon\in(0,\varepsilon_0)}$ such that ${\rm dist}(\Omega_{\varepsilon},\mathcal B)\rightarrow 0$, $\lambda_2(\Omega_{\varepsilon})\rightarrow\lambda_2(\Omega^*)$ as $\varepsilon\rightarrow 0$, and there exists contants $c_1,c_2>0$ which do not depend on $\varepsilon>0$ and $\Omega^*$ such that [$$c_1\Phi({\rm dist}(\Omega_{\varepsilon},\mathcal B))\leq 1-\frac{\lambda_2(\Omega_{\varepsilon})}{\lambda_2(\Omega^*)}\leq c_2\Phi({\rm dist}(\Omega_{\varepsilon},\mathcal B)),$$]{} as $\varepsilon\rightarrow 0$. Note that, in our case, the distance function is given by the Fraenkel asymmetry ${\rm dist}(\Omega,\mathcal B)=\mathcal A(\Omega)$ while the modulus of continuity is $\Phi(t)=Kt^2$, for some $K>0$. By means of the construction introduced in [@brascosteklov; @brascopratelli], we prove in Section \[sharpness\_neumann\] that the quantitative Neumann inequality is sharp.
It is worth noting that in the Neumann Laplacian case in [@brascopratelli], the authors try, as a first guess, to consider ellipsoids as the family $\lbrace\Omega_{\varepsilon}\rbrace_{\varepsilon\in(0,\varepsilon_0)}$, with the ball $\Omega_0$ being the maximizer. Unfortunately, this is not a good family to prove sharpness; this can be explained observing that different directions of perturbation behave in a different way with respect to the fundamental tone. In particular, some directions are not “good enough” to see the sharpness (cf. [@brascopratelli Remark 5.2]). This phenomenon can be observed in our case as well: therefore we need to restrict our analysis by excluding some directions. See (\[perturbation\]) and Remark \[directions\].
The Steklov problem is of particular interest despite its recent introduction, since in [@buosoprovenzano] the authors show that it has a very strict relationship with the Neumann problem . Using a mass perturbation argument, they prove that the Steklov problem can in fact be viewed as a limiting Neumann problem where the mass is distributed only on the boundary. Note that this construction was already performed in [@lambertiprozisaac] for the Laplace operator, obtaining similar results (see also [@dallarivaproz; @lambertiproz] for the computation of the topological derivative). Moreover, this justifies the fact of thinking of Steklov problems in terms of vibrating objects (plates or membranes) where the mass lies only on the boundary (see [@steklov]). The authors also prove the quantitative inequality by adapting an argument due to Brock (see [@brock]) for the Steklov Laplacian to the biharmonic case in the refined version of [@brascosteklov]. However, they do not discuss its sharpness. The similarity of the variational characterization of Neumann and Steklov eigenvalues allows us to prove that inequality is sharp by an easy adaptation of the arguments used in the Neumann case.
The paper is organized as follows. In Section \[preliminaries\], we give some preliminary results and introduce the notation. Section \[proof\_neumann\_quantitative\] is devoted to the Neumann quantitative isoperimetric inequality , the sharpness of which we prove in Section \[sharpness\_neumann\]. Finally, in Section \[sharpness\_steklov\] we prove that the Steklov inequality is sharp.
Preliminaries and notation {#preliminaries}
==========================
We introduce here the notation used throughout the paper and recall some fundamental results proved in [@chasman].
Let $B$ be the unit ball in $\mathbb{R}^N$ centered at the origin and $\omega_N$ be the Lebesgue measure $|B|$ of $B$.
We denote by $j_1$ and $i_1$ the ultraspherical and modified ultraspherical Bessel functions of the first kind and order $1$ respectively. They can be expressed in terms of standard Bessel and modified Bessel functions of the first kind $J_{\nu}, I_{\nu}$ as follows: $$j_1(z)=z^{1-N/2}J_{N/2}(z),\qquad i_1(z)=z^{1-N/2}I_{N/2}(z).$$ For more information on Bessel and modified Bessel functions, see, e.g., [@abram §9].
We will define trial functions in terms of the eigenfunctions corresponding to $\lambda_2(B)$ of the Neumann problem. For a fixed $\tau>0$, we take positive constants $a,b$ satisfying $a^2b^2=\lambda_2(B)$ and $b^2-a^2=\tau$. We set $$R(r)=j_1(ar)+\gamma i_1(br),\qquad\text{where}\qquad \gamma=-\frac{a^2 j_1''(a)}{b^2 i_1''(b)}.$$ We then define the function $\rho:[0,+\infty)\to[0,+\infty)$ as $$%\label{rho}
\rho(r)=\begin{cases}
R(r),&r\in[0,1)\\
R(1)+(r-1)R'(1),&r\in[1,+\infty).
\end{cases}$$
Let $u_k:\mathbb{R}^N\to\mathbb{R}$ be defined by $$\label{uk}
u_k(x):=\rho(|x|)\frac{x_k}{|x|},$$ for $k=1,\dots,N$. The functions ${u_k}_{|_{B}}$ are in fact the eigenfunctions associated with the eigenvalue $\lambda_2(B)$ of the Neumann problem on the unit ball $B$. Recall that $\lambda_2(B)$ has multiplicity $N$ (see [@chasman11 Theorem 3]). Moreover, we have (see [@chasman p. 437]) $$\begin{aligned}
%\label{relation_1}
\sum_{k=1}^N |u_k|^2&=\rho(|x|)^2,\\
%\label{relation_2}
\sum_{k=1}^N|D u_k|^2&=\frac{N-1}{|x|^2}\rho(|x|)^2+(\rho'(|x|))^2,\\
%\label{relation_3}
\sum_{k=1}^N|D^2 u_k|^2&=(\rho''(|x|))^2+\frac{3(N-1)}{|x|^4}(\rho(|x|)-|x|\rho'(|x|))^2.\end{aligned}$$ We denote by $N[\rho]$ the quantity $$N[\rho]:=\sum_{k=1}^N |D^2u_k|^2+\tau|D u_k|^2.$$
We recall some properties enjoyed by the functions $\rho$ and $N[\rho]$ which were proved in [@chasman].
\[pro\] The function $\rho$ satisfies the following properties.
i) $\rho''(r)\leq 0$ for all $r\geq 0$, therefore $\rho'$ is non-increasing.
ii) $\rho(r)-r\rho'(r)\geq 0$, equality holding only for $r=0$.
iii) The function $\rho(r)^2$ is strictly increasing.
iv) The function ${\rho(r)^2}/{r^2}$ is decreasing.
v) The function ${3(\rho(r)-r\rho'(r))^2}/{r^4}+\tau{\rho^2(r)}/{r^2}$ is decreasing.
vi) $N[\rho(r_1)]>N[\rho(r_2)]$ for any $r_1\in [0,1)$, $r_2\in [1,+\infty)$.
vii) For all $r\geq 0$ we have $$N[\rho(r)]=(\rho''(r))^2+\frac{3(N-1)(\rho(r)-r\rho'(r))^2}{r^4}+\tau(N-1)\frac{\rho^2(r)}{r^2}+\tau(\rho'(r))^2.$$
viii) For all $r\geq 1$, $N[\rho(r)]$ is decreasing.
To conclude this section, let us recall the definition of the Fraenkel asymmetry $\mathcal A(\Omega)$ of a set $\Omega\subset\mathbb{R}^N$: $$\label{fra}
\mathcal A(\Omega):=\inf\left\{\frac{|\Omega\triangle {B}|}{|\Omega|}: {B}\text{\ is a ball such that}\ |{B}|=|\Omega|\right\}.$$
Quantitative isoperimetric inequality for the Neumann problem {#proof_neumann_quantitative}
=============================================================
In this section we state and prove the [quantitative isoperimetric inequality for the fundamental tone of the Neumann problem ]{}.
\[NeumannQI\] For every bounded domain $\Omega$ in $\mathbb{R}^N$ of class $C^1$ the following estimate holds $$\label{quantitative_neumann}
\lambda_2(\Omega)\leq\lambda_2(\Omega^*)\left(1-\eta_{N,\tau,|\Omega|}\mathcal A(\Omega)^2\right),$$ where $\eta_{N,\tau,|\Omega|}>0$, $\Omega^*$ is a ball such that $|\Omega^*|=|\Omega|$, and $\lambda_2(\Omega)$, $\lambda_2(\Omega^*)$ are the first positive eigenvalues of problem on $\Omega$, $\Omega^*$ respectively.
Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ of class $C^1$ with the same measure as the unit ball $B$. We recall the variational characterization of the second eigenvalue $\lambda_2(\Omega)$ of on $\Omega$: $$\label{ray}
\lambda_2(\Omega)=\inf_{\substack{0\ne u\in H^2(\Omega)\\ \int_{\Omega}u dx=0}}\frac{\int_{\Omega}|D^2u|^2+\tau|Du|^2dx}{\int_{\Omega}u^2dx}.$$ Let $u_k(x)$, for $k=1,\dots,N$, be the eigenfunctions corresponding to $\lambda_2(B)$ defined in . Clearly ${u_k}_{|_{\Omega}}\in H^2(\Omega)$ by construction. It is possible to choose the origin of the coordinate axes in $\mathbb{R}^N$ in such a way that $\int_{\Omega}u_k dx=0$ for all $k=1,\dots,N$. With this choice, the functions $u_k$ are suitable trial functions for the Rayleigh quotient . Once we have fixed the origin, let $$\alpha:=\frac{|\Omega\triangle B|}{|\Omega|}.$$ By definition of Fraenkel asymmetry, we have $$\label{A}
\mathcal A(\Omega)\leq\alpha\leq 2.$$ From the variational characterization , it follows that for each $k=1,\dots,N$, $$\lambda_2(\Omega)\leq\frac{\int_{\Omega}|D^2u_k|^2+\tau|D u_k|^2dx}{\int_{\Omega}u_k^2 dx}.$$ We multiply both sides by $\int_{\Omega}u_k^2 dx$ and sum over $k=1,\dots,N$, obtaining $$\label{n1}
\lambda_2(\Omega)\leq\frac{\int_{\Omega}N[\rho]dx}{\int_{\Omega}\rho^2 dx}.$$ The same procedure for $\lambda_2(B)$ clearly yields $$\label{b1}
\lambda_2(B)=\frac{\int_{B}N[\rho]dx}{\int_{B}\rho^2 dx}.$$ From and , it follows that $$\label{ineq1}
\lambda_2(B)\int_B \rho^2 dx-\lambda_2(\Omega)\int_{\Omega}\rho^2 dx\geq\int_B N[\rho]dx-\int_{\Omega}N[\rho]dx\geq 0,$$ where the last inequality follows from Lemma \[pro\], [*iv)*]{} and [@chasman Lemma 14].
Now we consider the two balls $B_1$ and $B_2$ centered at the origin with radii $r_1,r_2$ taken such that $|\Omega\cap B|=|B_1|=\omega_N r_1^N$ and $|\Omega\setminus B|=|B_2\setminus B|=\omega_N(r_2^N-1)$. Then $|B_2|=\omega_N r_2^N$, and by construction $$\label{aster}
1-r_1^N=\frac{\alpha}{2}=r_2^N-1.$$ This is due to the fact that $|\Omega|+|B|=|\Omega\triangle B|+2 |\Omega\cap B|$, and then $1-r_1^N=\alpha/2$. Similarly, $|\Omega\setminus B|+|\Omega\cap B|=|\Omega|$, hence $r_1^N=2-r_2^N$, and then $r_2^N-1=\alpha/2$.
Now we observe, again by Lemma \[pro\], [*vi)*]{} and [*viii)*]{}, that $$\int_{\Omega}N[\rho]dx\leq\int_{B_1}N[\rho]dx+\int_{B_2\setminus B}N[\rho]dx.$$ From this and , we obtain $$\begin{aligned}
\label{ineq2}
\lambda_2(B)\int_B\rho^2 dx-\lambda_2(\Omega)\int_{\Omega}\rho^2 dx&\geq\int_B N[\rho]dx-\int_{\Omega}N[\rho]dx\\
&\geq\int_{B\setminus B_1}N[\rho]dx-\int_{B_2\setminus B}N[\rho]dx.\nonumber\end{aligned}$$ Since the function $\rho(r)^2$ is strictly increasing by Lemma \[pro\], [*iii)*]{}, we have $$%\label{den}
\int_{\Omega}\rho^2 dx\geq\int_B\rho^2 dx=N\omega_N\int_0^1\rho^2(r)r^{N-1}dr=:C^{(1)}_{N,\tau},$$ hence, $$\begin{aligned}
\label{passoA}
\lambda_2(B)&\int_B\rho^2 dx-\lambda_2(\Omega)\int_{\Omega}\rho^2 dx\\
&\leq\left(\lambda_2(B)-\lambda_2(\Omega)\right)\int_B\rho^2 dx +\lambda_2(\Omega)\left(\int_B\rho^2 dx-\int_{\Omega}\rho^2 dx\right)\nonumber\\
&\leq C^{(1)}_{N,\tau}\left(\lambda_2(B)-\lambda_2(\Omega)\right).\nonumber\end{aligned}$$
Now we consider the right-hand side of . We write $N[\rho]$ more explicitly in terms of $\rho$, obtaining: $$\begin{aligned}
\label{passoB1}
\int_{B\setminus B_1}&N[\rho]dx=N\omega_N\int_{r_1}^1\Big((\rho''(r))^2+\frac{3(N-1)(\rho(r)-r\rho'(r))^2}{r^4}\\
&\qquad\qquad\qquad\qquad+\tau(\rho'(r))^2+\frac{\tau (N-1)}{r^2}\rho(r)^2\Big)r^{N-1}dr\nonumber\\
&\geq N\omega_N\int_{r_1}^1\left(\frac{3(N-1)(\rho(r)-r\rho'(r))^2}{r^4}+\tau(\rho'(r))^2+\frac{\tau (N-1)}{r^2}\rho(r)^2\right)r^{N-1}dr\nonumber\\
&\geq \omega_N\left(3(N-1)(R(1)-R'(1))^2+\tau R'(1)^2+\tau (N-1)R(1)^2\right)(1-r_1^N),\nonumber\end{aligned}$$ where in the last inequality, we used the fact that $N[\rho]-(\rho'')^2$ is non-increasing in $r$ (see Lemma \[pro\], [*i)*]{} and [*v)*]{}). Moreover, $$\begin{aligned}
\label{passoB2}
\int_{B_2\setminus B}&N[\rho]dx\\
&=N\omega_N\int_1^{r_2}\left(\frac{3(N-1)}{r^4}(R(1)-R'(1))^2+\tau R'(1)^2\right.\nonumber\\
&\qquad\qquad+\frac{\tau(N-1)}{r^2}\Big((R(1)-R'(1))^2+2rR'(1)(R(1)-R'(1))\Big)\nonumber\\
&\qquad\qquad\left.+\frac{\tau(N-1)}{r^2}\Big(r^2R'(1)^2\Big)\right)r^{N-1}dr\nonumber\\
&\leq N\omega_N\int_1^{r_2}\left(N\tau R'(1)^2+\frac{N-1}{r}\left((3+\tau)(R(1)-R'(1))^2\right.\right.\nonumber\\
&\qquad\qquad\left.+2\tau R'(1)(R(1)-R'(1))\right)\Big)r^{N-1}dr\nonumber\\
&=N\omega_N\tau R'(1)^2(r_2^N-1)+N\omega_N\left((3+\tau)(R(1)-R'(1))^2\right.\nonumber\\
&\qquad\left.+2\tau R'(1)(R(1)-R'(1))\right)(r_2^{N-1}-1),\nonumber\end{aligned}$$ where we have estimated the quantities ${1}/{r^2}$ and ${1}/{r^4}$ by ${1}/{r}$. We note that $r_2=\left(1+{\alpha}/{2}\right)^{{1}/{N}}$ and $0\leq\alpha\leq 2$. Using the Taylor expansion up to order $1$ and remainder in Lagrange form, we obtain $$\begin{aligned}
\label{1n}
r_2^{N-1}&=1+\frac{N-1}{N}\frac{\alpha}{2}-\frac{(N-1)\left(1+\frac{\xi}{2}\right)^{\frac{N-1}{N}-2}}{8N^2}\alpha^2\\
&\leq 1+\frac{N-1}{N}\frac{\alpha}{2}-\frac{(N-1)2^{\frac{N-1}{N}-2}}{8N^2}\alpha^2=1+\frac{N-1}{N}\frac{\alpha}{2}-c_{N}\alpha^2,\nonumber\end{aligned}$$ for some $\xi\in(0,\alpha)$, where $c_{N}$ is a positive constant which depends only on $N$. Using , , , and , in the right-hand side of , we obtain: $$\begin{aligned}
\label{tofinal1}
\int_{B\setminus B_1}&N[\rho]dx-\int_{B_2\setminus B}N[\rho]dx \\
&\ge-N\omega_N\Big((3+\tau)(R(1)-R'(1))^2+2\tau R'(1)(R(1)-R'(1))\Big)\left(\frac{N-1}{N}\frac{\alpha}{2}- c_{N}\alpha^2\right)\nonumber\\
&\qquad+\omega_N\left(3(N-1)(R(1)-R'(1))^2+\tau R'(1)^2+\tau (N-1) R(1)^2\right)\frac{\alpha}{2}\nonumber\\
&\qquad -N\omega_N\tau R'(1)^2\frac{\alpha}{2}\nonumber\\
&=:C^{(2)}_{N,\tau}\alpha^2,\nonumber\end{aligned}$$ where the constant $C^{(2)}_{N,\tau}>0$ is given by $$C^{(2)}_{N,\tau}=N\omega_N\left((3+\tau)(R(1)-R'(1))^2+2\tau R'(1)(R(1)-R'(1))\right)c_{N}.$$ From , , , and , it follows that $$\lambda_2(B)-\lambda_2(\Omega)\geq\frac{C^{(2)}_{N,\tau}}{C^{(1)}_{N,\tau}}\mathcal A(\Omega)^2,$$ and therefore, $$\label{quantN-1}
\lambda_2(\Omega)\leq\lambda_2(B)\left(1-\frac{C_{N,\tau}^{(2)}}{\lambda_2(B)C_{N,\tau}^{(1)}}\mathcal A(\Omega)^2\right).$$ The isoperimetric inequality is thus proved in the case of $\Omega$ with the same measure as the unit ball. The inequality for a generic domain $\Omega$ follows from scaling properties of the eigenvalues of problem . Writing our eigenvalues as $\lambda_2(\tau,\Omega)$ to make explicit the dependence on the parameter $\tau$, we have $$\label{scaling}
\lambda_2(\tau,\Omega)=s^4\lambda_2(s^{-2}\tau,s\Omega),$$ for all $s>0$. From and taking $s=(\omega_N/|\Omega|)^{1/N}$ in , it follows that for every $\Omega$ in $\mathbb R^N$ of class $C^1$ we have $$\begin{aligned}
\lambda_2(\tau,\Omega)&=s^4\lambda_2(s^{-2}\tau,s\Omega)\\
&\leq s^4\lambda_2(s^{-2}\tau,B)\left(1-\frac{C^{(2)}_{N,s^{-2}\tau}}{\lambda_2(s^{-2}\tau,B)C^{(1)}_{N,s^{-2}\tau}}\mathcal A(s\Omega)\right)\\
&=\lambda_2(\tau,\Omega^*)\left(1-\frac{C^{(2)}_{N,s^{-2}\tau}}{\lambda_2(s^{-2}\tau,B)C^{(1)}_{N,s^{-2}\tau}}\mathcal A(\Omega)\right).\end{aligned}$$ We set $$\eta_{N,\tau,|\Omega|}:=\frac{C^{(2)}_{N,s^{-2}\tau}}{\lambda_2(s^{-2}\tau,B)C^{(1)}_{N,s^{-2}\tau}}.$$ This concludes the proof of the theorem.
One generalization of the [biharmonic Neumann problem ]{} is to consider the case where the plate is made of a material with a nonzero Poisson’s ratio $\sigma$, which replaces the term $|D^2u|^2$ in the Rayleigh quotient by $(1-\sigma)|D^2u|^2+\sigma(\Delta u)^2$. A partial result towards the non-quantitative form of the isopermetric inequality has been obtained for certain values of $\tau>0$ and $\sigma\in(-1/(N-1),1)$, proved by the second author in [@chasmanpreprint] (see also [@buoso15; @prozkalamata]). In this case, the proof of Theorem \[NeumannQI\] can be easily adapted, yielding $$\lambda_2(B)-\lambda_2(\Omega)\geq \frac{C^{(3)}_{N,\tau}}{C^{(1)}_{N,\tau}}\mathcal{A}(\Omega)+\frac{C^{(2)}_{N,\tau}}{C^{(1)}_{N,\tau}}\mathcal{A}(\Omega)^2,$$ where $C^{(1)}_{N,\tau}$, $C^{(2)}_{N,\tau}$ are as in the proof of Theorem \[NeumannQI\], and $$C^{(3)}_{N,\tau}=\frac{1}{2}(R(1)-R'(1))^2(N-1)\sigma\Big(\sigma(N-1)(\sigma-2)+N-2\Big).$$ This result is not particularly satisfying, since it carries all of the same limitations of the non-quantitative result (only being valid for certain $\tau$ and $\sigma$), and in some cases it is strictly worse, since $C^{(3)}_{N,\tau}$ is non-negative only when $0\leq \sigma\leq 1-1/\sqrt{N-1}$.
Even though we are able to give a quantitative isoperimetric inequality for the fundamental tone of problem , very little is known in this regard for higher eigenvalues. To the best of our knowledge, only criticality results are available in the literature, where the ball is shown to be a critical domain under volume constraint (see, e.g., [@buoso15; @buosolamberti15; @buosoprovenzano]). However, as in the second-order case, the ball is not expected to be an optimizer for higher eigenvalues.
Sharpness of the Neumann inequality {#sharpness_neumann}
===================================
In this section, we prove the sharpness of inequality .
\[theorem2\] Let $B$ be the unit ball in $\mathbb{R}^N$ centered at zero. There exist a family $\left\{\Omega_{\epsilon}\right\}_{\epsilon>0}$ of smooth domains and positive constants $c_1,c_2,c_3,c_4$ and $r_1, r_2,r_3,r_4$ independent of $\epsilon>0$ such that $$\label{sharp1}
r_1\epsilon^2\le\Big||\Omega_{\epsilon}|-|B|\Big|\leq r_2\epsilon^2,$$ $$\label{sharp2}
c_1\epsilon\leq c_2\mathcal A(\Omega_{\epsilon})\leq\frac{|\Omega_{\epsilon}\triangle B|}{|\Omega_{\epsilon}|}\leq c_3\mathcal A(\Omega_{\epsilon})\leq c_4\epsilon,$$ and $$\label{sharp3}
r_3\epsilon^2\le\left|\lambda_2(\Omega_{\epsilon})-\lambda_2(B)\right|\leq r_4\epsilon^2,$$ for all $\epsilon\in(0,\epsilon_0)$, where $\epsilon_0>0$ is sufficiently small, [and $\lambda_2(\Omega_\epsilon)$, $\lambda_2(B)$ are the first positive eigenvalues of problem on $\Omega_\epsilon$, $B$ respectively]{}.
In order to prove Theorem \[theorem2\], we start by defining the family of domains $\left\{\Omega_{\epsilon}\right\}_{\epsilon>0}$ as follows (see Figure \[figura\]): $$\label{family}
\Omega_{\epsilon}=\left\{x\in\mathbb R^N: x=0 {\text\ or\ }|x|<1+\epsilon\psi\left(\frac{x}{|x|}\right)\right\},$$ where $\psi$ is a function belonging to the following class: $$\label{perturbation}
\mathcal{P}=\left\{\psi\in C^{\infty}(\partial B):\int_{\partial B}\psi d\sigma=\int_{\partial B}(a\cdot x)\psi d\sigma=\int_{\partial B}(a\cdot x)^2\psi d\sigma=0,\ \forall a\in\mathbb{R}^N\right\}.$$
![Domains $\Omega_{\varepsilon}$ defined by with a given $\psi\in\mathcal P$.[]{data-label="figura"}](good_domains.pdf){width="\textwidth"}
Under our choice of $\Omega_\epsilon$, the existence of constants $r_1,r_2,c_1,\dots,c_4$ satisfying inequalities and follow immediately from [@brascosteklov Lemma 6.2]. Thus, we need only prove .
Let $\lambda_2(\Omega_{\epsilon})$ be the first positive eigenvalue of the Neumann problem on $\Omega_{\epsilon}$, and let $u_{\epsilon}$ be an associated eigenfunction normalized by $\|u_{\epsilon}\|_{L^2(\Omega_{\epsilon})}=1$, so that $$\int_{\Omega_{\epsilon}}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2\,dx=\lambda_2(\Omega_{\epsilon}).$$ By standard elliptic regularity (see e.g., [@gazzola §2.4.3]), since $\Omega_{\epsilon}$ is of class $C^{\infty}$ by construction, we may take a sufficiently small $\epsilon_0>0$ so that $u_{\epsilon} \in C^{\infty}(\overline\Omega_{\epsilon})$ for all $\epsilon\in(0,\epsilon_0)$. Moreover, for all $k\in\mathbb N$, the sets $\Omega_{\epsilon}$ are of class $C^k$ uniformly in $\epsilon\in(0,\epsilon_0)$, which means that there exist constants $H_k>0$ independent of $\epsilon$ that satisfy $$\label{regularity}
\|u_{\epsilon} \|_{C^k(\overline{\Omega_{\epsilon}})}\leq H_k.$$ Now let $\tilde{u}_{\epsilon} $ be a $C^4$ extension of $u_{\epsilon}$ to some open neighborhood $A$ of $B\cup\Omega_{\epsilon}$. Then, there exists $K_A>0$ independent of $\epsilon>0$ for which $$\label{regularity_extension}
\|\tilde{u}_{\epsilon} \|_{C^4(\overline A)}\leq K_A\|u_{\epsilon} \|_{C^4(\overline{\Omega_{\epsilon}})}\leq K_A H_4.$$ From the fact that $\int_{\Omega_{\epsilon}}u_{\epsilon} \,dx=0$ and $|B\setminus\Omega_{\epsilon}|,|\Omega_{\epsilon}\setminus B|\in O(\epsilon)$ as $\epsilon\rightarrow 0$, it follows that the quantity $\delta:=\frac{1}{|B|}\int_B\tilde{u}_{\epsilon}\,dx$ satisfies $$\label{delta_bound}
\delta=\frac{1}{|B|}\int_B\tilde{u}_{\epsilon} \,dx=\frac{1}{|B|}\left(\int_{B\setminus \Omega_{\epsilon}}\tilde{u}_{\epsilon}\,dx-\int_{\Omega_{\epsilon}\setminus B}u_{\epsilon} \,dx\right)\leq c\epsilon,$$ where $c>0$ does not depend on $\epsilon\in(0,\epsilon_0)$. Now let us set $$\label{test}
v_{\epsilon} :={\tilde u_{\epsilon|_B}}-\delta.$$ The function $v_{\epsilon}$ is of class $C^4(\overline B)$ with $\int_B v_{\epsilon} \,dx=0$ and $$\label{regularity_v}
\|v_{\epsilon} \|_{C^4(\overline B)}\leq K_1$$ for some constant $K_1>0$ independent of $\epsilon\in(0,\epsilon_0)$. Therefore, $v_{\epsilon}$ is a suitable trial function for the Rayleigh quotient of $\lambda_2(B)$ (see formula ). Thus, $$\label{minmax_1}
\lambda_2(B)\leq\frac{\int_B |D^2 v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2\,dx}{\int_B {v_{\epsilon} }^2\,dx}.$$ We now consider the quantity $\left|\int_B v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2\,dx\right|$. We have $$\begin{aligned}
\label{ineq_1}
\left|\int_B v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2\,dx\right|&=\left|\int_B \delta^2-2\delta\tilde{u}_{\epsilon}\,dx\right|=\left|\int_B\delta(v_{\epsilon}-\tilde{u}_{\epsilon})\,dx\right|\\
%&=\left|\int_B\frac{1}{|B|^2}\left(\int_B \tilde{u}_{\epsilon} dy\right)^2-\frac{2}{|B|}\left(\int_B \tilde{u}_{\epsilon} dy\right)\tilde{u}_{\epsilon}\,dx\right|=\frac{1}{|B|}\left(\int_B \tilde{u}_{\epsilon} \right)^2\leq K_2\epsilon^2,\nonumber
&=\frac{1}{|B|}\left(\int_B \tilde{u}_{\epsilon}\,dx \right)^2\leq K_2\epsilon^2,\nonumber\end{aligned}$$ where $K_2>0$ is a positive constant independent of $\epsilon\in(0,\epsilon_0)$. Moreover, by and , we have that $$\begin{aligned}
\label{ineq_2}
\left|\int_{B\setminus\Omega_{\epsilon}}v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2dx\right|&\leq \int_{B\setminus\Omega_{\epsilon}}|v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2|dx\leq K_3\int_{B\setminus\Omega_{\epsilon}}|v_{\epsilon} -\tilde{u}_{\epsilon} |dx\\
&=K_3\frac{|B\setminus\Omega_{\epsilon}|}{|B|}\left|\int_B\tilde{u}_{\epsilon}\,dx\right|\leq K_4\epsilon^2,\nonumber\end{aligned}$$ where $K_3,K_4>0$ are positive constants independent of $\epsilon\in(0,\epsilon_0)$. Therefore, from , , and , it follows that $$\begin{aligned}
\label{estimate_1}
\lambda_2(B)&\leq\frac{\int_{B\cap\Omega_{\epsilon}}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2dx+\int_{B\setminus\Omega_{\epsilon}}|D^2 v_{\epsilon}|^2+\tau|D v_{\epsilon}|^2\,dx}{\int_B\tilde{u}_{\epsilon} ^2dx-K_2\epsilon^2}\\
&\le\frac{\lambda_2(\Omega_{\epsilon})+\int_{B\setminus\Omega_{\epsilon}}|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2dx-\int_{\Omega_{\epsilon}\setminus B}|D^2 u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2\,dx}{1+\int_{B\setminus\Omega_{\epsilon}}v_{\epsilon} ^2dx-\int_{\Omega_{\epsilon}\setminus B}u_{\epsilon}^2dx-(K_2+K_4)\epsilon^2}.\nonumber\end{aligned}$$ We introduce now the two error terms $R_1(\epsilon)$ and $R_2(\epsilon)$ defined by $$R_1(\epsilon):=\int_{B\setminus\Omega_{\epsilon}}|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2dx-\int_{\Omega_{\epsilon}\setminus B}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2\,dx$$ and $$R_2(\epsilon):=\int_{B\setminus\Omega_{\epsilon}}v_{\epsilon} ^2dx-\int_{\Omega_{\epsilon}\setminus B}u_{\epsilon}^2dx.$$ Then inequality can be rewritten as $$\label{estimate_2}
\lambda_2(B)\leq\frac{\lambda_2(\Omega_{\epsilon})+R_1(\epsilon)}{1+R_2(\epsilon)-K_5\epsilon^2}.$$ From the uniform estimates and on $u_{\epsilon}$ and $v_{\epsilon}$, it easily follows that $R_1,R_2\in O(\epsilon)$ as $\epsilon\rightarrow 0$, which together with immediately yields $\lambda_2(B)\leq\lambda_2(\Omega_{\epsilon})+C\epsilon$ for some constant $C>0$ independent of $\epsilon\in(0,\epsilon_0)$ (taking $\epsilon_0>0$ smaller if necessary).
We observe that, due to the strict relation of $R_1(\epsilon)$ and $R_2(\epsilon)$ with the difference $\lambda_2(B)-\lambda_2(\Omega_{\epsilon})$, a better estimate for $R_1(\epsilon)$ and $R_2(\epsilon)$ provides a better estimate for $\lambda_2(B)-\lambda_2(\Omega_{\epsilon})$. More precisely, we have the following
\[lemma\_refinement\] Let $\omega:[0,1]\rightarrow[0,+\infty)$ be a continuous function such that $t^2/K\leq\omega(t)\leq K t$, for some $K>0$. If there exists a constant $C>0$ such that $|R_1(\epsilon)|$, $|R_2(\epsilon)|\leq C\omega(\epsilon)$, then there exists a constant $C'>0$ such that $$\lambda_2(B)\leq\lambda_2(\Omega_{\epsilon})+C'\omega(\epsilon)$$ for every sufficient small $\epsilon>0$.
We refer to [@brascopratelli Lemma 6.2] for the proof (see also [@brascosteklov Lemma 6.7]).
We also need the following
\[lemma2\] Let $\omega$ be a function as in Lemma \[lemma\_refinement\], and let $v_{\epsilon}$ be as in . Suppose that there exists $C>0$ such that for all $\epsilon>0$ sufficiently small we have $|R_1(\epsilon)|, |R_2(\epsilon)|\leq C\omega(\epsilon)$. Then there exists an eigenfunction $\xi_{\epsilon}$ associated with $\lambda_2(B)$ such that $$\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline B)}\leq\tilde C\sqrt{\omega(\epsilon)}$$ for some $\tilde C>0$ independent of $\epsilon>0$.
Take $\{\xi_n\}_{n\geq 1}$ to be an orthonormal basis of $L^2(B)$ consisting of eigenfunctions of problem on the unit ball $B$. Note that from such a normalization, we have $$\int_B|D^2\xi_n|^2+\tau|D\xi_n|^2\,dx=\lambda_n(B)\,\quad\forall n\in\mathbb{N}.$$
We may write $v_{\epsilon} =\sum_{n=1}^{+\infty}a_n(\epsilon)\xi_n$. Note that $a_1(\epsilon)\equiv 0$, since $v_{\epsilon}$ has zero integral mean over $B$ and $\xi_1$ is a constant. We have $$\begin{aligned}
\sum_{n=2}^{+\infty}a_n(\epsilon)^2-1&=\|v_{\epsilon} \|_{L^2(B)}^2-1=\int_Bv_{\epsilon} ^2dx-\int_{\Omega_{\epsilon}}u_{\epsilon}^2dx\\
&=\int_B(v_{\epsilon}^2-\tilde{u}_{\epsilon}^2)dx-\int_{B\setminus\Omega_{\epsilon}}(v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2)dx+R_2(\epsilon).\end{aligned}$$ Then by using , , we obtain $$\label{asterisco}
\left|\sum_{n=2}^{+\infty}a_n(\epsilon)^2-1\right|\leq K_5\epsilon^2+C\omega(\epsilon)\leq C_1\omega(\epsilon).$$ We may now write $$\begin{aligned}
\lambda_2(\Omega_{\epsilon})&=\int_{\Omega_{\epsilon}}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2dx\\
&=\int_B|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2\,dx+\int_{\Omega_{\epsilon}\setminus B}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2dx\\
&\qquad-\int_{B\setminus\Omega_{\epsilon}}|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2\,dx\\
&=\sum_{n=2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)-R_1(\epsilon). \end{aligned}$$ From Lemma \[lemma\_refinement\], it follows that $$|\lambda_2(B)-\lambda_2(\Omega_{\epsilon})|\leq C'\omega(\epsilon),$$ and therefore, $$\label{star1}
\left|\sum_{n=2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)-\lambda_2(B)\right|=|\lambda_2(\Omega_{\epsilon})+R_1(\epsilon)-\lambda_2(B)|\leq C_2\omega(\epsilon).$$
By the symmetry of the ball, the first nonzero eigenvalue $\lambda_2(B)$ has multiplicity $N$, and so $\lambda_2(B)=\lambda_3(B)=\cdots=\lambda_{N+1}(B)<\lambda_{N+2}(B)$. Therefore, $$\begin{aligned}
C_2\omega(\epsilon)
&\geq\left|\sum_{n=2}^{N+1}a_n(\epsilon)^2\lambda_2(B)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)-\lambda_2(B)\right|\\
&=\left|\lambda_2(B)\left(\sum_{n=2}^{+\infty}a_n(\epsilon)^2-1\right)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\left(\lambda_n(B)-\lambda_2(B)\right)\right|\\
&\geq\left(\lambda_{N+2}(B)-\lambda_2(B)\right)\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2-\lambda_2(B)C_1\omega(\epsilon),\end{aligned}$$ which yields $$\label{C3}
\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\leq C_3\omega(\epsilon),$$ and hence by , $$\label{C4}
\left|\sum_{n=2}^{N+1}a_n(\epsilon)^2-1\right|\leq C_4\omega(\epsilon).$$ Revisiting , we see that $$\begin{aligned}
C_2\omega(\varepsilon)&\geq\left|\sum_{n=2}^{N+1}a_n(\epsilon)^2\lambda_2(B)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)-\lambda_2(B)\right|\\
&=\left|\lambda_2(B)\left(\sum_{n=2}^{N+1}a_n(\epsilon)^2-1\right)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)\right|\\
&\geq\lambda_2(B)\left(\sum_{n=2}^{N+1}a_n(\epsilon)^2-1\right)+\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B),\end{aligned}$$ which, together with and , yields $$\label{star2}
\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2\lambda_n(B)\leq C_2\omega(\epsilon)-\lambda_2(B)\left(\sum_{n=2}^{N+1}a_n(\epsilon)^2-1\right)\leq C_5\omega(\epsilon).$$
Now set $\varphi:=\sum_{n=2}^{N+1}a_n(\epsilon)\xi_n$ and define the norm $\|\cdot\|_{H^2_{\tau}(B)}$ by $$\|h\|_{H^2_{\tau}(B)}^2:=\int_B |D^2h|^2+\tau|Dh|^2+h^2\,dx,\qquad \forall h\in H^2(B).$$ This norm is equivalent to the standard $H^2(B)$-norm by coercivity of the bilinear form.
We now estimate the quantity $\|v_{\epsilon} -\varphi\|_{H^{2}_{\tau}(B)}$. We have $$\begin{aligned}
\|v_{\epsilon} -\varphi\|^2_{H^2_{\tau}(B)}&=\int_B|D^2(v_{\epsilon} -\varphi)|^2+\tau|D(v_{\epsilon} -\varphi)|^2+(v_{\epsilon} -\varphi)^2dx\\
&=\int_B\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2(|D^2\xi_n|^2+\tau|D\xi_n|^2+\xi_n^2)dx\\
&=\sum_{n=N+2}^{+\infty}a_n(\epsilon)^2(1+\lambda_n(B))\leq C_6\omega(\epsilon),\end{aligned}$$ where the last inequality follows from and . Thus the function $v_{\epsilon}$ is $\sqrt{\omega(\epsilon)}$-close to $\varphi$ in the $H^2_{\tau}(B)$-norm.
We want to bound the $C^3(\overline B)$-norm with the $H^2_{\tau}(B)$-norm. To do so, we use standard elliptic regularity estimates for the biharmonic operator. We have that, in $B\cap\Omega_{\epsilon}$, $$\Delta^2 v_{\epsilon} -\tau\Delta v_{\epsilon} =\Delta^2u_{\epsilon} -\tau\Delta u_{\epsilon} =\lambda_2(\Omega_{\epsilon})u_{\epsilon} =\lambda_2(\Omega_{\epsilon})(v_{\epsilon} +\delta).$$ Recall that $\delta\in O(\epsilon)$ as $\epsilon\rightarrow 0$ from . We set $$%\label{feps}
f_{\epsilon}:=\Delta^2v_{\epsilon} -\tau\Delta v_{\epsilon} .$$ Note that, in particular, $f_{\epsilon}=\lambda_2(\Omega_{\epsilon})(v_{\epsilon} +\delta)$ on $B\cap\Omega_{\epsilon}$. Then defining the functions $g_{\epsilon}^{(1)}$ and $g_{\epsilon}^{(2)}$ on $\partial B$ by $g_{\epsilon}^{(1)}:=\frac{\partial^2v_{\epsilon} }{\partial\nu^2}$ and $g_{\epsilon}^{(2)}:=\tau\frac{\partial v_{\epsilon} }{\partial\nu}-{\rm div}_{\partial B}(D^2 v_{\epsilon} \cdot\nu)-\frac{\partial\Delta v_{\epsilon} }{\partial\nu}$, we see that the function $v_{\epsilon}$ uniquely solves the problem $$%\label{auxiliarypb}
\begin{cases}
\Delta^2u-\tau\Delta u=f_{\epsilon}, & {\rm in}\ B,\\
\frac{\partial^2u}{\partial\nu^2}=g_{\epsilon}^{(1)}, & {\rm on}\ \partial B,\\
\tau\frac{\partial u}{\partial\nu}-{\rm div}_{\partial B}(D^2u\cdot\nu)-\frac{\partial\Delta u}{\partial\nu}=g_{\epsilon}^{(2)}, & {\rm on}\ \partial B,\\
\int_{B}udx=0.
\end{cases}$$ Now let $f:=\lambda_2(B)\varphi$. Then by definition, the function $\varphi$ is the unique solution of $$\begin{cases}
\Delta^2 u-\tau\Delta u=f, & {\rm in}\ B,\\
\frac{\partial^2u}{\partial\nu^2}=0, & {\rm on}\ \partial B,\\
\tau\frac{\partial u}{\partial\nu}-{\rm div}_{\partial B}(D^2u\cdot\nu)-\frac{\partial\Delta u}{\partial\nu}=0, & {\rm on}\ \partial B,\\
\int_B u\,dx=0.
\end{cases}$$ Finally, define the function $w:=v_{\epsilon} -\varphi$, which is the unique solution of $$\begin{cases}
\Delta^2 w-\tau\Delta w=f_{\epsilon}-f, & {\rm in}\ B,\\
\frac{\partial^2w}{\partial\nu^2}=g_{\epsilon}^{(1)}, & {\rm on}\ \partial B,\\
\tau\frac{\partial w}{\partial\nu}-{\rm div}_{\partial B}(D^2w\cdot\nu)-\frac{\partial\Delta w}{\partial\nu}=g_{\epsilon}^{(2)}, & {\rm on}\ \partial B,\\
\int_B w\,dx=0.
\end{cases}$$ For any $p>N$, we have (see e.g., [@gazzola Theorem 2.20]) $$\label{gazzola_estimate}
\|w\|_{W^{4,p}(B)}\leq C\left(\|f_{\epsilon}-f\|_{L^p(B)}+\|g_{\epsilon}^{(1)}\|_{W^{2-\frac{1}{p},p}(\partial B)}+\|g_{\epsilon}^{(2)}\|_{W^{1-\frac{1}{p},p}(\partial B)}\right).$$ We consider separately the three summands in the right-hand side of . We start from the first summand. Recall that for any $x\in B\cap \Omega_{\epsilon}$, we have (see ) $$f_{\epsilon}(x)=\lambda_2(\Omega_{\epsilon})(v_{\epsilon} (x)+\delta).$$ Since $\delta\in O(\epsilon)$ and $\lambda_2(\Omega_\epsilon)$ is bounded from above and from below, we have that $f_{\epsilon}(x)=\lambda_2(\Omega_{\epsilon})v_{\epsilon}(x)+O(\epsilon)$, and thus, as $\epsilon\rightarrow 0$, for any $p>N$, we have (cf. Lemma \[lemma\_refinement\]) $$\begin{aligned}
\label{gaz1}
\|f_{\epsilon}-f\|_{L^p(B)}&=\|\lambda_2(\Omega_{\epsilon})v_{\epsilon} -\lambda_2(B)\varphi\|_{L^p(B)}+O(\epsilon)\\
&\leq |\lambda_2(\Omega_{\epsilon})-\lambda_2(B)|\|v_{\epsilon} \|_{L^p(B)}+|\lambda_2(B)|\|v_{\epsilon} -\varphi\|_{L^p(B)}+O(\epsilon)\nonumber\\
&\leq C_7\omega(\epsilon)+C_8\sqrt{\omega(\epsilon)}+O(\epsilon)\leq C_9\sqrt{\omega(\epsilon)}.\nonumber\end{aligned}$$
Now we consider the second summand in the right-hand side of . Since $g_{\epsilon}^{(1)}=\frac{\partial^2v_{\epsilon} }{\partial\nu^2}$ and $v_{\epsilon}$ is an extension of $u_{\epsilon}$, by the regularity of both $u_{\epsilon}$ and $v_{\epsilon}$ (see , ) and from the fact that $\frac{\partial^2u_{\epsilon} }{\partial\nu^2}=0$ on $\partial\Omega_{\epsilon}$, we may conclude $$\label{gaz2}
\|g_{\epsilon}^{(1)}\|_{W^{2-\frac{1}{p},p}(\partial B)}\leq C\epsilon.$$ For the same reason, for the third summand in the right-hand side of we have $$\label{gaz3}
\|g_{\epsilon}^{(2)}\|_{W^{1-\frac{1}{p},p}(\partial B)}\leq C\epsilon.$$
From and the bounds , , and , it follows that, for any $p>N$, $$\|v_{\epsilon} -\varphi\|_{W^{4,p}(B)}\leq C_{10}\sqrt{\omega(\epsilon)},$$ and thus, from the Sobolev embedding theorem, $$\|v_{\epsilon} -\varphi\|_{C^{3}(\overline B)}\leq \tilde C\sqrt{\omega(\epsilon)}.$$ The proof is concluded by setting $\xi_{\varepsilon}=\varphi$.
The next lemma gives us refined bounds on $|R_1(\epsilon)|$ and $|R_2(\epsilon)|$.
\[lemma3\] Let $\omega(t), v_{\epsilon}$ be as in Lemma \[lemma\_refinement\]. Suppose that for all $\epsilon>0$ small enough there exists an eigenfunction $\xi_{\epsilon}$ associated with $\lambda_2(B)$ such that $$\label{hypo}
\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline B)}\leq C \sqrt{\omega(\epsilon)},$$ for some $C>0$ which does not depend on $\epsilon>0$. Then there exists $\tilde C>0$ which does not depend on $\epsilon$ such that $|R_1(\epsilon)|,|R_2(\epsilon)|\leq\tilde C\epsilon\sqrt{\omega(\epsilon)}$.
It is convenient to use spherical coordinates $(r,\theta)\in\mathbb{R}_{+}\times\mathbb{S}^{N-1}$ in $\mathbb R^N$ and the corresponding change of variables $x=\phi(r,\theta)$. We denote by $\mathcal D$ and $\tilde{\mathcal D}$ the sets $\mathcal D:=\partial(\Omega_{\epsilon}\setminus B)\cap\partial B$ and $\tilde{\mathcal D}=\partial(B\setminus \Omega_{\epsilon})\cap\partial B$. Observe that $\psi\geq0$ on $\mathcal D$ and $\psi\le0$ on $\tilde{\mathcal D}$.
Thanks to the regularity of $u_{\epsilon}$ and $\tilde{u}_{\epsilon}$ by , on $\Omega_{\epsilon}\setminus B$ we have $$\begin{aligned}
D^2u_{\epsilon} \circ\phi(1+\epsilon\psi,\theta)&=&D^2u_{\epsilon} \circ\phi(1,\theta)+O(\epsilon),\\
Du_{\epsilon} \circ\phi(1+\epsilon\psi,\theta)&=&Du_{\epsilon} \circ\phi(1,\theta)+O(\epsilon),\end{aligned}$$ as $\epsilon\rightarrow 0$. Therefore, integrating with respect to the radius $r$ and applying the definition of $v_{\epsilon}$ , we see $$\begin{aligned}
\int_{\Omega_{\epsilon}\setminus B}|D^2u_{\epsilon}|^2+\tau|Du_{\epsilon}|^2dx
&=\epsilon\int_{\mathcal D }\psi\left(\left|D^2u_{\epsilon} \right|^2+\tau \left|Du_{\epsilon} \right|^2\right)d\sigma+O(\epsilon^2)\\
&=\epsilon\int_{\mathcal D }\psi\left(\left|D^2v_{\epsilon} \right|^2+\tau \left|Dv_{\epsilon} \right|^2\right)d\sigma+O(\epsilon^2),\end{aligned}$$ as $\epsilon\rightarrow 0$. Similarly, $$\int_{B\setminus\Omega_{\epsilon}}|D^2v_{\epsilon}|^2+\tau|Dv_{\epsilon}|^2dx
=-\epsilon\int_{\tilde{\mathcal D}}\psi\left(\left|D^2v_{\epsilon} \right|^2+\tau \left|Dv_{\epsilon} \right|^2\right)d\sigma+O(\epsilon^2),$$ as $\epsilon\rightarrow 0$. From these and hypothesis , we see $$\begin{aligned}
\label{ineqR1}
|R_1(\epsilon)|&\leq \epsilon\left|\int_{\partial B}\psi\left(\left|D^2v_{\epsilon} \right|^2+\tau \left|Dv_{\epsilon} \right|^2\right)d\sigma\right|+O(\epsilon^2)\\
&\leq \epsilon\left|\int_{\partial B}\psi\left(\left|D^2\xi_{\epsilon}\right|^2+\tau \left|D\xi_{\epsilon}\right|^2\right)d\sigma\right|+C\epsilon\sqrt{\omega(\epsilon)}+O(\epsilon^2)\nonumber\\
&\le \tilde{C}\epsilon\sqrt{\omega(\epsilon)},\nonumber\end{aligned}$$ as $\epsilon\rightarrow 0$. In the last inequality, we have used the following identity for eigenfunctions of $\lambda_2(B)$: $$\label{spherical_harmonic}
\left.\left(\left|D^2\xi_{\epsilon}\right|^2+\tau \left|D\xi_{\epsilon}\right|^2\right)\right|_{\partial B}=(a\cdot x)^2$$ for some $a\in\mathbb R^N$ (cf. ).
By following the same scheme, we can prove the analogue of for $R_2(\epsilon)$. This concludes the proof.
We can now proceed to complete the proof of Theorem \[theorem2\].
Let $\omega_0(\epsilon):=|R_1(\epsilon)|+|R_2(\epsilon)|$. This function is continuous in $\epsilon$ and, moreover, has the property $$\frac{\epsilon^2}{K}\leq\omega_0(\epsilon)\leq K\epsilon.$$ The first inequality follows from Theorem \[NeumannQI\], while the latter follows from the fact that $R_1,R_2\in O(\epsilon)$. By Lemma \[lemma2\], it follows that there exists an eigenfunction $\xi_{\epsilon}$ of the Neumann problem on $B$ associated with eigenvalue $\lambda_2(B)$ such that $$\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline{B})}\leq C\sqrt{\omega_0(\epsilon)}.$$ Now we apply Lemma \[lemma3\], obtaining $$\omega_0(\epsilon)\leq2\tilde C\epsilon\sqrt{\omega_0(\epsilon)},$$ and therefore $$\sqrt{\omega_0(\epsilon)}=\frac{|R_1(\epsilon)|+|R_2(\epsilon)|}{\sqrt{\omega_0(\epsilon)}}\leq 2\tilde C\epsilon.$$ From this, it follows that $\omega_0(\epsilon)\leq 4\tilde C^2\epsilon^2$, and hence both $|R_1(\epsilon)|,|R_2(\epsilon)|\leq 4\tilde C^2\epsilon^2 $. Finally, we apply Lemma \[lemma\_refinement\] and obtain $$\lambda_2(B)\leq\lambda_2(\Omega_{\epsilon})+\mathcal C\epsilon^2$$ for a constant $\mathcal C>0$ independent of $\epsilon\in(0,\epsilon_0)$. This concludes the proof of Theorem \[theorem2\].
\[directions\] In [@brascopratelli], the authors provided an explicit construction of a family $\{\Omega_{\epsilon}\}_{\epsilon}$ in $\mathbb R^2$ suitable for proving the sharpness of their [quantitative isoperimetric inequality for the fundamental tone of the Neumann Laplacian]{}. On the other hand, in [@brascosteklov], the authors gave only sufficient conditions to generate the family $\{\Omega_{\epsilon}\}_{\epsilon}$, which are exactly those we apply in . We observe that the first two conditions, namely $$\label{ellipses}
\int_{\partial B}\psi d\sigma=\int_{\partial B}(a\cdot x)\psi d\sigma=0,$$ have a purely geometrical meaning, and are used to prove inequalities and (cf. [@brascosteklov Lemma 6.2]). The latter has a stricter relation with the problem, since any function $\xi$ belonging to the eigenspace associated with $\lambda_2(B)$ satisfies equality . This is due to the fact that $\xi$ can be expressed as a radial part times a spherical harmonic polynomial of degree $1$. This also tells us that the correct conditions to impose when considering the Steklov problem are still . In particular, as pointed out in [@brascosteklov Remark 6.9], ellipsoids satisfy conditions , and hence inequalities and hold, but miss the final condition, and therefore are not a suitable family for this problem. Note that for the Dirichlet Laplacian case in [@brasco2015], ellipsoids are a suitable family for proving the sharpness, and therefore conditions are sufficient.
We also observe that in [@brasco2015], the construction is somewhat more general (cf. [@brasco2015 Theorem 3.3, pp. 1788-1789]), while the perturbation used in [@brascopratelli] does not belong to . This means that it is possible to state less-restrictive conditions which would produce families of domains achieving the sharpness.
Sharpness of the Steklov inequality {#sharpness_steklov}
===================================
In this section, we prove the sharpness of inequality . Due to the strong similarities between the Steklov problem and the Neumann problem , we shall maintain the same notation as in the previous section.
Let $B$ be the unit ball in $\mathbb{R}^N$ centered at zero. There exist a family $\left\{\Omega_{\epsilon}\right\}_{\epsilon>0}$ of smooth domains and positive constants $c_1,c_2,c_3,c_4$ and $r_1, r_2,r_3,r_4$ independent of $\epsilon>0$ such that $$%\label{sharp1}
r_1\epsilon^2\le\Big||\Omega_{\epsilon}|-|B|\Big|\leq r_2\epsilon^2,$$ $$%\label{sharp2}
c_1\epsilon\leq c_2\mathcal A(\Omega_{\epsilon})\leq\frac{|\Omega_{\epsilon}\triangle B|}{|\Omega_{\epsilon}|}\leq c_3\mathcal A(\Omega_{\epsilon})\leq c_4\epsilon,$$ and $$\label{sharp4}
r_3\epsilon^2\le\left|\lambda_2(\Omega_{\epsilon})-\lambda_2(B)\right|\leq r_4\epsilon^2,$$ for all $\epsilon\in(0,\epsilon_0)$, where $\epsilon_0>0$ is sufficiently small, [and $\lambda_2(\Omega_\epsilon)$, $\lambda_2(B)$ is the first positive eigenvalue of problem on $\Omega_\epsilon$, $B$ respectively]{}.
To prove this theorem, we begin by defining the family $\left\{\Omega_{\epsilon} \right\}_{\epsilon>0}$ as in . Thus it remains only to prove .
We remind the reader of the variational characterization of the first positive eigenvalue of the Steklov problem on a domain $\Omega$: $$\label{steklov-ray}
\lambda_2(\Omega)=\inf_{\substack{0\ne u\in H^2(\Omega)\\ \int_{\partial\Omega}u \,d\sigma=0}}\frac{\int_{\Omega}|D^2u|^2+\tau|Du|^2\,dx}{\int_{\partial\Omega}u^2\,d\sigma}.$$
We take the first positive eigenvalue $\lambda_2(\Omega_{\epsilon} )$ of the Steklov problem on $\Omega_{\epsilon} $, and let $u_{\epsilon} $ be an associated eigenfunction, normalized by $$%\label{normalization1}
\int_{\partial\Omega_{\epsilon} }u_{\epsilon} ^2dx=1.$$ Then by the variational characterization , $$%\label{normalization2}
\int_{\Omega_{\epsilon} }|D^2u_{\epsilon} |^2+\tau|\nabla u_{\epsilon} |^2 dx=\lambda_2(\Omega_{\epsilon} ).$$ By standard elliptic regularity (see e.g., [@gazzola §2.4.3]), since $\Omega_{\epsilon} $ is of class $C^{\infty}$ by construction, we have that $u_{\epsilon} \in C^{\infty}(\overline{\Omega_{\epsilon}})$ for all $\epsilon\in(0,\epsilon_0)$. Moreover, for all $k\in\mathbb N$, the sets $\Omega_{\epsilon} $ are of class $C^k$ uniformly in $\epsilon\in(0,\epsilon_0)$, which means that there exist constants $H_k>0$ independent of $\epsilon$ such that $$%\label{regularity_s}
\|u_{\epsilon} \|_{C^k(\overline{\Omega_{\epsilon}} )}\leq H_k.$$ Let now $\tilde{u}_{\epsilon} $ be a $C^4$ extension of $u_{\epsilon} $ to an open neighborhood $A$ of $B\cup\Omega_{\epsilon} $. Then, there exists $K_A>0$ independent of $\epsilon>0$ such that $$%\label{regularity_extension_s}
\|\tilde{u}_{\epsilon} \|_{C^4(\overline A)}\leq K_A\|u_{\epsilon} \|_{C^4(\overline{\Omega_{\epsilon}} )}\leq K_A H_4.$$ Analogous to the Neumann case, take $\delta:=\frac{1}{|\partial B|}\int_{\partial B}\tilde{u}_{\epsilon} \,d\sigma$ to be the mean of $\tilde{u}_{\epsilon}$ over $\partial B$. From the fact that $\int_{\partial\Omega_{\epsilon} }u_{\epsilon} dx=0$ and $|B\setminus\Omega_{\epsilon} |,|\Omega_{\epsilon} \setminus B|\in O(\epsilon)$ as $\epsilon\rightarrow 0$, it follows that, as $\epsilon\rightarrow0$ (see also [@brascosteklov formula (6.15)]), $$\delta=\frac{1}{|\partial B|}\int_{\partial B}\tilde{u}_{\epsilon} \,d\sigma\in O(\epsilon).$$ Now let us set $v_{\epsilon}:=\tilde u_{\epsilon|_B}-\delta$. This function is of class $C^4(\overline B)$, satisfies $\int_{\partial B} v_{\epsilon} \,d\sigma=0$, and $$%\label{regularity_v_s}
\|v_{\epsilon} \|_{C^4(\overline B)}\leq K'$$ for a constant $K'>0$ independent of $\epsilon\in(0,\epsilon_0)$. Therefore, $v_{\epsilon} $ is a suitable trial function for the Rayleigh quotient of $\lambda_2(B)$, hence, $$%\label{minmax_1_s}
\lambda_2(B)\leq\frac{\int_B |D^2 v_{\epsilon} |^2+\tau|\nabla v_{\epsilon} |^2 dx}{\int_{\partial B} {v_{\epsilon} }^2 \,d\sigma}.$$ On the other hand, $$%\label{ineq_1}
\left|\int_{\partial B} v_{\epsilon} ^2-\tilde{u}_{\epsilon} ^2 \,d\sigma\right|=\left|\int_{\partial B} \delta^2-2\delta\tilde{u}_{\epsilon} \,d\sigma\right|\leq K''\epsilon^2,$$ where $K''>0$ is a positive constant independent of $\epsilon\in(0,\epsilon_0)$. Therefore, we may write $$%\label{estimate_2}
\lambda_2(B)\leq\frac{\lambda_2(\Omega_{\epsilon} )+R_1(\epsilon)}{1+R_2(\epsilon)-\tilde K\epsilon^2},$$ where we have once again defined the error terms $$%\label{R1}
R_1(\epsilon):=\int_{B\setminus\Omega_{\epsilon} }|D^2v_{\epsilon} |^2+\tau|\nabla v_{\epsilon} |^2dx-\int_{\Omega_{\epsilon} \setminus B}|D^2u_{\epsilon} |^2+\tau|\nabla u_{\epsilon} |^2 dx,$$ and $$R_2(\epsilon):=\int_{\partial B}v_{\epsilon} ^2\,d\sigma-\int_{\partial \Omega_{\epsilon} }u_{\epsilon} ^2\,d\sigma.$$
At this point, we note that the observations made in Section \[sharpness\_neumann\] remain valid here. Therefore, in order to conclude the proof of , we need only a few lemmas.
Let $\omega$ be as in Lemma \[lemma\_refinement\]. If there exists a constant $C>0$ such that $|R_1(\epsilon)|, |R_2(\epsilon)|\leq C\omega(\epsilon)$, then there exists a constant $C'>0$ such that $$\lambda_2(B)\leq\lambda_2(\Omega_{\epsilon} )+C'\omega(\epsilon)$$ for every $\epsilon>0$ sufficiently small.
See [@brascosteklov Lemma 6.7].
Let $\omega$ be as in Lemma \[lemma\_refinement\]. Suppose that there exists $C>0$ such that for all $\epsilon>0$ sufficiently small we have $|R_1(\epsilon)|, |R_2(\epsilon)|\leq C\omega(\epsilon)$. Then there exists an eigenfunction $\xi_{\epsilon}$ associated with $\lambda_2(B)$ such that $$\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline B)}\leq\tilde C\sqrt{\omega(\epsilon)},$$ for some $\tilde C>0$ independent of $\epsilon>0$.
The proof is essentially identical to that of Lemma \[lemma2\] and hence the details are omitted. Some small changes are necessary since $L^2(\Omega)$-norms have to be replaced by $L^2(\partial\Omega)$-norms, since we are considering the Steklov problem.
Let $\omega$ be as in Lemma \[lemma\_refinement\]. Suppose that for all $\epsilon>0$ sufficiently small there exists an eigenfunction $\xi_{\epsilon}$ associated with $\lambda_2(B)$ such that $$%\label{hypo}
\|v_{\epsilon} -\xi_{\epsilon}\|_{C^3(\overline B)}\leq C \sqrt{\omega(\epsilon)},$$ for some $C>0$ independent of $\epsilon>0$. Then there exists $\tilde C>0$ independent of $\epsilon$ such that $|R_1(\epsilon)|,|R_2(\epsilon)|\leq\tilde C\epsilon\sqrt{\omega(\epsilon)}$.
Regarding the bound on $R_1$, we refer to the proof of Lemma \[lemma3\]. For $R_2$, we refer to [@brascosteklov Lemma 6.8, p. 4701], observing that if $\xi_{\epsilon}$ is an eigenfunction associated with $\lambda_2(B)$, then on $\partial B$, $${\rm div}_{\partial B}(D^2\xi_{\epsilon}\cdot\nu)+\frac{\partial\Delta\xi_{\epsilon}}{\partial\nu}=0,$$ and therefore the second boundary condition in reads as $\partial\xi_{\epsilon}/\partial\nu=\lambda_2(B)\xi_{\epsilon}$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The first and the third author wish to thank Berardo Ruffini for discussions on his paper [@brascosteklov]. The first author has been partially supported by the research project FIR (Futuro in Ricerca) 2013 ‘Geometrical and qualitative aspects of PDE’s’. The third author acknowledges financial support from the research project ‘Singular perturbation problems for differential operators’ Progetto di Ateneo of the University of Padova and from the research project ‘INdAM GNAMPA Project 2015 - Un approccio funzionale analitico per problemi di perturbazione singolare e di omogeneizzazione’. The first and the third author are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
| ArXiv |
---
abstract: 'Recently a novel type of epithelial cell has been discovered and dubbed the “scutoid”. It is induced by curvature of the bounding surfaces. We show by simulations and experiments that such cells are to be found in a dry foam subjected to this boundary condition.'
author:
-
title: 'Demonstration and interpretation of “scutoid” cells in a quasi-2D soap froth'
---
quasi-2D; foams; scutoid; epithelial cells
Introduction
============
Recently G[ó]{}mez-G[á]{}lvez [*et al.*]{} [@gomez2018scutoids] have described epithelial cells of a previously unreported form which they have called *scutoid*; they appear when the bounding surfaces are *curved*. The distinguishing feature of such a cell is a triangular face attached to one of the bounding surfaces. Here we offer a simple illustration of this phenomenon, which is derived from the physics of foams [@weaire2001physics], consisting of a computer simulation together with preliminary experimental observations.
In an ideal dry foam, bubbles enclose gas (which is treated as incompressible) and the energy is proportional to their total surface area. Alternatively, the soap films may be considered to be in equilibrium under a constant surface tension and the gas pressure of the neighbouring cells. Plateau’s rules [@plateau1873statique], more than a century old, place restrictions on the topology of a *dry* foam (one of negligible liquid content), which is the only case considered here.
From the earliest intrusion of physics into biology, this elementary soap froth model has attracted attention to account for the shape and development of cells [@thompson1942growth; @dormer1980fundamental]. More sophisticated attempts to adopt it to that purpose persist today [@merks2005cell; @bi2014energy; @graner2017forms]. In the present context we show that the model largely accounts for the appearance of scutoids, in very simple and semi-quantitative terms, broadly consistent with the description in the original paper [@gomez2018scutoids].
Topology of dry foams
=====================
The relevance of foams to biology is apparent from the pioneering work of the botanist Edwin Matzke [@matzke1946three]. Inspired by the resemblance in shape between bubbles in foam and cells in tissues, Matzke sought to understand the forces that may be common to both. His approach was to painstakingly and exhaustively catalogue bubble shapes observed in a dry monodisperse foam, confined within a cylindrical jar. Matzke distinguished between peripheral bubbles (i.e. bubbles in contact with the walls of the cylindrical jar) and central bubbles (i.e. bubbles inside the bulk foam). Amongst the peripheral bubbles are listed two scutoids: the $(1,3,3,1)$ (see Figure 9-8 of [@matzke1946three]) and $(1,4,2,1,1)$ polyhedrons (as identified in Matzke’s notation). Not a single triangular face was found amongst the central (i.e. bulk) bubbles.
Quasi-2D foam sandwich
======================
Cyril Stanley Smith [@Smith52] first introduced the experimental quasi-2D foam that is formed between two glass plates. The plates are close enough together that all bubble cells span both boundaries, so that there are no internal bubbles and the internal soap films meet the glass plates at right angles (see Figure \[quasi\]). The quasi 2D foam between flat parallel plates is often taken as the experimental counterpart of the ideal 2D foam - which consists of polygonal 2D cells, with (in general curved) edges meeting three at a time (only) at $120^o$. Such a finite foam sandwich presents *two* such patterns on its two boundaries, and indeed on any plane taken parallel to them. However, if the plate separation is increased, this structure is overtaken by an instability, described and analysed by Cox [*et al.*]{}[@cox2002transition], in which individual cells cease to span the two plates. This instability is not directly relevant to scutoid formation but places limitations on experiment and theory.
![A quasi-2D foam showing a single layer of bubbles confined between two flat parallel glass plates (plate separation 8mm, average bubble diameter 2-3cm). Internal films meet the plates at right angles. The polygonal cells on both glass plates are identical.[]{data-label="quasi"}](15.png){width="0.75\columnwidth"}
The novel element that is brought into consideration by the work of G[ó]{}mez-G[á]{}lvez [*et al.*]{} is the introduction of *curved* boundaries which may be represented by two concentric cylinders or a portion thereof. While there has been some work on the effects of curvature of one or both plates [@roth2012coarsening; @mughal2017curvature], it did not address the case considered by G[ó]{}mez-G[á]{}lvez et al which consists of two concentric boundaries. As the separation between the two cylinders is increased, the 2D patterns on the inner and outer surfaces become distorted, the inner one being compressed in the circumferential direction, with respect to the outer one. Eventually, this should lead to the vanishing of a 2D cell edge, and hence to a topological change, as in Figure \[T1\]. This is the so-called T1 process [@weaire1984soap]. It necessarily entails the creation of a *scutoid* feature within the bulk of the foam (as illustrated in sections \[s:simulations\] and \[s:expts\]). However, its appearance may be only transitory, as it may provoke a similar effect of the other surface, in a double-$T1$ process that restores the original columnar structure. The geometry required by Plateau’s rules makes it obvious that this must be the case if the gap between the cylinders is very small. Increasing the gap is expected to allow stable scutoids to persist, provided we do not encounter the other type of instability mentioned above.
![A schematic of a T1 transition in an ideal 2D foam [@weaire1984soap]. The edge shared between bubbles A and B gradually shrinks and vanishes, the resulting fourfold vertex is in violation of Plateau’s laws and the system transitions to a new arrangement. As a result, bubbles A and B are no longer neighbours, while C and D (which were previously unconnected) now share a boundary.[]{data-label="T1"}](t1process.png){width="0.8\columnwidth"}
These arguments leave room for doubt as to whether such scutoid features can really be found in the foam sandwich. Both simulations and experiments, described in the following section, have yielded positive results.
Simulations {#s:simulations}
===========
![Cells in a Surface Evolver simulation of a polydisperse foam confined between two concentric cylinders. (a) In the initial state the 2D pattern on both boundaries is purely hexagonal (only the pattern on the substrate is shown). Red and blue bubbles are not in contact while the green bubble is in contact with a fourth neighbouring bubble (not shown for clarity). (b) The foam after a T1 transition on the substrate, resulting in four stable scutoid cells. The pattern on the substrate contains two five-sided and two seven-sided regions, while the pattern on the superstrate remains purely hexagonal. The cells are shown slightly separated for clarity. (c) The two types of scutoids cells (pentagonal and heptagonal) are shown separately. (d) A combined view showing the scutoids and the surrounding foam cells. []{data-label="SEscutoid"}](scutoid_four_images.jpg){width="0.95\columnwidth"}
As in the simulations of [@gomez2018scutoids], we start from a Voronoi partition of the gap between two concentric cylinders, to give a collection of hexagonal prismatic cells. This structure is imported into the Surface Evolver software [@brakke1992surface], which permits the minimization of surface energy (here equivalent to surface area, as in the ideal foam model) subject to fixed cell volumes. We employ a periodic boundary condition in the direction of the axis of the cylinders to reduce the effect of the finite size of the simulation. Cell volumes are assigned fixed values within a restricted range so that the initial structure is polydisperse but still hexagonal. In the example shown in Fig \[SEscutoid\]a, the cylinder has axis length 5.2 units, the cylinder radii are 2.8 and 4.3 units and there are 144 cells. To allow the cell walls to develop realistic curvature, we tessellate each face with small triangles and perform a standard Surface Evolver minimization of the surface area.
In this preliminary exploration topological changes were triggered using the Surface Evolver software. A number of stable scutoids were identified of which one example is shown in Figure \[SEscutoid\]. In the continuation of this work we expect to map out the parameter space in which such stable scutoids are to be found.
Experiments with soap bubbles {#s:expts}
=============================
We performed preliminary experiments with soap bubbles between curved surfaces, using a glass cylinder of diameter $21$mm as a substrate and a hollow half cylinder (made from perspex) with inner diameter $39$mm as a superstrate. The bubbles (approximate equivalent sphere diameter 8 mm) were produced using a simple aquarium pump with flow control and commercial dish-washing solution. Rather than placing the two cylinders upright into the vessel containing the solution we placed them on their long axis, creating an approximately 7mm wide gap between them, which was initially about half-way filled with liquid. We then used a syringe needle attached to the pump to blow gas into this gap, leading to the formation of a quasi-2D foam sandwich. By reducing the water level we arrive at bubbles which are in contact with both cylinder surfaces, some of them forming scutoids, see Figure \[scutoid-photo\]. The present process involves a measure of trial and error: repeated raising and lowering the water level allows for repeated bubble rearrangements which increases the chance of finding scutoids.
![Photograph of scutoids in a quasi-2d foam sandwich. The bubble on the left features a hexagon in contact with the outer cylinder and a pentagon in contact with the inner cylinder while the bubble on the right shows a heptagon on the outer and a hexagon on the inner cylinder. Also visible is the small triangular face separating these two bubbles. (diameter of inner cylinder $21$mm, internal diameter of hollow outer cylinder $39$mm, spacing about $7$mm, approximate equivalent sphere diameter of the bubbles 8 mm.) []{data-label="scutoid-photo"}](scutoid-photo.jpg){width="0.8\columnwidth"}
Conclusion
==========
Both simulation and experiment have confirmed that stable scutoid configurations are to be found in a dry foam sandwich between cylindrically curved faces. It remains for future work to identify the conditions for this in terms of geometrical parameters.
The foam model is well established in the description of biological cells and the processes by which they change their arrangements, but is at best a rough first approximation. In the present case we have noted that epithelial cells may be relatively elongated. If greater realism is called for, further energy terms may be added, stiffening the cell walls.
Acknowledgements
================
This research was supported in part by a research grant from Science Foundation Ireland (SFI) under grant number 13/IA/1926. A. Mughal acknowledges the Trinity College Dublin Visiting Professorships and Fellowships Benefaction Fund. We thank B. Haffner for providing the photograph of Figure \[quasi\].
[10]{}
P. G[ó]{}mez-G[á]{}lvez, P. Vicente-Munuera, A. Tagua, C. Forja, A.M. Castro, M. Letr[á]{}n, A. Valencia-Exp[ó]{}sito, C. Grima, M. Berm[ú]{}dez-Gallardo, [Ó]{}. Serrano-P[é]{}rez-Higueras, F. Cavodeassi, S. Sotillos, M.D. Mart[í]{}n-Bermudo, A. M[á]{}rquez, J. Buceta, and L.M. Escudero, Nat. Commun. 9 (2018) p. 2960.
D. Weaire and S. Hutzler, *The physics of foams*, Clarendon Press, Oxford, 1999.
J.A.F. Plateau, *Statique Expérimentale et Théorique des Liquides soumis aux seules Forces Moléculaires*, Gauthier-Villars, Paris, 1873.
D.W. Thompson, *On growth and form*, Cambridge University Press, 1917.
K. Dormer, *Fundamental tissue geometry for biologists*, Cambridge University Press, Cambridge, 1980.
R.M. Merks and J.A. Glazier, Phys. A 352 (2005) p. 113.
D. Bi, J.H. Lopez, J. Schwarz, and M.L. Manning, Soft Matter 10 (2014) p. 1885.
F. Graner and D. Riveline, Development 144 (2017) p. 4226.
E.B. Matzke, Am. J. Bot. 33 (1946) p. 58.
C. Smith, Metal Interfaces (ASM Cleveland) (1952) p. 65.
S. Cox, D. Weaire, and M.F. Vaz, Eur. Phys. J. E 7 (2002) p. 311.
A. Roth, C. Jones, and D.J. Durian, Phy. Rev. E 86 (2012) p. 021402.
A. Mughal, S. Cox, and G. Schr[ö]{}der-Turk, Interface focus 7 (2017) p. 20160106.
D. Weaire and N. Rivier, Contemp. Phys. 25 (1984) p. 59.
K.A. Brakke, Exp. Math. 1 (1992) p. 141.
| ArXiv |
---
abstract: 'We present a new concept for a multi-stage Zeeman decelerator that is optimized particularly for applications in molecular beam scattering experiments. The decelerator consists of a series of alternating hexapoles and solenoids, that effectively decouple the transverse focusing and longitudinal deceleration properties of the decelerator. It can be operated in a deceleration and acceleration mode, as well as in a hybrid mode that makes it possible to guide a particle beam through the decelerator at constant speed. The deceleration features phase stability, with a relatively large six-dimensional phase-space acceptance. The separated focusing and deceleration elements result in an unequal partitioning of this acceptance between the longitudinal and transverse directions. This is ideal in scattering experiments, which typically benefit from a large longitudinal acceptance combined with narrow transverse distributions. We demonstrate the successful experimental implementation of this concept using a Zeeman decelerator consisting of an array of 25 hexapoles and 24 solenoids. The performance of the decelerator in acceleration, deceleration and guiding modes is characterized using beams of metastable Helium ($^3S$) atoms. Up to 60% of the kinetic energy was removed for He atoms that have an initial velocity of 520 m/s. The hexapoles consist of permanent magnets, whereas the solenoids are produced from a single hollow copper capillary through which cooling liquid is passed. The solenoid design allows for excellent thermal properties, and enables the use of readily available and cheap electronics components to pulse high currents through the solenoids. The Zeeman decelerator demonstrated here is mechanically easy to build, can be operated with cost-effective electronics, and can run at repetition rates up to 10 Hz.'
author:
- Theo Cremers
- Simon Chefdeville
- Niek Janssen
- Edwin Sweers
- Sven Koot
- Peter Claus
- 'Sebastiaan Y.T. van de Meerakker'
title: 'A new concept multi-stage Zeeman decelerator'
---
Introduction {#sec:intro}
============
In the last two decades, tremendous progress has been made in manipulating the motion of molecules in a molecular beam. Using methods that are inspired by concepts from charged particle accelerator physics, complete control over the velocity of molecules in a beam can be achieved. In particular, Stark and Zeeman decelerators have been developed to control the motion of molecules that possess an electric and magnetic dipole moment using time-varying electric and magnetic fields, respectively. Since the first experimental demonstration of Stark deceleration in 1998 [@Bethlem:PRL83:1558], several decelerators ranging in size and complexity have been constructed [@Meerakker:CR112:4828; @Narevicius:ChemRev112:4879; @Hogan:PCCP13:18705]. Applications of these controlled molecular beams are found in high-resolution spectroscopy, the trapping of molecules at low temperature, and advanced scattering experiments that exploit the unprecedented state-purity and/or velocity control of the packets of molecules emerging from the decelerator [@Carr:NJP11:055049; @Bell:MolPhys107:99; @Jankunas:ARPC66:241; @Stuhl:ARPC65:501; @Brouard:CSR43:7279; @Krems:ColdMolecules].
Essential in any experiment that uses a Stark or Zeeman decelerator is a high particle density of the decelerated packet. For this, it is imperative that the molecules are decelerated with minimal losses, i.e., molecules within a certain volume in six-dimensional (6D) phase-space should be kept together throughout the deceleration process [@Bethlem:PRL84:5744]. It is a formidable challenge, however, to engineer decelerators that exhibit this so-called phase stability. The problem lies in the intrinsic field geometries that are used to manipulate the beam. In a multi-stage Zeeman (Stark) decelerator a series of solenoids (high-voltage electrodes) yields the deceleration force as well as the transverse focusing force. This can result in a strong coupling between the longitudinal (forward) and transverse oscillatory motions; parametric amplification of the molecular trajectories can occur, leading to losses of particle density [@Meerakker:PRA73:023401; @Sawyer:EPJD48:197].
For Stark decelerators, the occurrence of instabilities can be avoided without changing the electrode design. By operating the decelerator in the so-called $s=3$ mode [@Meerakker:PRA71:053409], in which only one third of the electrode pairs are used for deceleration while the remaining pairs are used for transverse focusing, instabilities are effectively eliminated [@Meerakker:PRA73:023401; @Scharfenberg:PRA79:023410]. The high particle densities afforded by this method have recently enabled a number of high-resolution crossed beam scattering experiments, for instance [@Gilijamse:Science313:1617; @Kirste:Sience338:1060; @Zastrow:NatChem6:216; @Vogels:SCIENCE350:787]. For multi-stage Zeeman decelerators, several advanced switching protocols have been proposed and tested to mitigate losses. Wiederkehr *et al.* extensively investigated phase stability in a Zeeman decelerator, particularly including the role of the nonzero rise and fall times of the current pulses, as well as the influence of the operation phase angle [@Wiederkehr:JCP135:214202; @Wiederkehr:PRA82:043428]. Evolutionary algorithms were developed to optimize the switching pulse sequence, significantly increasing the number of particles that exit from the decelerator. Furthermore, inspired by the $s=3$ mode of a Stark decelerator, alternative strategies for solenoid arrangements were investigated numerically [@Wiederkehr:PRA82:043428]. Dulitz *et al.* developed a model for the overall 6D phase-space acceptance of a Zeeman decelerator, from which optimal parameter sets can be derived to operate the decelerator at minimum loss [@Dulitz:PRA91:013409]. Dulitz *et al.* also proposed and implemented schemes to improve the transverse focusing properties of a Zeeman decelerator by applying reversed current pulses to selected solenoids [@Dulitz:JCP140:104201]. Yet, despite the substantial improvements these methods can offer, the phase-stable operation of a multi-stage Zeeman decelerator over a large range of velocities remains challenging.
Recently, a very elegant approach emerged that can be used to overcome these intrinsic limitations of multi-stage decelerators. So-called traveling wave decelerators employ spatially moving electrostatic or magnetic traps to confine part of the molecular beam in one or multiple wells that start traveling at the speed of the molecular beam pulse and are subsequently gradually slowed down. In this approach the molecules are confined in genuine potential wells, and stay confined in these wells until the final velocity is reached. Consequently, these decelerators are inherently phase stable, and no losses occur due to couplings of motions during the deceleration process. The acceptances are almost equal in both the longitudinal and transverse directions, which appears to be particularly advantageous for experiments that are designed to spatially trap the molecules at the end of the decelerator. Both traveling wave Stark [@Osterwalder:PRA81:051401; @vandenBerg:JMS300:201422] and Zeeman [@Trimeche:EPJD65:263; @Lavert-Ofir:NJP13:103030; @Lavert-Ofir:PCCP13:18948; @Akerman:NJP17:065015] decelerators have been successfully demonstrated. Recently, first experiments in which the decelerated molecules are subsequently loaded into static traps have been conducted [@Quintero:PRL110:133003; @Jansen:PRA88:043424].
These traveling wave decelerators typically feature a large overall 6D acceptance. This acceptance is almost equally partitioned between the longitudinal and both transverse directions. For high-resolution scattering experiments, however, there are rather different requirements for the beam than for trapping. Certainly, phase-stable operation of the decelerator—and the resulting production of molecular packets with high number densities—is essential. In addition, tunability over a wide range of final velocities is important, but the ability to reach very low final velocities approaching zero meters per second is often inconsequential. More important is the shape of the emerging packet in phase-space, i.e., the spatial and velocity distributions in both the longitudinal and transverse directions.
Ideally, for scattering experiments the longitudinal acceptance of the decelerator should be relatively large, whereas it should be small in the transverse directions. A broad longitudinal distribution—in the order of a few tens of mm spatially and 10–20 m/s in velocity—is typically required to yield sufficiently long interaction times with the target beam or sample, and to ensure the capture of a significant part of the molecular beam pulse that is available for scattering. In addition, a large longitudinal velocity acceptance allows for the application of advanced phase-space manipulation techniques such as bunch compression and longitudinal cooling to further improve the resolution of the experiment [@Crompvoets:PRL89:093004]. By contrast, much narrower distributions are desired in the transverse directions. Here, the spatial diameter of the beam should be matched to the size of the target beam and the detection volume; typically a diameter of several mm is sufficient. Finally, the transverse velocity distribution should be narrow to minimize the divergence of the beam. These desiderata on beam distributions are unfortunately not met by traveling wave decelerators, where the resulting longitudinal (spatial) distributions are smaller and the transverse distributions are larger than what may be considered ideal for scattering experiments.
Here, we describe a new concept for a multi-stage Zeeman decelerator that is optimized for applications in scattering experiments. The decelerator consists of an array of alternating magnetic hexapoles and solenoids, used to effectively decouple the longitudinal and transverse motions of the molecules inside the decelerator. We analyze in detail the performance of the decelerator using numerical trajectory calculations, and we will show that the decelerator exhibits phase stability, with a spatial and velocity acceptance that is much larger in the longitudinal than in the transverse directions. We show that the decelerator is able to both decelerate and accelerate, as well as to guide a packet of molecules through the decelerator at constant speed. We present the successful experimental implementation of the concept, using a multi-stage Zeeman decelerator consisting of 24 solenoids and 25 hexapoles. The performance of the decelerator in acceleration, deceleration and guiding modes is characterized using a beam of metastable helium atoms.
In the decelerator presented here, we use copper capillary material in a new type of solenoid that allows for direct contact of the solenoid material with cooling liquid. The solenoid is easily placed inside vacuum, it offers excellent thermal properties and it allows for the use of low-voltage electronic components that are readily available and cost effective. Together, this results in a multi-stage Zeeman decelerator that is relatively easy and cheap to build, and that can be operated at repetition rates up to 10 Hz.
This paper is organized as follows. In section \[sec:concept\] we first describe the concept of the multi-stage Zeeman decelerator and characterize its inherent performance with numerical simulations. For this, we use decelerators of arbitrary length and the NH ($X\,^3 \Sigma^-$) radical as an example, as this molecule is one of our target molecules for future scattering experiments. In the simulations, we use the field geometry as induced by the experimentally proven solenoid used in the Zeeman decelerator at ETH Z[ü]{}rich [@Wiederkehr:JCP135:214202]. In section \[sec:experiment\], we describe in a proof-of-principle experiment the successful implementation of the concept. Here, we use metastable helium atoms, as this species can be decelerated significantly using the relatively short decelerator presently available.
Zeeman decelerator concept and design {#sec:concept}
=====================================
The multi-stage Zeeman decelerator we propose consists of a series of alternating hexapoles and solenoids, as is shown schematically in Figure \[fig:mode-explain\]. The length of the hexapoles and solenoids are almost identical. To simulate the magnetic field generated by the solenoids, we choose parameters that are similar to the ones used in the experiments by Wiederkehr *et al.* [@Wiederkehr:JCP135:214202]. We assume a solenoid with a length of 7.5 mm, an inner and outer diameter of 7 and 11 mm, respectively, through which we run maximum currents of 300 A. Furthermore, we set the inner diameter to 3 mm for molecules to pass through. These solenoids can, for instance, be produced by winding enameled wire in multiple layers, and the current through these solenoids can be switched using commercially available high-current switches. With these levels of current, this solenoid can create a magnetic field strength on the molecular beam axis as shown in Figure \[fig:coilhexafield\]*a*; the radial profiles of the field strength at a few positions $z$ along the beam axis are shown in panel *b*. It is shown that the solenoid creates a concave field distribution near the center of the solenoid, whereas a mildly convex shape is produced outside the solenoid.
The hexapoles have a length of 8.0 mm, are separated by a distance $D=4$ mm from the solenoids, and produce a magnetic field that is zero on the molecular beam axis but that increases quadratically as a function of the radial off-axis position $r$ (see Figure \[fig:coilhexafield\]*c*). We assume that the maximum magnetic field strength amounts to 0.5 T at a radial distance $r=1.5$ mm from the beam axis. Such magnetic field strengths are readily produced by arrangements of current carrying wires, permanent magnets [@Watanabe:EPJD38:219; @Osterwalder:EPJ-TI2:10], or a combination of both [@Poel:NJP17:055012].
The key idea behind this Zeeman decelerator concept is to effectively decouple the longitudinal and transverse motions of the molecules inside the decelerator. The fields generated by the solenoids are used to decelerate or accelerate the beam, but their mild transverse focusing and defocusing forces are almost negligible compared to the strong focusing effects of the hexapoles. These hexapoles, in turn, hardly contribute to the longitudinal deceleration forces. As we will discuss more quantitatively in the next sections, this stabilizes molecular trajectories and results in phase stability.
Decelerators in which dedicated and spatially separated elements are used for transverse focusing and longitudinal deceleration have been considered before [@Kalnins:RSI73:2557; @Sawyer:EPJD48:197]. In charged particle accelerators, such separation is common practice, and the detrimental effects of elements that affect simultaneously the longitudinal and transverse particle motions are well known [@Lee:AccPhys:2004]. The insertion of focusing elements between the mechanically coupled deceleration electrodes in a Stark decelerator appears technically impractical, however. By contrast, the relatively open structure of individually connected solenoids in a Zeeman decelerator allows for the easy addition of focusing elements. In addition, magnetic fields generated by adjacent elements are additive; shielding effects of nearby electrodes that are a common problem when designing electric field geometries do not occur.
The insertion of hexapoles further opens up the possibility to operate the Zeeman decelerator in three distinct modes that allow for either deceleration, acceleration, or guiding the molecular packet through the decelerator at constant speed. These operation modes are schematically illustrated in the lower half of Figure \[fig:mode-explain\]. In the description of the decelerator, we use the concepts of an equilibrium phase angle $\phi_0$ and a synchronous molecule from the conventions used to describe Stark decelerators [@Bethlem:PRL83:1558; @Bethlem:PRA65:053416]. The definition of $\phi_0$ in each of the modes is illustrated in Figure \[fig:mode-explain\], where zero degrees is defined as the relative position along the beam axis where the magnetic field reaches half the strength it has at the solenoid center. In deceleration mode, the solenoids are switched on before the synchronous molecule arrives in the solenoid, and switched off when the synchronous molecule has reached the position corresponding to $\phi_0$. In acceleration mode, the solenoid is switched on when the synchronous molecules has reached the position corresponding to $\phi_0$, and it is only switched off when the synchronous molecule no longer experiences the field induced by the solenoid. In hybrid mode, two adjacent solenoids are simultaneously activated to create a symmetric potential in the longitudinal direction. For this, each solenoid is activated twice: once when the synchronous molecule approaches, and once when the synchronous molecule exits the solenoid.
In this description we neglected the nonzero switching time of the current in the solenoids. In our decelerator, however, the current pulses feature a rise time of about 8 $\mu$s, as will be explained in more detail in section \[subsec:simulations\]. In the simulations, the full current profile is taken into account; we will adopt the convention that the current has reached half of the maximum value when the synchronous particle reaches the $\phi_0$ position. This switching protocol ensures that in hybrid mode with $\phi_0=0^{\circ}$, the molecules will receive an equal amount of acceleration and deceleration, in analogy with operation of a Stark decelerator with $\phi_0=0^{\circ}$.
The kinetic energy change $\Delta K$ that the synchronous molecule experiences per stage is shown for each mode in Figure \[fig:acceptance-overview\]*a*. In this calculation we assume the NH radical in its electronic ground state, that has a 2-$\mu_B$ magnetic dipole moment (*vide infra*). In the deceleration and acceleration modes, the full range of $\phi_0$ ($-90^{\circ}$ to $90^{\circ}$) can be used to reduce and increase the kinetic energy, respectively. In hybrid mode, deceleration and acceleration are achieved for $0^{\circ} < \phi_0 \leq 90^{\circ}$ and $-90^{\circ}\leq \phi_0 < 0^{\circ}$, respectively, whereas the packet is transported through the decelerator at constant speed for $\phi_0=0^{\circ}$. The maximum value for $\Delta K$ that can be achieved amounts to approximately 1.5 cm$^{-1}$.
Numerical trajectory simulations {#subsec:simulations}
--------------------------------
The operation characteristics of the Zeeman decelerator are extensively tested using numerical trajectory simulations. In these simulations, it is essential to take the temporal profile of the current pulses into account. Unless stated otherwise, we assume single pulse profiles as illustrated in Figure \[fig:NHZeemanshift\]*a*. The current pulses feature a rise time of approximately 8 $\mu$s, then a variable hold-on time during which the current has a constant value of 300 A. The current exponentially decays to a lingering current of 15 A with a characteristic decay time of 5 $\mu$s, as can be created by switching the current to a simple resistor in the electronic drive unit. This lingering current is only switched off at much later times, and is introduced to prevent Majorana transitions as will be explained in section \[subsec:Majorana\]. Furthermore, we assume that the hexapoles are always active when molecules are in their proximity.
In these simulations, we use NH radicals in the $X\,^3\Sigma^-, N=0, J=1$ rotational ground state throughout. The Zeeman effect of this state is shown in Figure \[fig:NHZeemanshift\]*b*. NH radicals in the low-field seeking $M=1$ component possess a magnetic moment of 2 $\mu_B$, and experience a linear Zeeman shift. NH radicals in this state have a relatively small mass-to-magnetic moment ratio of 7.5 amu/$\mu_B$, making NH a prime candidate for Zeeman deceleration experiments. Our findings are easily translated to other species by appropriate scaling of this ratio, in particular for species that also have a linear Zeeman shift (such as metastable helium, for instance).
The inherent 6D phase-space acceptance of the decelerator is investigated by uniformly filling a block-shaped area in 6D phase-space, and by propagating each molecule within this volume through a decelerator that consists of 100 solenoids and 100 hexapoles. In the range of negative $\phi_0$ in deceleration mode and positive $\phi_0$ for acceleration mode we instead used 200 pairs of solenoids and hexapoles to spatially separate the molecules within the phase stable area from the remainder of the distribution. This is explained in the appendix. The uniform distributions are produced using six unique Van der Corput sequences [@Corput:PAWA38:813]. For each of the three operation modes, the resulting longitudinal phase-space distributions of the molecules in the last solenoid of the decelerator are shown in Figure \[fig:phasespace3D\] for three different $\phi_0$. The separatrices that follow from the 1D model for phase stability that explicitly takes the temporal profiles of the currents into account, as described in detail by Dulitz *et al.* [@Dulitz:PRA91:013409], are given as a cyan overlay.
In each simulation, the synchronous molecule has an initial velocity chosen such that the total flight time is approximately 4.8 ms. This results in velocity progressions of $[370 \rightarrow 625]$, $[390 \rightarrow 599]$ and $[421 \rightarrow 568]$ m/s in acceleration mode with $\phi_0=-60^{\circ}, -30^{\circ}$ and $0^{\circ}$, respectively; a progression of $[445 \rightarrow 550]$, $[500 \rightarrow 500]$ and $[550 \rightarrow 447]$ m/s in hybrid mode with $\phi_0=-30^{\circ}, 0^{\circ}$ and $30^{\circ}$; and finally a progression of $[570 \rightarrow 421]$, $[595 \rightarrow 399]$ and $[615 \rightarrow 383]$ m/s in deceleration mode corresponding to $\phi_0=0^{\circ}, 30^{\circ}$ and $60^{\circ}$.
It is shown that in all operation modes and for all values of $\phi_0$, the separatrices accurately describe the longitudinal acceptances of the decelerator. For larger values of $|\phi_0|$, the sizes of the separatrices are reduced, reflecting the smaller size and depth of the effective time-averaged potential wells. Note the symmetric shape of the separatrix when the decelerator is operated in hybrid mode with $\phi_0 = 0^{\circ}$, corresponding to guiding of the packet through the decelerator at constant speed. The transmitted particle density is slightly less in hybrid mode than in other modes, which indicates that the transverse acceptance is not completely independent of the solenoid fields. However, in each mode of operation the regions in phase-space accepted by the decelerator are homogeneously filled; no regions with a significantly reduced number of molecules are found. This is a strong indication that the decelerator indeed features phase stability.
The transverse acceptance is found to be rather independent of $\phi_0$, and is shown in Figure \[fig:transspace3D\] for $\phi_0=0^{\circ}$ only. It can be seen that the transverse acceptance is typically smaller than the longitudinal acceptance, in accordance with our desideratum for molecular beam scattering experiments. Note that the transverse (velocity) acceptance can be modified independently from the deceleration and acceleration properties of the decelerator, simply by adjusting the field strength of the hexapoles.
Additionally, trajectory simulations can be used to quantify the overall 6D acceptance of the decelerator. Because of the uniform initial distribution, all particles that are propagated represent a small but equal volume in phase-space. At the end of the decelerator, the particles within a predefined range with respect to the synchronous particle are counted, yielding the volume in phase-space occupied by these particles. In the simulations, the initial “block” distribution is widened until the number of counted particles increases no further. We define the corresponding phase-space volume as the acceptance of the decelerator. The resulting 6D acceptance is shown for each operation mode in panel *b* of Figure \[fig:acceptance-overview\].
Operating in hybrid mode results in the typical triangle-shaped acceptance curve as a function of $\phi_0$ that is also found for Stark decelerators. A maximum 6D phase-space acceptance of approximately $1.2 \cdot 10^6$ mm$^3$ (m/s)$^3$ is found for $\phi_0=0^{\circ}$, and drops below $10^5$ mm$^3$ (m/s)$^3$ at large $|\phi_0|$. A peculiar effect is seen in the deceleration and acceleration modes for $\phi_0<0^{\circ}$ and $\phi_0>0^{\circ}$, respectively. Here, the acceptance largely exceeds the acceptance for $\phi_0=0^{\circ}$, and approaches values of $6 \cdot 10^6$ mm$^3$ (m/s)$^3$. This is a special consequence of the continuously acting focusing forces of the hexapoles, and will be discussed in more detail in the Appendix.
Although one has to be careful to derive the merits of a decelerator from the 6D phase-space acceptance alone, it is instructive to compare these numbers to the phase-space acceptances found in other decelerators. Conceptually, the hybrid mode of our Zeeman decelerator is compared best to the $s=3$ mode of a Stark decelerator. For the latter, Scharfenberg *et al.* found a maximum phase-space acceptance of $3 \cdot 10^5$ mm$^3$ (m/s)$^3$ for OH ($X\,^2\Pi_{3/2}, J=3/2$) radicals, with a similar partitioning of this acceptance between the longitudinal and transverse coordinates as found here [@Scharfenberg:PRA79:023410]. In comparison, for a multi-stage Zeeman decelerator without hexapoles, Wiederkehr *et al.* found that the 6D acceptance peaks at about $2 \cdot 10^3$ mm$^3$ (m/s)$^3$ for Ne ($^3P_2$) atoms when equilibrium phase angles in the range $30^{\circ}$–$45^{\circ}$ degrees are used [@Wiederkehr:JCP135:214202]. The acceptance of the multi-stage Zeeman decelerator developed by Raizen and coworkers, also referred to as a magnetic coilgun, was reported to have an upper limit of $10^5$ mm$^3$ (m/s)$^3$ [@Narevicius:ChemRev112:4879]. The highest 6D acceptances to date are found in traveling wave decelerators, mostly thanks to the large transverse acceptances of these decelerators. The maximum acceptance of the traveling wave Zeeman decelerator of Narevicius and coworkers, for instance, amounts to $2 \cdot 10^7$ mm$^3$ (m/s)$^3$ for Ne ($^3P_2$) atoms [@Lavert-Ofir:PCCP13:18948].
Phase stability
---------------
The numerical trajectory simulations yield very strong indications that the molecules are transported through the Zeeman decelerator without loss, i.e., phase stable operation is ensured. We support this conjecture further by considering the equation of motion for the transverse trajectories, using a model that was originally developed to investigate phase stability in Stark decelerators [@Meerakker:PRA73:023401]. In this model, we consider a (nonsynchronous) molecule with initial longitudinal position $z_i$ relative to the synchronous molecule, which oscillates in longitudinal phase-space around the synchronous molecule with longitudinal frequency $\omega_z$. In other words, during this motion the relative longitudinal coordinate $\phi$ oscillates around the synchronous value $\phi_0$. In the transverse direction, the molecule oscillates around the beam axis with transverse frequency $\omega_r$, which changes with $\phi$. In Figure \[fig:frequencies\], the longitudinal and transverse oscillation frequencies are shown that are found when the Zeeman decelerator is operated in hybrid mode with $\phi_0=0^{\circ}$. For deceleration and acceleration modes rather similar frequencies are found (data not shown). It can be seen that the transverse oscillation frequency largely exceeds the longitudinal oscillation frequency. As we will show below, this eliminates the instabilities that has deteriorated the phase-space acceptance of multi-stage Stark and Zeeman decelerators in the past [@Meerakker:PRA73:023401; @Sawyer:EPJD48:197; @Wiederkehr:JCP135:214202; @Wiederkehr:PRA82:043428].
During its motion, a molecule experiences a time-dependent transverse oscillation frequency that is given by [@Meerakker:PRA73:023401]: $$\omega_r(t) = \omega^2_0-A \cos(2\omega_z t),
\label{eq:trans}$$ where $\omega_0$ and $A$ are constants that characterize the oscillatory function. The resulting transverse equation of motion is given by the Mathieu differential equation: $$\frac{d^2 r}{d\tau^2}+[a-2q\cos(2\tau)]r=0,$$ with: $$a=\left(\frac{\omega_0}{\omega_z}\right)^2, \qquad q=\frac{A}{2\omega^2_z}, \qquad \tau=\omega_z t.
\label{eq:param}$$ Depending on the values of $a$ and $q$, the solution of this equation exhibits stable or unstable behavior. This is illustrated in Figure \[fig:stability\] that displays the Mathieu stability diagram. Stable and unstable solutions exist for combinations of $a$ and $q$ within the white and gray areas, respectively. For each operation mode of the decelerator, and for a given phase angle $\phi_0$, the values for the parameters $a$ and $q$ can be determined from the longitudinal and transverse oscillation frequencies of Figure \[fig:frequencies\]. The resulting values for the parameters $q$ and $a$ as a function of $z_i$ are shown in panel (*a*) and (*b*) of Figure \[fig:stability\], for the decelerator running in hybrid mode with $\phi_0=0^{\circ}$. The ($a,q$) combinations that govern the molecular trajectories for this operation mode are included as a solid red line in the stability diagram shown in panel (*c*). Clearly, the red line circumvents all unstable regions, and only passes through the unavoidable “vertical tongues” where they have negligible width. These narrow strips do not cause unstable behavior for decelerators of realistic length. The unstable areas in the Mathieu diagram are avoided because of the high values of the parameter $a$. This same result was found for the other operation modes and equilibrium phase angles. We thus conclude that the insertion of hexapoles effectively decouples the transverse motion from the longitudinal motion; the Zeeman decelerator we propose is inherently phase stable, and can in principle be realized with arbitrary length.
Prevention of Majorana losses {#subsec:Majorana}
-----------------------------
An important requirement in devices that manipulate the motion of molecules using externally applied fields, is that the molecules remain in a given quantum state as they spend time in the device. As the field strength approaches zero, different quantum states may become (almost) energetically degenerate, resulting in a possibility for nonadiabatic transitions. This may lead to loss of particles, which is often referred to as Majorana losses.
The occurrence of nonadiabatic transitions has been studied extensively for neutral molecules in electric traps [@Kirste:PRA79:051401], as well as for miniaturized Stark decelerators integrated on a chip [@Meek:PRA83:033413]. Tarbutt and coworkers developed a theoretical model based on the time-dependent Hamiltonian for the field-molecule interaction, and quantitatively investigated the transition probability as the field strength comes close to zero, and/or if the field vector rotates quickly relative to the decelerated particles [@Wall:PRA81:033414].
In the multi-stage Zeeman decelerators that are currently operational, losses due to nonadiabatic transitions can play a significant role [@Hogan:PRA76:023412]. Specifically, when switching off a solenoid right as the particle bunch is near the solenoid center, there will be a moment in time where no well-defined magnetic quantization field is present. In previous multi-stage Zeeman decelerator designs, this was compensated by introducing a temporal overlap between the current pulses of adjacent solenoids, effectively eliminating nonadiabatic transitions [@Hogan:PRA76:023412]. In the Zeeman decelerator concept presented in this manuscript, this solution is not available, since adjacent solenoids are separated by hexapole elements. The hexapoles induce only marginal fringe fields, and do not contribute any magnetic field strength on the molecular beam axis.
Referring back to Figure \[fig:NHZeemanshift\]*a*, we introduce a quantization field throughout the hexapole-solenoid array by switching each solenoid to a low-level lingering current when the high current pulse is switched off. Since the fringe field of a solenoid extends beyond the geometric center of adjacent hexapoles, and since in the center of the solenoid the maximum magnetic field per unit of current is created, a lingering current of approximately 15 A is sufficient to provide a minimum quantization field of 0.1 T. The resulting sequences of current profiles through the solenoids with number $n$, $n+1$ and $n+2$ are shown in the upper half of Figure \[fig:Majorana-currents\] for the deceleration (panel *a*) and hybrid modes (panel *b*). The profiles for acceleration mode are not shown here, but they feature the low current before switching to full current, instead of a low current after. The lingering current exponentially decays to its final value, and lasts until the next solenoid is switched off. In panels *c* and *d* the corresponding magnetic field strength is shown that is experienced by the synchronous molecule as it propagates through the decelerator (blue curves), together with the field that would have resulted if the solenoid were switched off with a conventional ramp time (red curves). Clearly, the low level current effectively eliminates the zero-field regions.
From model calculations similar to the ones developed by Tarbutt and coworkers [@Wall:PRA81:033414], we expect that the magnetic field vector inside the solenoids will not rotate fast enough to induce nonadiabatic transitions, provided that all solenoid fields are oriented in the same direction. We therefore conclude that the probability for nonadiabatic transitions is expected to be negligible for the Zeeman decelerator concept proposed here.
One may wonder how the addition of the slowly decaying lingering current affects the ability to efficiently accelerate or decelerate the molecules. This is illustrated in panels (*e*) and (*f*) of Figure \[fig:Majorana-currents\] that displays the acceleration rate experienced by the synchronous molecule. The acceleration follows from $-(\vec{\triangledown} U_{\textrm{Z}})/m$, where $U_{\textrm{Z}}$ is the Zeeman energy for NH ($X\,^3\Sigma^-, N=0, J=1, M=1$) induced by the time-varying magnetic field B(T), and $m$ is the mass of the NH radical. It is seen that the lingering current only marginally affects the acceleration force; a slight additional deceleration at early times is compensated by a small acceleration when the synchronous molecule exits the solenoid. Overall, the resulting values for $\Delta K$ with or without lingering current, as obtained by integrating the curves in panels (*e*) and (*f*), are almost identical (data not shown).
Excessive focusing at low velocities {#subsec:low-velocities}
------------------------------------
A common problem in multi-stage decelerators is the occurrence of losses due to excessive focusing at low forward velocities. This effect has been studied and observed in multi-stage Stark decelerators that operate in the $s=1$ or $s=3$ modes, where losses occur below approximately 50 or 150 m/s, respectively [@Sawyer:EPJD48:197; @Scharfenberg:PRA79:023410]. Our concept for a multi-stage Zeeman decelerator shares these over-focusing effects at low final velocities, which may be considered a disadvantage over traveling wave decelerators, which are phase stable down to near-zero velocities.
At relatively high velocities, the hexapole focusing forces can be seen as a continuously acting averaged force, keeping the molecules confined to the beam axis. However, at low velocities this approximation is no longer valid, and the molecules can drift from the beam axis between adjacent hexapoles. We investigate the expected losses using similar numerical trajectory simulations as discussed in section \[subsec:simulations\], i.e., we again assume a Zeeman decelerator consisting of 100 hexapole-solenoid-pairs. We assume packets of molecules with five different mean initial velocities ranging between $v_{\textrm{in}}=$ 350 m/s and 550 m/s, and these packets are subsequently propagated through the decelerator. The decelerator is operated in hybrid mode, and can be used with different values for $\phi_0$. Since we assume a 100-stage decelerator throughout, the packets emerge from the decelerator with different final velocities.
In Figure \[fig:over-focusing\] we show the number of decelerated particles that are expected at the end of the decelerator as a function of $\phi_0$ (panel *a*), or as a function of the final velocity (panel *b*). For low values of $\phi_0$, the transmitted number of molecules is (almost) equal for all curves; the slightly higher transmission for higher values of $v_{\textrm{in}}$ is related to the shorter flight time of the molecules in the decelerator. Consequently, molecules that are not within the inherent 6D phase-space acceptance of the decelerator can still make it to the end of the decelerator, and are counted in the simulations. For higher values of $\phi_0$, the transmitted number of molecules decreases, reflecting the reduction of the phase-space acceptance for these phase angles. This is particularly clear for the blue and green curves ($v_{\textrm{in}}=$ 550 and 500 m/s, respectively), which follow the 6D phase-space acceptance curve from Figure \[fig:acceptance-overview\]*b*. The three other curves feature a drop in transmission that occurs when the velocity drops below approximately 160 m/s, as is indicated by the dashed vertical lines. Obviously, for lower values of $v_{\textrm{in}}$, this velocity is reached at lower values of $\phi_0$ (see panel (*a*)).
The production of final velocities below this drop-off velocity is not a prime requirement in crossed beam scattering experiments, as the collision energy is determined by the velocities of both beams and the crossing angle between the beams. Very low collision energies can be reached using small crossing angles, relaxing the requirements on the final velocities of the reagent beams. For these applications we therefore see no direct need to combat these over-focusing effects. However, there are several promising options to mitigate these effects if desired. The first option is to employ hexapoles with a variable strength, such that the transverse oscillation frequency can be tuned along with the decreasing velocity of the molecular packet. Similarly, permanent hexapoles with different magnetization can be installed to modify the focusing properties. Finally, it appears possible to merge a hexapole and solenoid into a single element, by superimposing a hexapole arrangement on the outer diameter of the solenoid. Although technically more challenging, this approach will provide an almost continuously acting transverse focusing force, while keeping the possibility to apply current pulses to the solenoids. Preliminary trajectory simulations suggest that indeed a significant improvement can be achieved, but the validity of these approaches will need to be investigated further if near-zero final velocities are required.
Experimental implementation {#sec:experiment}
===========================
Multi-stage Zeeman decelerator
------------------------------
An overview of the experiment is shown in Figure \[fig:schematic\_setup\]. The generation and detection of the metastable helium beam will be discussed in section \[subsec:He\]; in this section we will first describe the decelerator itself, starting with a description of the solenoids and associated electronics.
An essential part in a multi-stage Zeeman decelerator is the design of the deceleration solenoids, and the cooling strategy to remove the dissipated energy. A variety of solenoid designs have been implemented successfully in multi-stage Zeeman decelerators before. Merkt and coworkers utilized tightly-wound solenoids of insulated copper wire that were thermally connected to water-cooled ceramics [@Vanhaecke:PRA75:031402]. Later, similar solenoids were placed outside a vacuum tube, and submerged in a bath of cooling water [@Hogan:JPB41:081005]. This improved the cooling capacity, and enabled the experiment to operate at repetition rates of 10 Hz. Raizen and coworkers also developed a multi-stage Zeeman decelerator, referred to as the atomic or molecular coilgun, that is based on solenoids encased in high permeability material to increase the on-axis maximum magnetic field strength [@Narevicius:PRA77:051401; @Liu:PRA91:021403]. Recently, different types of traveling wave Zeeman decelerators have been developed, which consist of numerous spatially overlapping quadrupole solenoids [@Lavert-Ofir:PCCP13:18948], or a helical wire arrangement to produce the desired magnetic field [@Trimeche:EPJD65:263].
In the decelerator presented here, we use a new type of solenoid that is placed inside vacuum, but that allows for direct contact of the solenoid material with cooling liquid. The solenoids consist of 4 windings of a copper capillary that is wound around a 3 mm bore diameter. The capillary has an inner diameter of 0.6 mm and an outer diameter of 1.5 mm, and cooling liquid is circulated directly through the capillary. The solenoid is wound such that the first and last windings end with a straight section of the capillary, as is shown in a photograph of a single solenoid in Figure \[fig:setup\_photo\]*b*. These straight sections are glued into an aluminum mounting flange, as will be further discussed later. The inherent magnetic field profile generated by this solenoid is very similar to the solenoids as used in the simulations presented in section \[sec:concept\].
The use of a single layer of rather thick copper capillary as solenoid material in a Zeeman decelerator is unconventional, but it has some definite advantages. Because of the low-resistance copper capillary, small operating voltages (24 V) are sufficient to generate currents of approximately 4.5 kA that produce a maximum field of 2.2 T on the solenoid axis. This in turn allows for the use of FET-based electronics components to switch these currents, which are considerably cheaper than their high voltage IGBT-based counterparts. The same holds for the power supplies that deliver the current. By running cooling liquid directly through the solenoid capillary, the solenoids are efficiently cooled. The low operation voltage ensures that the cooling liquid does not conduct any significant electricity.
The current pulses are provided by specially designed circuit boards; one such board is displayed in Figure \[fig:electronics\]*b*. Each solenoid is connected to a single board, that is mounted directly onto the solenoid-flange feedthroughs in order to minimize power loss between board and solenoid. Brass strips are used to mechanically clamp the board to the capillary material. The simplified electronic circuit is shown schematically in Figure \[fig:electronics\]*a*. The circuit board is mostly occupied by a parallel array of capacitors, with a total capacitance of 70 mF. The capacitors are charged by a 24 V power supply and then discharged through the connected solenoid. The solenoids have a very low resistance $R_C$ of about 1 m$\Omega$ and self-inductance $L_C$ of about 50 nH, even compared to the electronic circuit itself. The capacitors are discharged via two possible pathways indicated in red and green, respectively, by activating the two independent gates S1 and S2. Closing gate S1 will allow electrons to flow through the solenoid, generating a maximum current of about 4.5 kA. Closing gate S2 will send the flow through both the solenoid and a 100 m$\Omega$ resistor that limits the current to about 150 A. When both gates are opened any remaining power in the solenoid will either dissipate in the electrical components along pathway 3 (in blue) or return to the capacitors. The electronic configuration is able to apply up to two consecutive pulses to each solenoid, as is required for the hybrid mode of operation.
As an example, the current profiles for a single pulse or double pulse are shown in Figure \[fig:electronics\]*c* and \[fig:electronics\]*d*, respectively, together with the trigger pulses that activate gates S1 and S2. These profiles were obtained from the induced voltage over a miniature solenoid that was placed inside the center of a decelerating solenoid [@Wiederkehr:JCP135:214202]. The current pulse is initiated by closing gate S1, after which the solenoid current shows a rapid rise to a maximal current of approximately 4.5 kA. After reopening gate S1, the current exponentially decreases with a time constant of 10 $\mu$s. Gate S2 is programmed to close automatically for a fixed duration of 50 $\mu$s, starting 30 $\mu$s after the reopening of S1. While gate S2 is closed, a low-level lingering current is maintained in the solenoid to prevent Majorana transitions (see section \[subsec:Majorana\]), providing a quantization field for atoms or molecules that are near the solenoid.
The solenoids and electronics boards are actively cooled using a closed-cycle cooling system. An approximately 10-cm-long capillary section is soldered onto each electronics board, and each capillary is connected in series to its connecting solenoid using silicon tubes. All board-solenoid pairs are individually connected to a mains and return cooling line, using the same flexible silicon material. Each electronics board is additionally cooled by a small fan. Using this cooling system, relatively low operation temperatures are maintained despite the high currents that are passed through the solenoid. In the experiments shown here, the Zeeman decelerator is routinely operated with a repetition frequency of 10 Hz, while the temperature of the solenoids is kept below 40 degrees Celsius.
The solenoids are pulsed in a predefined time sequence designed to control the longitudinal velocity of a specific paramagnetic particle. This time sequence is calculated while taking current profiles into account that are modeled after the measured profiles shown in Figure \[fig:electronics\]*b*. The resulting pulse sequence for gates S1 and S2 is programmed into a pattern generator (Spincore PulseBlaster PB24-100-4k), which sends pulse signals to each individual circuit board. The temperature of each solenoid is continuously monitored via a thermocouple on the connecting clamps of the circuit board. When the temperature of the solenoid exceeds a user-set threshold, operation of the decelerator is interrupted.
The magnetic hexapoles consist of six wedge-shaped permanent magnets in a ring, as seen in Figure \[fig:setup\_photo\]*a*. Adjacent magnets in the ring have opposite radial remanence. The inner diameter of the hexapole is 3 mm and the length is 8 mm, such that these dimensions match approximately to the corresponding solenoid dimensions. The magnets used in this experiment are based on NdFeB (grade N42SH) with a remanence of approximately 450 mT. The advantage of using hexapoles consisting of permanent magnets is twofold: first, implementation is mechanically straightforward, and second, no additional electronics are needed to generate the focusing fields. However, this approach lacks any tunability of the field strength. This can in part be overcome by selectively removing hexapoles from the decelerator, or by exchanging the magnets for ones with a different magnetization. If required, electromagnetic hexapoles that allow for tunability of the field strength can be used instead. We have built and successfully operated hexapoles that are made of the solenoid capillary material, and could optimize their focusing strength by simply adjusting the time these hexapoles are switched on. However, we found that similar beam densities were achieved using the permanent hexapoles, and experiments with electromagnetic hexapoles are not further discussed here.
The decelerator contains 24 solenoids and 25 hexapoles that are placed with a center-to-center distance of 11 mm inside a vacuum chamber. The chamber consist of a hollow aluminum block of length 600 mm with a squared cross section of side lengths 40 mm. This chamber is made by machining the sides of standard aluminum pipe material with an inner diameter of 20 mm. Solenoids and hexapoles are mounted on separate flanges, as can be seen in Figure \[fig:setup\_photo\], such that each element can be installed or removed separately. The first and last element of the decelerator is a hexapole to provide transverse focusing forces at the entrance and exit of the decelerator, respectively. Openings for the individual flanges on the decelerator housing spiral along the sides between subsequent elements, with clockwise 90 degree rotations. In this way there is enough space on each side of the decelerator to accommodate the electronics boards of the solenoids, which have a 42 mm height. In addition, since subsequent solenoids are rotated by 180 degrees in the decelerator, any asymmetry in the magnetic field because of the relatively coarse winding geometry is compensated.
Vacuum inside the decelerator housing is maintained by a vacuum pump installed under the detection chamber, which has an open connection to the decelerator housing. Only a minor pressure increase in the chamber is observed if the solenoids are operational, reflecting the relatively low operational temperature of the solenoids. Although for long decelerators additional pumping capacity inside the decelerator is advantageous, we find that for the relatively short decelerator used here the beam density is hardly deteriorated by collisions with background gas provided the repetition rate of the experiment is below 5 Hz. Under these conditions, the pressure in the decelerator maintains below $5 \cdot 10^{-7}$ mbar.
Metastable helium beam {#subsec:He}
----------------------
A beam of helium in the metastable (1s)(2s) $^3S$ ($m_S = 1$) state (from this point He\*) was used to test the performance of the Zeeman decelerator. This species was chosen for two main reasons. First, He\* has a small mass-to-magnetic-moment ratio (2.0 amu/$\mu_B$) with a large Zeeman shift, which allows for effective manipulation of the atom with magnetic fields. This allows us to significantly vary the mean velocity of the beam despite the relatively low number of solenoids. Second, He\* can be measured directly with a micro-channel plate (MCP) detector, without the need for an ionizing laser, such that full time-of-flight (TOF) profiles can be recorded in a single shot. This allows for a real-time view of TOF profiles when settings of the decelerator are changed, and greatly facilitates optimization procedures.
The beam of He\* is generated by expanding a pulse of neat He atoms into vacuum using a modified Even-Lavie valve (ELV) [@Even:JCP112:8068] that is cooled to about 16 K using a commercially available cold-head (Oerlikon Leybold). At this temperature the mean thermal velocity of helium is about 460 m/s. The ELV nozzle is replaced by a discharge source consisting of alternating isolated and conducting plates, similar to the source described by Ploenes *et al.* [@Ploenes:RSI87:053305]. The discharge occurs between the conducting plates, where the front plate is kept at -600 V and the back plate is grounded. To ignite the discharge, a hot filament running 3 A of current is used. The voltage applied to the front plate is pulsed (20-30 $\mu$s duration) to reduce the total energy dissipation in the discharge. Under optimal conditions, a beam of He\* is formed, with a mean velocity just above 500 m/s. Unless stated otherwise, in the experiments presented here, the decelerator is programmed to select a packet of He\* with an initial velocity of 520 m/s.
The beam of He\* passes through a 3 mm diameter skimmer (Beam Dynamics, model 50.8) into the decelerator housing. The first element (a hexapole) is positioned about 70 mm behind the skimmer orifice. The beam is detected by an MCP detector that is positioned 128 mm downstream from the exit of the decelerator. This MCP is used to directly record the integrated signal from the impinging He\* atoms.
Results and Discussion {#sec:results}
======================
Longitudinal velocity control
-----------------------------
As explained in section \[sec:concept\], the decelerator can be operated in three distinct modes of operation: in deceleration or acceleration modes, the atoms are most efficiently decelerated or accelerated, respectively, whereas in the so-called hybrid mode of operation, the beam can be transported or guided through the decelerator at constant speed (some mild deceleration or acceleration is in principle also possible in this mode). In this section, we present experimental results for all three modes of operation.
We will start with the regular deceleration mode. In Figure \[fig:decelTOF\], TOF profiles for He\* atoms exiting the decelerator are shown that are obtained when the decelerator is operated in deceleration mode, using different values for the equilibrium phase angle $\phi_0$. In the corresponding pulse sequences, the synchronous atom is decelerated from 520 m/s to 365 m/s, 347 m/s and 333 m/s, corresponding to effective equilibrium phase angles of 30$^{\circ}$, 45$^{\circ}$ and 60$^{\circ}$, respectively. The corresponding loss of kinetic energy amounts to 23 cm$^{-1}$, 25 cm$^{-1}$ and 27 cm$^{-1}$. The arrival time of the synchronous atom in the graphs is indicated by the vertical green lines. Black traces show the measured profiles; the gray traces that are shown as an overlay are obtained when the decelerator was not operated, i.e., the solenoids are all inactive but the permanent hexapole magnets are still present to focus the beam transversely.
The experimental TOF profiles are compared with profiles that result from three dimensional trajectory simulations. In these simulations, an initial beam distribution is assumed that closely resembles the He\* pulse generated by the modified ELV. The resulting TOF profiles are shown in red, vertically offset from the measured profiles for clarity. The simulated profiles show good agreement with the experiment, both in relative intensity and arrival time of the peaks. However, it must be noted that the relative intensities are very sensitive to the chosen parameters of the initial He\* pulse. By virtue of the supersonic expansion and discharge processes, these distributions are often not precisely known, and may vary from day to day. Nevertheless, the agreement obtained here, in particular regarding the overall shape of the TOF profiles and the predicted arrival times of the decelerated beam, suggests that the trajectory simulations accurately describe the motion of atoms inside the decelerator. No indications are found for unexpected loss of atoms during the deceleration process, or for behavior that is not described by the simulations.
The profiles presented in Figure \[fig:decelTOF\] show more features than the decelerated packets alone. In particular, there is an additional decelerated peak in each of the graphs that is more intense but slightly faster than the decelerated packet. We use the three dimensional trajectory simulations to study the origin of this feature. In Figure \[fig:decelphase\], the longitudinal phase-space distributions are shown that result from these simulations at the entrance (upper panel), middle (central panel), and exit (lower panel) of the decelerator. The simulation pertains to the situation that results in the TOF profiles presented in Figure \[fig:decelTOF\]*a*, i.e., the decelerator is operated in deceleration mode with $\phi_0=30^{\circ}$.
In these phase-space distributions, the grey contour lines depict the predicted trajectories considering the time-averaged Zeeman potential energy. The separatrix of the stable phase-space is highlighted with a cyan overlay. From this evolution of the longitudinal phase-space distribution, we can understand the origin of various pronounced features in the TOF profiles. The first peak in each TOF profile is a collection of the fastest particles in the initial beam distribution. These particles are hardly affected by the solenoids, and propagate to the detector almost in free flight. However, the part of the beam that is initially slower than the synchronous molecule is strongly affected by the solenoids. This part eventually gains in velocity relative to the decelerated bunch, resulting in an ensemble of particles with a relatively high density. This part arrives at the detector just before the decelerated He\* atoms, resulting in the second intense peak in the TOF profiles of Figure \[fig:decelTOF\]. It is noted that this peak appears intense because our decelerator is rather short, leaving insufficient time for the decelerated bunch to fully separate from the initial beam distribution. For longer decelerators, the part of the beam that is not enclosed by the separatrix will gradually spread out, and its signature in the TOF profiles will weaken.
The phase-space distributions that are found at the end of the decelerator may also be used to determine the velocity width of the decelerated packet of atoms. For the examples of Figure \[fig:decelphase\], these widths are about 25 m/s.
For completeness, we also measure a TOF profile when the decelerator is operated in acceleration mode. Figure \[fig:accelTOF\]*a* shows the TOF profile for the acceleration of He\* atoms from an initial velocity of 560 m/s to a final velocity of 676 m/s. The simulated profile (red trace) shows good agreement with the experimental profile (black trace). Again, the vertical green line indicates the expected arrival time of the accelerated bunch. The sequence selects the fastest atoms in the beam, which is why no additional peaks are visible.
Finally, we study the performance of the decelerator in hybrid mode. This mode of operation allows for guiding of the beam through the decelerator at constant speed. In Figure \[fig:accelTOF\]*b* a TOF profile is shown when the decelerator is operated in hybrid mode and $\phi_0=0^{\circ}$, selecting an initial velocity of 520 m/s. The simulated TOF profile (red trace) again shows good agreement with the experimental TOF profile (black trace), although the intensity ratios between the guided part and the wings of the distributions are slightly different in the simulations than in the experiment. This is attributed to the idealized initial atom distribution that are assumed in the simulations.
Presence of metastable helium molecules
---------------------------------------
While our experiment is designed to decelerate He\* atoms in the $^3S$ state, other types of particles may be created in the discharged beam as well. Specifically, formation of metastable He$_2$ molecules in the a$^3\Sigma$ state (from here on He$_2$\*) is expected, as is also observed in the experiments by Motsch *et al.* and Jansen *et al.* that use a similar discharge source [@Motsch:PRA89:043420; @Jansen:PRL115:133202]. However, He$_2$\* is indistinguishable from He\* in our detection system. In order to probe both species separately, mass selective detection using a non-resonant laser ionization detection scheme is used. Ultraviolet (UV) laser radiation with a wavelength of 243 nm is produced by doubling the light from an Nd:YAG-pumped pulsed dye laser running with Coumarin 480 dye, and focused into the molecular beam close to the exit of the decelerator. The resulting ions are extracted with an electric field of about 1 kV/cm and accelerated towards an MCP detector, where the arrival time of the ions reflect their mass over charge ratio.
We used this detection scheme to investigate the chemical composition of the beam that exits the Zeeman decelerator. In Figure \[fig:Hespec\], ion TOF spectra (i.e., the arrival times of the ions at the MCP detector with respect to the laser pulse) are shown. The black trace shows the ion TOF spectrum if the beam of He\* atoms is passed through the decelerator without operating the solenoids. The UV laser is fired at the mean arrival time of the beam in the laser ionization region. Two peaks are clearly visible corresponding to the expected arrival time of He$^+$ and He$_2^+$, confirming that indeed He$_2$ molecules are created in the discharge. He atoms and molecules are detected in an 8:1 ratio in the neutral beam.
The green trace in Figure \[fig:Hespec\] shows the ion TOF spectra that is recorded when the solenoids are operated for a typical deceleration sequence similar to the ones used to generate Figure \[fig:decelTOF\]. This trace was taken when the UV laser selectively detects the decelerated part of the He\* beam. Here, only He$^+$ is present in the ion TOF spectrum. Although He$_2$\* has the same magnetic moment as He\* and will thus experience the same force, the double mass of the molecule results in only half the acceleration. He$_2$\* is therefore not decelerated at the same rate as He\*, and will not exit the decelerator at the same time as the decelerated He\* atoms. In conclusion, the Zeeman decelerator is quite effective in separating He\* from the He$_2$\*; the decelerated bunch only contains those species and/or particles in the quantum level for which the deceleration sequence was calculated.
Referring back to Figures \[fig:decelTOF\] and \[fig:accelTOF\] that were recorded without laser-based mass spectroscopic detection, one may wonder how the presence of He$_2$ molecules in the beam affect the recorded TOF profiles. Figure \[fig:Hematch\] revisits the measurement from Figure \[fig:decelTOF\]*a*, but taking also He$_2$ molecules into account with the appropriate ratio to generate the simulated TOF profile. The resulting TOF profile for He$_2$\* molecules is shown by the green trace, and is seen to fill the part of the TOF that was under represented by the original simulations (indicated by the vertical green arrow for the experimental trace).
Conclusions and Outlook
=======================
We have presented a new type of multi-stage Zeeman decelerator that is specifically optimized for scattering experiments. The decelerator consists of an array of alternating solenoids and hexapoles, that effectively decouples the longitudinal deceleration and transverse focusing forces. This ensures that phase-stable operation of the decelerator is possible over a wide range of velocities. For applications in scattering experiments, this decelerator concept has a number of advantages over existing and experimentally demonstrated Zeeman decelerators. The decelerator can be operated in three distinct modes that make either acceleration, deceleration, or guiding at constant speed possible, enabling the production of molecular packets with a continuously tunable velocity over a wide range of final velocities. Phase stability ensures that molecules can be transported through the decelerator with minimal loss, resulting in relatively high overall 6D phase-space acceptance. Most importantly, this acceptance is distributed unequally between the longitudinal and transverse directions. Both the spatial and velocity acceptances are much larger in the longitudinal than in the transverse directions, which meets the requirements for beam distributions in scattering experiments in an optimal way. At low final velocities, however, losses due to over-focusing occur. In crossed beam scattering experiments this appears inconsequential, but for trapping experiments—where low final velocities are essential—the use of the concept presented here should be carefully considered. We have discussed various promising options for combating these losses using alternative hexapole designs in the last section of the decelerator. Additionally, Zhang *et al.* recently proposed a new operation scheme in a Stark decelerator that optimizes the transmitted particle numbers and velocity distributions, which could potentially be translated to a Zeeman decelerator [@Zhang:PRA93:023408]. The validity of these approaches will need to be investigated further, especially if near-zero final velocities are required.
In a proof-of-principle experiment, we demonstrated the successful experimental implementation of a new concept presented here, using a decelerator that consist of 24 solenoids and 25 hexapoles. The performance of the decelerator was experimentally tested using beams of metastable helium atoms. Both deceleration, acceleration, and guiding of a beam at constant speed have been demonstrated. The experimental TOF profiles of the atoms exiting the decelerator show excellent agreement with the profiles that result from numerical trajectory simulations. Although the decelerator presented here is relatively short, up to 60% of the kinetic energy of He\* atoms that travel with an initial velocity of about 520 m/s could be removed.
In the Zeeman decelerator presented here, we utilize a rather unconventional solenoid design that uses a thick copper capillary through which cooling liquid is circulated. The solenoid design allows for the switching of high currents up to 4.5 kA, using readily available and cheap low-voltage electronics components. The design is mechanically simple, and can be built at relatively low cost. We are currently developing an improved version of the decelerator, that is fully modular, and which can be extended to arbitrary length. The modules can be connected to each other without mechanically disrupting the solenoid-hexapole sequence, while the housing design will allow for the installation of sufficient pumping capacity to maintain excellent vacuum conditions throughout the decelerator. Operation of the Zeeman decelerator consisting of 100 solenoids and 100 hexapoles at repetition rates up to 30 Hz appears technically feasible.
Acknowledgments
===============
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013/ERC grant agreement nr. 335646 MOLBIL). This work is part of the research program of the Netherlands Organization for Scientific Research (NWO). We thank Katrin Dulitz, Paul Janssen, Hansj[ü]{}rg Schmutz and Fr[é]{}d[é]{}ric Merkt for stimulating discussions on Zeeman deceleration, solenoid focusing properties and current switching protocols. We thank Rick Bethlem and Fr[é]{}d[é]{}ric Merkt for carefully reading the manuscript and for valuable suggestions for textual improvements. We thank Gerben Wulterkens for the design of prototypes.
Appendix: Extreme equilibrium phase angles in deceleration mode {#Appendix-unbound}
===============================================================
As can be seen in Figure \[fig:acceptance-overview\]*b*, the highest acceptance is found with $\phi_0 = -90^{\circ}$ in deceleration mode or $\phi_0 = 90^{\circ}$ in acceleration mode. This is a surprising result if we consider conventional multi-stage Zeeman decelerators. In these decelerators, the inherent transverse defocusing fields outside the solenoids prevent the effective use of these extreme values of $\phi_0$[@Wiederkehr:JCP135:214202]. However, with the addition of magnetic hexapoles this limitation no longer exists. Indeed, the total acceptance changes almost solely with the longitudinal acceptance. This acceptance increases in deceleration and acceleration mode with lower and higher $\phi_0$, respectively. We show this for deceleration mode in Figure \[fig:unbound\]*a*.
In the negative $\phi_0$ range of deceleration mode the solenoids are turned off early, resulting in only a small amount of deceleration per stage. This is reflected in the kinetic energy change with this mode shown in Figure \[fig:acceptance-overview\]*a*. With lower $\phi_0$, less of the slope of the solenoid field is used to decelerate, and more of it is available for longitudinal focusing of the particle beam.
Moreover, the minimum value of $\phi_0 = -90^{\circ}$ is an arbitrary limit, as even lower values of $\phi_0$ would produce even less deceleration and more longitudinal acceptance. Nevertheless, it is important to remember that this is a theoretical prediction, with the important assumption that the decelerator is of sufficient length that the slower particles have sufficient time to catch up with the synchronous particle. With less deceleration of the synchronous particle per solenoid, this catch-up time will increase. This is reflected in the difference in longitudinal phase-space distributions after 100 and 200 stages in Figures \[fig:unbound\] (b) and (c), respectively. In these simulations (similar to those shown in Figure \[fig:phasespace3D\]) a block distribution of NH($X\,^3\Sigma^-, N=0, J=1$) particles was used that well exceeded the predicted longitudinal separatrix. After 100 stages, the (deformed) corners of the initial block distribution are still visible as they revolve around the synchronous particle in longitudinal phase-space. Only after 200 stages of deceleration have these unaccepted particles had enough time to spatially separate from the particles with stable trajectories. This graph also shows that the prediction of the separatrix is quite accurate, and the uniformity of the particle distribution within is evidence of transverse phase stability, even with these extreme values of $\phi_0$. In acceleration mode, a similar rise in acceptance can be found with increasing $\phi_0$, which is also visible in Figure \[fig:acceptance-overview\]*b*.
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P. Jansen, L. Semeria, L.E. Hofer, S. Scheidegger, J.A. Agner, H. Schmutz and F. Merkt, Phys. Rev. Lett. 115, 133202 (2015).
| ArXiv |
---
abstract: 'A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However, there do exist computable UFDs (in fact, polynomial rings over computable fields) where the set of prime/irreducible elements is not computable. Outside of the class of UFDs, the notions of irreducible and prime may not coincide. We demonstrate how different these concepts can be by constructing computable integral domains where the set of irreducible elements is computable while the set of prime elements is not, and vice versa. Along the way, we will generalize Kronecker’s method for computing irreducibles and factorizations in $\mathbb{Z}[x]$.'
address:
- |
Department of Mathematics and Statistics\
Grinnell College\
Grinnell, Iowa 50112 U.S.A.
- |
Department of Mathematics and Statistics\
Grinnell College\
Grinnell, Iowa 50112 U.S.A.
- |
Department of Mathematics and Statistics\
Grinnell College\
Grinnell, Iowa 50112 U.S.A.
author:
- Leigh Evron
- 'Joseph R. Mileti'
- 'Ethan Ratliff-Crain'
title: Irreducibles and Primes in Computable Integral Domains
---
[^1]
Introduction
============
In an integral domain, there are two natural definitions of basic “atomic" elements: irreducibles and primes. We recall these standard algebraic definitions.
Let $A$ be an integral domain, i.e. a commutative ring with $1 \neq 0$ and with no zero divisors (so $ab = 0$ implies either $a = 0$ or $b=0$). Recall the following definitions.
1. An element $u \in A$ is a [*unit*]{} if there exists $w \in A$ with $uw = 1$. We denote the set of units by $U(A)$. Notice that $U(A)$ is a multiplicative group.
2. Given $a,b \in A$, we say that $a$ and $b$ are [*associates*]{} if there exists $u \in U(A)$ with $au = b$.
3. An element $p \in A$ is [*irreducible*]{} if it nonzero, not a unit, and has the property that whenever $p = ab$, either $a$ is a unit or $b$ is a unit. An equivalent definition is that $p \in A$ is irreducible if it is nonzero, not a unit, and its divisors are precisely the units and the associates of $p$.
4. An element $p \in A$ is [*prime*]{} if it nonzero, not a unit, and has the property that whenever $p \mid ab$, either $p \mid a$ or $p \mid b$.
5. $A$ is a [*unique factorization domain*]{}, or [*UFD*]{}, if it has the following two properties:
- For each $a \in A$ such that $a$ is nonzero and not a unit, there exist irreducible elements $r_1,r_2,\dots,r_n \in A$ with $a = r_1r_2 \cdots r_n$.
- If $r_1,r_2,\dots,r_n,q_1,q_2,\dots,q_m \in A$ are all irreducible and $r_1r_2 \cdots r_n = q_1q_2 \cdots q_m$, then $n = m$ and there exists a permutation $\sigma$ of $\{1,2,\dots,n\}$ such that $r_i$ and $q_{\sigma(i)}$ are associates for all $i$.
It is a simple fact that if $A$ is an integral domain, then every prime element of $A$ is irreducible. Although the converse is true in any UFD, it does fail for general integral domains. For example, in the integral domain $\mathbb{Z}[\sqrt{-5}]$, there are two different factorizations of $6$ into irreducibles: $$2 \cdot 3 = 6 = (1 + \sqrt{-5})(1 - \sqrt{-5}).$$ Since $U(\mathbb{Z}(\sqrt{-5})) = \{1,-1\}$, these two factorizations are indeed distinct. This example also shows that $2$ is an irreducible element that is not prime because $2 \mid (1 + \sqrt{-5})(1 - \sqrt{-5})$ but $2 \nmid 1 + \sqrt{-5}$ and $2 \nmid 1 - \sqrt{-5}$. In fact, all four of the above irreducible factors are not prime.
For another example that will be particularly relevant for our purposes, let $A$ be the subring of $\mathbb{Q}[x]$ consisting of those polynomials whose constant term and coefficient of $x$ are both integers, i.e. $$A = \{a_0 + a_1x + a_2x^2 + \dots + a_nx^n \in \mathbb{Q}[x] : a_0 \in \mathbb{Z} \text{ and } a_1 \in \mathbb{Z}\}.$$ In this integral domain, all of the normal integer primes are still irreducible (by a simple degree argument), but none of them are prime in $A$ because given any integer prime $p \in \mathbb{Z}$, we have that $p \mid x^2$ since $\frac{x^2}{p} \in A$, but $p \nmid x$ as $\frac{x}{p} \notin A$.
We are interested in the extent to which the irreducible and prime elements can differ in an integral domain. As just discussed, the set of prime elements is always a subset of the set of irreducible elements, but it may be a proper subset. Can one of these sets be significantly more complicated than the other? We approach this question from the point of view of computability theory. We begin with the following fundamental definition.
A [*computable ring*]{} is a ring whose underlying set is a computable set $A \subseteq \mathbb{N}$, with the property that $+$ and $\cdot$ are computable functions from $A \times A$ to $A$.
For a general overview of results about computable rings and fields, see [@SHTucker]. Computable fields together with computable factorizations in polynomial rings over those fields have received a great deal of attention ([@FrohlichShep], [@MetakidesNerode], [@Rabin]), and [@MillerNotices] provides an excellent overview of work in this area. In particular, there exists a computable field $F$ such that the set of primes in $F[x]$ is not computable (see [@MillerNotices Lemma 3.4] or [@SHTucker Section 3.2] for an example). Moreover, there is a computable UFD such that the set of primes is as complicated as possible in the arithmetical hierarchy (see [@JoeDamir]). For our purposes, we will only need the first level of this hierarchy (see [@Soare Chapter 4] for more information).
Let $Z \subseteq \mathbb{N}$.
- We say that $Z$ is a $\Sigma_1^0$ set, or [*computably enumerable*]{}, if there exists a computable $R \subseteq \mathbb{N}^2$ such that $$i \in Z \Longleftrightarrow (\exists x) R(x,i).$$
- We say that $Z$ is a $\Pi_1^0$ set if there exists a computable $R \subseteq \mathbb{N}^2$ such that $$i \in Z \Longleftrightarrow (\forall x) R(x,i).$$
Notice that the complement of $\Sigma_1^0$ set is a $\Pi_1^0$ set, and the complement of $\Pi_1^0$ set is a $\Sigma_1^0$ set. Although every computable set is both a $\Sigma_1^0$ set and $\Pi_1^0$ set, there exists a $\Sigma_1^0$ set that is not computable, such as the set of natural numbers coding programs that halt. The complement of a noncomputable $\Sigma_1^0$ set is a noncomputable $\Pi_1^0$ set. We will use the following standard fact (see [@Soare Section II.1])
\[p:Sigma1IffRangeComputable\] An infinite set $Z \subseteq \mathbb{N}$ is $\Sigma_1^0$ if and only if there exists a computable injective function $\alpha \colon \mathbb{N} \to \mathbb{N}$ such that $\text{range}(\alpha) = Z$.
We will prove that there exists a computable integral domain where the set of irreducible elements is computable while the set of prime elements is not, and also there exists a computable integral domain where the set of prime elements is computable while the set of irreducible elements is not. Thus, these two notions can be wildly different. Our approach will be to code an arbitrary $\Pi_1^0$ set into the set of irreducible (resp. prime) elements while maintaining control over the set of prime (reps. irreducible) elements. Moreover, our integral domains will extend $\mathbb{Z}$ and we will perform our noncomputable coding into the normal integer primes as in [@JoeDamir].
Strongly Computable Finite Factorization Domains
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In Section 3, we will build a computable integral domain $A$ such that the set of irreducible elements of $A$ is computable but the set of prime elements of $A$ is not computable. The idea is that we will turn off the primeness of a normal integer prime $p_i$ in response to a $\Sigma_1^0$ event (such as program $i$ halting) by introducing a new element $x$ with $p_i \mid x^2$ but $p_i \nmid x$. In doing this, we will expand $A$ and we will want to ensure that we can compute the irreducible elements in the resulting integral domain. Since we are adding a new element, this construction will be analogous to expanding our original $A$ to the polynomial ring $A[x]$. However, there is a potential problem here in that even if the irreducible elements of an integral domain $A$ are computable, it need not be the case the the irreducible elements of $A[x]$ are computable. In fact, as mentioned in the introduction, there are computable fields $F$ (where the irreducibles are trivially computable because no element is irreducible) such that the irreducibles of $F[x]$ are not computable.
To remedy this situation, we will ensure that the integral domains in our construction have a stronger property. As motivation, we first summarize Kronecker’s method for finding the divisors of an element $\mathbb{Z}[x]$, and hence for determining whether an element is irreducible. Let $f(x) \in \mathbb{Z}[x]$ be nonzero, and let $n = \deg(f(x))$. We try to restrict the set of possible divisors to a finite set that we need to check. Since the degree function is additive, notice that any divisor of $f(x)$ has degree at most $n$. Now perform the following:
- Notice that if $g(x) \in \mathbb{Z}[x]$ and $g(x) \mid f(x)$ in $\mathbb{Z}[x]$, then $g(a) \mid f(a)$ for all $a \in \mathbb{Z}$.
- Find $n+1$ many points $a \in \mathbb{Z}$ with $f(a) \neq 0$ (which exist because $f(x)$ has at most $n$ roots). Notice that each such $f(a)$ has only finitely many divisors in $\mathbb{Z}$.
- For each of the possible choices of the divisors of these values in $\mathbb{Z}$, find the unique interpolating polynomial in $\mathbb{Q}[x]$ of degree at most $n$.
- Check if any of these polynomials are in $\mathbb{Z}[x]$, and if so, check if they divide $f(x)$ in $\mathbb{Z}[x]$.
- Compile the resulting list of divisors.
Therefore, we can compute the finite set of divisors of any element of $\mathbb{Z}[x]$. Since we know the units of $\mathbb{Z}[x]$, it follows that we can computably determine if an element of $\mathbb{Z}[x]$ is irreducible.
The key algebraic fact that makes Kronecker’s method work is that every nonzero element of $\mathbb{Z}$ has only finitely many divisors. Integral domains with this property were defined and studied in [@AAZ-Factor1; @AAZ-Factor2; @AndersonMullins].
Let $A$ be an integral domain.
- $A$ is a [*finite factorization domain*]{}, or FFD, if every nonzero element has only finitely many divisors up to associates.
- $A$ is a [*strong finite factorization domain*]{} if every nonzero element has only finitely many divisors.
We now define an effective analogue of strong finite factorization domains. In addition to wanting our ring to be computable, we also want the stronger property that we can compute the finite set of divisors of any nonzero element. Instead of using the word “strong" twice, we adopt the following definition.
A [*strongly computable finite factorization domain*]{}, or SCFFD, is a computable integral domain $A$ equipped with a computable function $D$ such that for all $a \in A \backslash \{0\}$, we have that $D(a)$ is (a canonical index for) the finite set of divisors of $a$ in $A$.
Let $A$ be an SCFFD equipped with divisor function $D$.
1. The set $U(A)$ is a finite set that can be computed from $A$.
2. The set of irreducible elements of $A$ is computable.
For the first claim, simply notice that $U(A) = D(1)$. For the second, given any $a \in A$, we have that $a$ is irreducible if and only it nonzero, not a unit, and its only divisors are units and associates. Suppose then that we are given an arbitrary $a \in A$. We can check whether $a$ is zero or a unit (by part 1), and if either is true, then $a$ is not irreducible. Otherwise, then since $a \neq 0$, we can compute the finite set $D(a)$ of divisors of $a$. Since we can also compute the finite set $U(A)$, we can examine each $b \in D(a)$ in turn to determine whether $b \in U(A)$ or whether there exists $u \in U(A)$ with $b = au$. If this is true for all $b \in D(a)$, then $a$ is irreducible in $A$, and otherwise it is not.
If we include an additional assumption that $A$ is a UFD, then we have a converse to the previous result.
Let $A$ be a computable integral domain with the following properties:
- $A$ is a UFD.
- $U(A)$ is finite.
- The set of irreducible elements of $A$ is computable.
We can then equip $A$ with a computable function $D$ so that $A$ becomes an SCFFD.
We first argue that we can computably factor elements of $A$ into irreducibles. Let $a \in A$ be nonzero and not a unit. Since the set of irreducibles of $A$ is computable, we can check whether $a$ is irreducible. If not, we search until we find two nonzero nonunit elements of $A$ whose product is $a$. We can now check if these factors are irreducible, and if not we can repeat to factor them. Notice that this process must eventually produce finitely many irreducibles whose product is $a$ by König’s Lemma together with the fact that there are no infinite descending chains of strict divisibilities in a UFD.
We now define our function $D$. Let $a \in A \backslash \{0\}$ be arbitrary. Check if $a \in U(A)$ (which is possible because $U(A)$ is finite and computable from $A$), and if so, define $D(a)$ to equal $U(A)$. If $a \notin U(A)$, then we we can computably factor it into irreducibles $q_i$ so that $a = q_1q_2 \dots q_n$. Since $U(A)$ is finite, we can now computably check if any of the $q_i$ are associates of each other, and if so we can find witnessing units. Thus, we can write $a = up_1^{k_1} \cdots p_m^{k_m}$ where $w \in U(A)$, each $p_i$ is irreducible, each $k_i \in \mathbb{N}^+$, and $p_i$ and $p_j$ are not associates whenever $i \neq j$. Since $A$ is a UFD, we then have that the set of divisors of $a$ equals the set of elements of the form $wp_1^{\ell_1} \cdots p_m^{\ell_m}$ where $w \in U(A)$ and $0 \leq \ell_i \leq k_i$ for all $i$. Thus, we can define $D(a)$ to be this finite set.
In contrast, there are SCFFDs that are not UFDs, such as $\mathbb{Z}[\sqrt{-5}]$. More generally, the ring of integers in any imaginary quadratic number field is an SCFFD. To see this, Let $K$ be an imaginary quadratic number field, and fix an integral basis of $\mathcal{O}_K$. Using this integral basis, we can view $\mathcal{O}_K$ as a computable integral domain in such a way that the norm function and divisibility relation are both computable on $\mathcal{O}_K$ (see [@JoeDamir Proposition 1.4]). Given any $n \in \mathbb{N}$, there are only finitely many elements of norm $n$, and moreover we can compute the finite set of such elements. Now given any nonzero $a \in A$, we can compute $N(a)$, examine all elements of norm dividing $N(a)$, and check which of them divide $a$ (since the divisibility relation is computable) to compute the set of divisors of $a$.
Let $A$ be a computable integral domain and let $F$ be the field of fractions of $A$. Recall that elements of $F$ are equivalence classes of pairs of elements of $A$. If we were to allow multiple representations of elements, we can of course work with pairs of elements of $A$ and define addition and multiplication on these elements computably. Nonetheless, a computable ring is defined in a way that forbids such multiple representations, so it is not immediately obvious that we can view $F$ as a computable field. However, since a computable integral domain is coded as a subset of $\mathbb{N}$, we can view pairs of elements $(a,b) \in A^2$ with $b \neq 0$ as being coded by elements of $\mathbb{N}^2$, which in turn can be coded by elements of $\mathbb{N}$. Thus, we can view the field of fractions $F$ as a computable field by working only with pairs $(a,b)$ such that there is no strictly smaller pair $(c,d)$ in the usual ordering of $\mathbb{N}$ with $ad = bc$. In this way, we can still define addition and multiplication computably be searching back for the smallest equivalent representative.
In general, for a computable integral domain $A$, it may not be possible to build the field of fractions as a computable extension of $A$, because it may not be possible to determine when an element $\frac{a}{b} \in F$ is actually an element of $A$. The issue is that we may not be able to determine if $b \mid a$ because the divisibility relation may not be computable. However, we have the following.
If $A$ is an SCFFD, then the field of fractions of $A$ is a computable field, and we can computably build it as an extension of $A$.
Notice that in the field of fractions of $A$, we have that $\frac{a}{b} \in A$ if and only if $b \mid a$, which is if and only if $b \in D(a)$. Now since $A$ is a computable integral domain, it is coded as a subset of $\mathbb{N}$. We can now add on minimal pairs $(a,b)$ such that $b \nmid a$. With this, we can define addition and multiplication
In fact, we can computably “reduce" fractions over an SCFFD to lowest terms, as we now show.
\[p:ReduceFractionsOverSCFFD\] Let $A$ be an SCFFD and let $F$ be the field of fractions of $A$. Given an arbitrary pair of elements $a,b \in R$ with $b \neq 0$, we can computably find a pair of elements $c,d \in R$ with $d \neq 0$, with $\frac{c}{d} = \frac{a}{b}$ in $F$, and such that the only common divisors of $c$ and $d$ are the units of $A$.
First notice that if $a = 0$, then we may take $c = 0$ and $d = 1$. Suppose then that $a \neq 0$. Since we also have that $b \neq 0$, we can now computably determine the finite set of divisors of each of $a$ and $b$, and thus can computably build the finite set $S$ of common divisors of $a$ and $b$, i.e. $S = D(a) \cap D(b)$. For each $r \in S$, we can computably determine the number $|\{s \in S : s \mid r\}| = |D(r) \cap S|$. Fix an $r \in S$ such that $|\{s \in S : s \mid r\}|$ is as large as possible. Since $r$ is a common divisor of $a$ and $b$, we can computably search for $c,d \in A$ such that $rc = a$ and $rd = b$. Notice that $d \neq 0$ (because $b \neq 0$) and $\frac{a}{b}= \frac{c}{d}$. Suppose now that $t$ is a common divisor of $c$ and $d$. We then have that $rt$ is a common divisor of $a$ and $b$, so $rt \in S$. By definition of $R$, this implies that $|\{s \in S : s \mid rt\}| \leq |\{s \in S : s \mid r\}|$. Since $\{s \in S : s \mid r\} \subseteq \{s \in S : s \mid rt\}$, it follows that $\{s \in S : s \mid r\} \subseteq \{s \in S : s \mid rt\}$. Thus ,$|\{s \in S : s \mid rt\}| = |\{s \in S : s \mid r\}|$. In particular, we must have $rt \mid r$, so $t \in U(A)$.
Notice this reduction need not be unique, even up to units. In the SCFFD $\mathbb{Z}[\sqrt{-5}]$ we have that $$\frac{2}{1+\sqrt{-5}} = \frac{1-\sqrt{-5}}{3}$$ where there are no nonunit common factors for the numerator and denominator of either side.
By [@AAZ-Factor1 Proposition 5.3] and [@AndersonMullins Theorem 5], if $A$ is a (strong) finite factorization domain, then so is $A[x]$. We now prove an effective analogue of this result. Notice first that if $A$ is a finite integral domain, then $A$ is a finite field, and $A[x]$ is trivially an SCFFD because given $f(x) \in A[x] \backslash \{0\}$, every divisor $g(x)$ of $f(x)$ must satisfy $\deg(g(x)) \leq \deg(f(x))$, and so we need only check each of the finitely many possibilities (which is possible because we can computably search for quotients and remainders). We now handle the infinite case.
\[t:PolynomialRingOverSCFFDisSCFFD\] If $A$ is an infinite SCFFD, then so is $A[x]$. Moreover, given an index for a function $D$ witnessing that $A$ is an SCFFD, we can computably obtain an index for a function $D'$ extending $D$ to witness the fact that $A[x]$ is an SCFFD.
Before jumping into the proof, we give two lemmas.
\[l:InterpoteAndCheckIfInA\] Let $A$ be an SCFFD, let $n \in \mathbb{N}^+$, let $a_0, a_1, \dots, a_n \in A$ be distinct and let $b_0, b_1, \dots, b_n \in R$. Let $F$ be the field of fractions of $A$. There is exactly one polynomial $p(x) \in F[x]$ of degree at most $n$ with $p(a_i) = b_i$ for all $i$. Furthermore, we can computably construct $p(x)$ in $F[x]$, and can computably determine if $p(x) \in A[x]$.\
Uniqueness follows from that fact that if two polynomials over a field having degree at most $n$ agree at $n+1$ points, then they must be the same polynomial. For existence, using Lagrange’s method of interpolation for $n+1$ distinct points of the form $(a_i, b_i)$ will result in a polynomial of the following form: $$p(x) = \sum_{i=0}^{n} b_i \cdot \frac{(x-a_0) \cdots (x-a_{i-1}) (x-a_{i+1}) \cdots (x-a_n)}{(a_i-a_0) \cdots (a_i-a_{i-1}) (a_i-a_{i+1})\cdots (a_i-a_n)}$$ Notice that the denominator is nonzero because $A$ is an integral domain and $a_i \neq a_j$ whenever $i \neq j$. We can computably expand $p(x)$ to write it as $p(x) = \sum_{i=0}^n \frac{c_i}{d_i} x^i$. We then have that $p(x) \in A[x]$ if and only if $d_i \mid c_i$ for all $i$, which we can verify by checking if $d_i \in D(c_i)$ for all $i$.
\[l:DivisibilityRelationOnPolyRingIsComputable\] Suppose that $A$ is an SCFFD. The divisibility relation on $A[x]$ is computable, i.e. given $f(x),g(x) \in A[x]$, we can computably determine if $f(x) \mid g(x)$ in $A[x]$.
Let $f(x),g(x) \in A[x]$ be arbitrary. If $g(x) = 0$, then trivially we have $f(x) \mid g(x)$. Suppose then that both $g(x)$ is nonzero. Perform polynomial long division (or search) to find $q(x),r(x) \in F[x]$ with $f(x) = q(x)g(x) + r(x)$ and either $r(x) = 0$ or $\deg(r(x)) < \deg(g(x))$. Since quotients and remainders are unique in $F[x]$, we have that $g(x) \mid f(x)$ in $A[x]$ if and only if $q(x) \in A[x]$ and $r(x) = 0$. Since we can computably determine if an element of $F[x]$ is in $A[x]$ as in Lemma \[l:InterpoteAndCheckIfInA\], this completes the proof.
Let $f(x) \in A[x]$ be arbitrary, and let $n = \deg(f(x))$. Suppose that $g(x) \in A[x]$ is such that $g(x) \mid f(x)$. First notice that $\deg(g(x)) \leq n$ because the degree function is additive (as $A$ is an integral domain). Now if we fix $h(x) \in A[x]$ with $g(x)h(x) = f(x)$, we then have $g(a)h(a) = f(a)$ for all $a \in A$, so since $f(a),g(a),h(a) \in A$ for all $a \in A$, we have that $g(a) \mid f(a)$ for all $a \in A$.
Search until we find $n+1$ many distinct elements $a_0,a_1,\dots,a_n \in A$ such that $f(a_i) \neq 0$ for all $i$ (such $a_i$ exist because $A$ is infinite and $f(x)$ has at most $n$ roots in $A$). Since $A$ is an SCFFD, we have that $f(a_i)$ has only finitely many divisors for each $i$, and we can compute the finite sets $D(f(a_i))$. Suppose that we pick elements $b_i \in D(f(a_i))$ for each $i$. From Lemma \[l:InterpoteAndCheckIfInA\], there is a unique element $p(x) \in F[x]$ with $\deg(p(x)) \leq n$ and $p(a_i) = b_i$ for all $i$, and we can compute this polynomial $p(x)$ and determine if $p(x) \in A[x]$. As we do this for each choice of the $b_i$, we obtain a finite subset of $A[x]$ of all possible divisors of $f(x)$. Now using Lemma \[l:DivisibilityRelationOnPolyRingIsComputable\], we can thin out this set to form the actual finite set of divisors of $f(x)$.
Irreducibles Computable and Primes Noncomputable
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Let $A$ be an integral domain that is an SCFFD and suppose that $q$ is a prime of $A$. Suppose that we want to destroy the primeness of $q$ while maintaining its irreducibility (say in response to a $\Sigma_1^0$ event such as the halting of a program). The idea is to introduce a new element $x$ so that $q \mid x^2$ but $q \nmid x$. If we let $F$ be the field of fractions of $A$, then we can accomplish this by working in $F[x]$, and extending $A$ to the subring $A[\frac{x^2}{q}]$ of $F[x]$. More explicitly, $A[\frac{x^2}{q}]$ is the set of all polynomials of the form $$a_0 + a_1x + \frac{a_2}{q} \cdot x^2 + \frac{a_3}{q} \cdot x^3 + \frac{a_4}{q^2} \cdot x^4 + \frac{a_5}{q^2} \cdot x^5 + \dots + \frac{a_n}{q^{\lfloor n/2 \rfloor}} \cdot x^n$$ where each $a_i \in A$. Although this works, we will find it more convenient notationally to work with subring $B$ of $F[x]$ consisting of those polynomials of the form $$a_0 + a_1x + \frac{a_2}{q^2} \cdot x^2 + \frac{a_3}{q^3} \cdot x^3 + \frac{a_4}{q^4} \cdot x^4 + \frac{a_5}{q^5} \cdot x^5 + \dots + \frac{a_n}{q^n} \cdot x^n$$ where each $a_i \in A$.
\[t:PropoertiesAfterDestroyingOnePrime\] Let $A$ be an SCFFD and let $q \in A$ be prime. Let $F$ be the field of fractions of $A$ and let $B$ be the subring of $F[x]$ consisting of those polynomials of the form $$a_0 + a_1x + \frac{a_2}{q^2} \cdot x^2 + \frac{a_3}{q^3} \cdot x^3 + \frac{a_4}{q^4} \cdot x^4 + \frac{a_5}{q^5} \cdot x^5 + \dots + \frac{a_n}{q^n} \cdot x^n$$ where each $a_i \in A$. We then have the following.
1. \[e:DivisorsPreservedInPrimeDestruction\] For any $a \in A$, the set of divisors of $a$ in $A$ equals the set of divisors of $a$ in $B$.
2. $B$ is an SCFFD. Moreover, given $A$, $q$, and an index for a function $D$ witnessing that $A$ is an SCFFD, we can computably build $B$ as an extension of $A$ and obtain an index for a function $D'$ witnessing that $B$ is an SCFFD with the property that $D'(a) = D(a)$ for all $a \in A$.
3. \[e:UnitsPreservedInPrimeDestruction\] $U(B) = U(A)$.
4. If $p$ is irreducible in $A$, then $p$ is irreducible in $B$.
5. If $p_1,p_2 \in A$ are irreducibles that are not associates in $A$, then they are not associates in $B$.
6. $q$ is not prime in $B$.
7. If $p$ is a prime of $A$ that is not an associate of $q$, then $p$ is prime in $B$.
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1. Let $a \in A$. Clearly, if an element of $A$ divides $a$ in $A$, then it divides $a$ in $B$. For the converse, since the degree function is additive on $F[x]$, if $f(x),g(x) \in B$ are such that $a = f(x)g(x)$, then we must have $\deg(f(x)) = 0 = \deg(g(x))$, and hence $f(x),g(x) \in A$.
2. Notice first that we computably build $B$ as an extension of $A$ trivially, because if $\frac{a}{q^k} = \frac{b}{q^k}$, then $a = b$ (so there is no issue of distinct representations). The proof that $B$ is an SCFFD is analogous to the proof of Theorem \[t:PolynomialRingOverSCFFDisSCFFD\], with a few straightforward modifications. Given $f(x) \in B$ with $\deg(f(x)) = n$, to determine the divisors of $f(x)$ in $B$, we note the following:
- Notice that if $f(x) \in B$ and $a \in A$, then in general it need not be the case that $f(a) \in A$. However, we will only plug in values $q^i$ for $i \geq n$ to avoid this issue. Suppose then that $g(x) \in B$ with $g(x) \mid f(x)$, and fix $h(x) \in B$ with $g(x)h(x) = f(x)$. We then have that $\deg(g(x)) \leq n$ and $\deg(h(x)) \leq n$. Thus, for any $i \geq n$, we have $f(q^i), g(q^i), h(q^i) \in A$, and so $g(q^i) \mid f(q^i)$ in $A$. Since there are infinitely many $i \geq n$, and these $q^i$ provide an infinite supply of distinct elements (because $A$ is an integral domain), we can plug in $n+1$ many such values with $f(q^i) \neq 0$ to form the basis for our Lagrange interpolations.
- We can computably determine if an element $p(x) \in F[x]$ is actually an element of $B$. The key question is given $a,b \in A$ with $b \neq 0$ and a $k \geq 2$, can we determine if we can write an element $\frac{a}{b}$ of $F$ in the form $\frac{c}{q^k}$. Notice that this is possible if and only if there exists $c \in A$ with $aq^k = bc$, which is if and only if $b \mid aq^k$. Since $A$ is an SCFFD, we can computably determine if $b \in D(aq^k)$, and furthermore in this case we can computably find $c$ with $bc = aq^k$ and hence $\frac{a}{b}= \frac{c}{q^k}$, Thus, we can determine if an element of $F[x]$ is an element of $B$, and if so write it in the above form.
- The divisibility relation is computable on $B$ as in Lemma \[l:DivisibilityRelationOnPolyRingIsComputable\], because we can computably determine if an element of $F[x]$ is an element of $B$ as just mentioned.
This shows that $B$ is an SCFFD and allows us to compute $D'$ uniformly from $A$ and $D$. Finally, notice that $D'$ extends $D$ by \[e:DivisorsPreservedInPrimeDestruction\].
3. Immediate from \[e:DivisorsPreservedInPrimeDestruction\] and the fact that $U(B) = D(1)$.
4. This follow from \[e:DivisorsPreservedInPrimeDestruction\] and \[e:UnitsPreservedInPrimeDestruction\].
5. Immediate from \[e:UnitsPreservedInPrimeDestruction\].
6. Notice that $q$ is nonzero and not a unit by \[e:UnitsPreservedInPrimeDestruction\]. We have that $q \mid x^2$ in $B$ because $\frac{1}{q} \cdot x^2 = \frac{q}{q^2} \cdot x^2 \in B$, but $q \nmid x$ because $\frac{1}{q} \cdot x \notin B$ as $q$ is not a unit (and this is the only possible witness for divisibility because $F[x]$ is an integral domain). Therefore, $q$ is not prime in $B$.
7. Let $p$ be a prime of $A$ that is not an associate of $q$. Notice that $p$ is nonzero and not a unit of $B$ by \[e:UnitsPreservedInPrimeDestruction\]. Let $f(x),g(x) \in B$, and suppose that $p \mid f(x)g(x)$ in $B$. Write out $$\begin{aligned}
f(x) & = a_0 + a_1x + \frac{a_2}{q^2} \cdot x^2 + \frac{a_3}{q^3} \cdot x^3 + \dots + \frac{a_n}{q^n} \cdot x^n \\
g(x) & = b_0 + b_1x + \frac{b_2}{q^2} \cdot x^2 + \frac{b_3}{q^3} \cdot x^3 + \dots + \frac{b_n}{q^n} \cdot x^n \\
f(x)g(x) & = c_0 + c_1x + \frac{c_2}{q^2} \cdot x^2 + \frac{c_3}{q^3} \cdot x^3 + \dots + \frac{c_n}{q^n} \cdot x^n\end{aligned}$$ Since $p \mid f(x)g(x)$ in $B$, we have that $p \mid c_i$ in $A$ for all $i$. Assume that $p \nmid f(x)$ and $p \nmid g(x)$ in $B$. Then there must exist $i$ and $j$ such that $p \nmid a_i$ in $A$ and $p \nmid b_j$ in $A$. Let $k$ and $\ell$ be largest possible such that $p \nmid a_k$ in $A$ and $p \nmid b_{\ell}$ in $A$. Now element $c_{k+\ell}$ will be a sum of terms, one of which will be $a_kb_{\ell}q^j$ for some $j \in \{0,1\}$, while other terms will be divisible by $p$ in $A$. Since $p$ divides $c_{k+\ell}$, it follows that $p \mid a_kb_{\ell}q^j$ in $A$. However, this is a contradiction because $p$ is prime in $A$ but divides none of $a_k$, $b_{\ell}$, or $q$ (the last because $p$ is not an associate of $q$ in $A$).
We now show that we can code an arbitrary $\Pi_1^0$ set into the primes of an integral domain $A$ while maintaining the computability of the irreducible elements. In fact, we perform our coding within the normal integer primes and can make the resulting integral domain an SCFFD.
\[t:Pi1ControlOfPrimes\] Let $S$ be a $\Sigma_1^0$ set, and let $p_0,p_1,p_2,\dots$ list the usual primes from $\mathbb{N}$ in increasing order. There exists an SCFFD $A$ such that:
- $\mathbb{Z}$ is a subring of $A$.
- $U(A) = \{1,-1\}$.
- Every $p_i$ is irreducible in $A$.
- $p_i$ is prime in $A$ if and only if $i \notin S$.
If $S = \emptyset$, this is trivial by letting $A = \mathbb{Z}$. Assume then that $S \neq \emptyset$. If $S$ is finite, say $|S| = n$, then we can trivially fix a computable injective function $\alpha \colon \{1,2,\dots,n\} \to \mathbb{N}$ with $\text{range}(\alpha) = S$. If $S$ is infinite, then we can fix a computable injective function $\alpha \colon \mathbb{N} \to \mathbb{N}$ with $\text{range}(\alpha) = S$ by Proposition \[p:Sigma1IffRangeComputable\].
We build our computable SCFFD $A$ in stages, starting by letting $A_0 = \mathbb{Z}$ and letting $D_0(a)$ be the finite set of divisors of $a$ for all $a \in \mathbb{Z} \backslash \{0\}$. Suppose that we are at a stage $k$ and have constructed an SCFFD $A_k$ together with witnessing function $D_k$. We now extend $A_k$ to $A_{k+1}$ by destroying the primality of $p_{\alpha(k)}$ as in the construction of Theorem \[t:PropoertiesAfterDestroyingOnePrime\] using a new indeterminate $x_k$. In other words, letting $F_k$ be the field of fractions of $A_k$, we let $A_{k+1}$ be the subring of $F_k[x]$ consisting of those polynomials of the form $$a_0 + a_1x + \frac{a_2}{p_{\alpha(k)}^2} \cdot x_k^2 + \frac{a_3}{p_{\alpha(k)}^3} \cdot x_k^3 + \frac{a_4}{p_{\alpha(k)}^4} \cdot x_k^4 + \dots + \frac{a_n}{p_{\alpha(k)}^n} \cdot x_k^n$$ where each $a_i \in A_k$. We continue this process through the construction of $A_n$ if $|S| = n$, and infinitely often if $S$ is infinite. Using Theorem \[t:PropoertiesAfterDestroyingOnePrime\], the following properties hold by induction on $k$:
- $A_k$ is an SCFFD with witnessing function $D_k$ extending $D_i$ for all $i < k$.
- $U(A_k) = \{1,-1\}$.
- Every $p_i$ is irreducible in $A_k$.
- $p_i$ is prime in $A_k$ if and only if $i \notin \{\alpha(1),\alpha(2),\dots,\alpha(k)\}$.
Now if $S$ is finite, say $|S| = n$, then it follows that the integral domain $A_n$ has the required properties.
Suppose then that $S$ is infinite, and let $A = A_{\infty} = \bigcup_{k=0}^{\infty} A_k$. Also, let $D = \bigcup_{k=1}^{\infty} D_k$, which makes sense because the $D_i$ extend each other as functions. Notice that $D$ is a computable function and that for any $a \in A_k$, we have that the set of divisors of $a$ in $A$ equals the set of divisors of $a$ in $A_k$, so $D(a) = D_k(a)$ is the finite set of divisors of $a$ in $A$. Therefore, $A$ is an SCFFD as witnessed by $D$. Since $U(A_k) = \{1,-1\}$ for all $k \in \mathbb{N}$, it follows that $U(A) = \{1,-1\}$. Since we maintain the units and divisibility at each stage, it also follows that every $p_i$ is irreducible in $A$.
We now show that $p_i$ is prime in $A$ if and only if $i \notin S$. First notice that each $p_i$ is nonzero and not a unit of $A$.
- Suppose first that $i \notin S$. We then have that $i \notin \text{range}(\alpha)$, so $p_i$ is prime in every $A_k$ by the last property above. Let $a,b \in A$, and suppose that $p_i \mid ab$ in $A$. Fix $c \in A$ with $p_ic = ab$. Go to a point $k$ where each of $p_i,a,b,c$ exist. We then have that $p_i \mid ab$ in $A_k$, so as $p_i$ is prime in $A_k$, either $p_i \mid a$ in $A_k$ or $p_i \mid b$ in $A_k$. Therefore, either $p_i \mid a$ in $A$ or $p_i \mid b$ in $A$. It follows that $p_i$ is prime in $A_k$.
- Suppose now that $i \in S$. Thus, we can fix $k \in \mathbb{N}$ with $\alpha(k) = i$. We then have that $p_i$ is not prime in $A_{k+1}$ by the last property above. Fix $a,b \in A_{k+1}$ such that $p_i \mid ab$ in $A_{k+1}$ but $p_i \nmid a$ in $A_{k+1}$ and $p_i \nmid b$ in $A_{k+1}$. Since the $D_i$ extend each other as functions, and $A$ is an SCFFD as witnessed by $D$, it follow that $p_i \mid ab$ in $A$ but $p_i \nmid a$ in $A$ and $p_i \nmid b$ in $A$. Therefore, $p_i$ is not prime in $A$.
There exists a computable integral domain $A$ such that the set of irreducible elements of $A$ is computable but the set of prime elements of $A$ is not computable.
Fix a noncomputable $\Sigma_1^0$ set $S$, and let $A$ be the SCFFD given by Theorem \[t:Pi1ControlOfPrimes\]. Since $A$ is an SCFFD, it is a computable integral domain and the set of irreducible elements of $A$ is computable. However, the set of prime elements of $A$ is not computable, because if we could compute it, then we could compute $S$, which is a contradiction.
Primes Computable and Irreducibles Noncomputable
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Consider the subring $A = \mathbb{Z} + x\mathbb{Z} + x^2\mathbb{Q}[x]$ of $\mathbb{Q}[x]$. In other words, $A$ is the set of polynomials of the form $q_0 + q_1x + q_2x^2 + \dots + q_nx^n$ where $q_0 \in \mathbb{Z}$ and $q_1 \in \mathbb{Z}$. As mentioned in the introduction, each normal integer prime is irreducible in $A$ but is not prime in $A$. It is also a standard fact for $p(x) \in A$, we have that $p(x)$ is prime in $A$ if and only if $p(x)$ is irreducible in $\mathbb{Q}[x]$ and $p(0) \in \{1,-1\}$.
We will generalize this construction by replacing $\mathbb{Z}$ with an arbitrary integral domain. Suppose that $R$ is an integral domain, and let $F$ be its field of fractions. Consider the subring $A = R + xR + x^2F[x]$ of $F[x]$, i.e. $A$ is the set of polynomials of the form $q_0 + q_1x + q_2x^2 + \dots + q_nx^n$ where $q_0 \in R$ and $q_1 \in R$. Such an integral domain $A$ is particularly nice from our perspective because the irreducibles in $R$ will remain irreducible in $A$ (so all of the complexity of irreducibles remain), but no element of $R$ is prime in $A$ (so any complexity of primes is “erased"). Moreover, we can reduce the complexity of primality of elements of $A$ to that of irreducibles in the polynomial ring over a field, about which a great deal is understood.
\[l:PrimesInAAreNonconstantAndIrreducible\] Let $R$ be an integral domain with field of fractions $F$. Consider the subring $A = R + xR + x^2F[x]$ of $F[x]$. Let $p(x) \in A$. If $p(x)$ is prime in $A$, then $p(x)$ is non-constant and irreducible in $F[x]$.
We prove the contrapositive, i.e. if $p(x) \in A$ is either constant or not irreducible, then $p(x)$ is not prime in $A$.
Suppose first that $p(x)$ is a constant, and fix $k \in R$ with $p(x) = k$. If $k \in \{0\} \cup U(R)$, then $k$ is either zero or a unit, so $k$ is not prime in $A$ by definition. Suppose then that $k \notin \{0\} \cup U(R)$. Notice that $k \mid x^2$ in $A$ because $\frac{1}{k} \cdot x^2 \in A$, but $k \nmid x$ in $A$ because $\frac{1}{k} \cdot x \notin A$. Therefore, $p(x) = k$ is not prime in $A$.
Suppose now that $p(x) \in A$ is non-constant and not irreducible in $F[x]$. Since $p(x)$ is non-constant, it is not a unit in $F[x]$. Fix $g(x),h(x) \in F[x]$ with $p(x) = g(x)h(x)$ and such that $0 < \deg(g(x)) < \deg(p(x))$ and $0 < \deg(h(x)) < \deg(p(x))$. Now since $g(x),h(x) \in F[x]$, the constant terms and coefficients of $x$ in these polynomials need not be in $R$. Let $b$ be the product of the denominators of these coefficients in $g(x)$, and let $c$ be the product of the denominators of these coefficients in $h(x)$. We then have that $p(x) \cdot bc = (b \cdot g(x)) \cdot (c \cdot h(x))$ where both $b \cdot g(x) \in A$ and $c \cdot h(x) \in A$. Since $bc \in R \subseteq A$, we have that $p(x) \mid (b \cdot g(x)) \cdot (c \cdot h(x))$ in $A$. However, notice that $p(x) \nmid b \cdot g(x)$ in $A$ because $\deg(b \cdot g(x)) < \deg(p(x))$ and $p(x) \nmid c \cdot h(x)$ because $\deg(c \cdot h(x)) < \deg(p(x))$. Therefore, $p(x)$ is not prime in $A$.
\[l:CharacterizePrimesInA\] Let $R$ be an integral domain with field of fractions $F$. Consider the subring $A = R + xR + x^2F[x]$ of $F[x]$. Let $p(x) \in A$ and suppose that $p(x)$ is irreducible in $F[x]$. The following are equivalent.
1. $p(x)$ is prime in $A$.
2. For all $f(x) \in F[x]$, if $p(x)f(x) \in A$, then $f(x) \in A$.
3. For all $g(x) \in A$ such that $p(x) \mid g(x)$ in $F[x]$, we have that $p(x) \mid g(x)$ in $A$.
$(1) \rightarrow (2)$: Suppose first that $p(x)$ is prime in $A$. We know that no constants are prime in $A$ from above, so $p(x)$ is non-constant. Let $f(x) \in F[x]$ be such that $p(x)f(x) \in A$. We prove that $f(x) \in A$. Write $f(x) = q_0 + q_1x + \dots + q_nx^n$ where each $q_i \in F$. Let $d$ be the product of the denominators of $q_0$ and $q_1$. Now $d \in R \subseteq A$ and $d \cdot f(x) \in A$, hence $p(x) \mid p(x) \cdot d \cdot f(x)$ in $A$, i.e. $p(x) \mid d \cdot (p(x)f(x))$ in $A$. Since $p(x)$ is prime in $A$, either $p(x) \mid d$ in $A$ or $p(x) \mid p(x)f(x)$ in $A$. The former is impossible because $p(x)$ is non-constant, so we must have that $p(x) \mid p(x)f(x)$ in $A$. Fix $h(x) \in A$ with $p(x)h(x) = p(x)f(x)$. Since $F[x]$ is an integral domain, we conclude that $f(x) = h(x) \in A$.
$(2) \rightarrow (3)$: Immediate.
$(3) \rightarrow (1)$: Let $g(x),h(x) \in A$ and suppose that $p(x) \mid g(x)h(x)$ in $A$. Since $A$ is a subring of $F[x]$, we then have that $p(x) \mid g(x)h(x)$ in $F[x]$. Now $p(x)$ is irreducible in $F[x]$, so since $F[x]$ is a UFD, we know that $p(x)$ is prime in $F[x]$. Thus, either $p(x) \mid g(x)$ in $F[x]$ or $f(x) \mid h(x)$ in $F[x]$. Using $(3)$, we conclude that either $p(x) \mid g(x)$ in $A$ or $p(x) \mid h(x)$ in $A$. Therefore, $p(x)$ is prime in $A$.
\[p:ClassifyPrimesInSubringOfFX\] Let $R$ be an integral domain that is not a field, and let $F$ be its field of fractions. Consider the subring $A = R + xR + x^2F[x]$ of $F[x]$. An element $p(x) \in A$ is prime in $A$ if and only if $p(x)$ is irreducible in $F[x]$ and $p(0) \in U(R)$.
We first prove that if $p(x) \in A$ does not satisfy $p(0) \notin U(R)$, then $p(x)$ is not prime in $A$. If $p(0) = 0$, then fixing any nonzero nonunit $b \in R$ (which exists because $R$ is not a field), we have $p(x) \cdot \frac{x}{b} \in A$ but $\frac{x}{b} \notin A$, so $p(x)$ is not prime in $A$ by Lemma \[l:CharacterizePrimesInA\]. Suppose then that $p(0) \notin \{0\} \cup U(R)$. Write $p(x) = q_nx^n + \dots + q_2x^2 + ax + b$ where $a,b \in R$ and $b \notin \{0\} \cup U(R)$. We have $$\begin{aligned}
p(x) \cdot \left(\frac{1}{b} \cdot x\right) & = (q_nx^n + \dots + q_2x^2 + ax + b) \cdot \left(\frac{1}{b} \cdot x\right) \\
& = \left(\frac{q_n}{b}\right) \cdot x^{n+1} + \dots + \left(\frac{q_2}{b}\right) \cdot x^3 + \left(\frac{a}{b}\right) \cdot x^2+x\end{aligned}$$ Thus, $f(x) \cdot \frac{1}{b} \cdot x \in A$ but $\frac{1}{b} \cdot x \notin A$, so $f(x)$ is not prime in $A$ by Lemma \[l:CharacterizePrimesInA\].
We have just shown that $p(x) \in A$ is prime in $A$, then $p(0) \in U(R)$. We also know that if $p(x) \in A$ is prime in $A$, then $p(x)$ is irreducible in $F[x]$ by Lemma \[l:PrimesInAAreNonconstantAndIrreducible\].
Suppose conversely that $p(x)$ is irreducible in $F[x]$ and that $p(0) \in U(R)$. Using Lemma \[l:CharacterizePrimesInA\], to show that $p(x)$ is prime in $A$ it suffices to show that whenever $f(x) \in F[x]$ is such that $p(x)f(x) \in A$, then we must have $f(x) \in A$. Suppose then that $f(x) \in F[x]$ and $p(x)f(x) \in A$. Write $$\begin{aligned}
f(x) & = q_0 + q_1x + q_2x^2 + \dots + q_nx^n \\
p(x) & = a_0 + a_1x + r_2x^2 + \dots + r_nx^n\end{aligned}$$ where $a_0 \in U(R)$, $a_1 \in R$, each $q_i \in F$, and each $r_i \in F$. We then have that $p(x)f(x) \in F[x]$ with $$p(x)f(x) = q_0a_0 + (q_0a_1 + a_0q_1) x + \dots$$ As $p(x)f(x) \in A$, we know that $q_0a_0 \in R$ and $q_0a_1 + a_0q_1 \in R$. Since $q_0a_0 \in R$ and $a_0 \in U(R)$, it follows that $q_0 \in R$. Using this together with the facts that $a_1 \in R$ and $q_0a_1 + a_0q_1 \in R$, it follows that $a_0q_1 \in R$. Applying again the fact that $a_0 \in U(R)$, we conclude that $q_1 \in R$. Since $q_0,q_1 \in R$, it follows that $p(x) \in A$.
With these results in hand, we now proceed to construct an integral domain $R$ with a complicated set of irreducible elements. We will want our $R$ to have a “nice" field of fractions $F$ in the sense that the irreducibles of $F[y]$ will be computable.
\[l:DestorySigma1PrimesLemma\] Let $S$ be a $\Sigma_1^0$ set, and let $p_0,p_1,p_2,\dots$ list the usual primes from $\mathbb{N}$ in increasing order. There exists a computable UFD $R$ such that:
- $\mathbb{Z}$ is a subring of $R$, and in fact $$\mathbb{Z}[x_1,x_2,\dots] \subseteq R \subseteq \mathbb{Q}(x_1,x_2,\dots),$$ where there are infinitely many indeterminates if $S$ is infinite, and exactly $n$ of them if $|S| = n$.
- $U(R) = \{1,-1\}$.
- $p_i$ is irreducible in $R$ if and only if $i \notin S$.
If $S = \emptyset$, this is trivial by letting $A = \mathbb{Z}$. Assume then that $S \neq \emptyset$. If $S$ is finite, say $|S| = n$, then we can trivially fix a computable injective function $\alpha \colon \{1,,2\dots,n\} \to \mathbb{N}$ with $\text{range}(\alpha) = S$. If $S$ is infinite, then we can fix a computable injective function $\alpha \colon \mathbb{N} \to \mathbb{N}$ with $\text{range}(\alpha) = S$ by Proposition \[p:Sigma1IffRangeComputable\].
We build our computable UFD $R$ in stages, starting by letting $R_0 = \mathbb{Z}$. Suppose that we are at a stage $k$ and have constructed through the integral domain $R_k$. We now destroy the irreducibility of $p_{\alpha(k)}$ by letting $R_{k+1} = R_k[x_k,\frac{p_{\alpha(k)}}{x_k}]$ as in [@JoeDamir Section 3]. We continue this process through the construction of $R_{n+1}$ if $|S| = n$, and infinitely often if $S$ is infinite. Using [@JoeDamir Proposition 3.3 and Theorem 3.10], the following properties hold by induction on $k$:
- $R_k$ is a Noetherian UFD.
- $\mathbb{Z}[x_1,x_2,\dots,x_k] \subseteq R_k \subseteq \mathbb{Q}(x_1,x_2,\dots,x_k)$.
- $U(R_k) = \{1,-1\}$.
- $p_i$ is irreducible in $R_k$ if and only if $i \notin \{\alpha(1),\alpha(2),\dots,\alpha(k)\}$.
Now if $S$ is finite, say $|S| = n$, then it follows that the integral domain $R_n$ has the required properties.
Suppose then that $S$ is infinite, and let $R = R_{\infty} = \bigcup_{k=0}^{\infty} R_k$. We then have that $R$ has the required properties by the proofs in [@JoeDamir Section 4] (although they are significantly easier in this case because we never change the units).
\[t:Pi1ControlOfIrreducibles\] Let $S$ be a $\Sigma_1^0$ set, and let $p_0,p_1,p_2,\dots$ list the usual primes from $\mathbb{N}$ in increasing order. There exists a computable integral domain $A$ such that:
- $\mathbb{Z}$ is a subring of $A$.
- $U(A) = \{1,-1\}$.
- No $p_i$ is prime in $A$.
- The set of prime elements of $A$ is computable.
- $p_i$ is irreducible in $A$ if and only if $i \notin S$.
Let $R$ be the integral domain given by Lemma \[l:DestorySigma1PrimesLemma\]. Let $F$ be the field of fractions of $R$. Since $$\mathbb{Z}[x_1,x_2,\dots] \subseteq R \subseteq \mathbb{Q}(x_1,x_2,\dots)$$ (where there are infinitely many indeterminates if $S$ is infinite, and exactly $n$ of them if $|S| = n$) and the field of fractions of $\mathbb{Z}[x_1,x_2,\dots]$ is $\mathbb{Q}(x_1,x_2,\dots)$, it follows that $F = \mathbb{Q}(x_1,x_2,\dots)$. Let $A$ be the subring $R + yR + y^2F[y]$ of $F[y]$. Now we clearly have that $\mathbb{Z}$ is a subring of $A$ and $U(A) = \{1,-1\}$. Also, each $p_i$ is a constant polynomial in $A$, so is not prime in $A$ by Lemma \[l:PrimesInAAreNonconstantAndIrreducible\]. By [@FrohlichShep Theorem 4.5], the set of irreducible elements of $F[y]$ is computable, so since $U(R) = \{1,-1\}$, we may use Proposition \[p:ClassifyPrimesInSubringOfFX\] to conclude that the set of prime elements of $A$ is computable.
Finally, by Lemma \[l:DestorySigma1PrimesLemma\], we have that $p_i$ is irreducible in $R$ if and only if $i \notin S$. Now $R$ is the subring of $A$ consisting of the constant polynomials, so as $U(A) = U(R)$ and divisors of the constant polynomials in $A$ must be constants, it follows that $p_i$ is irreducible in $A$ if and only $p_i$ is irreducible in $R$, which is if and only if $i \notin S$.
There exists a computable integral domain $A$ such that the set of prime elements of $A$ is computable but the set of irreducible elements of $A$ is not computable.
Fix a noncomputable $\Sigma_1^0$ set $S$, and let $A$ be the integral domain give by Theorem \[t:Pi1ControlOfIrreducibles\]. We then have the set of prime elements of $A$ is computable. However, the set of irreducible elements of $A$ is not computable, because if we could compute it, then we could compute $S$, which is a contradiction.
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[^1]: The authors thank Grinnell College for its generous support through the MAP program for research with undergraduates.
| ArXiv |
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abstract: |
This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy of $F$. We first resolve the most fundamental question of minimizing the number of cliques of size $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait, and Timmons. We also go further and prove a corresponding stability result. Next we minimize the number of cycles of length $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, and classify the extremal graphs for most values of $r$, answering another question of Kritschgau, Methuku, Tait, and Timmons for most $r$.
We then move on to a central and longstanding conjecture in graph saturation made by Tuza, which states that for every graph $F$, the limit $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, F)}{n}$ exists, where $\operatorname{sat}(n, F)$ denotes the minimum number of edges in an $n$-vertex $F$-saturated graph. Pikhurko made progress in the negative direction by considering families of graphs instead of a single graph, and proved that there exists a graph family $\mathcal{F}$ of size $4$ for which $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, \mathcal{F})}{n}$ does not exist (for a family of graphs $\mathcal{F}$, a graph $G$ is called $\mathcal{F}$-saturated if $G$ does not contain a copy of any graph in $\mathcal{F}$, but the addition of any edge creates a copy of a graph in $\mathcal{F}$, and $\operatorname{sat}(n, \mathcal{F})$ is defined similarly). We make the first improvement in 15 years by showing that there exist infinitely many graph families of size $3$ where this limit does not exist. Our construction also extends to the generalized saturation problem when we minimize the number of fixed-size cliques. We also show an example of a graph $F_r$ for which there is irregular behavior in the minimum number of $C_r$’s in an $n$-vertex $F_r$-saturated graph.
author:
- 'Debsoumya Chakraborti[^1] and Po-Shen Loh[^2]'
title: 'Minimizing the numbers of cliques and cycles of fixed size in an $F$-saturated graph'
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Introduction
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Extremal graph theory focuses on finding the extremal values of certain parameters of graphs under certain natural conditions. One of the most well-studied conditions is $F$-freeness. For graphs $G$ and $F$, we say that $G$ is $F$-free if $G$ does not contain a subgraph isomorphic to $F$. This gives rise to the most fundamental question of finding the Turán number $\operatorname{ex}(n, F)$, which asks for the maximum number of edges in an $n$-vertex $F$-free graph. The asymptotic answer is known for most graphs $F$, with the exception of bipartite $F$ where the most intricate and unsolved cases appear (see, e.g., [@FS] and [@S] for nice surveys). Recently, Alon and Shikhelman [@AS] introduced a natural generalization of the Turán number. They systematically studied $\operatorname{ex}(n, H, F)$, which denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Note that the case $H = K_2$ is the standard Turán problem, i.e., $\operatorname{ex}(n, K_2, F) = \operatorname{ex}(n, F)$.
While the Turán number asks for the maximum number of edges in an $F$-free graph, another very classical problem concerns the minimum number of edges in an $F$-free graph with a fixed number of vertices. This problem is not interesting as stated because the empty graph is the obvious answer. In much of the research, this issue is resolved by imposing the additional condition that adding any edge to $G$ will create a copy of $F$. With this additional condition, we say that $G$ is $F$-saturated. A moment’s thought will convince the reader that when maximizing the number of edges, this additional condition does not change the problem at all. On the other hand, this new condition makes the edge minimization problem very interesting, and this area of research is commonly known as graph saturation. Let the saturation function $\operatorname{sat}(n, F)$ denote the minimum number of edges in an $n$-vertex $F$-saturated graph. Erdős, Hajnal, and Moon [@EHM] started the investigation of this area with the following beautiful result.
\[Erdős, Hajnal, and Moon 1964\] \[EHM\] For every $n \ge s \ge 2$, the saturation number $$\operatorname{sat}(n, K_s) = (s -2)(n-s+2) + \binom{s-2}{2}.$$ Furthermore, there is a unique $K_s$-saturated graph on $n$ vertices with $\operatorname{sat}(n, K_s)$ edges: the join of a clique with $s-2$ vertices and an independent set with $n-s+2$ vertices.
The *join* $G_1 \ast G_2$ of two graphs $G_1$ and $G_2$ is obtained by taking the disjoint union of $G_1$ and $G_2$ and adding all the edges between them. Erdős, Hajnal, and Moon proved Theorem \[EHM\] by using a clever induction argument. A novel approach to prove this theorem is due to Bollobás [@B65], who developed an interesting tool based on systems of intersecting sets. Graph saturation has been studied extensively since Theorem \[EHM\] appeared half a century ago (see, e.g., [@FFS] for a very informative survey). Alon and Shikhelman’s generalization of the Turán number motivated Kritschgau, Methuku, Tait, and Timmons [@KMTT] to start the systematic study of the function $\operatorname{sat}(n, H, F)$, which denotes the minimum number of copies of $H$ in an $n$-vertex $F$-saturated graph. Here again note that $\operatorname{sat}(n, K_2, F) = \operatorname{sat}(n, F)$. Historically, a natural generalization of counting the number of edges ($K_2$) is to count the number of cliques ($K_r$) of a fixed size, see e.g., [@B76], [@E], and [@Z], where the authors answered the generalized extremal question of finding the maximum number of $K_r$’s in a $K_s$-free graph with fixed number of vertices. Towards generalizing Theorem \[EHM\] in a similar fashion, Kritschgau, Methuku, Tait, and Timmons proved the following lower and upper bounds, which differ by a factor of about $r-1$, and conjectured that the upper bound (achieved by the same construction given in Theorem \[EHM\]) is correct.
\[Kritschgau, Methuku, Tait, and Timmons 2018\] \[tait\] For every $s > r \ge 3$, there exists a constant $n_{r,s}$ such that for all $n \ge n_{r,s}$, $$\begin{aligned}
\max \left\{\frac{\binom{s-2}{r-1}}{r-1} \cdot n - 2 \binom{s-2}{r-1}, \frac{\binom{s-2}{r-1} + \binom{s-3}{r-2}}{r} \cdot n\right\} &\le \operatorname{sat}(n, K_r, K_s)
\\ &\le (n - s + 2) \binom{s-2}{r-1} + \binom{s-2}{r} .\end{aligned}$$
Our first main contribution confirms their conjecture for sufficiently large $n$ by showing that the upper bound is indeed the correct answer. We also show that the natural construction is the unique extremal graph for this generalized saturation problem for large enough $n$. Furthermore, we prove a corresponding stability result for sufficiently large $n$ which shows that even if we allow up to some $cn$ more copies of $K_r$ than $\operatorname{sat}(n, K_r, K_s)$ in an $n$-vertex $K_s$-saturated graph, the extremal graph will still be the same and unique. It is worth noting that there are relatively few stability results in the area of graph saturation, essentially only [@AFGS] by Amin, Faudree, Gould, and Sidorowicz, and [@BFP] by Bohman, Fonoberova, and Pikhurko. In the notation of joins, the extremal graph in our problem is $K_{s-2} \ast \overline{K}_{n-s+2}$, i.e., the join of a clique with $s-2$ vertices and an independent set with $n-s+2$ vertices.
\[sat\] For every $s > r \ge 2$, there exists a constant $n_{r,s}$ such that for all $n \ge n_{r,s}$, we have $\operatorname{sat}(n, K_r, K_s) = (n-s+2) \binom{s-2}{r-1} + \binom{s-2}{r}$. Moreover, there exists a constant $c_{r,s} > 0$ such that the only $K_s$-saturated graph with up to $\operatorname{sat}(n, K_r, K_s) + c_{r,s} n$ many copies of $K_r$ is $K_{s-2} \ast \overline{K}_{n-s+2}$.
The moreover part of this theorem is tight in the sense that Theorem \[sat\] fails for $c_{r,s} = \binom{s-3}{r-2}$. To see that consider the graph $G$ on $n$ vertices which is the join of two graphs $G_1$ and $G_2$, where $G_1$ is $K_{s-1}$ minus an edge, and $G_2$ is an independent set on $n-s+1$ vertices. Clearly, $G$ is $K_s$-saturated, with $\left(2 \binom{s-3}{r-2} + \binom{s-3}{r-1}\right) (n-s+1) + 2\binom{s-3}{r-1} + \binom{s-3}{r}$ many copies of $K_r$.
In the process of proving Theorem \[sat\], we consider a more general setting and prove and use an intermediate result, which may also be of independent interest. The condition that $G$ is $F$-saturated can be weakened by removing the condition that $G$ is $F$-free (as also studied in [@B78] and [@T92]). Perhaps counterintuitively, despite the fact that this is a weaker condition, the literature calls $G$ *strongly $F$-saturated* if adding any edge to $G$ creates a new copy of $F$. Following the notation in the literature, we write $\operatorname{ssat}(n, H, F)$ to denote the minimum number of copies of $H$ in an $n$-vertex strongly $F$-saturated graph. It is obvious that $\operatorname{ssat}(n, H, F) \le \operatorname{sat}(n, H, F)$. We have the following asymptotic result for the function $\operatorname{ssat}$ for cliques.
\[ssat\] For every $s > r \ge 2$, we have $\operatorname{ssat}(n, K_r, K_s) = n \binom{s-2}{r-1} - o(n)$.
Kritschgau, Methuku, Tait, and Timmons [@KMTT] showed an interesting result, which says that for any natural number $m$, there are graphs $H$ and $F$ such that $\operatorname{sat}(n,H,F) = \Theta(n^m)$. They showed this as an implication of the following bounds that they proved on the minimum number of $C_r$’s in an $n$-vertex $K_s$-saturated graph.
\[Kritschgau, Methuku, Tait, and Timmons 2018\] \[tait2\] For $s \ge 5$ and $r \le 2s - 4$, $\operatorname{sat}(n, C_r, K_s) = \Theta(n^{\floor{\frac{r}{2}}})$. More precisely, $$\begin{aligned}
& \left(1 - o(1)\right) \frac{n^k (s-2)_k}{4 k} \le \operatorname{sat}(n, C_r, K_s) \le \left(1 + o(1)\right) \frac{n^k (s-2)_k}{2 k} &&\text{ if } 2 \mid r
\\& \left(1 - o(1)\right) \frac{n^k (s-2)_{k+1} (k-2)!}{r (r-3) (r)_k (s-1)} \le \operatorname{sat}(n, C_r, K_s) \le \left(1 + o(1)\right) \frac{n^k (s-2)_{k+1}}{2} &&\text{ if } 2 \nmid r\end{aligned}$$ where $k = \floor{\frac{r}{2}}$ and $(m)_k = m(m-1) \cdots (m-k+1)$.
Note that the same construction as in Theorem \[EHM\] proves the upper bound in Theorem \[tait2\]. We explain the counting for the upper bound in the proof of our Theorem \[cycle\] in Section 4. We show that for all sufficiently large $n$ the same natural construction is indeed the unique extremal graph for most $r$.
\[cycle\] For every $s \ge 4$ and odd $r$ with $r \ge 7$ or even $r$ with $r \ge 4 \sqrt{s-2}$, there exists a constant $n_{r,s}$ such that for all $n \ge n_{r,s}$, $K_{s-2} \ast \overline{K}_{n-s+2}$ has the minimum number of copies of $C_r$ among $n$-vertex $K_s$-saturated graphs. Moreover, when also $r \le 2s - 4$ this is the unique such graph.
We remark here that for any $r,s$ that do not satisfy the assumptions that $s \ge 4$ and $ r \le 2s-4$, we have $\operatorname{sat}(n, C_r, K_s) = 0$, which can be seen from the same extremal graph $K_{s-2} \ast \overline{K}_{n-s+2}$. In Theorem \[cycle\], we could write the explicit value of $\operatorname{sat}(n, C_r, K_s)$, which is just the number of cycles of length $r$ in the graph $K_{s-2} \ast \overline{K}_{n-s+2}$. We chose not to do so because the explicit number is not particularly elegant. Also, it turns out that we are able to find the correct asymptotic answers for $r=4$ and $r=5$, which we include in the sections proving Theorem \[cycle\].
Next we turn our attention to a long-standing, yet very fundamental conjecture made by Tuza [@T86; @T88]. In contrast to the Turán number, one of the inherent challenges in studying the saturation number $\operatorname{sat}(n, H)$ for general graphs $H$ is that this function lacks monotonicity properties that one might hope for. For example, Pikhurko [@P] showed that there is a pair of connected graphs $F_1 \subset F_2$ on the same vertex set such that $\operatorname{sat}(n, F_1) > \operatorname{sat}(n, F_2)$ for large $n$, violating monotonicity in the second parameter. Regarding non-monotonicity in the first parameter, Kászonyi and Tuza [@KT] observed that $\operatorname{sat}(2k-1, P_3) = k+1 > k = \operatorname{sat}(2k, P_3)$ where $P_3$ is the path with $3$ edges. Moreover, Pikhurko showed a wide variety of examples of irregular behavior of the saturation function in [@P]. All of this non-monotonicity makes proving statements about the saturation function difficult, in particular because inductive arguments generally do not work. However, in order to find some smooth behavior of the saturation function Tuza conjectured the following.
\[Tuza 1986\] \[con\] For every graph $F$, the limit $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, F)}{n}$ exists.
Not much progress has been made towards settling the conjecture. The closest positive attempt was made by Truszczyński and Tuza [@TT], who showed that for every graph $F$, if $\lim \inf_{n \rightarrow \infty} \frac{\operatorname{sat}(n, F)}{n} < 1$, then $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, F)}{n}$ exists and is equal to $1 - \frac{1}{p}$ for some positive integer $p$.
Pikhurko considered the saturation number for graph families to make progress in the negative direction of Conjecture \[con\]. For a family of graphs $\mathcal{F}$, the saturation number $\operatorname{sat}(n, \mathcal{F})$ is defined to be the minimum number of edges in an $n$-vertex $\mathcal{F}$-saturated graph, where a graph $G$ is called $\mathcal{F}$-saturated if $G$ does not contain a copy of any graph in $\mathcal{F}$ and adding any edge to $G$ will create a copy of a graph in $\mathcal{F}$. Pikhurko first showed in [@P01] that there exists an infinite family $\mathcal{F}$ of graphs for which $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, \mathcal{F})}{n}$ does not exist, and later in [@P] proved the same for a graph family of size only $4$. We make the first progress in 15 years, moving one step closer.
\[pro\] There exist infinitely many graph families $\mathcal{F}$ of size $3$ such that the ratio $\frac{\operatorname{sat}(n,\mathcal{F})}{n}$ does not converge as $n$ tends to infinity.
In the spirit of considering the generalized saturation number, it is natural to ask the more general question of whether $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, K_r, F)}{n}$ exists for every graph $F$. We remark that this problem is interesting since the order of $\operatorname{sat}(n, K_r, \mathcal{F})$ is linear in $n$ for every graph family $\F$, which can easily be shown by considering the same construction used by Kászonyi and Tuza in [@KT], who showed the same for $r=2$. We show that our construction of graph families of size $3$ can be extended to this scenario.
\[construction\] For every $r \ge 3$, there exist infinitely many graph families $\mathcal{F}$ of size $3$ such that the ratio $\frac{\operatorname{sat}(n, K_r, \mathcal{F})}{n}$ does not tend to a limit as $n$ tends to infinity.
We next show an example of a graph $F_r$ for which the function $\operatorname{sat}(n, C_r, F_r)$ behaves irregularly. To be precise, we show that for certain $F_r$, the value of the saturation function depends on certain divisibility conditions of $n$, and the sequence $\operatorname{sat}(n, C_r, F_r)$ oscillates.
\[easy\_cons\] For every $r \ge 5$, there exists a graph $F_r$ such that $\operatorname{sat}(n, C_r, F_r)$ is zero for infinitely many values of $n$ and also positive infinitely often.
The remainder of this paper is organized as follows. In the next section we prove an asymptotically tight lower bound on $\operatorname{ssat}(n, K_r, K_s)$. Then, we use the results and notations of that section to determine $\operatorname{sat}(n, K_r, K_s)$ exactly for sufficiently large $n$ in Section $3$. Next in Section $4$, we prove a few lemmas which will be useful for computing $\operatorname{sat}(n, C_r, K_s)$, i.e., Theorem \[cycle\]. We handle the cases of even and odd $r$ in Theorem \[cycle\] separately, and those will be proved in the subsequent two sections. In Section $7$, we construct infinitely many graph families $\mathcal{F}$ of size $3$ for which the ratio $\frac{\operatorname{sat}(n,\mathcal{F})}{n}$ does not converge. We then extend this construction, with the help of Theorem \[ssat\], in Section $8$ in order to prove Theorem \[construction\]. We prove Theorem \[easy\_cons\] in Section $9$. We finish with a few open problems and concluding remarks in Section $10$.
Asymptotic result for $\boldsymbol{\operatorname{ssat}(n, K_r, K_s)}$
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In this section, we prove Theorem \[ssat\]. Let $G = (V,E)$ be an $n$-vertex strongly $K_s$-saturated graph such that the number of $K_r$’s in $G$ is $\operatorname{ssat}(n, K_r, K_s)$. Our aim is to find a lower bound on the number of $K_r$ in $G$. Note that if there is an edge $e \in E$ such that $e$ is not in a copy of $K_r$, then $e$ does not contribute to the number of copies of $K_r$. It turns out that a careful analysis of the edges which are in a copy of $K_r$ saves us the required factor of $r-1$ when we compare against the previous best result (Theorem \[tait\]). So, it is natural to split the edge set $E$ into two parts in the following manner. Let $E_1$ denote the set of edges which are at least in one copy of $K_r$. Let $E_2 = E \setminus E_1$ be the remaining edges in $G$. Now we will prove a simple but powerful lemma which will be useful throughout the current and next sections.
\[trivial\] Every edge of $E_2$ would not be in a copy of $K_s$ even if any non-edge were added to $G$.
Fix an arbitrary edge $uv$ of $E_2$ and an arbitrary non-edge $ab$ of $G$. Note that the sets $\{u,v\}$ and $\{a,b\}$ can overlap, but without loss of generality $b \not \in \{u,v\}$. Assume for the sake of contradiction that after adding the missing edge $ab$ we create a copy of $K_s$ containing both $u$ and $v$. Now if we remove the vertex $b$ from the created copy of $K_s$, we will find a copy of $K_{s-1}$ in $G$ which contains both $u$ and $v$. So $uv$ is in a copy of $K_{s-1}$ in $G$, which contradicts the fact that $uv$ is not in a copy of $K_r$, because $r \le s-1$.
It will be convenient to define a couple of sets which we will use throughout this section and the next section. For $i = 1,2$, let $G_i$ denote the graph on the same vertex set $V$ with the edge set $E_i$. For a graph $H$, it will be convenient to use the notation $d_{H}(v)$ to denote the degree of $v$ in $H$. It will be useful to split the vertices according to their degree in $G_1$, so we define $$A = \{v \in V : d_{G_1}(v) \le n^{\frac{1}{3}}\}. \label{large}$$
We can observe that $A$ consists of almost all vertices of $G$, i.e., $|A| = n - o(n)$. This is because $|E_1| \le \binom{r}{2} \operatorname{ssat}(n, K_r, K_s) \le \binom{r}{2} \operatorname{sat}(n, K_r, K_s) = O(n)$, where the last equality follows from the upper bound in Theorem \[tait\], and so $|V \setminus A| = O(n^{\frac{2}{3}})$. Now our aim is to show that almost every vertex of $A$ is in a copy of $K_{s-1}$ which has only one vertex of $A$. Note that the extremal graph $K_{s-2} \ast \overline{K}_{n-s+2}$ has this property. Formally, we define the following: $$\begin{aligned}
B = \{v \in A : \exists a_1, \dots, a_{s-2} \in V \setminus A \text{ such that } v, a_1, \dots, a_{s-2} \text{ induce a copy of } K_{s-1}\}. \label{select}\end{aligned}$$
\[main\] Almost all vertices are in $B$, in the sense that $|B| = n - o(n)$.
Let $R$ denote the set of vertices in $A$ with degree more than $|A| - 2n^{\frac{2}{3}}$ in the induced subgraph of $G_2$ on $A$. Now we claim that $R$ has at most $2rn^{\frac{2}{3}}$ vertices. Assume for the sake of contradiction that $|R| > 2rn^{\frac{2}{3}}$; then with a simple greedy process we will find a copy of $K_r$ in $G_2$. Start with any vertex $v_1 \in R$, and let $R_1 \subseteq R$ denote the set of vertices in $R$ which are neighbors of $v_1$. Clearly, $|R_1| > 2(r-1)n^{\frac{2}{3}}$ because $v_1$ has less than $2n^{\frac{2}{3}}$ non-neighbors in $R$. For $2 \le i \le r$, we continue this process, i.e., at step $i$ we take a vertex $v_i \in R_{i-1}$, and let $R_i \subseteq R_{i-1}$ denote the set of vertices in $R_{i-1}$ which are neighbors of $v_i$. Clearly, $|R_i| > 2(r-i)n^{\frac{2}{3}}$. Now observe that $v_1, v_2, \dots, v_r$ induce a copy of $K_r$ in $G_2$ which is the desired contradiction. So $|R| \le 2rn^{\frac{2}{3}}$.
Now our aim is to show that $A \setminus R \subseteq B$, which will be sufficient to finish the proof of this lemma. To this end, fix an arbitrary vertex $v \in A \setminus R$. We will first show that there is $w \in A$ such that $vw$ is not an edge of $G$ and there is no $z \in A$ such that $vz$ and $zw$ are both in $E_1$. This is because there are at most $|A| - 2n^{\frac{2}{3}}$ many $E_2$-neighbors of $v$ in $A$ (which follows from the definition of $R$), and in the induced graph of $G_1$ on $A$, there can be at most $n^{\frac{1}{3}} \left(n^{\frac{1}{3}} - 1\right) = n^{\frac{2}{3}} - n^{\frac{1}{3}}$ vertices at distance $2$ from $v$ (which follows from ). So, there are at least $$|A| - 1 - \left(|A| - 2n^{\frac{2}{3}}\right) - n^{\frac{1}{3}} - \left(n^{\frac{2}{3}} - n^{\frac{1}{3}}\right) = n^{\frac{2}{3}} - 1$$ choices for $w$. Fix such a vertex $w$. As $G$ is $K_s$-saturated, if we added the edge $vw$, then we would create a copy of $K_s$. Furthermore, that $K_s$ cannot contain any vertex from $A$ except $v$ and $w$, because if it contained some $z \in A$, then at least one of $vz$ or $zw$ is in $E_2$, contradicting Lemma \[trivial\]. Hence there is a copy of $K_{s-1}$ induced by $v$ together with $s-2$ vertices from $V \setminus A$, and so $v \in B$. Therefore, $|B| \ge |A| - |R| \ge n - o(n)$.
For an arbitrary vertex $v \in B$, the number of $K_r$’s induced by $v$ together with $r-1$ vertices from $V \setminus A \subseteq V \setminus B$ is at least $\binom{s-2}{r-1}$ from . So by Lemma \[main\], the number of $K_r$’s in $G$ is at least $\binom{s-2}{r-1} |B| = \binom{s-2}{r-1}n - o(n)$. This matches the upper bound from Theorem \[tait\], completing the proof of Theorem \[ssat\].
Note that by defining the set $A$ in optimally, the best lower bound we can achieve with this argument is that $\operatorname{ssat}(n, K_r, K_s) \ge \binom{s-2}{r-1}n - O\left(\sqrt{n}\right)$. Also note that Theorem \[ssat\] already proves an asymptotically tight lower bound on $\operatorname{sat}(n, K_r, K_s)$, because: $$n \binom{s-2}{r-1} - o(n) \le \operatorname{ssat}(n, K_r, K_s) \le \operatorname{sat}(n, K_r, K_s) \le (n-s+2) \binom{s-2}{r-1} + \binom{s-2}{r}.$$
Exact result for $\boldsymbol{\operatorname{sat}(n, K_r, K_s)}$
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In this section, we will find the exact value of $\operatorname{sat}(n, K_r, K_s)$ for all sufficiently large $n$, proving Theorem \[sat\]. The same argument will also show that the graph $K_{s-2} \ast \overline{K}_{n-s+2}$ is the unique extremal graph. Moreover, we will prove a stability result, i.e., the same graph is also the unique graph among $K_s$-saturated graphs even if we allow up to some $cn$ more copies of $K_r$ than $\operatorname{sat}(n, K_r, K_s)$. We will start with the structural knowledge we developed in the last section and successively deduce more structure to finally reach the exact structure.
Define $c = \frac{1}{4r^2}$ and consider an $n$-vertex $K_s$-saturated graph $G$ with at most $\operatorname{sat}(n, K_r, K_s) + cn$ copies of $K_r$. By defining the sets $A$ and $B$ as in and and applying the same arguments we can make the same structural deductions about $G$ as in the last section. In particular, the number of $K_r$’s with one vertex in $B$ and $r-1$ vertices in $V \setminus A$ is at least $$n \binom{s-2}{r-1} - o(n). \label{count}$$
Next, define $$C = \{v \in B : d_{G_1} (v) > s-2\}. \label{problem}$$ For $v \in C$, fix $s-2$ neighbors of $v$ in $V \setminus A$ such that those neighbors along with $v$ induce a copy of $K_{s-1}$ in $G$. For each $v \in C$, pick an edge $vw \in E_1$ such that $w$ is not among the $s-2$ fixed neighbors. Note that the same edge $vw$ can be picked at most once more. Each of these particular edges is in $E_1$, hence these edges are contained in some $K_r$, which is not counted in . After counting for multiplicity, these extra edges will constitute at least an extra $\frac{|C|}{2\binom{r}{2}}$ many copies of $K_r$. Hence, for sufficiently large $n$, $\frac{|C|}{2\binom{r}{2}} \le 2cn$, which implies that $|C| \le \frac{n}{2}$.
So, the set $B \setminus C$ is non-empty for large enough $n$. We will now prove two more structural lemmas.
\[tight\] Let $v$ be an arbitrary vertex in $B \setminus C$, and suppose $x_1, x_2, \dots, x_{s-2}$ are vertices in $G$ such that $\{v, x_1, \dots, x_{s-2}\}$ induces a copy of $K_{s-1}$. Then for all $u \in V \setminus \{v\}$ such that $uv$ is not an edge, $u$ is adjacent to all of $x_1, \dots, x_{s-2}$.
Since $\{v, x_1, \dots, x_{s-2}\}$ induces $K_{s-1}$ and $s-1 \geq r$, every edge $vx_i$ is in $E_1$. As $v \in B \setminus C$, $v$ has no more $E_1$-edges. If we add the non-edge $uv$, we must create a copy of $K_s$. If some vertex $w \not \in \{u, v, x_1, \dots, x_{s-2}\}$ participates in the created copy of $K_s$, then we know that $vw$ must be in $E_2$ since $v$ has no more $E_1$-edges, contradicting Lemma \[trivial\]. So, the only choice for the remaining $s-2$ vertices of the created copy of $K_s$ would be $x_1, \dots, x_{s-2}$. Thus $u$ must be adjacent to all of $x_1, \dots, x_{s-2}$.
\[T\] All vertices of $B \setminus C$ have no incident edges from $E_2$.
Assume for the sake of contradiction that $uv \in E_2$, where $v \in B \setminus C$. Since $G$ is $K_s$-saturated, $u$ is in a copy $S$ of $K_{s-1} \supseteq K_r$. Since $uv \in E_2$, $v \not \in S$ by Lemma \[trivial\]. Furthermore, $v$ cannot be adjacent to all the vertices in $S$, or else there would be a copy of $K_s$. Similarly, $v$ is in a copy of $K_{s-1}$, and $u$ is not adjacent to the full set of those vertices. Let $a_1, \dots, a_k$, $b_1, \dots, b_k$ and $c_{k+1}, \dots, c_{s-2}$ be distinct vertices such that $\{u, a_1, \dots, a_k, c_{k+1}, \dots, c_{s-2}\}$ and $\{v, b_1, \dots, b_k, c_{k+1}, \dots, c_{s-2}\}$ both induce $K_{s-1}$. The above argument shows that $k \ge 1$. Now we claim that there must be at least two non-edges between $v$ and the set $\{a_1, \dots, a_k\}$, otherwise the neighbors of $v$ in $\{a_1, \dots, a_k\}$ along with $u, v, c_{k+1}, \dots, c_{s-2}$ will induce a clique of order at least $s-1 \ge r$, which contradicts the fact that $uv \in E_2$. Without loss of generality, $v$ is not adjacent to both $a_1$ and $a_2$. Now by applying Lemma \[tight\] with $v \in B \setminus C$, and $b_1, \dots, b_k, c_{k+1}, \dots, c_{s-2}$ as $x_1, \dots, x_{s-2}$, and $a_1$ as $u$, we see that $a_1$ is adjacent to all of $b_1, \dots, b_k, c_{k+1}, \dots, c_{s-2}$. The same is true of $a_2$. So, $a_1, a_2, b_1, \dots, b_k, c_{k+1}, \dots, c_{s-2}$ induce a copy of $K_s$ in $G$ which is impossible.
\[Proof of Theorem \[sat\]\] Fix any vertex $v \in B \setminus C$. There exists a set $S$ of $s-2$ vertices such that $S \cup \{v\}$ induces a copy of $K_{s-1}$. Since $v \in B \setminus C$, there are no more $E_1$ edges incident to $v$ other than those to $S$. By Lemma \[T\], there are no $E_2$ edges either. By Lemma \[tight\], every vertex $u \not \in S \cup \{v\}$ must be adjacent to all vertices in $S$. This is already the graph $K_{s-2} \ast \overline{K}_{n-s+2}$, which is $K_s$-saturated, so $G$ is precisely $K_{s-2} \ast \overline{K}_{n-s+2}$.
Preparation to compute $\boldsymbol{\operatorname{sat}(n, C_r, K_s)}$
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In this section, we state a few lemmas which will be helpful to prove Theorem \[cycle\] in the subsequent two sections. Our proof is inspired by the proof in [@KMTT]. Compared to that paper, we count the number of cycles more carefully to avoid double-counting, which helps us to get the exact answer. We first find asymptotically the number of cycles of length $r$ in the graph $K_{s-2} \ast \overline{K}_{n-s+2}$. Let $\boldsymbol{k = \floor{\frac{r}{2}}}$ throughout the current and next two sections. There are $\binom{n-s+2}{k}$ many independent sets of order $k$ in the subgraph $\overline{K}_{n-s+2}$. If $r$ is even, then for an arbitrary $k$-vertex independent set $A$, the number of copies of $C_r$ containing $A$ is $\frac{(s-2)_k (k-1)!}{2}$, and each copy of $C_r$ is counted exactly once. If $r$ is odd, then for an arbitrary $k$-vertex independent set $A$, the number of copies of $C_r$ containing $A$ is $\frac{(s-2)_{k+1} k!}{2}$, and each copy of $C_r$ is counted exactly once. Furthermore, there is no copy of $C_r$ with more than $k$ vertices in $\overline{K}_{n-s+2}$ because the maximum independent set of $C_r$ has $k$ vertices. Hence, we have the upper bounds of Theorem \[tait2\], i.e.,
$$\begin{aligned}
& \operatorname{sat}(n, C_r, K_s) \le \frac{(s-2)_k}{2 k} \cdot n^k + O(n^{k-1}) &&\text{ if } 2 \mid r \label{optimal1}
\\& \operatorname{sat}(n, C_r, K_s) \le \frac{(s-2)_{k+1}}{2} \cdot n^k + O(n^{k-1}) &&\text{ if } 2 \nmid r \label{optimal2}\end{aligned}$$
We will use the standard notation $\Theta$ in the next few sections. For two functions $f(n)$ and $g(n)$, we call $f(n) = \Theta(g(n))$ if $0 < \lim\inf_{n \rightarrow \infty} \frac{g(n)}{f(n)} \le \lim\sup_{n \rightarrow \infty} \frac{g(n)}{f(n)} < \infty$.
\[ramsey\] For every fixed $l$, there are $\Theta(n^l)$ independent sets of order $l$ in every $n$-vertex $K_s$-free graph.
Consider an $n$-vertex $K_s$-free graph $G$. It is obvious that the number of independent sets of order $l$ in an $n$-vertex graph is at most $\binom{n}{l} = \Theta(n^l)$. From the most classical result [@ES] in Ramsey theory, we know that $R(l,s)$ exists, where $R(l,s)$ denotes the minimum number $N$ such that every graph of order $N$ contains an independent set of order $l$ or a clique of order $s$. So, for each $R(l,s)$-vertex subset $A$ of $G$, the subgraph induced by $A$ must contain an independent set of order $l$ because $A$ does not contain a copy of $K_s$. Now an independent set of order $l$ can be counted at most $\binom{n-l}{R(l,s)-l}$ times. Accounting for multiple-counts, the number of independent sets of order $l$ in $G$ is at least $\frac{\binom{n}{R(l,s)}}{\binom{n-l}{R(l,s)-l}} = \Theta(n^l)$.
Next we give an upper bound on the number of edges of any $n$-vertex $K_s$-saturated graph minimizing the number of copies of $C_r$ with $r \le 2s-4$. It is shown in [@KMTT] that for every fixed even $r$, there are $o(n^2)$ many edges in an $n$-vertex $K_s$-saturated graph with minimal number of copies of $C_r$. Next we prove the same for all $r \le 2s-4$. We prove a stronger result for odd $r \le 2s-4$, and repeat the proof for even $r$ from [@KMTT] for the sake of completion.
\[upperbound\] For every $n$-vertex $K_s$-saturated graph $G$ minimizing the number of copies of $C_r$, and for any function $f(n)$ such that $f(n) \rightarrow \infty$ as $n \rightarrow \infty$:
- For odd $r \le 2s-4$, $G$ has $O\left(n f(n)\right)$ many edges.
- For even $r$, $G$ has $o(n^2)$ many edges.
In the case of even $r$, if we could prove that $G$ has $o\left(n^{\frac{3}{2}}\right)$ many edges, then we could follow the proof for odd $r$ and would not have the condition $r \ge 4 \sqrt{s-2}$ in Theorem \[cycle\]. We have briefly mentioned this again in the concluding remarks.
*Case 1: $r$ is odd and $r \le 2s-4$.* We can assume that the function $f(n)$ is such that $f(n) = O(\log n)$. Let $G$ be an $n$-vertex $K_s$-saturated graph minimizing the number of copies of $C_r$. For the sake of contradiction, assume that $G$ has more than $n f(n)$ edges. Let $B$ denote the set of all vertices of $G$ with degree more than $f(n)$. A simple counting implies that $\sum_{v \in B} d(v) \ge n f(n)$. To prove Lemma \[upperbound\], it is enough to show that for all $v \in B$, there are at least $\Theta\left(n^{k-1} d(v)\right)$ cycles containing $v$. In this case, the total number of cycles will be at least $\Theta\left(\sum_{v \in B} n^{k-1} d(v)\right) \ge \Theta\left(n^k f(n)\right)$, contradicting for all sufficiently large $n$. To show this, consider a vertex $v \in B$. Consider an arbitrary independent set $I = \{v_1, \cdots, v_{k-1}\}$ of order $k-1$ in $V(G) \setminus \{v\}$. For every $i \in [k-2]$, choose a set $V_{i,i+1}$ of $s-2$ vertices such that adding the edge $v_iv_{i+1}$ would create a copy of $K_s$ on $\{v_i,v_{i+1}\} \cup V_{i,i+1}$. Let $V_1$ denote an empty set if $vv_1$ is an edge, else set it to be a set of $s-2$ vertices such that adding the edge $vv_1$ would create a copy of $K_s$ on $\{v,v_1\} \cup V_1$. Let $U = I \cup V_1 \cup V_{1,2} \cup \cdots \cup V_{k-2,k-1}$. Let $V'$ denote the set of neighbors of $v$ outside of $U$. Note that $|V'| \ge d(v) - ks \ge \frac{1}{2} \cdot d(v)$ for large enough $n$ (remember that $d(v) \ge f(n)$). For each $a \in V'$, we will show the existence of a cycle of length $r$ containing $a$, $v$, and all vertices in $I$, proving that there are at least $\frac{1}{2} \cdot d(v)$ many copies of $C_r$ containing $I$.\
*Subcase 1: $av_{k-1}$ and $vv_1$ both are edges.* Pick $k$ distinct vertices $u_1, u_1^*, u_1^{**} \in V_{1,2}, u_2 \in V_{2,3}, \cdots, u_{k-2} \in V_{k-2,k-1}$. This is clearly possible because $|V_{i,i+1}| = s-2$ for all $i$ and $k < s-2$. So, $v v_1 u_1 u_1^* u_1^{**} v_2 u_2 v_3 u_3 \cdots v_{k-2} u_{k-2} v_{k-1} a v$ forms a cycle of length $r$ in $G$.\
*Subcase 2: $av_{k-1}$ is an edge, but $vv_1$ is not an edge.* Pick $k$ distinct vertices $w, w^* \in V_1, u_1 \in V_{1,2}, \cdots, u_{k-2} \in V_{k-2,k-1}$. This is clearly possible because $|V_1| = s-2$, $|V_{i,i+1}| = s-2$ for all $i$, and $k < s-2$. So, $v w w^* v_1 u_1 v_2 u_2 \cdots v_{k-2} u_{k-2} v_{k-1} a v$ forms a cycle of length $r$ in $G$.\
*Subcase 3: $av_{k-1}$ is not an edge, but $vv_1$ is an edge.* Pick $k-1$ distinct vertices $u_1, u_1^* \in V_{1,2}, u_2 \in V_{2,3}, \cdots, u_{k-2} \in V_{k-2,k-1}$. This is clearly possible because $|V_{i,i+1}| = s-2$ for all $i$ and $k-1 < s-2$. Choose a set $S$ of $s-2$ vertices such that adding the edge $av_{k-1}$ would create a copy of $K_s$ on $\{a,v_{k-1}\} \cup S$. Now as $I$ is an independent set, no vertex from $I \setminus \{v_{k-1}\}$ can be in $S$, so there is a vertex $c \in S$ that is not in the set $I \cup \{v, u_1, u_1^*, \cdots, u_{k-2}\}$. Hence, $v v_1 u_1 u_1^* v_2 u_2 \cdots v_{k-2} u_{k-2} v_{k-1} c a v$ forms a cycle of length $r$ in $G$.\
*Subcase 4: $av_{k-1}$ and $vv_1$ both are not edges.* Pick $k-1$ distinct vertices $w \in V_1, u_1 \in V_{1,2}, \cdots, u_{k-2} \in V_{k-2,k-1}$. This is clearly possible because $|V_1| = s-2$, $|V_{i,i+1}| = s-2$ for all $i$, and $k-1 < s-2$. Choose a set $S$ of $s-2$ vertices such that adding the edge $av_{k-1}$ would create a copy of $K_s$ on $\{a,v_{k-1}\} \cup S$. Now as $I$ is an independent set, no vertex from $I \setminus \{v_{k-1}\}$ can be in $S$, so there is a vertex $c \in S$ that is not in the set $I \cup \{v, w, u_1, \cdots, u_{k-2}\}$. Hence, $v w v_1 u_1 v_2 u_2 \cdots v_{k-2} u_{k-2} v_{k-1} c a v$ forms a cycle of length $r$ in $G$.\
From Lemma \[ramsey\], we know that there are $\Theta(n^{k-1})$ many independent sets of order $k-1$ in the induced graph $G \setminus \{v\}$ for any vertex $v$, and for each $v \in B$ and such an independent set, we have $\frac{1}{2} \cdot d(v)$ many copies of $C_r$ containing $v$ and the independent set. It is clear that a copy of $C_r$ in $G$ can be counted at most only a constant (depending on $k$) times in this way. So, the number of $C_r$’s in $G$ is at least $\Theta\left(\sum_{v \in B} n^{k-1} d(v)\right) = \Theta\left(n^k f(n)\right)$, contradicting for all sufficiently large $n$.\
*Case 2: $r$ is even.* By Theorem $1^{**}$ in [@ES83], there exists $c, c' > 0$ such that for any graph $G$ with more than $c n^{2 - \frac{2}{r}}$ edges, there exists $$c'n^r \left(\frac{|E(G)|}{n^2}\right)^{\frac{r^2}{4}}$$ copies of $K_{\frac{r}{2},\frac{r}{2}}$. Therefore, if the number of edges of $G$ is $\epsilon n^2$ for some $\epsilon > 0$ and sufficiently large $n$, then there are $\Theta(n^r)$ copies of $C_r$, contradicting .\
Since all the cases give contradictions, we are done.
It is also shown in [@KMTT] that for every even $r$, there are $(1 - o(1)) \binom{n}{k}$ many independent sets of order $k$ in an $n$-vertex $K_s$-saturated graph with minimal number of copies of $C_r$, with an application of the Moon-Moser theorem [@MM]. Next we prove the same for all $r \le 2s-4$ by using Lemma \[upperbound\] and the following lemma which is equivalent to the problem appeared in Exercise 40(b) in Chapter 10 of [@L].
\[mm\] Let $G$ be a graph on $n$ vertices with $\frac{1}{\tau} \binom{n}{2}$ many edges, where $\tau$ is a positive real number. Let $l$ be a positive integer such that $l \le \tau + 1$. Then, the number of independent sets of order $l$ in $G$ is at least $\binom{\tau}{l} \left(\frac{n}{\tau}\right)^l$.
\[indep\] For every $n$-vertex $K_s$-saturated graph $G$ minimizing the number of copies of $C_r$ for some $r \le 2s-4$, $G$ has $(1 - o(1)) \binom{n}{k}$ many independent sets of order $k$.
Consider an $n$-vertex $K_s$-saturated graph $G$ minimizing the number of copies of $C_r$. The number of edges in $G$ is $o(n^2)$ from Lemma \[upperbound\], so we can apply Lemma \[mm\] to conclude that $G$ has $(1 - o(1)) \binom{n}{k}$ many independent sets of order $k$.
Notice that the arguments for the even cycles and the odd cycles are bit different in Lemma \[upperbound\]. It turns out that the proof of Theorem \[cycle\] for the cases of even and odd $r$ is very different. So, we split the cases in two subsequent sections.
Few copies of $\boldsymbol{C_r}$ in $\boldsymbol{K_s}$-saturated graphs for odd $\boldsymbol{r}$
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Let $G$ be an $n$-vertex $K_s$-saturated graph minimizing the number of copies of $C_r$. Similar to the proof of Theorem \[ssat\] in Section 2, we define $$A = \{ v \in V : d_G(v) \le n^{\frac{1}{3}}\}. \label{redefine}$$ We know that $G$ has $O\left(n \log n\right)$ edges from Lemma \[upperbound\], so $|A| = n - o(n)$.
Recall that $r \le 2s-4$ and $k = \floor{\frac{r}{2}}$. Consider the collection of independent sets $I$ of order $k$ in $A$ such that for all $v_1, v_2 \in I$, there is no common neighbor of $v_1$ and $v_2$ in $A$. Denote this collection of such independent sets by $\mathcal{I}$. Clearly, there will be $(1 - o(1)) \binom{n}{k}$ independent sets in $\mathcal{I}$. Now consider an arbitrary independent set $I = \{v_1, \cdots, v_k\} \in \mathcal{I}$. For every $i,j \in [k]$, there exists a set $V_{i,j} \subseteq V \setminus A$ of $s-2$ vertices such that adding the edge $v_iv_j$ would create a copy of $K_s$ on $\{v_i,v_j\} \cup V_{i,j}$. Now an easy but cumbersome calculation (similar to the calculation for ) tells us that the number of copies of $C_r$ containing $I$ and $k+1$ vertices from $V \setminus A$ is at least $\frac{(s-2)_{k+1} k!}{2}$, where equality holds if and only if all $V_{i,j}$’s are the same and $v_i, v_j$ do not have any common neighbor in $(V \setminus A) \setminus V_{ij}$. At this point, we can conclude that the upper bound in equation is asymptotically tight for all odd $r$. Moreover, we can safely say that there are $(1 - o(1)) \binom{n}{k}$ independent sets $I = \{v_1, \cdots, v_k\} \in \mathcal{I}$ for which $V_{i,j}$’s are the same and $v_i, v_j$ do not have any common neighbor in $(V \setminus A) \setminus V_{ij}$, otherwise $G$ will have more copies of $C_r$ than the upper bound in , which is a contradiction. Let $\mathcal{J}$ denote the collection of independent sets for which the above holds.
Although the statement of Theorem \[cycle\] is only for odd $r \ge 7$, the above argument actually asymptotically finds the value of $\operatorname{sat}(n, C_r, K_s)$ for $r = 5$ as well.
\[maximum\_independent\] For odd $r$ with $7 \le r \le 2s-4$, there is an independent set of order $n - o(n)$ in $G$ such that there is a copy $T$ of $K_{s-2}$ in $G$ with the property that every vertex in $T$ is a neighbor of every vertex of the independent set.
From the fact that $|\mathcal{J}| = \binom{n}{k} - o(n^k)$, we can say that there exist two vertices $u,v \in A$ such that there are $\binom{n}{k-2} - o(n^{k-2})$ independent sets in $\mathcal{J}$ where each of them contains both $u$ and $v$. Let $\mathcal{K}$ denote the collection of independent sets in $\mathcal{J}$ containing both $u$ and $v$. Let $T \subseteq V \setminus A$ be a set of $s-2$ vertices such that adding the edge $uv$ would create a copy of $K_s$ on $\{u,v\} \cup T$. By the definition of $\mathcal{J}$, all the vertices appearing in an independent set in $\mathcal{K}$ should be neighbors of all the vertices in $T$, hence they will form an independent set (because $G$ does not have a copy of $K_s$). For $r \ge 7$, equivalently for $k \ge 3$, it is easy to check that the number of such vertices is $n - o(n)$ (note that this is not true for $k=2$). So, we are done.
Consider the maximum size independent set $I$ in $G$ such that there exists a copy $T$ of $K_{s-2}$ in $G$ such that every vertex in $T$ is a neighbor of every vertex of the independent set. Let $|I| = n - m$. We know that $m = o(n)$ from Lemma \[maximum\_independent\]. Let $S$ denote the set of all vertices outside of $I$ and $T$. For the sake of contradiction, assume that $S$ is non-empty. Now if we let $m' = |S|$, clearly $m' = m - s + 2 = o(n)$. We claim that any $v \in I$ has at least one neighbor in $S$, which will imply that there are at least $n-m$ edges between $I$ and $S$. If there is some $v \in I$ with no neighbor in $S$, then for any $u \in S$, the copy of $K_s$ created by adding the edge $uv$ cannot contain any vertex from $I$ or $S$ except $u$ and $v$, hence $u$ is neighbor of all the vertices of $T$, which in turn tells us that $u$ cannot have any neighbor in $I$, contradicting the maximal choice of $I$. Thus, every vertex in $I$ has at least one neighbor in $S$.
Let $z$ be the number such that $(z)_k = \sqrt{n(m+k)} \cdot n^{k-1}$. We will show that all the vertices in $S$ can have at most $z$ neighbors in $I$ for sufficiently large $n$. Suppose for contradiction that $v \in S$ has more than $z = o(n)$ neighbors in $I$. We already know that the number of copies of $C_r$ in the induced subgraph on $T \cup I$ is at least $\frac{(s-2)_{k+1}}{2} \cdot (n-m)_k$. Now for any set of $k$ vertices from the neighbors of $v$, there is at least a copy of $C_r$ containing those vertices, together with the vertex $v$ and $k$ vertices from $T$. Clearly, there will be at least $\binom{z}{k}$ such copies of $C_r$. This implies that the number of copies of $C_r$ in $G$ is at least $\frac{(s-2)_{k+1}}{2} \cdot (n-m)_k + \binom{z}{k}$, which contradicts for all large $n$ because of the following.
$$\begin{aligned}
\frac{(s-2)_{k+1}}{2} &\cdot (n-m)_k + \binom{z}{k} \\
&\ge \frac{(s-2)_{k+1}}{2} \cdot (n-m-k)^k + \frac{(z)_k}{k!} \\
&\ge \frac{(s-2)_{k+1}}{2} \cdot n^k - \frac{(s-2)_{k+1}}{2} \cdot k (m+k) n^{k-1} + \frac{1}{k!} \sqrt{n(m+k)} \cdot n^{k-1} \\
&\ge \frac{(s-2)_{k+1}}{2} \cdot n^k + \frac{1}{2k!} \sqrt{n(m+k)} \cdot n^{k-1} \\
&\ge \frac{(s-2)_{k+1}}{2} \cdot n^k + \Theta\left(n^{k-\frac{1}{2}}\right)\end{aligned}$$
We now return to our analysis of $S$, which we assumed to be non-empty for the sake of contradiction. If $m \le \log n$, then the number of edges between $S$ and $I$ is at most $(m - s + 2) z < n - m$ for all sufficiently large $n$, which is a contradiction. The only case remaining to handle is when $m > \log n$. For a vertex $v \in S$, pick a set $B$ of $k-1$ vertices from $I$ which are not neighbors of $v$. Fix an order $v_1, v_2, \cdots, v_{k-1}$ of the vertices in $B$, and choose $(s-2)$-element sets $V_i$ such that adding $vv_i$ creates a copy of $K_s$ on $\{v,v_i\} \cup V_i$. Note that as $v$ is not adjacent to all the vertices in $T$ (this follows from the maximality of $I$), $V_i$ cannot be equal to $T$ for any $i$. Consider copies of $C_r$ containing $v, v_1, \cdots, v_{k-1}$ in that order such that the vertices (one or two) between $v$ and $v_1$ are from $V_1$, the vertices (one or two) between $v$ and $v_{k-1}$ are from $V_{k-1}$, and the rest of the vertices are from $T$. Call these cycles *good*. The number of good cycles is at least $\frac{(s-2)_{k+1} k}{2} + 1$. Then there are at least $\left(\frac{(s-2)_{k+1} k}{2} + 1\right) \cdot (k-1)! \ge \frac{(s-2)_{k+1} k!}{2} + 1$ many copies of $C_r$ of good type containing $v$ and all the vertices in $B$. So, if there is no over-counting, then the number of copies of $C_r$ containing $k-1$ vertices from $I$ and one vertex from $S$ is at least $\binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_{k+1} k!}{2} + 1\right)$. To show that there is no over-counting, consider a vertex $v \in S$, $k-1$ non-neighbors $v_1, \cdots, v_{k-1}$ of $v$ in $I$ and sets $V_i$ for which adding $vv_i$ creates a copy of $K_s$ on $\{v,v_i\} \cup V_i$. The good cycles containing exactly one vertex from $V_1$ and one from $V_{k-1}$ cannot be counted twice because there is a unique independent set consisting of one vertex in $S$ and $k-1$ in $I$ in this kind of cycle. Now consider a good cycle with two vertices in $V_1$ or $V_{k-1}$. Without loss of generality, the cycle is of the form $vuu'v_1u_1v_2 \cdots u_{k-2}v_{k-1}wv$ with $u,u' \in V_1$. There is again a unique independent set consisting of one vertex in $S$ and $k-1$ in $I$ in this kind of cycle, because $u,v_1, \cdots, v_{k-1}$ cannot be an independent set due to the fact that $u$ and $v_1$ are adjacent. Hence, the total number of copies of $C_r$ in $G$ is at least $\frac{(s-2)_{k+1}}{2} \cdot (n-m)_k + \binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_{k+1} k!}{2} + 1 \right)$. Noting that $z = o(n)$ and $m > \log n$, we can see that this contradicts due to the following:
$$\begin{aligned}
\begin{aligned}[t]
&\frac{(s-2)_{k+1}}{2} \cdot (n-m)_k + \binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_{k+1} k!}{2} + 1 \right) \\
&\ge \frac{(s-2)_{k+1}}{2} \cdot n^k - \frac{(s-2)_{k+1}}{2} \cdot k (m+k) n^{k-1} + \frac{(s-2)_{k+1}}{2} \cdot k m (n-z-k)^{k-1} \\
& \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; + \binom{n - z}{k-1} m - \Theta\left(n^{k-1}\right) \\
&\ge \frac{(s-2)_{k+1}}{2} \cdot n^k - \frac{(s-2)_{k+1}}{2} \cdot k m n^{k-1} - \Theta\left(n^{k-1}\right) + \frac{(s-2)_{k+1}}{2} \cdot k m n^{k-1} \\
& \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; - \Theta\left(mzn^{k-2}\right) + \Theta\left(mn^{k-1}\right) \\
&\ge \frac{(s-2)_{k+1}}{2} \cdot n^k + \Theta\left(mn^{k-1}\right).
\end{aligned}\end{aligned}$$
This shows that $S$ is an empty set and so, $G$ is the union of $I$ (an independent set with maximum size) and $T$ (which is a $K_{s-2}$) where every vertex in $T$ is incident to every vertex in $I$. This finishes the proof of Theorem \[cycle\] for odd $r$.
Few copies of $\boldsymbol{C_r}$ in $\boldsymbol{K_s}$-saturated graphs for even $\boldsymbol{r}$
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Let $G$ be an $n$-vertex $K_s$-saturated graph minimizing the number of copies of $C_r$. Let $I$ be an arbitrary independent set of order $k = \frac{r}{2}$ in $G$. Our goal is to count the number of copies of $C_r$ in $G$ containing $I$. There are $\frac{(k-1)!}{2}$ circular permutations of the vertices of $I$, accounting for the directional symmetry of a cycle. Fix such an order $v_1, v_2, \cdots, v_k$. For every distinct $i,j \in [k]$, choose a set $V_{i,j}$ of $s-2$ vertices such that adding the edge $v_iv_j$ would create a copy of $K_s$ on $\{v_i,v_j\} \cup V_{i,j}$. For $i \in [k-1]$, we can iteratively pick a common neighbor $u_i$ of $v_i$ and $v_{i+1}$ among the $s-2$ vertices in $V_{i,i+1}$, and finally pick a common neighbor $u_k$ of $v_1$ and $v_k$ from $V_{1,k}$, thus forming a cycle $v_1 u_1 v_2 u_2 \cdots v_k u_k v_1$. Clearly, the number of ways to do this is at least $(s-2)_k$, so there are at least $\frac{(s-2)_k (k-1)!}{2}$ many copies of $C_r$ containing $I$. But there may be over-counting due to the fact that a cycle of length $r$ has two independent set of order $k$. So, to efficiently account for this double-counting, let us define a notion of ‘essential count’. The idea is to count a copy of $C_r$ containing two independent sets of order $k$ as half, so that the double-counting will make the count exactly one. So, we have two categories of $C_r$ containing $I$, (i) with two independent sets of order $k$, and (ii) with exactly one independent set of order $k$. Now, if there are $x$ copies of $C_r$ containing $I$ of type (i) and $y$ copies of $C_r$ containing $I$ with type (ii), then we say the essential count of the number of copies of $C_r$ containing $I$ is $\frac{x}{2} + y$. For a fixed $k$-independent set $I = \{v_1, \cdots, v_k\}$, we now want to find the essential count of the number of copies of $C_r$ containing $I$ in the order $v_1, v_2, \cdots, v_k$. As before, for every distinct $i,j \in [k]$, choose a set $V_{i,j}$ of $s-2$ vertices such that adding the edge $v_iv_j$ would create a copy of $K_s$ on $\{v_i,v_j\} \cup V_{i,j}$. Define the sets $A_j = V_{j,j+1} \setminus \bigcup_{i \neq j} V_{i,i+1}$, where $V_{k,k+1} = V_{k,1}$. Now observe that for all $j$ when we pick a common neighbor $u_j \in V_{j,j+1}$ of $v_j$ and $v_j$ to count the cycle $v_1u_1v_2 \cdots v_ku_kv_1$, the vertices $u_1, u_2, \cdots, u_k$ will form an independent set in $G$ if and only if $u_j \in A_j$ for all $j \in [k]$. So, if $s_j = |A_j|$, then the essential count is at least the following: $$f(s_1, s_2, \cdots, s_k) = \frac{1}{2} \prod_{j = 1}^k s_j + \sum_{\substack{J \subseteq [k] \\ |J| \neq 0}} \prod_{j \notin J} s_j \prod_{j \in J} (s-2-s_j - \iota(J,j)),$$ where $\iota(J,j)$ denotes the number of elements in $J$ smaller than $j$.
\[calculus\] For any $k \ge 2 \sqrt{s-2}$, the function $f(s_1, s_2, \cdots, s_k)$ attains its minimum uniquely at $(0,0, \cdots, 0)$ over the region $\{0, 1, \cdots, s-2\}^k$.
For a fixed $j$, if we fix all the variables except $s_j$ and vary $s_j$, $f$ is a linear function with respect to $s_j$. So, the minimum will occur either at $s_j = 0$ or $s_j = s-2$, when other variables are fixed. Hence, applying the same argument for all variables, we can conclude that the minimum can occur only at the vertices of the cube $[0,s-2]^k$. It is easy to check that if we evaluate $f$ at a vertex with at least one co-ordinate $0$ and one co-ordinate $s-2$, then the value will be strictly greater than $f(0,0, \cdots, 0)$. Now, the only thing we need to verify is that $f(s-2,s-2, \cdots, s-2) > f(0,0, \cdots, 0)$, which is equivalent to $\frac{1}{2} (s-2)^k > (s-2)_k$. This holds for $k \ge 2 \sqrt{s-2}$, because: $$\begin{aligned}
\frac{(s-2)_k}{(s-2)^k} = 1 \left(1 - \frac{1}{s-2}\right) \cdots \left(1 - \frac{k-1}{s-2}\right)
< e^{-\left(0 + \frac{1}{s-2} + \cdots + \frac{k-1}{s-2}\right)}
= e^{-\frac{k(k-1)}{2(s-2)}}
< \frac{1}{2}.\end{aligned}$$ The function $f$ takes strictly greater values at all vertices in $[0,s-2]^k$ than $f(0,0, \cdots, 0)$, so $f(s_1, s_2, \cdots, s_k)$ is strictly greater than $f(0,0, \cdots, 0)$ for all $(s_1, \cdots, s_k) \neq (0, \cdots, 0)$. As there are finitely many points in $\{0, 1, \cdots, s-2\}^k$, there exists some constant $\epsilon > 0$ (that does not depend on $n$, but may depend on $s$ and $k$) such that $f(s_1, s_2, \cdots, s_k) - f(0, 0, \cdots, 0) \ge \epsilon$ for all $(s_1, \cdots, s_k) \neq (0, \cdots, 0)$.
The rest of the proof is similar to the odd $r$ case, and we provide an outline here for completeness. Corollary \[indep\] and Lemma \[calculus\] imply that the number of copies of $C_r$ in $G$ is at least $(1-o(1)) \frac{(s-2)_k}{2k} \cdot n^k$, which shows that is asymptotically tight for $r \ge 4 \sqrt{s-2}$. Now by a similar argument to the odd $r$ case, the number of independent sets $I = \{v_1, \cdots, v_k\}$ of order $k$, for which there is a set $V' \subseteq V(G)$ of size $s-2$ such that for all $i \neq j$, $v$ is a common neighbor of $v_i$ and $v_j$ if and only if $v \in V'$, is $(1-o(1))\binom{n}{k}$. Next we have the following lemma whose proof is the same as Lemma \[maximum\_independent\].
\[maximum\_independent1\] For even $r$ with $4 \sqrt{s-2} \le r \le 2s-4$, there is an independent set $I$ of order $n - o(n)$ in $G$ such that there is a copy $T$ of $K_{s-2}$ in $G$ with the property that every vertex in $T$ is a neighbor of every vertex of $I$.
Define the sets $I$, $T$ and $S$, and the numbers $m$, $m'$ and $z$ as after Lemma \[maximum\_independent\]. Following the proof in the last section, we can show that all the vertices in $S$ can have at most $z$ neighbors in $I$ for sufficiently large $n$. For the sake of contradiction, assume that $S$ is non-empty. As before, the case when $m \le \log n$ leads to a contradiction, and $m > \log n$ remains the only case to resolve. For an arbitrary vertex $v \in S$ and an arbitrary set $B$ of $k-1$ vertices from $I$ that are not neighbors of $I$, consider the good cycles (as defined in the last section) containing $v$ and all the vertices in $B$. Like before, if there is no over-counting, then the number of copies of good $C_r$ containing $k-1$ vertices from $I$ and one vertex from $S$ is at least $\binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_k (k-1)!}{2} + 1\right)$. To show that there is no over-counting, it turns out that the situation is simpler in this case compared to the odd $r$ case, which follows from the fact that the good cycles always have a unique independent set consisting of one vertex in $S$ and $k-1$ in $I$. Hence, the total number of copies of $C_r$ in $G$ is at least $\frac{(s-2)_k}{2k} \cdot (n-m)_k + \binom{n - z}{k-1} (m-s+2) \left(\frac{(s-2)_k (k-1)!}{2} + 1 \right) \ge \frac{(s-2)_k}{2k} \cdot n^k + \Theta\left(mn^{k-1}\right)$, contradicting . So, we have completed the proof of Theorem \[cycle\].
Having completed the proof of Theorem \[cycle\], we also solve the problem asymptotically for $r = 4$.
For every $s \ge 4$, we have the following: $$\operatorname{sat}(n, C_4, K_s) = (1 + o(1)) \binom{s-2}{2} \binom{n}{2} = (1 + o(1)) \frac{n^2 (s-2)(s-3)}{4}.$$
The upper bound follows from . For the lower bound, consider an $n$-vertex $K_s$-saturated graph $G$ minimizing the number of copies of $C_4$. For a non-edge $uv \in E(G)$, choose a set $T \subseteq V(G)$ of $s-2$ vertices such that adding the edge $uv$ would create a copy of $K_s$ on $\{u,v\} \cup T$. Hence the number of copies of $K_4 \setminus e$ (which is the graph after removing an edge from a complete graph on 4 vertices) containing the non-edge $uv$ is at least $\binom{s-2}{2}$. By Lemma \[upperbound\], we can conclude that the number of copies of $K_4 \setminus e$ is at least $(1 - o(1)) \binom{s-2}{2} \binom{n}{2}$ (it is a routine to check that we are not doing any multiple-counting). Hence, $\operatorname{sat}(n, C_4, K_s) = (1 + o(1)) \binom{s-2}{2} \binom{n}{2}$.
Family of size 3 with non-converging saturation ratio
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In this section, we prove Theorem \[pro\]. We begin by stating the families of graphs that we will use for the construction.
\[def\] For every positive integer $m \ge 4$, let $\F_m$ be the family of the following three graphs.
- Let $B_{m,m}$ be the disjoint union of two copies of $K_m$ plus one edge joining them (often called a “dumb-bell").
- Let $V_m$ be a copy of $K_m$ plus two more edges incident to a single vertex of the $K_m$.
- Let $\Lambda_m$ be a copy of $K_m$ plus a single vertex with exactly two edges incident to the $K_m$.
The proof of Theorem \[pro\] boils down to the fact that the behavior of $\operatorname{sat}(n, \F_m)$ depends on whether or not $n$ is divisible by $m$. The following two lemmas constitute the proof.
\[easy\] For every $n$ divisible by $m$, we have $\operatorname{sat}(n, \F_m) \le \frac{n}{m} \binom{m}{2}$.
Since $n$ is divisible by $m$, the graph $G$ consisting of the disjoint union of $\frac{n}{m}$ many copies of $K_m$ is clearly $\mathcal{F}_m$-saturated, and the number of edges in $G$ is $\frac{n}{m} \binom{m}{2}$, which proves the result.
\[family\] For every $n \ge m \ge 4$ where $n$ is not divisible by $m$, we have $\operatorname{sat}(n, \F_m) \ge \frac{n-m}{m} \left(\binom{m}{2} + 1\right)$.
The proof of Lemma \[family\] will easily follow from the next three lemmas about the structure of $\F_m$-saturated graphs. Let $G$ be an $\F_m$-saturated graph on $n$ vertices. Let $B$ be the set of all vertices of $G$ which are contained in any copy of $K_m$.
\[B\] The subgraph induced by $B$ is only a disjoint union of $K_m$’s.
First, note that no subgraph of $G$ is isomorphic to $F_{m,j}$ for any $j \in \{1, 2, \dots, m-1\}$, where $F_{m,j}$ denotes the union of $2$ copies of $K_m$ overlapping in exactly $j$ common vertices. This is because each $F_{m,j}$ contains a copy of $V_m$ or $\Lambda_m$. As $B$ does not have any copies of $F_{m,j}$ for all $j$, all copies of $K_m$ induced by $B$ are pairwise disjoint. Furthermore, the subgraph of $G$ that $B$ induces is just a disjoint union of $K_m$’s, because any other edge would create a copy of $B_{m,m}$ in $G$.
Now let $A$ be the set of all vertices not in $B$. Since the structure in $B$ is so simple, our lower bound will follow by independently lower-bounding the number of edges induced by $A$, and the number of edges between $A$ and $B$. We start with $A$.
\[A\] The set $A$ has at most $m$ vertices, or $A$ is $K_m$-saturated.
If $A$ is complete, then the number of vertices in $A$ is at most $m$, or else $G$ contains a copy of $K_{m+1}$, and hence $G$ contains a copy of $\Lambda_m$, which is a contradiction.
So, suppose $A$ is not complete. We claim that adding any edge to the induced graph on $A$ must create a copy of $K_m$ in $G$. Suppose for the sake of contradiction that there is a non-edge $uv$ with $u,v \in A$ such that adding $uv$ does not create a copy of $K_m$ in $G$. However, it must create a copy of one of the graphs $B_{m,m}$, $V_m$, or $\Lambda_m$, hence one of $u$ or $v$ must be in a copy of a $K_m$ in $G$ (because these three graphs have the property that for all edges $ab$, either $a$ or $b$ is in a copy of $K_m$), which contradicts the definition of $A$.
Finally, we show that adding any edge to the induced graph on $A$ creates a copy of $K_m$ which entirely lies in $A$. Suppose for the sake of contradiction that there is a non-edge $uv$ in the induced graph on $A$ which, if added, would create a copy of $K_m$ which intersects $B$. Let $w \in B$ be a vertex which lies in a created copy of $K_m$ after adding the edge $uv$. That means that $G$ has the edges $uw$ and $vw$, and the copy of $K_m$ in $B$ containing $w$, together with the edges $uw$ and $vw$, creates a copy of $V_m$. So, this is not possible, and we conclude that the induced subgraph on $A$ is indeed $K_m$-saturated.
It only remains to bound the number of edges between $A$ and $B$. We have the following structural lemma.
\[AB\] If $A$ is non-empty, then each copy of $K_m$ in $B$ has at least one edge to $A$.
Assume for the sake of contradiction that there is a copy $U$ of $K_m$ in $B$ which does not have an edge to $A$. Consider arbitrary vertices $u \in U$ and $v \in A$. Then one of the following situations must happen.\
*Case 1: Adding $uv$ creates a copy of $K_m$.* Then it is easy to check that $U$ has an edge to $A$, which is a contradiction.\
*Case 2: Adding $uv$ creates a copy of $B_{m,m}$ with $uv$ being the middle edge connecting the copies of $K_m$.* This would imply that $v$ is in a copy of $K_m$, which is a contradiction.\
*Case 3: Adding $uv$ creates a copy of $V_m$ or $\Lambda_m$ with $uv$ being one of the two extra edges outside of the copy of $K_m$.* Then if $uv$ becomes one of the extra edges, the other extra edge should already be there and will connect $U$ and $A$, giving a contradiction.\
Since all the cases give contradictions, we are done.
We now combine the previous three lemmas to prove Lemma \[family\], which then finishes the proof of Theorem \[pro\].
Let $n$ and $m$ satisfy the conditions of Lemma \[family\]. Clearly $A$ must be non-empty because the number of vertices in $B$ is a multiple of $m$ by Lemma \[B\], and so Lemma \[AB\] implies that there are at least $k$ edges between $B$ and $A$, where $k$ is the number of disjoint copies of $K_m$ in $B$. Now from Lemma \[A\], we have two situations. When $A$ has at most $m$ vertices, using Lemmas \[B\] and \[AB\], the number of edges in $G$ is at least $\big\lfloor\frac{n}{m}\big\rfloor \binom{m}{2} + \big\lfloor\frac{n}{m}\big\rfloor \ge \frac{n-m}{m} \left(\binom{m}{2} + 1\right)$. Otherwise, $A$ is $K_m$-saturated, so Theorem \[EHM\] implies that for all $m \ge 4$ the number of edges in $G$ is at least:
$$\begin{aligned}
& k \binom{m}{2} + k + \left(n - (k+1)m + 2\right) (m-2) \\
& > (km) \frac{m-1}{2} + k + \left(n - (k+1)m\right) \left(\frac{m-1}{2} + \frac{1}{m}\right) \\
& = (km) \frac{m-1}{2} + \left(n - (k+1)m\right) \frac{m-1}{2} + k + \left(n - (k+1)m\right) \frac{1}{m} \\
& = \frac{n-m}{m} \left(\binom{m}{2} + 1\right).\end{aligned}$$
This completes the proof.
Family of size 3 for generalized saturation ratio
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Inspired by the construction for Theorem \[pro\], we extend the construction to prove Theorem \[construction\]. One of the key challenges is to find an appropriate extension of Theorem \[EHM\]. Fortunately, our Theorem \[ssat\] rescues us. We start by stating the families of graphs that we will use for the construction, which are not quite the straightforward generalizations of the families used in Theorem \[pro\]. For notational brevity, let $r \ge 2$ be a fixed integer for the remainder of this section.
For every positive integer $m \ge 2r^2 + 2r$, let $\F_m$ be the family of the following three graphs.
- Let $B_{m,m}$ be the same “dumb-bell" graph from Definition \[def\].
- Let $V_{m,r}$ be the union of a copy of $K_m$ and a copy of $K_{m-r+1}$ overlapping in exactly one common vertex.
- Let $\Lambda_{m,r}$ be a copy of $K_m$ plus a single vertex with exactly $r$ edges incident to the $K_m$.
Note that for $r = 2$, we have $\Lambda_{m,2} = \Lambda_m$. However, $V_{m,r}$ is not quite a generalization of $V_m$, and in fact $V_m$ is a subgraph of $V_{m,2}$. We considered $V_m$ instead of $V_{m,2}$ in the case of $r = 2$ to make the analysis simpler and more elegant. So, the above construction actually gives different families of three graphs with non-converging saturation ratio for $r = 2$.
We proceed to the proof of Theorem \[construction\]. It turns out that the behavior of $\operatorname{sat}(n, K_r, \F_m)$ is similar to before, i.e., it depends on whether or not $n$ is divisible by $m$. The following two lemmas constitute the proof.
For every $n$ divisible by $m$, we have $\operatorname{sat}(n, K_r, \F_m) \le \frac{n}{m} \binom{m}{r}$.
The same graph used in the proof of Lemma \[easy\], i.e., the disjoint union of $\frac{n}{m}$ many copies of $K_m$, gives us the desired upper bound.
\[family’\] For every $n \ge m \ge 2r^2 + 2r$ where $n$ is not divisible by $m$, we have that $\operatorname{sat}(n, K_r, \F_m) \ge \frac{n}{m} \left(\binom{m}{r} + 1\right) - o(n)$.
Similarly to Lemma \[family\], the proof of Lemma \[family’\] will follow from the next couple of structural lemmas about $\F_m$-saturated graphs. Let $G$ be an $\F_m$-saturated graph on $n$ vertices. Let $B$ be the set of all vertices of $G$ which are contained in any copy of $K_m$. The subgraph induced by $B$ is only a disjoint union of $K_m$’s, by essentially the same proof as Lemma \[B\]. Now let $A$ be the set of all vertices not in $B$. Motivated by Lemma \[A\], we have the following lemma.
\[A’\] $A$ has at most $m$ vertices, or $A$ is strongly $K_{m-r}$-saturated.
This is in contrast to Lemma \[A\], which got that the induced graph on $A$ was $K_m$-saturated. Here we only get strongly $K_{m-r}$-saturated (recall that despite its counterintuitive name, strong saturation is a weaker condition), but we can later use our Theorem \[ssat\] to lower-bound the number of copies of $K_r$ in $A$.
If $A$ is complete, then the number of vertices in $A$ is at most $m$, or else $G$ contains a copy of $K_{m+1}$, and hence $G$ contains a copy of $\Lambda_{m,r}$, which is a contradiction. So, suppose $A$ is not complete. Fix a non-edge $uv$ in the induced graph on $A$. We consider two cases.\
*Case 1: Adding $uv$ would create a copy of $K_m$ in $G$.* We will show that the copy of $K_m$ would lie entirely in $A$, giving the required $K_{m-r}$ in $A$. Indeed, assume for the sake of contradiction that there is a non-edge $uv$ in the induced graph on $A$ which, if added, would create a copy of $K_m$ which intersects $B$. That implies that there is a copy $T$ of $K_{m-1}$ which contains the vertex $u$ and intersects $B$. Clearly $T$ can intersect only a single copy $U$ of $K_m$ in $B$, because the induced graph on $B$ is just a disjoint union of $K_m$’s. Now, if $|T \cap U| \ge r$, then $T \cup U$ contains a copy of $\Lambda_{m,r}$, which is a contradiction. Otherwise, $|T \cap U| < r$, and so $T \cup U$ contains a copy of $V_{m,r}$, which is also a contradiction.\
*Case 2: Adding $uv$ would not create a copy of $K_m$ in $G$.* If adding $uv$ creates a copy of $B_{m,m}$ or $\Lambda_{m,r}$ in $G$, then one of $u$ or $v$ must be in a copy of a $K_m$ in $G$, which contradicts the definition of $A$. Alternatively, if adding $uv$ creates a copy of $V_{m,r}$ in $G$, then that copy of $V_{m,r}$ would contain a copy of $K_m$ in $B$, together with $m-r$ vertices in $A$. These $m-r$ vertices would clearly induce a copy of $K_{m-r}$ after adding $uv$. Hence we are done.
Next, following the proof of Lemma \[family\], we bound the number of $K_r$’s that intersect both $A$ and $B$.
\[AB’\] Suppose $m \ge 2r + 1$. If $A$ is non-empty, then for each copy $U$ of $K_m$ in $B$, there is at least one copy of $K_r$ intersecting both $U$ and $A$.
Assume for the sake of contradiction that there is a copy $U$ of $K_m$ in $B$ for which there is no copy of $K_r$ intersecting both $U$ and $A$. Consider arbitrary non-adjacent vertices $u \in U$ and $v \in A$. One of the following situations must happen.\
*Case 1: Adding $uv$ creates a copy $T$ of $K_{m-r}$.* Then it is easy to check that there is a copy of $K_{m-r-1}$ (and hence a copy of $K_r$ if $m \ge 2r + 1$) intersecting both $U$ and $A$, which is a contradiction.\
*Case 2: Adding $uv$ creates a copy of $B_{m,m}$ with $uv$ being the middle edge connecting the copies of $K_m$.* This case is exactly the same as before, i.e., $v$ is in a copy of $K_m$, which is a contradiction.\
*Case 3: Adding $uv$ creates a copy of $\Lambda_{m,r}$ with $uv$ being one of the $r$ extra edges outside of the copy of $K_m$.* Then if $uv$ becomes one of the extra $r$ edges, the $r-1$ endpoints in $U$ of the remaining $r-1$ extra edges, together with the vertex $v$, induce a copy of $K_r$, giving a contradiction.\
Since all the cases give contradictions, we are done.
Let $n$ and $m$ satisfy the conditions of Lemma \[family’\]. Clearly $A$ must be non-empty because the number of vertices in $B$ is a multiple of $m$, and so Lemma \[AB’\] implies that there are at least $k$ copies of $K_r$ intersecting both $B$ and $A$, where $k$ is the number of disjoint copies of $K_m$ in $B$. Now from Lemma \[A’\], we have two situations. When $A$ has at most $m$ vertices, by Lemma \[AB’\], the number of copies of $K_r$ in $G$ is at least $\big\lfloor\frac{n}{m}\big\rfloor \binom{m}{r} + \big\lfloor\frac{n}{m}\big\rfloor \ge \frac{n-m}{m} \left(\binom{m}{r} + 1\right)$. Otherwise, $A$ is strongly $K_{m-r}$-saturated, so Theorem \[ssat\] implies that $A$ induces at least $\binom{m-r-2}{r-1} (n-km) - o(n)$ many copies of $K_r$, and so for all $m \ge 2r^2 + 2r$ the number of copies of $K_r$ in $G$ is at least: $$\begin{aligned}
k \binom{m}{r} + k + & \binom{m-r-2}{r-1} (n-km) - o(n). \label{binomial}\end{aligned}$$ To get the required lower bound, we next prove the simple claim that $\binom{m-r-2}{r-1} \ge \frac{1}{m} \left(\binom{m}{r} + 1\right)$ for all $m \ge 2r^2 + 2r$ and $r \ge 2$. The most convenient way to do this is to show that $m \binom{m-r-2}{r-1} > \binom{m}{r}$, since both sides of this last inequality are integers. Indeed, let $m$ and $r$ satisfy the conditions we just mentioned. Then, $$\frac{m-1}{m-r-2} \le \frac{m-2}{m-r-3} \le \cdots \le \frac{m-r+1}{m-2r} \le \frac{2r^2 + r + 1}{2r^2}.$$ Hence, $$\frac{\binom{m}{r}}{m \binom{m-r-2}{r-1}} \le \frac{1}{r} \left(1 + \frac{r + 1}{2r^2}\right)^{r-1} \le \frac{1}{r} \cdot e^{\frac{(r+1)(r-1)}{2r^2}} \le \frac{1}{r} \cdot \sqrt{e} < 1,$$ which establishes the claim that $\binom{m-r-2}{r-1} \ge \frac{1}{m} \left(\binom{m}{r} + 1\right)$. Using this, we get that is at least $k \binom{m}{r} + k + \frac{1}{m} \left(\binom{m}{r} + 1\right) (n-km) - o(n) \ge \frac{n}{m} \left(\binom{m}{r} + 1\right) - o(n)$, completing the proof.
Irregular behavior of $\boldsymbol{\operatorname{sat}(n, C_r, F_r)}$
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In this section, we prove Theorem \[easy\_cons\]. In particular, we will prove that for every $r \ge 4$, $\lim \inf_{n \rightarrow \infty} \operatorname{sat}(n, C_{r+1}, B_{r,r}) = 0$, and $\lim \sup_{n \rightarrow \infty} \operatorname{sat}(n, C_{r+1}, B_{r,r}) > 0$, where $B_{r,r}$ is the same “dumb-bell” graph from Definition \[def\]. We remark here that this statement is false for $r = 2$ and $r = 3$, which we show in Proposition \[excess\] at the end of this section. The following two lemmas constitute the entire proof of Theorem \[easy\_cons\].
\[si\] For every $n$ divisible by $r$, we have $\operatorname{sat}(n, C_{r+1}, B_{r,r}) = 0$.
The same graph used in the proof of Lemma \[easy\], i.e., the disjoint union of $\frac{n}{r}$ many copies of $K_r$, is $B_{r,r}$-saturated but has no copies of $C_{r+1}$, proving the lemma.
\[cycle\_sat\] For every $n \ge 2r$, and $r \ge 4$ such that $n$ is not divisible by $r$, we have $\operatorname{sat}(n, C_{r+1}, B_{r,r}) \ge 1$.
Let $G$ be a $B_{r,r}$-saturated graph on $n$ vertices. We show that there is a cycle of length $r+1$ in $G$ if the conditions of Lemma \[cycle\_sat\] are met. We divide the proof in two cases.\
*Case 1: There is a copy of $F_{r,j}$ in $G$ for some $j \in \{1, 2, \cdots, r-1\}$, where $F_{r,j}$ denotes the union of $2$ copies of $K_r$ overlapping in exactly $j$ common vertices.* It is easy to check that $F_{r,j}$ contains a copy of $C_{r+1}$ for every $j \ge 2$. Hence, if $G$ contains a copy of $F_{r,j}$ for some $j \ge 2$, then there is already a cycle of length $r+1$ in $G$. So, we can assume that $G$ contains a copy of $F_{r,1}$. Assume that $w, u_1, u_2, \cdots, u_{r-1}, v_1, v_2, \cdots, v_{r-1}$ are distinct vertices such that $\{w, u_1, \cdots, u_{r-1}\}$ and $\{w, v_1, \cdots, v_{r-1}\}$ both induce $K_r$. Note that if there is an edge $u_iv_j$ for some $i,j$, then it is easy to find a copy of $C_{r+1}$ using the edge $u_iv_j$. For example, if $u_1v_1$ is an edge, then $w v_1 u_1 u_2 \cdots u_{r-1} w$ forms a $C_{r+1}$. So, we can assume that there is no edge $u_iv_j$ for any $i,j$. Now one of the following situations must happen.\
*Subcase 1: Adding $u_1v_1$ creates a copy of $K_r$.* So, $u_1$ and $v_1$ must have at least $r-2$ common neighbors. If $r \ge 4$, then among $r-2 \ge 2$ common neighbors of $u_1$ and $v_1$, we can pick a vertex $x$ which is distinct from $w$. Now it is easy to check that $w v_1 x u_1 u_2 \cdots u_{r-2} w$ forms a cycle of length $r+1$.\
*Subcase 2: Adding $u_1v_1$ creates a copy of $B_{r,r}$ with $u_1v_1$ being the middle edge connecting the copies of $K_m$.* Hence, there is either a copy of $K_r$ containing $u_1$ and not containing any vertex in $\{w, v_1, \cdots, v_{r-1}\}$, or a copy of $K_r$ containing $v_1$ and not containing any vertex in $\{w, u_1, \cdots, u_{r-1}\}$. Due to symmetry, it is enough to check the first situation. If there is a copy of $K_r$ containing $u_1$ and not containing any vertex in $\{w, v_1, \cdots, v_{r-1}\}$, then that copy of $K_r$ along with the copy of $K_r$ induced by $\{w, v_1, \cdots, v_{r-1}\}$ and the edge $u_1w$ forms a copy of $B_{r,r}$, which is a contradiction.\
*Case 2: There is no copy of $F_{r,j}$ in $G$ for any $j \in \{1, 2, \cdots, r-1\}$.* Let $B$ be the set of all vertices of $G$ which are contained in any copy of $K_r$. Firstly note that $B$ cannot be empty, because there are two disjoint copies of $K_r$ in the graph $B_{r,r}$ and it is not possible to create two disjoint copies of $K_r$ by adding one edge to $G$. The subgraph induced by $B$ is only a disjoint union of $K_r$’s, by the same proof as Lemma \[B\]. Now let $A$ be the set of all vertices not in $B$. Clearly $A$ must be non-empty, because the number of vertices in $B$ is a multiple of $r$, and $r$ does not divide $n$. Fix a copy $T = \{v_1, v_2, \cdots, v_r\}$ of $K_r$ in $B$. Now one of the following situations must happen.\
*Subcase 1: There is at most one vertex in $T$ which has edges to $A$.* If there is no edge between $T$ and $A$, then some easy case-checking (similar to before) implies that adding an edge $v_1 a$ (where $a \in A$) would not create a copy of $B_{r,r}$. Now assume that there is exactly one vertex (without loss of generality $v_1$) in $T$ which has edges to $A$. Again some easy case-checking will tell us that adding $v_2 a$ for any $a \in A$ would not create a copy of $K_r$ (because $r \ge 4$), so, the only way to create a copy of $B_{r,r}$ would be to become the middle edge connecting the copies of $K_r$ in $B_{r,r}$, but that would contradict the fact that $a \in A$ (remember that no vertices in $A$ are in a copy of $K_r$). These are all contradictions.\
*Subcase 2: There are at least two vertices in $T$ which have edges to $A$.* If there is $a \in A$ such that $a$ is adjacent to at least two vertices in $T$, then one can find a cycle of length $r+1$ (for example, without loss of generality $a$ is adjacent to both $v_1$ and $v_2$, so, $v_1 a v_2 v_3 \cdots v_r v_1$ forms a copy of $C_{r+1}$). So, we can assume that for all $a \in A$, the vertex $a$ is adjacent to at most one vertex in $T$. Without loss of generality, $v_1, v_2 \in T$ have edges to $A$. Let $v_1 a_1$ and $v_2 a_2$ be edges for some $a_1, a_2 \in A$. If $a_1 a_2$ is an edge, then $v_1 a_1 a_2 v_2 v_3 \cdots v_{r-1} v_1$ forms a cycle of length $r+1$. Now if $a_1 a_2$ is a non-edge, then adding $a_1 a_2$ would create a copy of $K_r$ (because it must create a copy of $B_{r,r}$, but $a_1 a_2$ could not become the middle edge in the created copy of $B_{r,r}$ due to the fact that $a_1$ is not in a copy of $K_r$ in $G$). Note that $a_1$ and $a_2$ cannot have a common neighbor in $T$, so they must have a common neighbor $x \not \in T$, which implies that $v_1 a_1 x a_2 v_2 \cdots v_{r-2}$ forms a cycle of length $r+1$ in $G$.\
Since all the cases either find a cycle of length $r+1$ or give contradictions, we are done.
\[excess\] For $r = 2$ and $r = 3$, we have $\operatorname{sat}(n, C_{r+1}, B_{r,r}) = 0$ for all $n \ge 2r$.
In Lemma \[si\], we have already seen that $\operatorname{sat}(n, C_{r+1}, B_{r,r}) = 0$ when $n$ is divisible by $r$. So, we have to prove Proposition \[excess\] when $n$ is not divisible by $r$.
The graph $B_{2,2}$ is the path with 3 edges, i.e., $P_3$. A graph which is a disjoint union of $P_2$’s and $P_1$’s is always $P_3$-saturated. If $n \ge 4$ is odd, then the graph consisting of the disjoint union of a copy of $P_2$ and $\frac{n-3}{2}$ many copies of $P_1$ is a $P_3$-saturated graph with no copy of $C_3$. So, we have $\operatorname{sat}(n, C_3, B_{2,2}) = 0$ for all $n \ge 4$.
For $r = 3$, we split into two cases depending on the value of $n \pmod 3$. If $n$ is of the form $3m + 1$ for some integer $m$, then the graph consisting of $m$ disjoint copies of $K_3$ together with $m$ edges connecting an extra vertex to each copies of $K_3$, is a $B_{3,3}$-saturated graph without any copy of $C_4$. So, for $n \equiv 1 \pmod 3$, we have $\operatorname{sat}(n, C_4, B_{3,3}) = 0$.
Now when $n$ is of the form $3m + 2$, we have a similar construction. Consider a graph $G$ on the vertex set $\{a,b\} \cup \{x_1, x_2, \cdots, x_m\} \cup \{y_1, \cdots, y_m\} \cup \{z_1, \cdots, z_m\}$, and the edge set $\{x_jy_j : j \in [m]\} \cup \{y_jz_j : j \in [m]\} \cup \{z_jx_j : j \in [m]\} \cup \{ax_j : j \in [m]\} \cup \{by_j : 2 \le j \le m\} \cup \{bx_1\}$. It is easy to verify that $G$ is $B_{3,3}$-saturated, and does not have a copy of $C_4$. Hence, we have $\operatorname{sat}(n, C_4, B_{3,3}) = 0$ for all $n \ge 6$.
Concluding remarks
==================
We end with some open problems. We determined the exact value of $\operatorname{sat}(n, K_r, K_s)$ for all sufficiently large $n$, but our arguments do not extend to find the value for small $n$. So, the following question still remains open.
For $s > r \ge 3$, determine the exact value of $\operatorname{sat}(n, K_r, K_s)$ for all $n$.
We have already made a remark on the maximum constant $c_r$ we can write in the stability result in Theorem \[sat\]. It would be interesting to determine that maximum constant.
For $s > r \ge 3$, what is the second smallest number of copies of $K_r$ in an $n$-vertex $K_s$-saturated graph?
It might be interesting to consider a more general problem of finding the spectrum (set of possible values) of the number of copies of $K_r$ in a $K_s$-saturated graph. The $r = 2$ case, i.e., the edge spectrum of $K_s$-saturated graphs, was completely solved in [@AFGS] and [@BCFF].
For $s > r \ge 3$, what are the possible numbers of copies of $K_r$ in an $n$-vertex $K_s$-saturated graph?
We could not extend our method to find the exact value of $\operatorname{sat}(n, C_r, K_s)$ for the situations when $r = 5$, and when $r$ is an even number with $r = O(\sqrt{s})$. So, it will be interesting to find the values of $\operatorname{sat}(n, C_r, K_s)$ for all $r$. We conjecture that the extremal graph $K_{s-2} \ast \overline{K}_{n-s+2}$ is the unique graph minimizing the number of cycles of length $r$ among all $n$-vertex $K_s$-saturated graphs. It is worth mentioning that in Lemma \[upperbound\], if we can prove that any $n$-vertex $K_s$-saturated graph with the minimal number of copies of $C_r$ has $o(n^{\frac{3}{2}})$ edges for even $r$, then the proof of Theorem \[cycle\] for odd $r$ in Section 5 can be adapted for even $r$ as well, and it will prove our conjecture for even $r \ge 6$.
For every $s \ge 4$ and $r \le 2s - 4$, compute the exact value of $\operatorname{sat}(n, C_r, K_s)$.
Theorem \[sat\] and Theorem \[cycle\] motivate us to ask the following general question.
Is there a graph $F$, for which $K_{s-2} \ast \overline{K}_{n-s+2}$ does not (uniquely) minimize the number of copies of $F$ among $n$-vertex $K_s$-saturated graphs for all sufficiently large $n$?
As we mentioned earlier, Conjecture \[con\] is still wide open and likely needs new ideas to settle it. It would be interesting to figure out if the size of the family in Theorem \[pro\] can be further reduced to $2$. Finally, as we briefly discussed before stating Theorem \[construction\], it would be interesting to consider Conjecture \[con\] for the generalized saturation problem.
For $r \ge 2$, does the limit $\lim_{n \rightarrow \infty} \frac{\operatorname{sat}(n, K_r, F)}{n}$ exist for every graph $F$?
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to the anonymous referees for their suggestions and comments to improve the exposition of this paper. In particular, we are thankful to them for pointing out a technical issue in Lemma \[upperbound\] in an earlier version of this paper.
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[^1]: Department of Mathematical Sciences, Carnegie Mellon University. Email: [[email protected]]{}. Research supported in part by National Science Foundation CAREER Grant DMS-1455125.
[^2]: Department of Mathematical Sciences, Carnegie Mellon University. Email: [[email protected]]{}. Research supported in part by National Science Foundation CAREER Grant DMS-1455125.
| ArXiv |
---
abstract: 'A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.'
address:
- |
M. Hitrik, Department of Mathematics\
UCLA\
Los Angeles\
CA 90095-1555\
USA
- |
K. Krupchyk, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
- |
P. Ola, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
- |
L. Päivärinta, Department of Mathematics and Statistics\
University of Helsinki\
P.O. Box 68\
FI-00014 Helsinki\
Finland
author:
- Michael Hitrik
- Katsiaryna Krupchyk
- Petri Ola
- Lassi Päivärinta
title: Transmission eigenvalues for elliptic operators
---
Introduction
============
Let $P_0(D)$ be an elliptic partial differential operator on ${\mathbb{R}}^n$, $n\ge 2$, of order $m\ge 2$ with constant real coefficients, $$P_0(D)=\sum_{|\alpha|\le m} a_{\alpha}D^\alpha, \quad a_\alpha\in{\mathbb{R}}, \quad D_j=-i\frac{\partial}{\partial x_j},\quad j=1,\dots,n.$$ Let $\Omega\subset {\mathbb{R}}^n$ be a bounded domain with a $C^\infty$-boundary and assume that $V\in C^\infty(\overline{\Omega}, {\mathbb{R}})$ with $V>0$ in $\overline{\Omega}$. The interior transmission problem associated to $P_0$ and $V$ is the following degenerate boundary value problem, $$\label{eq_TE_acoustic}
\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0-\lambda(1+ V))w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{m}_0(\Omega).
\end{aligned}$$ Here $H^m_0(\Omega)$ is the standard Sobolev space, defined as the closure of $C^\infty_0(\Omega)$ in the Sobolev space $H^m(\Omega)$. We say that $\lambda\in {\mathbb{C}}$ is a transmission eigenvalue if the problem has non-trivial solutions $0\ne v\in L^2_{\textrm{loc}}(\Omega)$ and $0\ne w\in L^2_{\textrm{loc}}(\Omega)$.
In the recent paper [@HitKruOlaPai], we have studied the interior transmission problem and transmission eigenvalues for multiplicative sign-definite perturbations of linear partial differential operators with constant real coefficients. Sufficient conditions for the discreteness of the set of transmission eigenvalues and for the existence of real transmission eigenvalues were obtained. In particular, in the elliptic case, the set of transmission eigenvalues is discrete and in [@HitKruOlaPai], the existence of real transmission eigenvalues was obtained for certain elliptic operators such as the biharmonic operator and the Dirac system in ${\mathbb{R}}^3$.
The purpose of the present note is to point out an approach to the study of the transmission eigenvalues in the elliptic case, based on a reduction to the eigenvalue problem for a compact non-selfadjoint operator. By an application of Lidskii’s theorem, we obtain sufficient conditions for the existence of (possibly complex) transmission eigenvalues, and the completeness of the set of the generalized eigenvectors, as well as demonstrate the generic existence of transmission eigenvalues. Let us mention explicitly that in this approach, we were directly inspired by the recent works [@AboRob; @ChaHelLap04; @HelRobWang; @Rob2004], where similar ideas in dealing with quadratic eigenvalue problems have been used to study hypoelliptic partial differential operators which are not analytic hypoelliptic.
The significance of transmission eigenvalues and of the interior transmission eigenvalue problem comes from inverse scattering theory, and originally, this problem was introduced in [@ColMonk88] in this context. The real transmission eigenvalues can be characterized as those values for which the scattering amplitude is not injective, see [@ColPaiSyl; @HitKruOlaPai]. Furthermore, in reconstruction algorithms of inverse scattering theory [@CakColbook; @ColKir96; @KirGribook], transmission eigenvalues correspond to frequencies that one needs to avoid in the reconstruction procedure.
Recently there has been a large number of works devoted to the interior transmission eigenvalue problem [@CakColGint_complex; @CakColHous10; @CakDroHou; @ColKirPai; @kir07; @paisyl08], with the major part being concerned with the case $P_0=-\Delta$. The existing results establish the discreteness of the set of transmission eigenvalues, [@ColKirPai], and give sufficient conditions for the existence of an infinite set of real transmission eigenvalues, [@CakDroHou; @paisyl08]. We would particularly like to mention the recent paper [@CakColGint_complex], where the existence of complex transmission eigenvalues was shown, assuming that the perturbation $V$ in is constant and sufficiently small.
In this note, we have chosen to base our presentation on the generalized acoustic wave equation $(P_0-\lambda(1+V))u=0$. Under the assumption that the full symbol of $P_0$ is non-negative, all the results could equally well have been derived for the following interior transmission problem associated to the Schrödinger equation $(P_0+V-\lambda)u=0$, $$\begin{aligned}
(P_0-\lambda)v=0 \quad &\text{in} \quad \Omega,\\
(P_0+V-\lambda)w=0 \quad &\text{in} \quad \Omega,\\
v-w \in H^{m}_0(\Omega).\end{aligned}$$
The structure of this note is as follows. In Section 2 we reduce the interior transmission problem to an eigenvalue problem for a compact non-selfadjoint operator in a suitable Schatten class. As a consequence of this reduction, in Section 3, we derive sufficient conditions for the existence of transmission eigenvalues and completeness of the generalized eigenstates. Finally, in Section 4, we show the generic existence of transmission eigenvalues in the trace class case.
Reduction to an eigenvalue problem for a non-selfadjoint compact operator
=========================================================================
From [@HitKruOlaPai], let us recall the following characterization of transmission eigenvalues.
\[thm\_equivalence\] Assume that $V\in C^\infty(\overline{\Omega}, {\mathbb{R}})$ with $V>0$ in $\overline{\Omega}$. A complex number $\lambda\ne 0$ is a transmission eigenvalue if and only if there exists $0\ne u\in H^{m}_0(\Omega)$ satisfying $$T_\lambda u:=(P_0-\lambda(1+V))\frac{1}{V}(P_0-\lambda)u=0\quad \text{in}\quad \mathcal{D}'(\Omega).$$
The question of deciding whether $0\ne \lambda\in {\mathbb{C}}$ is a transmission eigenvalue is therefore equivalent to finding a non-trivial solution $u\in H^m_0(\Omega)$ of the following quadratic eigenvalue problem $$\label{eq_quadratic}
T_\lambda u=(A-\lambda B +\lambda^2 C)u=0,$$ where $$A=P_0\frac{1}{V}P_0,\quad B=\frac{1}{V}P_0+P_0\frac{1}{V}+P_0,\quad C=1+\frac{1}{V}.$$ Consider the following factorization $$\begin{aligned}
T_\lambda=C^{1/2}L_\lambda C^{1/2}, \quad L_\lambda&=\tilde A-\lambda \tilde B+\lambda^2,\\
\tilde A&=C^{-1/2}AC^{-1/2}, \quad \tilde B=C^{-1/2}BC^{-1/2}.\end{aligned}$$ In [@HitKruOlaPai] it was proved that the operator $\tilde A$, equipped with the domain $$\mathcal{D}(\tilde A)=H^{2m}(\Omega)\cap H^m_0(\Omega),$$ is a self-adjoint operator on $L^2(\Omega)$ with a discrete spectrum. Here the regularity assumption on $V$ can be relaxed to $V\in C^N(\overline{\Omega})$, with $N$ being large enough but finite.
\[prop\_properties\]
- The operator $\tilde A$ is positive, and $ \mathcal{D}(\tilde A^{1/2})=H_0^m(\Omega)$.
- The operators $\tilde B\tilde A^{-1/2}$ and $\tilde A^{-1/2}\tilde B$ are bounded in $L^2(\Omega)$.
- The operator $\tilde A^{-1/2}$ is in the Schatten class $\mathcal{C}^p$ for $p>n/m$.
We refer to [@Sim_book] for the definition and properties of the Schatten class operators.
(i). Let $u\in \mathcal{D}(\tilde A)\subset H^m_0(\Omega)$. Then $$(\tilde Au,u)=\int_{\Omega} \frac{1}{V}|P_0C^{-1/2}u|^2dx\ge C_{\Omega,V} \|u\|^2,\quad C_{\Omega,V}>0.$$ Here the last inequality follows from the estimate [@horbookII Theorem 10.3.7] $$\|P_0(D)u\|\ge C_{\Omega}\|u\|, \quad u\in H^m_0(\Omega).$$ We know from [@HitKruOlaPai] that the form domain of the positive self-adjoint operator $\tilde A$ is $H_0^m(\Omega)$ and thus, $$\mathcal{D}(\tilde A^{1/2})=H_0^m(\Omega).$$
(iii). The claim follows from the fact that the inclusion map $$i:H^m_0(\Omega)\to L^2(\Omega)$$ is in the Schatten class $\mathcal{C}^p$ for $p>n/m$. The latter can be concluded from the fact that the operator $(1-\Delta)^{-m/2}$ is in the Schatten class $\mathcal{C}^p$ for $p>n/m$, on $L^2(\mathbb{T}^n)$, where $\mathbb{T}^n$ is the $n$-dimensional torus. This concludes the proof of the proposition, as (ii) is clear.
Notice that $0\ne \lambda\in {\mathbb{C}}$ is an eigenvalue of the quadratic eigenvalue problem $T_\lambda u=0$ with an eigenstate $u\in H^m_0(\Omega)$ if and only if $\lambda$ is an eigenvalue of the quadratic eigenvalue problem $L_\lambda v=0$ with $v=C^{1/2} u\in H^m_0(\Omega)$.
The holomorphic family $L_\lambda:\mathcal{D}(\tilde A)\to L^2(\Omega)$ is Fredholm of index $0$, invertible at $\lambda=0$. Thus, by the analytic Fredholm theory, $$L^{-1}_\lambda:L^2(\Omega)\to \mathcal{D}(\tilde A), \quad \lambda\in {\mathbb{C}},$$ is a meromorphic family of operators, with residues of finite rank.
Following [@Rob2004], consider the closed operator $$\mathcal{A}=\begin{pmatrix} 0 & 1\\
-\tilde A & \tilde B
\end{pmatrix},$$ acting in the Hilbert space $$\mathcal{K}=\mathcal{D}(\tilde A^{1/2})\times L^2(\Omega)=H^m_0(\Omega)\times L^2(\Omega),$$ equipped with the domain $$\mathcal{D}(\mathcal{A})=\mathcal{D}(\tilde A)\times \mathcal{D}(\tilde A^{1/2})=(H^{2m}(\Omega)\cap H^m_0(\Omega))\times H^m_0(\Omega).$$ The spectrum of $\mathcal{A}$ is discrete, and as $$(\mathcal{A}-\lambda)^{-1}=\begin{pmatrix}
L_\lambda^{-1}(\tilde B-\lambda) & - L_\lambda^{-1}\\
L_\lambda^{-1}\tilde A & - L_\lambda^{-1}\lambda
\end{pmatrix},$$ it follows that $0\ne \lambda\in {\mathbb{C}}$ is a transmission eigenvalue if and only if $\lambda$ is an eigenvalue of the operator $\mathcal{A}$. The latter is equivalent to the fact that $1/\lambda$ is an eigenvalue of the operator $$\mathcal{A}^{-1}=\begin{pmatrix}
\tilde A^{-1}\tilde B & -\tilde A^{-1}\\
1 & 0
\end{pmatrix}:\mathcal{K}\to \mathcal{K}.$$ Given Proposition \[prop\_properties\], it follows from [@Rob2004] that $\mathcal{A}^{-1}$ is in the Schatten class $\mathcal{C}^p$ on $\mathcal{K}$, for $p>n/m$.
It will be more convenient to work in the Hilbert space $L^2(\Omega)\times L^2(\Omega)$ rather than $\mathcal{K}$. To this end, we introduce the operator $$T=\begin{pmatrix} \tilde A^{1/2} & 0\\
0 & 1
\end{pmatrix},$$ which defines an isomorphism $$T:\mathcal{K}\to L^2(\Omega)\times L^2(\Omega),$$ and set $$\label{eq_operator_P}
\mathcal{D}=T\mathcal{A}^{-1}T^{-1}=
\begin{pmatrix} \tilde A^{-1/2} \tilde B\tilde A^{-1/2} & -\tilde A^{-1/2}\\
\tilde A^{-1/2} &0
\end{pmatrix}: L^2(\Omega)\times L^2(\Omega)\to L^2(\Omega)\times L^2(\Omega).$$ The operator $\mathcal{D}$ is in the Schatten class $\mathcal{C}^p$ on $L^2(\Omega)\times L^2(\Omega)$. We summarize this section in the following result.
\[prop\_mathcal[D]{}\]
A complex number $\lambda\ne 0$ is a transmission eigenvalue for [(\[eq\_TE\_acoustic\])]{} if and only if $1/\lambda$ is an eigenvalue of the operator $\mathcal{D}$ in [(\[eq\_operator\_P\])]{}.
It was shown in [@ColKirPai; @ColPaiSyl; @HitKruOlaPai] that the set of transmission eigenvalues is discrete. The proof relied upon the analytic Fredholm theory. Our reduction of the transmission eigenvalue problem to the eigenvalue problem for the compact operator $\mathcal{D}$ gives another proof of the discreteness of the set of transmission eigenvalues in the elliptic case.
Existence of transmission eigenvalues and completeness of transmission eigenstates
==================================================================================
In this section, we continue to work under the assumptions made in the beginning of the paper, namely that $P_0=P_0(D)$ is elliptic, $V\in C^\infty(\overline\Omega)$, $V>0$ on $\overline\Omega$, and $\p \Omega\in C^\infty$.
In the previous section, we have reduced the transmission eigenvalue problem to a spectral problem for the operator $\mathcal{D}\in \mathcal{C}^p$, $p>n/m$. Recall from [@Sim_book] that this implies that $\mathcal{D}^p$ is of trace class, provided that $p\in {\mathbb{N}}$.
The following result is our main criterion for the existence of transmission eigenvalues. It is based on an application of Lidskii’s theorem, which we recall for the convenience of the reader, see e.g. [@GohGolKaa]: let $\mathcal{A}$ be a trace class operator. Then $$\sum_j\mu_j(\mathcal{A})=\textrm{tr}(\mathcal{A}),$$ where $\mu_j(\mathcal{A})$ are the non-vanishing eigenvalues of $\mathcal{A}$ counted with their algebraic multiplicities. In particular, if the spectrum $\textrm{spec}(\mathcal{A})=\{0\}$, then $\textrm{tr}(\mathcal{A})=0$.
\[thm\_trace\_L\] Assume that $p>n/m$, $p\in {\mathbb{N}}$, and $\emph{\textrm{tr}}(\mathcal{D}^{p})\ne 0$. Then the set of transmission eigenvalues is non-empty.
Assume that spectrum $\textrm{spec}(\mathcal{D})=\{0\}$. Then $\textrm{spec}(\mathcal{D}^p)=\{0\}$, since $$r(\mathcal{D}^p)=\lim_{n\to\infty}\|\mathcal{D}^{pn}\|^{1/n}=\lim_{n\to\infty}\|\mathcal{D}^{n}\|^{p/n}=r(\mathcal{D})^p=0,$$ where $r(\mathcal{D})$ is the spectral radius of $\mathcal{D}$. By an application of Lidskii’s theorem, we get $\textrm{tr}(\mathcal{D}^{p})=0$, which contradicts the assumption of the proposition.
In the case when $m>n$, the operator $\mathcal{D}$ is of trace class on $L^2(\Omega)\times L^2(\Omega)$, and $\textrm{tr}(\mathcal{D})=\textrm{tr}(\tilde A^{-1/2}\tilde B\tilde A^{-1/2})=\textrm{tr}(\tilde B\tilde A^{-1})$.
In the case when $m>n/2$, the operator $\mathcal{D}$ is of Hilbert-Schmidt class and $$\textrm{tr}(\mathcal{D}^{-2})=\textrm{tr}(\tilde A^{-1/2}(\tilde B\tilde A^{-1}\tilde B-2)\tilde A^{-1/2}).$$
The question of completeness of the eigenstates for the transmission eigenvalue problem for the Helmholtz equation has been posed in [@CakDroHou]. To the best of our knowledge, this issue remains unresolved in general. We shall now give a sufficient condition for completeness.
Following [@Rob2004] and [@Markus], we define the generalized eigenspace $\mathcal{E}_{\lambda_0}$ for the transmission eigenvalue $\lambda_0\in {\mathbb{C}}$ as the closed linear space spanned by the vectors $(u_j)_{j=0}^\infty$, $u_j\in H^m_0(\Omega)$, where $$\begin{aligned}
&L_{\lambda_0}u_0=0, \quad u_0\ne 0,\\
& L_{\lambda_0}u_j+L'_{\lambda_0}u_{j-1}+\frac{1}{2}L''_{\lambda_0}u_{j-2}=0, \quad j=1,2,\dots.\end{aligned}$$ Here we set $u_{-1}=0$.
\[thm\_complete\] Assume that the set $$\begin{aligned}
\{\langle \tilde A^{-1/2} \tilde B \tilde A^{-1/2} u_0,u_0 \rangle_{L^2}-2i \emph{{\hbox{Im}\,}} \langle \tilde A^{-1/2}v_0, u_0 \rangle_{L^2},\\
u_0,v_0\in L^2(\Omega),\|(u_0,v_0)\|_{L^2\times L^2}=1
\}
\end{aligned}$$ lies in a closed angle with vertex at zero and opening $\pi/p$, $p>n/m$. Then the space $\bigoplus_{\lambda\in {\mathbb{C}}}\mathcal{E}_\lambda$ is complete in $L^2(\Omega)$.
It follows from [@Rob2004] and Proposition \[prop\_mathcal[D]{}\] that to show that the space $\bigoplus_{\lambda\in {\mathbb{C}}}\mathcal{E}_\lambda$ is complete in $L^2(\Omega)$, it suffices to verify that the space of generalized eigenvectors $\bigoplus_{\lambda}\mathcal{E}_\lambda[\mathcal{D}]$ of the operator $\mathcal{D}$ is complete in $L^2(\Omega)\times L^2(\Omega)$. The latter can be obtained by an application of [@GohGolKaa Theorem 3.1, Chapter X.3], which states that if the set $$\{\langle \mathcal{D}\varphi,\varphi \rangle_{L^2\times L^2}:\varphi\in L^2(\Omega)\times L^2(\Omega),\|\varphi\|_{L^2\times L^2}=1\}$$ lies in a closed angle with vertex at zero and opening $\pi/p$, then the system of generalized eigenvectors of $\mathcal{D}$ is complete. The claim follows.
\[rem\_constant\_potential\]
In the case when $m>n$ and the operator $$B = \frac{1}{V} P_0 + P_0 \frac{1}{V} + P_0$$ is non-negative on $H^m_0(\Omega)$, it follows from Proposition \[thm\_complete\] that the space $\bigoplus_{\lambda\in {\mathbb{C}}}\mathcal{E}_\lambda$ is complete in $L^2(\Omega)$. In particular, if $V=\textrm{const}>0$ in $\overline{\Omega}$ and $P_0(\xi)\ge 0$, $\xi\in {\mathbb{R}}^n$, an application of Proposition \[thm\_complete\] shows that there exist infinitely many transmission eigenvalues and the corresponding generalized transmission eigenstates form a complete system in $L^2(\Omega)$. Notice that when $P_0=\Delta^2$ on ${\mathbb{R}}^3$, the existence of infinitely many real transmission eigenvalues has been established in [@HitKruOlaPai]. The completeness of the generalized transmission eigenstates in the case of a constant potential for this operator seems to be a new observation.
According to Proposition \[prop\_mathcal[D]{}\], $0\ne \lambda\in {\mathbb{C}}$ is a transmission eigenvalue if and only if $1/\lambda$ is an eigenvalue of the operator $\mathcal{D}$. Let us make explicit the connection between the generalized eigenvectors of $\mathcal{D}$ and the generalized transmission eigenstates. When doing so, since $\mathcal{D}=T\mathcal{A}^{-1}T^{-1}$, it will be convenient to consider the generalized eigenvectors of $\mathcal{A}$ directly. Let $$\begin{pmatrix}
u_0\\
v_0
\end{pmatrix}\in H^m_0(\Omega)\times L^2(\Omega)$$ be an eigenvector of $\mathcal{A}$ corresponding to $\lambda$, i.e. $$(\mathcal{A}-\lambda)\begin{pmatrix}
u_0\\
v_0
\end{pmatrix}=0\ \Longleftrightarrow \
v_0=\lambda u_0,\ L_{\lambda}u_0=0,$$ i.e. $u_0\in \mathcal{E}_{\lambda}$. Let $$(\mathcal{A}-\lambda)^2\begin{pmatrix}
u_1\\
v_1
\end{pmatrix}=0.$$ This is equivalent to the fact that $$(\mathcal{A}-\lambda)\begin{pmatrix}
u_1\\
v_1
\end{pmatrix}=\begin{pmatrix}
u_0\\
v_0
\end{pmatrix}$$ is an eigenvector of $\mathcal{A}$. The latter is equivalent to the fact that $$v_1=u_0+\lambda u_1,\quad L_{\lambda}u_1+L'_{\lambda}u_0=0,$$ i.e. $u_1\in \mathcal{E}_{\lambda}$. Continuing in the same fashion, for $j=2,3,\dots$, we have $$(\mathcal{A}-\lambda)^{j+1}\begin{pmatrix}
u_j\\
v_j
\end{pmatrix}=0$$ is equivalent to $$v_j=u_{j-1}+\lambda u_j,\quad L_{\lambda_0}u_j+L'_{\lambda }u_{j-1}+u_{j-2}=0,$$ i.e. $u_j\in \mathcal{E}_{\lambda}$. This shows that the first components of the generalized eigenvectors of $\mathcal{A}$, corresponding to the eigenvalue $\lambda$, are given by the generalized transmission eigenstates, corresponding to the transmission eigenvalue $\lambda$, and vice versa.
Generic existence of transmission eigenvalues in the trace class case
=====================================================================
In this section, we let $P_0=P_0(D)$ be a formally selfadjoint elliptic operator with constant coefficients of order $m$, with $m>n$, $V\in C^N(\overline{\Omega})$ where $N$ is large enough fixed, and $\p \Omega\in C^\infty$. Let us introduce the following open connected subset of the real Banach space $C^N(\overline{\Omega}, {\mathbb{R}})$, $$\mathcal{E}=\{V\in C^N(\overline{\Omega},{\mathbb{R}}):V>0\}.$$ When $V\in \mathcal{E}$, we shall be concerned with the quantity $\textrm{tr}(\mathcal{D})=\textrm{tr}(\tilde B\tilde A^{-1})$. In order to indicate the dependence of the operators $\tilde A$ and $\tilde B$ on the potential, we shall write $$\begin{aligned}
q=\frac{1}{V},&\quad V\in \mathcal{E},\quad A_q=P_0qP_0,\quad B_q=qP_0+P_0q+P_0,\\
\tilde A=\tilde A_q&=(1+q)^{-1/2}A_q(1+q)^{-1/2},\quad \tilde B= \tilde B_q=(1+q)^{-1/2}B_q(1+q)^{-1/2}.\end{aligned}$$ Using the cyclicity property of the trace, we have $$\textrm{tr}(\tilde B_q \tilde A_q^{-1})=\textrm{tr}((1+q)^{-1/2}B_q A_q^{-1}(1+q)^{1/2})=\textrm{tr}(B_q A_q^{-1}).$$
Assume that $m>n$ and that $P_0(\xi)\ge 0$, $\xi\in {\mathbb{R}}^n$. Then the set $$\mathcal{F}=\{V\in \mathcal{E}:\emph{tr}(B_qA^{-1}_q)\ne 0\}$$ is open and dense in $\mathcal{E}$.
The theorem above and Proposition \[thm\_trace\_L\] imply the existence of transmission eigenvalues in the trace class case, for an open and dense set of potentials.
Let us first show that the set $\mathcal{F}$ is open. To this end it suffices to prove that the function $
V\mapsto \textrm{tr}(B_qA^{-1}_q)$ is continuous on $\mathcal{E}$ in the topology of $C^N(\overline{\Omega}, {\mathbb{R}})$. We shall show that the map $V\mapsto B_qA^{-1}_q$ is continuous, with values in the space of trace class operators.
Let $V_j\to V$ in $\mathcal{E}$. Then $\p^\alpha q_j\to \p^\alpha q$ uniformly on $\overline{\Omega}$ for $|\alpha|\le N$. Let us write $$\label{eq_op_1}
\begin{aligned}
B_{q_j}A^{-1}_{q_j}-B_qA^{-1}_q=(B_{q_j}-B_q)A^{-1}_{q_j}+B_q(A_{q_j}^{-1}-A^{-1}_q)
\end{aligned}$$ When treating the first term in the right hand side of , we have $$(B_{q_j}-B_q)A^{-1}_{q_j}=((q_j-q)P_0+P_0(q_j-q))A^{-1}_{q_j}.$$ Thus, $$\begin{aligned}
&\|(q_j-q)P_0A^{-1}_{q_j}\|_{\textrm{tr}}\le \|q_j-q\|_{L^\infty}\|P_0A^{-1/2}_{q_j}\|\|A^{-1/2}_{q_j}\|_{\textrm{tr}},\\
&\|P_0(q_j-q)A^{-1}_{q_j}\|_{\textrm{tr}}\le \|P_0\|_{H^m_0\to L^2} \|q_j-q\|_{H^m_0\to H^m_0}\|A^{-1/2}_{q_j}\|_{L^2\to H^m_0}
\|A^{-1/2}_{q_j}\|_{\textrm{tr}},\end{aligned}$$ and hence, both expressions tend to zero as $j\to \infty$, provided that $N\ge m$. When considering the second term in the right hand side of , we write, using the resolvent identity, $$\begin{aligned}
B_q(A_{q_j}^{-1}-A^{-1}_q)&=B_qA^{-1}_{q_j}(A_q-A_{q_j})A^{-1}_q\\
&=(B_qA^{-1/2}_{q_j})(A^{-1/2}_{q_j}P_0)(q-q_j)(P_0A^{-1/2}_q)A^{-1/2}_q.\end{aligned}$$ The trace class norm of the above expression is easily seen to vanish as $j\to\infty$. If follows that the set $\mathcal{F}$ is open.
Let us now show that the set $\mathcal{F}$ is dense in $\mathcal{E}$. Let $V_0\in \mathcal{E}$ be fixed. Then there exists a complex neighborhood $U\subset C^N(\overline{\Omega}, {\mathbb{C}})$ of $V_0$ such that the map $$\label{eq_complex_pot}
U\to {\mathbb{C}},\quad
V\mapsto\textrm{tr}(B_qA^{-1}_q)$$ is well-defined on $U$. This follows from the fact that the operator $$A_q:H^{2m}(\Omega)\cap H^m_0(\Omega)\to L^2(\Omega), \quad q=\frac{1}{V},$$ is bijective for $V\in U$, since the operator norm of $$P_0{\hbox{Im}\,}q P_0 A_{\mathrm{Re}\, q}^{-1}:L^2(\Omega)\to L^2(\Omega)$$ is small.
We claim that the map is analytic. Since the arguments above show that the map is continuous, it therefore suffices to check the weak analyticity, [@postru_book]. To this end let $q_1=1/V_1$, $V_1\in U$, and $q_2$ be arbitrary, and consider the function $$\label{eq_holom}
z\mapsto \textrm{tr}(B_{q_1+zq_2}A_{q_1+zq_2}^{-1})$$ for $z$ near $0\in {\mathbb{C}}$. We have the convergent power series expansion $$A^{-1}_{q_1+zq_2}=A^{-1}_{q_1}\sum_{k=0}^\infty(-z)^k(P_0q_2P_0A_{q_1}^{-1})^k$$ for $z$ near $0\in {\mathbb{C}}$. Since the operator $(B_{q_1}+zB_{q_2})A^{-1}_{q_1}$ is of trace class, the operator $$B_{q_1+zq_2}A_{q_1+zq_2}^{-1}$$ is given by a power series in $z$ which converges in the trace class norm. Thus, it follows that the map is holomorphic near $0\in {\mathbb{C}}$.
We therefore conclude that the map $$V\mapsto\textrm{tr}(B_qA^{-1}_q)$$ is real-analytic on $\mathcal{E}$. We furthermore know from Remark \[rem\_constant\_potential\] that it does not vanish identically, for it is positive at $V=1$. Since $\mathcal{E}$ is connected, given $V_0\in \mathcal{E}$ it follows that for any neighborhood of $V_0$ there are points $V$ for which $\textrm{tr}(B_qA^{-1}_q)\ne 0$. This completes the proof.
Finally, concerning counting estimates for transmission eigenvalues, we have the following simple result.
Let $m>n$. Then the number of transmission eigenvalues in the disk of radius $R$ is $\mathcal{O}(R^2)$.
Recall that $0\ne\lambda\in {\mathbb{C}}$ is a transmission eigenvalue if and only if $1/\lambda$ is an eigenvalue of the operator $\mathcal{D}$ given by . The latter is equivalent to the fact that the operator $$I-\lambda(\tilde A^{-1/2}\tilde B\tilde A^{-1/2}) +\lambda^2 \tilde A^{-1}:L^2(\Omega)\to L^2(\Omega)$$ is not invertible. Here $\tilde A^{-1/2}\tilde B\tilde A^{-1/2}$ and $\tilde A^{-1}$ are of trace class. Thus, the latter is equivalent to the fact that $$\det(I-\lambda(\tilde A^{-1/2}\tilde B\tilde A^{-1/2}) +\lambda^2 \tilde A^{-1})=0.$$ The function $$f(\lambda)=\det(I-\lambda(\tilde A^{-1/2}\tilde B\tilde A^{-1/2}) +\lambda^2 \tilde A^{-1})$$ is entire holomorphic. Therefore, the number $N(R/2)$ of its zeros in the disk of radius $R/2$ can be estimated by Jensen’s formula, $$N(R/2)\le \frac{1}{\log 2}(\max_{|\lambda|=R }\log|f(\lambda)|-\log|f(0)|)=\mathcal{O}(R^2).$$ Here we have used that $|f(\lambda)|\le e^{C|\lambda|^2}$ with some constant $C$ and $|\lambda|\ge 1$.
Acknowledgements
================
The research of M.H. was partially supported by the NSF grant DMS-0653275 and he is grateful to the Department of Mathematics and Statistics at the University of Helsinki for the hospitality. The research of K.K. was financially supported by the Academy of Finland (project 125599). The research of P.O. and L.P. was financially supported by Academy of Finland Center of Excellence programme 213476.
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| ArXiv |
---
abstract: 'The mass of the axion and its decay rate are known to depend only on the scale of Peccei-Quinn symmetry breaking, which is constrained by astrophysics and cosmology to be between $10^9$ and $10^{12}$ GeV. We propose a new mechanism such that this effective scale is preserved and yet the fundamental breaking scale of $U(1)_{PQ}$ is very small (a kind of inverse seesaw) in the context of large extra dimensions with an anomalous U(1) gauge symmetry in our brane. Unlike any other (invisible) axion model, there are now possible collider signatures in this scenario.'
---
plus 1pt ‘@=12 -0.5in 0.0in 0.0in 8.5in 6.5in
UCRHEP-T283\
July 2000
[**Low-Scale Axion from Large Extra Dimensions\
**]{}
Although CP violation has been observed in weak interactions [@cp1; @cp2] and it is required for an explanation of the baryon asymmetry of the universe [@asym], it becomes a problem in strong interactions. The reason is that the multiple vacua of quantum chromodynamics (QCD) connected by instantons [@insta] require the existence of the CP violating $\theta$ term [@theta] $${\cal L}_\theta = \theta_{QCD} {g_s^2 \over 32 \pi^2}
G_{\mu \nu}^a \widetilde G^{a \mu \nu} ,$$ where $g_s$ is the strong coupling constant, $G^a_{\mu \nu}$ is the gluonic field strength and $\tilde G^a_{\mu \nu}$ is its dual. Nonobservation of the electric dipole moment of the neutron [@edm] implies that $$\bar \theta = \theta_{QCD} - Arg ~Det ~M_u ~M_d < 10^{-10},$$ instead of the theoretically expected order of unity. In the above, $M_u$ and $M_d$ are the respective mass matrices of the charge 2/3 and $-1/3$ quarks of the standard model of particle interactions. This is commonly known as the strong CP problem.
The first and best motivated solution to the strong CP problem was proposed by Peccei and Quinn [@pq], in which the quarks acquire a dynamical phase from the spontaneous breaking of a new global symmetry \[$U(1)_{PQ}$\] and relaxes $\bar \theta$ to its natural minimun value of zero. As a result, there appears a Goldstone boson called the axion but it is not strictly massless [@ww] because it couples to two gluons (like the neutral pion) through the axial triangle anomaly [@anomal].
The scale of $U(1)_{PQ}$ breaking (which is conventionally identified with the axion decay constant $f_a$) determines the axion coupling to gluons, which is proportional to $1/f_a$. If $f_a$ is the electroweak symmetry breaking scale as originally proposed [@pq], then the model is already ruled out by laboratory experiments [@expt]. In fact, $f_a$ is now known to be constrained by astrophysical and cosmological arguments [@astro] to be between $10^9$ and $10^{12}$ GeV. Hence the axion must be an electroweak singlet or predominantly so. It may couple to the usual quarks and leptons through a suppressed mixing with the standard Higgs doublet [@dfsz], or it may couple only to other unknown colored fermions [@ksvz], or it may couple to gluinos [@dms] as well as all other supersymmetric particles.
Because the axion must necessarily mix with the $\pi$ and $\eta$ mesons, it must have a two-photon decay mode. This is the basis of all experimental attempts [@expt] to discover its existence. On the other hand, the accompanying new particles in all viable axion models to date are very heavy, i.e. of order $f_a$; hence they are completely inaccessible to experimental verification.
In the following we consider instead the possiblily that the $U(1)_{PQ}$ breaking scale is actually very small, but that $f_a$ is large because of a kind of inverse seesaw mechanism. We show how this scenario may be realized in the context of large extra dimensions with an anomalous U(1) gauge symmetry in our brane. The associated new physics now exists at around 1 TeV, with a number of interesting observable consequences at future colliders.
We assume a singlet scalar field $\chi$ with a nonzero PQ charge existing in the bulk of large extra dimensions [@extra]. The $shining$ [@distant] of this field in our brane is the source of spontaneous $U(1)_{PQ}$ breaking in our world (called a 3-brane). The idea is that $\chi$ gets a large vacuum expectation value (VEV) in a distant brane, but its effect on our brane is small because we are far away from it. (In the case of lepton number, this mechanism has been used recently to obtain small Majorana neutrino masses [@extnu].) To convert this small $\langle \chi \rangle$ to a large $f_a$, we need to assume an anomalous U(1) gauge symmetry in our brane at the TeV energy scale, as explained below.
In a theory of large extra dimensions with quantum gravity at the TeV scale, there is no large scale available for the axion. Since the behavior of Goldstone bosons depends not on the coupling but only on the scale of symmetry breaking in general, it is a problem which is not easily resolved [@others]. Here we find a new and novel solution to this apparent contradiction in the case where there is an anomalous U(1) gauge symmetry, which is of course well studied [@u1] as a possible manifestation of string theory near the string scale (now considered also at around a few TeV) and has well-known applications in quark and lepton Yukawa textures and supersymmetry breaking.
We extend the standard model of particle interactions to include an extra $U(1)_A$ gauge symmetry and an extra $U(1)_{PQ}$ global symmetry. All standard-model particles are trivial under these two new symmetries. We then introduce a new heavy quark singlet $\psi$ and two scalar singlets $\sigma$ and $\eta$ with $U(1)_A$ and $U(1)_{PQ}$ charges as shown in Table 1. All fields except $\chi$ are confined to our brane.
-------------------- ---------------------------------------- ---------- -------------
Fields $SU(3)_C \times SU(2)_L \times U(1)_Y$ $U(1)_A$ $U(1)_{PQ}$
$(u_i, d_i)_L$ (3,2,1/6) 0 0
$u_{iR}$ (3,1,2/3) 0 0
$d_{iR}$ (3,1,$-$1/3) 0 0
$(\nu_i, e_i)_L$ (1,2,$-$1/2) 0 0
$e_{iR} $ (1,1,$-$1) 0 0
$\psi_L$ (3,1,–1/3) 1 $k$
$\psi_R$ (3,1,–1/3) –1 $-k$
$(\phi^+, \phi^0)$ (1,2,1/2) 0 0
$\sigma$ (1,1,0) 2 $2k$
$\eta $ (1,1,0) 2 $2k-2$
$\chi$ (1,1,0) 0 2
-------------------- ---------------------------------------- ---------- -------------
: Peccei-Quinn charges of the fermions and scalars
Because of our chosen charge assignments, only the field $\sigma$ couples to the colored fermion $\psi$, i.e. $${\cal L}_Y = f \sigma \bar \psi_L \psi_R + h.c.$$ Hence it also couples to two gluons through the usual triangular loop. As $\sigma$ acquires a VEV, say $u$, of order 1 TeV, both $U(1)_A$ and $U(1)_{PQ}$ are broken, whereas the latter is broken by $\langle \chi \rangle = z$, and it induces a VEV also for $\eta$, i.e. $\langle \eta \rangle = w$. We will show in the following that given $z$ is small from its origin in the bulk, $w$ is also small. Now the longitudinal component of the $Z_A$ boson is mostly given by Im$\sigma$, so the axion is excluded to be mostly a linear combination of Im$\eta$ and Im$\chi$, but the latter two fields do not couple to the colored fermion $\psi$. As a result, the axion’s coupling to two gluons is now effectively $${1 \over f_a} = {w^2 \over u^2 \sqrt {w^2 + z^2}},$$ which can be thought of as a kind of inverse seesaw, i.e. the largeness of $f_a$ is explained by the smallness of $w$. Details will be given later.
Our brane ${\cal P}$ is located at a point $y=0$ in the bulk. Peccei-Quinn symmetry is broken maximally in a distant brane ${\cal P}'$, located at a point $y=y_*$ in the bulk. We assume for simplicity that the separation of the two branes is of order the radius of compactification of the extra space dimensions, i.e. $|y_*|=r$, which is only a few $\mu$m in magnitude. The fundamental scale $M_*$ in this theory is then related to the reduced Planck scale $M_P = 2.4 \times 10^{18}$ GeV by the relation $$r^n M_*^{n + 2} \sim M_P^2 .$$ The $U(1)_{PQ}$ symmetry breaking in the distant brane acts as a point source $J$, which induces an effective VEV, i.e. $z$, to the singlet bulk field $\chi$. Other effects which may perturb the $shined$ value of $\langle
\chi \rangle$ in our world are all included as boundary conditions to the source $J$, so that the effect of the field $\chi$ in our brane always appears in the combination $z(y=0) e^{i\varphi}$, where $\varphi(x)$ is a dynamical phase which transforms under $U(1)_{PQ}$ to preserve its invariance. This formulation has also been used for the spontaneous breaking of lepton number in the case of neutrinos [@extnu].
In our brane, the profile of $\chi$ is given by the Yukawa potential in the transverse dimensions $$\langle \chi(y = 0) \rangle = J(y=y_*) \Delta_n(r),$$ where $$\Delta_n(r) = {1 \over (2 \pi )^{n \over 2}
M_*^{n- 3}} ~\left( {m_\chi \over r} \right)^{n-2 \over 2}
~K_{n - 2 \over 2} \left( m_\chi r \right),$$ $K$ being the modified Bessel function. We consider the source to be dimensionless, which we take to be $J=1$. For the interesting case of $n> 2$ and $m_\chi r \ll 1$, the $shined$ value of $\chi$ is given by $$\langle \chi \rangle \approx \displaystyle{
{ \Gamma ( {n -2 \over 2} ) \over
4 \pi^{n \over 2} }{M_* \over (M_* r)^{n-2} } }
= \displaystyle{
{ \Gamma ( {n -2 \over 2} ) \over
4 \pi^{n \over 2} }~ M_* ~\left({M_* \over M_P}\right)^{2 - (4/n)} }.$$ For $n=3$ and $M_* = 10$ TeV, we get $\langle \chi \rangle \sim 0.2$ keV. This is the smallest value possible with our assumptions. However, if the distant brane is located at $y_*$ less than $r$, larger values of $\langle \chi \rangle$ may be obtained. As we will show, the range 1 keV to 1 MeV corresponds nicely to the axion decay constant of $10^{12}$ to $10^9$ GeV.
We express the bulk field as $$\chi = {1 \over \sqrt 2} ( \rho + z \sqrt 2) e^{i \varphi} .$$ Its self-interaction terms are now given by $$V(\chi) = \lambda z(y)^2 \rho(x,y)^2 + {1 \over \sqrt 2} \lambda
z(y) \rho(x,y)^3 + {1 \over 8} \lambda \rho(x,y)^4 .$$ This Lagrangian has the virtue of universality, i.e., $\lambda$ is unchanged, but $z$ can change depending on where our brane is from the distant brane. The invariance under $U(1)_{PQ}$, i.e. $\rho \to \rho$ and $\varphi
\to \varphi + 2 \theta$, is also maintained in the other interactions, as described below. The parameters in the potential of $\chi$ are thus guaranteed to be independent of the parameters of our brane.
The scalar potential in our brane excluding $V(\chi)$ is now given by $$\begin{aligned}
V &=& m_1^2 \Phi^\dagger \Phi + m_2^2 \sigma^\dagger \sigma + m_3^2
\eta^\dagger \eta + {1 \over 2} \lambda_1 (\Phi^\dagger \Phi)^2 + {1 \over 2}
\lambda_2 (\sigma^\dagger \sigma)^2 + {1 \over 2} \lambda_3 (\eta^\dagger
\eta)^2 \nonumber \\ && + \lambda_4 (\Phi^\dagger \Phi)(\sigma^\dagger \sigma)
+ \lambda_5 (\Phi^\dagger \Phi)(\eta^\dagger \eta) + \lambda_6
(\sigma^\dagger \sigma)(\eta^\dagger \eta) + (\mu z e^{i \varphi}
\sigma^\dagger \eta + h.c.),\end{aligned}$$ where $\mu$ has the dimension of mass and we assume that all mass parameters are of the same order of magnitude, i.e. 1 TeV.
The minimum of $V$ satisfies the following conditions: $$\begin{aligned}
m_1^2 + \lambda_1 v^2 + \lambda_4 u^2 + \lambda_5 w^2 &=& 0, \\
u(m_2^2 + \lambda_2 u^2 + \lambda_4 v^2 + \lambda_6 w^2) + \mu z w &=& 0, \\
w(m_3 ^2 + \lambda_3 w^2 + \lambda_5 v^2 + \lambda_6 u^2) + \mu z u &=& 0,\end{aligned}$$ where $\langle \phi^0 \rangle = v$. Hence $$\begin{aligned}
v^2 &\simeq& {-\lambda_2 m_1^2 + \lambda_4 m_2^2 \over \lambda_1 \lambda_2 -
\lambda_4^2}, \\ u^2 &\simeq& {-\lambda_1 m_2^2 + \lambda_4 m_1^2 \over
\lambda_1 \lambda_2 - \lambda_4^2},\end{aligned}$$ and $$w \simeq {- \mu z u \over m_3^2 + \lambda_5 v^2 + \lambda_6 u^2},$$ which is indeed of order $z$ as mentioned earlier.
Whereas Im$\phi^0$ becomes the longitudinal component of the usual $Z$ boson, $(u {\rm Im} \sigma + w {\rm Im} \eta)/\sqrt {u^2 + w^2}$ becomes that of the new $Z_A$ boson. Since the $3 \times 3$ mass matrix in the basis \[Im$\sigma$, Im$\eta$, $z\varphi$\] is given by $$\pmatrix{-\mu z w / u
& \mu z& \mu w \cr
\mu z & - \mu z u / w &
-\mu u \cr \mu w & -\mu u & - \mu u w / z },$$ the axion $a$ is identified as the following: $$\begin{aligned}
{a \over \sqrt 2} &=& {1 \over {N}} \left[ uw^2 {\rm Im} \sigma
- w u^2 {\rm Im} \eta + z (u^2 + w^2) z \varphi \right] \nonumber \\
&\simeq& {w^2 \over u \left( w^2 + z^2 \right)^{1/2} } {\rm Im} \sigma -
{w \over \left( w^2 + z^2 \right)^{1/2} } {\rm Im} \eta +
{z \over \left( w^2 + z^2 \right)^{1/2} } z \varphi,\end{aligned}$$ where $N= \left\{ w^2 u^2 (w^2 + u^2) + z^2 (w^2 + u^2)^2 \right\}^{1/2}$ is the normalization. Since only $\sigma$ couples to the colored fermion $\psi$ and the component of Im$\sigma$ in the axion is $u$ times a phase, the axion coupling to the gluons through $\psi$ is effectively as given by Eq. (4) as mentioned earlier. Using $u \sim 1$ TeV and $w \sim z \sim 1$ keV to 1 MeV, we see that $f_a$ is indeed in the range $10^{12}$ to $10^9$ GeV.
In Table 1, we have not specified the value of $k$ for the PQ charge of $\psi$. This is intentional because our model is independent of it. This ambiguity also helps us to understand its pattern of symmetry breaking. For example, if $\langle \chi \rangle = 0$, then $\langle \eta \rangle = 0$ also. In that case, there is no axion and the Peccei-Quinn symmetry disappears, i.e. $k=0$. Hence the true scale of $U(1)_{PQ}$ breaking is indeed small, i.e. $z$ from the bulk, as asserted.
To understand why we have an exception to the general rule that the axion coupling is inversely proportional to the scale of $U(1)_{PQ}$ breaking, we point out that the anomalous nature of $U(1)_A$ is crucial. If we attempt to make it free of the axial triangle anomaly, we need to add colored fermions with opposite $U(1)_A$ charges to $\psi_{L,R}$. They must then acquire mass through a new scalar field with opposite $U(1)_A$ charge to $\sigma$. The longitudinal component of $Z_A$ takes up a linear combination of the two imaginary parts, leaving free the other to be the axion, which now couples to the colored fermions with the same scale as $U(1)_A$ symmetry breaking. The above is of course the analog of what happens in the well-known original Peccei-Quinn proposal [@pq].
All axion models to date have no accompanying verifiable new physics other than the $a \to \gamma \gamma$ decay, and that depends on the axion being a component of dark matter. In our scenario, the possibility exists for this new physics to be at the TeV scale and be observable at future colliders.
\(1) The stable heavy colored fermion $\psi$ may be produced in pairs, i.e. $gg \to \psi \bar \psi$. Both $\psi$ and $\bar \psi$ carry light quarks and gluons with them and appear as jets, but when these jets hit the hadron calorimeter in a typical detector, a large part (i.e. 2$m_\psi$) of the initial collision energy is “frozen” in the mass and appears “lost”.
\(2) There is mixing between the standard-model Higgs boson Re$\phi^0$ with the new scalar Re$\sigma$ of order $v/u$, i.e. 0.1 or so. This means that the lighter (call it $h$) of the two physical scalar bosons has a small component of Re$\sigma$, but that only modifies its (small) $gg$ and $\gamma \gamma$ decay amplitudes through the $\psi$ loop. Hence $h$ behaves almost exactly like the standard-model Higgs boson.
\(3) The $U(1)_A$ gauge boson $Z_A$ may be produced by $gg^*$ fusion through the $\psi$ loop. If kinematically allowed, it will decay into Re$\eta$ + Im$\eta$. Since Im$\eta$ is partly ($w/\sqrt{w^2+z^2}$) the axion $a$ which will escape detection, this event has a lot of possible missing transverse momentum. The subsequent decay of Re$\eta$ is into $a$ and a virtual $Z_A$ which turns into $gg$. This adds more missing transverse momentum. The end result of the production and subsequent decay of $Z_A$ is thus two gluon jets and two axions. This is a distinctive signature of our scenario [@note]. It predicts collider events with large missing energy without the existence of supersymmetry.
[*Acknowledgement.*]{} This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837. One of us (U.S.) would like to thank the Physics Department, University of California, Riverside for hospitality.
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Re$\eta$ will also decay (through its mixing with Re$\phi^0$) into standard-model final states such as $ZZ$, $WW$, etc. However, this mixing is very small, i.e. of order $wv/\mu u$. Details of the phenomenology of this model will be discussed in a forthcoming paper.
| ArXiv |
---
abstract: 'We consider the behaviour of current fluctuations in the one-dimensional partially asymmetric zero-range process with open boundaries. Significantly, we find that the distribution of large current fluctuations does not satisfy the Gallavotti-Cohen symmetry and that such a breakdown can generally occur in systems with unbounded state space. We also discuss the dependence of the asymptotic current distribution on the initial state of the system.'
author:
- 'R. J. Harris[^1]'
- 'A. R[á]{}kos'
- 'G. M. Sch[ü]{}tz'
bibliography:
- 'allref.bib'
title: 'Breakdown of [G]{}allavotti-[C]{}ohen symmetry for stochastic dynamics'
---
Substantial progress in the understanding of nonequilibrium systems has been achieved recently through so-called fluctuation theorems [@Evans02b]. Specifically, the Gallavotti-Cohen fluctuation theorem (GCFT) can be loosely written as $$\frac{p(-\sigma,t)}{p(\sigma,t)} \sim e^{-\sigma t} \label{e:GCFT}$$ where $p(\sigma,t)$ is the probability to observe an average value $\sigma$ for the entropy production in time interval $t$ and $\sim$ denotes the limiting behaviour for large $t$. This theorem was first derived for deterministic systems [@Gallavotti95] (motivated by computer simulations of sheared fluids [@Evans93]) and subsequently for stochastic dynamics [@Kurchan98; @Lebowitz99]. From [@Ciliberto98] onwards there have been successful attempts at experimental verification, including for simple random processes such as the driven two-level system in [@Schuler05]. Strictly the GCFT is a property of non-equilibrium steady states but, for systems with a unique stationary state it is usually also expected to hold for arbitrary initial states (see, e.g., [@Cohen99; @Searles99] for discussion on this point). We will refer to this more general property of the large deviation function as “GC symmetry”. Some related issues have previously been discussed for Langevin dynamics [@Kurchan98; @Farago02; @Baiesi06]; we consider the more general case of stochastic *many-particle* systems.
Specifically, we explore the GC symmetry in the context of a paradigmatic non-equilibrium model—the zero-range process [@Spitzer70]. For certain parameter values, this interacting particle system exhibits a condensation phenomenon [@Evans00; @Jeon00b] in which a macroscopic proportion of particles pile up on a single site. Condensation transitions are well-known in colloidal and granular systems [@Shim04] and also occur in a variety of other physical and nonphysical contexts [@Evans05]. In [@Me05] it was argued that current fluctuations in the asymmetric zero-range process with open boundary conditions can become spatially-inhomogeneous for large fluctuations—a precursor of the condensation which occurs for strong boundary driving. Here, for a specialized case, we explicitly calculate the current distribution in this large-fluctuation regime and thus prove a breakdown of the symmetry relation . Significantly, we argue that our analytical approach predicts that this effect also occurs for more general models. Fianlly, we discuss the relation of our results to GCFT breakdowns found in some other works [@Bonetto05; @vanZon03; @vanZon04].
Let us begin by defining our model—the partially asymmetric zero-range process (PAZRP) on an open one-dimensional lattice of $L$ sites [@Levine04c]. Each site can contain any integer number of particles, the topmost of which hops randomly to a neighbouring site after an exponentially distributed waiting time. In the bulk particles move to the right (left) with rate $p w_n$ ($q w_n$) where $w_n$ depends only on the occupation number $n$ of the departure site. Particles are injected onto site 1 ($L$) with rate $\alpha$ ($\delta$) and removed with rate $\gamma w_n$ ($\beta w_n$). If the partition function has a finite radius of convergence (i.e, $\lim_{n\to\infty} w_n$ is finite) then for strong boundary driving a growing condensate occurs at one or both of the boundary sites [@Levine04c].
We are interested in the probability distribution of integrated current $J_l(t)$, i.e., the net number of particle jumps between sites $l$ and $l+1$ in time interval $[0,t]$. The long-time asymptotic behaviour of this distribution is characterized by the limit of the generating function $$e_l(\lambda) =\lim_{t \rightarrow \infty} - \frac{1}{t} \ln {\langle e^{-\lambda J_l(t)} \rangle}. \label{e:e_l}$$ which implies [@Lebowitz99] a large deviation property for the asymptotic probability distribution, $p_l(j,t)=\mathrm{Prob}(j_l=j,t)$, of the observed “average” current $j_l=J_l/t$ $$p_l(j,t) \sim e^{-t\hat{e}_l(j)} \label{e:pj}$$ where $\hat{e}_l(j)$ is the Legendre transformation of $e_l(\lambda)$, i.e., $
\hat{e}_l(j)=\max_{\lambda}\{e_l(\lambda)-\lambda j \}. \label{e:lang}
$ To calculate the current distribution we employ the quantum Hamiltonian formalism [@Schutz01] where the master equation for the probability vector $|P_t\rangle$ resembles a Schrödinger equation with Hamiltonian $H$ (see [@Levine04c] for details). The generating function $\langle e^{-\lambda J_l(t)} \rangle$ can then be written as $\langle s | e^{-\tilde{H}_l t} |P_0\rangle$ where $\tilde{H}_l$ is a modified Hamiltonian in which the terms in $H$ giving a unit increase/decrease in $J_l$ are multiplied by $e^{\mp\lambda}$ [@Me05]. Here $|P_0\rangle$ is the initial probability distribution and $\langle s|$ is a summation vector giving the average value over all configurations. For the current into the system from the left (which can be positive or negative) we consider $\tilde{H}_0$ with lowest eigenvalue $\tilde{e}_{0}(\lambda)$ and corresponding eigenvector $|\tilde{0}\rangle$. In the case where $\langle s | \tilde{0} \rangle$ and $\langle \tilde{0} | P_0 \rangle$ are finite, the long-time limiting behaviour is given by $$\langle e^{-\lambda J_0(t)} \rangle \sim \langle s |\tilde{0} \rangle \langle \tilde{0} | P_0\rangle e^{-\tilde{e}_0(\lambda) t} \label{e:pre}$$ In this case we have $e_0(\lambda)=\tilde{e}_0(\lambda)$ and the form of $\tilde{H}_0$ imposes the GC symmetry relation $$e_0(\lambda)=e_0(2E-\lambda) \label{e:GCFTe}$$ which leads, via , to the relationship with $\sigma=2Ej$ and effective field $E$ given by $e^{2E}= (\alpha \beta / \gamma\delta ) ( p/q )^{L-1}$. The field $E$ can be related to a force $F=2Ek_BT$.
While the ground-state eigenvalue calculated in [@Me05] is independent of $w_n$, the latter determine the form of the eigenvectors $\langle \tilde{0}|$ and $|\tilde{0} \rangle$. If $\lim_{n\to\infty} w_n$ is finite then $\langle s | \tilde{0} \rangle$ diverges for some values of $\lambda$. For a fixed initial particle configuration $\langle \tilde{0} | P_0 \rangle$ is always finite. However, for a normalized distribution over initial configurations (e.g., the steady-state) $\langle \tilde{0} | P_0 \rangle$ can also diverge (again in the case where $w_n$ is bounded) meaning that *the asymptotic current distribution retains a dependence on the initial state.* This has important consequences for measurement of the current fluctuations in simulation (or equivalent experiments). Suppose we start from a fixed initial particle configuration, e.g., the empty lattice, wait for some time $T_1$ and then measure the current over a time interval $T_2$. These are two noncommuting timescales—if we take $T_2 \to \infty$ faster than $T_1 \to \infty$ we will measure the asymptotic distribution of current fluctuations corresponding to the fixed initial condition which may differ from the asymptotic behaviour of steady-state current fluctuations obtained by taking $T_1 \to \infty$ before $T_2 \to \infty$.
We first specialize to the case of the single-site PAZRP, i.e, one site with “input” (left) and “output” (right) bonds. In this model explicit calculation of the matrix element $\langle s | e^{-\tilde{H}_0 t} |P_0 \rangle$ is possible. For simplicity we consider here $w_n=1$, anticipating qualitatively the same effects for any bounded $w_n$. We take the case $\alpha-\gamma<\beta-\delta$ in order to ensure a well-defined steady state and assume an initial Boltzmann distribution $$|P_0\rangle = \sum_{n=0}^\infty x^n (1-x) |n\rangle$$ where $|n\rangle$ denotes the state with site occupied by $n$ particles and the fugacity $x=e^{-\beta \mu}<1$. The steady state is $ x=(\alpha+\delta)/(\beta+\gamma)$ while $x \to 0$ gives the empty site. By ergodicity this gives the same asymptotic current distribution as any fixed initial particle number.
Explicit computations yield an integral form for the generating function of input current $$\begin{gathered}
\langle s | e^{-\tilde{H}_0 t} |P_0 \rangle = \frac{ x-1}{2\pi i} \biggl\{ \oint_{C_1} e^{-\varepsilon(z)t} \frac{1}{(z-1)(z- x)}\, dz \\
+ \oint_{C_2} e^{-\varepsilon(z)t} \frac{ x^{-1} [u_\lambda/v_\lambda - z u_\lambda/(\beta + \gamma)]}{(z-1)[z- x^{-1}u_\lambda/v_\lambda][z-u_\lambda/(\beta + \gamma)]}\, dz \biggr\} \label{e:intinb}\end{gathered}$$ with $$\varepsilon(z)=\alpha+\beta+\gamma+\delta-v_\lambda z- u_\lambda z^{-1}.$$ Here, for notational brevity we write $
u_\lambda \equiv {\alpha e^{-\lambda} +\delta}
$, $
v_\lambda \equiv {\beta + \gamma e^\lambda}.
$ and for later use also define the parameter combination $
\eta=\sqrt{[(\beta+\gamma)^2-\beta\delta-\alpha\gamma]^2-4\alpha\beta\gamma\delta}.
$ The contour $C_1$ ($C_2$) is an anti-clockwise circle of radius $ x+\epsilon$ ($\epsilon$) around the origin of the complex plane with $\epsilon \to 0$.
In order to extract the large-time behaviour from this integral representation we use a saddle-point method, taking careful account of the contributions from residues when the saddle-point contour is deformed through poles in the integrand. This yields changes in behaviour at the values of $\lambda$ given in Table \[t:lam\].
[ll]{} Values of $\lambda$ & Corresponding values of $j$\
$e^{\lambda_1} \equiv \frac{\alpha}{\beta+\gamma-\delta}$ & $j_a \equiv \frac{(\beta+\gamma-\delta)^2-\alpha\gamma}{\beta+\gamma-\delta}$, $j_b \equiv \frac{\beta(\beta+\gamma-\delta)^2-\alpha\gamma\delta}{(\beta+\gamma)(\beta+\gamma-\delta)}$\
$e^{\lambda_2} \equiv \frac{(\beta+\gamma)^2-\alpha\gamma-\beta\delta+\eta}{2\gamma\delta}$ & $j_c \equiv -\frac{\eta}{\beta+\gamma}$\
$e^{\lambda_3} \equiv \frac{\delta-\beta x^2+\sqrt{(\delta-\beta x^2)^2+4\alpha\gamma x^2}}{2\gamma x^2} $ & $j_d\equiv \frac{-(\delta-\beta x^2)}{ x}$\
$e^{\lambda_4} \equiv \frac{\beta(1- x)+\gamma}{\gamma x} $ & $j_e \equiv \frac{\alpha\beta\gamma x^2-\delta\left[\beta(1- x)+\gamma \right]^2}{ x(\beta+\gamma)\left[\beta(1- x)+\gamma \right]} $, $j_f \equiv \frac{\alpha\gamma-\left[\beta(1- x)+\gamma \right]^2}{\beta(1- x)+\gamma}$\
For $$x< x_c\equiv\frac{-\eta+(\beta+\gamma)^2-\alpha\gamma+\beta\delta}{2\beta(\beta+\gamma)}$$ we find $$e_0(\lambda)=
\begin{cases}
\alpha(1-e^{-\lambda})+\gamma(1-e^\lambda) & \lambda < \lambda_1 \\
\alpha+\delta-\frac{u_\lambda v_\lambda}{\beta+\gamma} & \lambda_1<\lambda<\lambda_2 \\
\alpha+\beta+\gamma+\delta-2\sqrt{u_\lambda v_\lambda} & \lambda_2<\lambda<\lambda_3 \\
\alpha+\beta+\gamma+\delta-v_\lambda x - u_\lambda x^{-1} & \lambda_3<\lambda
\end{cases}$$ whereas for $ x> x_c$ we get $$e_0(\lambda)=
\begin{cases}
\alpha(1-e^{-\lambda})+\gamma(1-e^\lambda) & \lambda < \lambda_1 \\
\alpha+\delta-\frac{u_\lambda v_\lambda}{\beta+\gamma} & \lambda_1 < \lambda < \lambda_4 \\
\alpha+\beta+\gamma+\delta-v_\lambda x - u_\lambda x^{-1} & \lambda_4 < \lambda.
\end{cases}$$ We note that the form of $e_0(\lambda)$ seen in the regime $\lambda_1<\lambda<\lambda_3$ ($\lambda_4$) is the groundstate eigenvalue of $\tilde{H}_0$ [@Me05]. At $\lambda=\lambda_2$ the spectrum of $\tilde{H}_0$ becomes gapless. The changes at $\lambda_1$ and $\lambda_3$ ($\lambda_4$) correspond to the divergence of $\langle s | \tilde{0} \rangle$ and $\langle \tilde{0} | P_0 \rangle$ respectively. One immediately sees that the symmetry relation is only obeyed for a limited range of $\lambda$.
Via Legendre transformation we obtain the large deviation behaviour of $j_0=J_0/t$. The resulting “phase diagram” is shown in Fig. \[f:pdj\] where
\[Cr\]\[Cl\][$j\quad$]{} \[Tc\]\[Bc\][$ x$]{} \[\]\[\][III]{} \[\]\[\][I]{} \[\]\[\][II]{} \[\]\[\][VI]{} \[\]\[\][V]{} \[\]\[\][ IV]{} \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\]
$\hat{e}_0(j)$ has the following forms in the different regions: $$\hat{e}_0(j)=
\begin{cases}
f_j(\alpha,\gamma) & \text{I} \\
g_j\!\!\left(\frac{(\alpha-\beta-\gamma+\delta)(\beta-\delta)}{\beta+\gamma-\delta},\frac{\beta+\gamma-\delta}{\alpha}\right) & \text{II} \\
f_j\!\!\left(\frac{\alpha\beta}{\beta+\gamma},\frac{\gamma\delta}{\beta+\gamma}\right) & \text{III} \\
f_j(\alpha,\gamma)+f_j(\beta,\delta) & \text{IV} \\
f_j(\alpha,\gamma)+g_j(\beta(1- x)+\delta(1- x^{-1}), x) & \text{V} \\
g_j\!\!\left(\frac{(1- x)\left\{\alpha\beta x-\delta\left[\beta(1- x)+\gamma\right]\right\}}{ x\left[\beta(1- x)+\gamma\right]},\frac{\gamma x}{\beta(1- x)+\gamma} \right) & \text{VI} \\
\end{cases} \label{e:ejres} \\$$ with $$\begin{aligned}
f_j(a,b)&=a+b-\sqrt{j^2+4ab}+j \ln \frac{j+\sqrt{j^2+4ab}}{2a} \\
g_j(a,b)&=a+j \ln b.\end{aligned}$$ The function $f_j(a,b)$ is the “random walk” current distribution of a single bond with Poissonian jumps of rate $a$ to the right and $b$ to the left. The straightline function $g_j(a,b)$ gives an exponential decay of $p_0(j,t)$ with increasing $j$. We now give some brief remarks on the physical interpretation of these behaviours.
In region III, the current across the input bond is dependent on the current across the output bond, resulting in a distribution with mean $(\alpha\beta-\gamma\delta)/(\beta+\gamma)$ and diffusion constant $(\alpha\beta+\gamma\delta)/(\beta+\gamma)$. In IV ($j$ large and negative) there is a temporary build-up of particles on the site (an “instantaneous condensate”[@Me05]) and so to see $j_0=j$, requires a current of $j$ across both bonds independently. In I ($j$ large and positive) the piling-up of particles on the site means the input bond does not feel the presence of the output bond. The $ x$-dependence in region V arises from the possibility of an arbitrarily large initial occupation. II and VI are transition regimes involving linear combinations of two different behaviours. They correspond to values of $\lambda$ where $e_0(\lambda)$ has a discontinuous derivative (cf. a first order phase transition). Analogous results for $e_1(\lambda)$ and $\hat{e}_1(j)$ which characterize the distribution of outgoing current are obtained by the replacements $\alpha \leftrightarrow \delta$, $\beta \leftrightarrow \gamma$, $p \leftrightarrow q$, $\lambda \leftrightarrow -\lambda$, $j \leftrightarrow -j$.
\[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Tc\]\[Bc\][$j$]{} \[Bc\]\[Tc\][$\hat{e}(-j)-\hat{e}(j)$]{} \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\]
The GC symmetry states that, $\hat{e}(-j)-\hat{e}(j)$ should be a straight line (of slope $\log[(\alpha\beta)/(\gamma\delta)]$ in this single-site case) but the results imply that this only holds for small $j$ (specifically in the shaded region of Fig. \[f:pdj\]). In Fig. \[f:GCFT\] we test this prediction against simulation for both input and output bonds. The Monte Carlo simulation results were obtained using an efficient event-driven (continuous time) algorithm; for steady-state results the number of histories with each initial occupation was weighted according to the known steady-state distribution [@Levine04c]. For increasing measurement times the simulation data converges towards the long-time limits predicted by our theory rather than the straight line predicted by GC symmetry.
Unfortunately, since for increasing times it becomes exponentially more unlikely to measure a current fluctuation away from the mean, it is difficult to get long-time simulation data for a large range of $j$. A further check is provided by numerical evaluation of the integral followed by numerical Fourier transform to give the finite time distribution of $p(j,t)$—for small $t$ this gives excellent agreement with the simulation data; for larger $t$ the integrals converge too slowly for the method to be useful.
We now turn to numerical results for a larger system with a different choice of bounded $w_n$, see Fig. \[f:bnd\].
\[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Tc\]\[Bc\][$j$]{} \[Bc\]\[Tc\][$\hat{e}(-j)-\hat{e}(j)$]{} \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\]
In the finite-time simulation regime one again sees indications of violation of GC symmetry with bond-dependent form. Physically, we argue that the inhomogeneity of the fluctuations across the two different bonds in the single-site PAZRP and the associated violation of the GC symmetry is a result of the temporary build-up of particles on the site. In general, this possibility is expected to occur in any open-boundary zero-range process with $\lim_{n\to\infty} w_n$ finite (even when the boundary parameters are chosen so that there is a well-defined steady state, i.e., no permanent condensation).
Mathematically, the observed breakdown of the GC symmetry results from the divergence of $\langle s | \tilde{0} \rangle$ and $\langle \tilde{0} | P_0 \rangle$. For models where the number of particle configurations $N$ is limited, these quantities are finite and the relation holds for any initial state. However, the limit $N \to \infty$ does not necessarily commute with the $t\to\infty$ limit taken (implicitly) in and (explicitly) in . This non-commutation of limits leads in some cases to the violation of *even for steady-state initial conditions*. This and the initial state dependence (due to non-commuting timescales) are the main issues highlighted by our work.
We now give a more general explanation of this GC breakdown and highlight some connections to previous works. Firstly, consider the observed bond dependence. For systems with bounded state space, currents across different bonds differ by finite boundary terms which vanish in the long-time limit so any combination of currents has the same large deviation behaviour. In contrast, for systems with unbounded state space, current fluctuations can be spatially inhomogeneous and the boundary terms non-vanishing. However, there is always a specific weighted sum of currents for which these boundary terms cancel, giving an action functional analogous to heat production (see [@Lebowitz99]). For the choice $w_n=1$ this is $
W=2\sum_{l=0}^L E_l J_l
$ where $E_l$ is the effective field across each bond, e.g., for the single-site PAZRP we have $e^{2E_0}=\alpha/\gamma$ and $e^{2E_1}=\delta/\beta$.
However, it can readily be seen that the large deviations of $W$ still do not satisfy the GC symmetry. This is due to the presence of further non-vanishing boundary terms. Consider instead the modified action functional (again for $w_n=1$) $$W'=2\sum_{l=0}^L E_l J_l - \ln \frac{P_0(\{n\}(t))}{P_0(\{n\}(0))}$$ where $\{n\}(t)$ represents the configuration of particles at time $t$ and $P_0$ is the initial distribution. The fluctuations of this quantity do satisfy the relationship *even for finite times*—this is a statement of the transient fluctuation theorem of Evans and Searles [@Evans94; @Searles99] (see also [@Carberry04; @Wang05c] for recent experimental tests). Only for bounded state space (finite potentials) do the boundary terms containing the initial distribution vanish in the long-time limit leading to recovery of the GC symmetry and the steady-state theorem.
Note that if one measures only a single current (e.g., $J_0$ or $J_1$) but starts with an initial distribution corresponding to detailed balance across that bond, the boundary terms cancel and the GC symmetry *is* observed. A particular example is the zero-current case $\alpha\beta=\gamma\delta$ with an initial equilibrium distribution, $ x=(\alpha+\delta)/(\beta+\gamma)$—the current fluctuations across both bonds become symmetric $\hat{e}(j)=\hat{e}(-j)$ as predicted by the GCFT with $E \to 0$. This also implies the usual Green-Kubo formula and Onsager reciprocity relations [@Lebowitz99]. For other values of $x$ a breakdown of is still predicted in the $E \to 0$ limit (despite the system’s ergodicity).
An analogous apparent breakdown of the GCFT in models with *deterministic* dynamics and unbounded potentials was discussed by Bonnetto et al. [@Bonetto05]. They argue for the restoration of the symmetry by removal of the “unphysical” singular terms An earlier study of a model with both deterministic and stochastic forces [@vanZon03; @vanZon04] (see [@Garnier05] for experimental realization) found a modified form of heat fluctuation theorem for large fluctuations. In contrast to [@Bonetto05; @vanZon03; @vanZon04], we do not find a constant value for the ratio of probabilities for large forward and backward fluctuations.
A. Rákos acknowledges financial support from the Israel Science Foundation.
[^1]: E-mail:
| ArXiv |
---
abstract: 'A code for the numerical evaluation of hyperelliptic theta-functions is presented. Characteristic quantities of the underlying Riemann surface such as its periods are determined with the help of spectral methods. The code is optimized for solutions of the Ernst equation where the branch points of the Riemann surface are parameterized by the physical coordinates. An exploration of the whole parameter space of the solution is thus only possible with an efficient code. The use of spectral approximations allows for an efficient calculation of all quantities in the solution with high precision. The case of almost degenerate Riemann surfaces is addressed. Tests of the numerics using identities for periods on the Riemann surface and integral identities for the Ernst potential and its derivatives are performed. It is shown that an accuracy of the order of machine precision can be achieved. These accurate solutions are used to provide boundary conditions for a code which solves the axisymmetric stationary Einstein equations. The resulting solution agrees with the theta-functional solution to very high precision.'
address:
- 'Institut für Astronomie und Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany'
- 'LUTh, Observatoire de Paris, 92195 Meudon Cedex, France'
author:
- 'J. Frauendiener'
- 'C. Klein'
title: ' Hyperelliptic Theta-Functions and Spectral Methods '
---
Introduction
============
Solutions to integrable differential equations in terms of theta-functions were introduced with the works of Novikov, Dubrovin, Matveev, Its, Krichever, …(see [@DubNov75; @ItsMat75; @Kric78; @algebro]) for the Korteweg-de Vries (KdV) equation. Such solutions to e.g. the KdV, the Sine-Gordon, and the Non-linear Schrödinger equation describe periodic or quasi-periodic solutions, see [@dubrovin81; @algebro]. They are given explicitly in terms of Riemann theta-functions defined on some Riemann surface. Though all quantities entering the solution are in general given in explicit form via integrals on the Riemann surface, the work with theta-functional solutions admittedly has not reached the importance of soliton solutions.
The main reason for the more widespread use of solitons is that they are given in terms of algebraic or exponential functions. On the other hand the parameterization of theta-functions by the underlying Riemann surface is very implicit. The main parameters, typically the branch points of the Riemann surface, enter the solutions as parameters in integrals on the Riemann surface. A full understanding of the functional dependence on these parameters seems to be only possible numerically. In recent years algorithms have been developed to establish such relations for rather general Riemann surfaces as in [@tretkoff84] or via Schottky uniformization (see [@algebro]), which have been incorporated successively in numerical and symbolic codes, see [@seppala94; @hoeij94; @gianni98; @deconinck01; @deconinck03] and references therein (the last two references are distributed along with Maple 6, respectively Maple 8, and as a Java implementation at [@riemann]). For an approach to express periods of hyperelliptic Riemann surfaces via theta constants see [@enoric2003].
These codes are convenient to study theta-functional solutions of equations of KdV-type where the considered Riemann surfaces are ‘static’, i.e., independent of the physical coordinates. In these cases the characteristic quantities of the Riemann surface have to be calculated once, just the comparatively fast summation in the approximation of the theta series via a finite sum as e.g.in [@deconinck03] has to be carried out in dependence of the space-time coordinates.
The purpose of this article is to study numerically theta-functional solutions of the Ernst equation [@ernst] which were given by Korotkin [@Koro88]. In this case the branch points of the underlying hyperelliptic Riemann surface are parameterized by the physical coordinates, the spectral curve of the Ernst equation is in this sense ‘dynamical’. The solutions are thus not studied on a single Riemann surface but on a whole family of surfaces. This implies that the time-consuming calculation of the periods of the Riemann surface has to be carried out for each point in the space-time. This includes limiting cases where the surface is almost degenerate. In addition the theta-functional solutions should be calculated to high precision in order to be able to test numerical solutions for rapidly rotating neutron stars such as provided e.g. by the spectral code `LORENE` [@Lorene]. This requires a very efficient code of high precision.
We present here a numerical code for hyperelliptic surfaces where the integrals entering the solution are calculated by expanding the integrands with a Fast Cosine Transformation in MATLAB. The precision of the numerical evaluation is tested by checking identities for periods on Riemann surfaces and by comparison with exact solutions. The code is in principle able to deal with general (non-singular) hyperelliptic surfaces, but is optimized for a genus 2 solution to the Ernst equation which was constructed in [@prl2; @prd3]. We show that an accuracy of the order of machine precision ($\sim 10^{-14}$) can be achieved at a space-time point in general position with 32 polynomials and in the case of almost degenerate surfaces which occurs e.g., when the point approaches the symmetry axis with at most 256 polynomials. Global tests of the numerical accuracy of the solutions to the Ernst equation are provided by integral identities for the Ernst potential and its derivatives: the equality of the Arnowitt-Deser-Misner (ADM) mass and the Komar mass (see [@komar; @wald]) and a generalization of the Newtonian virial theorem as derived in [@virial]. We use the so determined numerical data for the theta-functions to provide ‘exact’ boundary values on a sphere for the program library `LORENE` [@Lorene] which was developed for a numerical treatment of rapidly rotating neutron stars. `LORENE` solves the boundary value problem for the stationary axisymmetric Einstein equations with spectral methods. We show that the theta-functional solution is reproduced to the order of $10^{-11}$ and better.
The paper is organized as follows: in section \[sec:ernsteq\] we collect useful facts on the Ernst equation and hyperelliptic Riemann surfaces, in section \[sec:spectral\] we summarize basic features of spectral methods and explain our implementation of various quantities. The calculation of the periods of the hyperelliptic surface and the non-Abelian line integrals entering the solution is performed together with tests of the precision of the numerics. In section \[sec:integrals\] we check integral identities for the Ernst potential. The test of the spectral code `LORENE` is presented in section \[sec:lorene\]. In section \[sec:concl\] we add some concluding remarks.
Ernst equation and hyperelliptic Riemann surfaces {#sec:ernsteq}
=================================================
The Ernst equation for the complex valued potential $\mathcal{E}$ (we denote the real and the imaginary part of $\mathcal{E}$ with $f$ and $b$ respectively) depending on the two coordinates $(\rho,\zeta)$ can be written in the form $$\Re \mathcal{E}\left(\mathcal{E}_{\rho\rho}+\frac{1}{\rho}
\mathcal{E}_{\rho}+\mathcal{E}_{\zeta\zeta}\right)=
\mathcal{E}_{\rho}^{2}+\mathcal{E}_{\zeta}^{2}
\label{ernst1}.$$ The equation has a physical interpretation as the stationary axisymmetric Einstein equations in vacuum (see appendix and references given therein). Its complete integrability was shown by Maison [@maison] and Belinski-Zakharov [@belzak]. For real Ernst potential, the Ernst equation reduces to the axisymmetric Laplace equation for $\ln \mathcal{E}$. The corresponding solutions are static and belong to the so called Weyl class, see [@exac].
Algebro-geometric solutions to the Ernst equation were given by Korotkin [@Koro88]. The solutions are defined on a family of hyperelliptic surfaces $\mathcal{L}(\xi,\bar{\xi})$ with $\xi=\zeta-i\rho$ corresponding to the plane algebraic curve $$\mu^{2}=(K-\xi)(K-\bar{\xi})\prod_{i=1}^{g}(K-E_{i})(K-F_{i})
\label{hyper1},$$ where $g$ is the genus of the surface and where the branch points $E_{i}$, $F_{i}$ are independent of the physical coordinates and for each $n$ subject to the reality condition $E_{n}=\bar{F}_{n}$ or $E_{n},F_{n}\in \mathbb{R}$.
Hyperelliptic Riemann surfaces are important since they show up in the context of algebro-geometric solutions of various integrable equations as KdV, Sine-Gordon and Ernst. Whereas it is a non-trivial problem to find a basis for the holomorphic differentials on general surfaces (see e.g. [@deconinck01]), it is given in the hyperelliptic case (see e.g. [@algebro]) by $$d\nu_k = \left( \frac{dK}{\mu}, \frac{KdK}{\mu},\ldots,
\frac{K^{g-1}dK}{\mu} \right)
\label{basis},$$ which is the main simplification in the use of these surfaces. We introduce on $\mathcal{L}$ a canonical basis of cycles $(a_{k},b_{k})$, $k=1,\ldots,n$. The holomorphic differentials $d\omega_k$ are normalized by the condition on the $a$-periods $$\int_{a_{l}}^{}d\omega_{k}=2\pi i \delta_{lk}.
\label{normholo}$$ The matrix of $b$-periods is given by $\mathbf{B}_{ik} =
\int_{b_{i}}^{}d\omega_{k}$. The matrix $\mathbf{B}$ is a so-called Riemann matrix, i.e. it is symmetric and has a negative definite real part. The Abel map $\omega: \mathcal{L} \to \mbox{Jac}(\mathcal{L}) $ with base point $E_{1}$ is defined as $\omega(P)=\int_{E_{1}}^{P}d\omega_k$, where $\mbox{Jac}(\mathcal{L})$is the Jacobian of $\mathcal{L}$. The theta-function with characteristics corresponding to the curve $\mathcal{L}$ is given by $$\Theta_{\mathbf{p}\mathbf{q}}(\mathbf{x}|\mathbf{B})=
\sum_{\mathbf{n}\in\mathbb{Z}^{g}}^{}\exp\left\{\frac{1}{2}
\langle\mathbf{B}(\mathbf{p}+\mathbf{n}),(\mathbf{p}+\mathbf{n})
\rangle+\langle\mathbf{p}+\mathbf{n},2i\pi\mathbf{q}+\mathbf{x}
\rangle\right\}
\label{theta},$$ where $\mathbf{x}\in \mathbb{C}^{g}$ is the argument and $\mathbf{p},\mathbf{q}\in \mathbb{C}^{g}$ are the characteristics. We will only consider half-integer characteristics in the following. The theta-function with characteristics is, up to an exponential factor, equivalent to the theta-function with zero characteristic (the Riemann theta-function is denoted with $\Theta$) and shifted argument, $$\Theta_{\mathbf{p}\mathbf{q}}(\mathbf{x}|\mathbf{B})=
\Theta(\mathbf{x}+\mathbf{B}\mathbf{p}+2i\pi\mathbf{q})\exp\left\{
\frac{1}{2}\langle\mathbf{B}\mathbf{p},\mathbf{p}
\rangle+\langle\mathbf{p},2i\pi\mathbf{q}+\mathbf{x} \rangle\right\}.
\label{theta2}$$
We denote by $d\omega_{PQ}$ a differential of the third kind, i.e., a 1-form which has poles in $P,Q\in \mathcal{L}$ with respective. residues $+1$ and $-1$. This singularity structure characterizes the differentials only up to an arbitrary linear combination of holomorphic differentials. The meromorphic differentials can be normalized by the condition that all $a$-periods vanish. We use the notation $\infty^{\pm}$ for the infinite points on different sheets of the curve $\mathcal{L}$, namely $\mu/K^{g+1}\to \pm 1$ as $K\to
\infty^{\pm}$. The differential $d\omega_{\infty^{+}\infty^{-}}$ is given up to holomorphic differentials by $-K^{g}dK/\mu$. It is well known that the $b$-periods of normalized differentials of the third kind can be expressed in terms of the Abel map (see e.g. [@dubrovin81]), $$\int_{b_{k}}^{}d\omega_{PQ}=\omega_{k}(P)-\omega_{k}(Q), \quad
k=1,\ldots,g
\label{period}.$$
In [@prl; @prd2] a physically interesting subclass of Korotkin’s solution was identified which can be written in the form $$\mathcal{E}=\frac{\Theta_{\mathbf{p}\mathbf{q}}(\omega(\infty^{+})+\mathbf{u})}{
\Theta_{\mathbf{p}\mathbf{q}}(\omega(\infty^{-})+\mathbf{u})}\cdot
e^{I}
\label{ernst2},$$ where $\mathbf{u}=(u_k)\in\mathbb{C}^g$ and where $$I=\frac{1}{2\pi i}\int_{\Gamma}^{}\ln G(K)\,d\omega_{\infty^{+}
\infty^{-}}(K), \qquad u_k=\frac{1}{2\pi i}
\int_{\Gamma}^{}\ln G(K)\,d\omega_k.
\label{path1}$$ $\Gamma$ is a piece-wise smooth contour on $\mathcal{L}$ and $G(K)$ is a non-zero Hölder-continuous function on $\Gamma$. The contour $\Gamma$ and the function $G$ have to satisfy the reality conditions that with $K\in \Gamma$ also $\bar{K}\in \Gamma$ and $\bar{G}(\bar{K})=G(K)$; both are independent of the physical coordinates.
In the following we will discuss the example of the solution constructed in [@prl2; @prd3] which can be interpreted as a disk of collisionless matter. For a physical interpretation see [@prd4]. The solution is given on a surface of the form (\[hyper1\]) with genus 2. The branch points independent of the physical coordinates are related through the relations $E_{i}=\bar{F}_{i}$, $i=1,2$ and $E_{1}=-F_{2}$. The branch points are parameterized by two real parameters $\lambda$ and $\delta$. Writing $E_{1}^{2}=\alpha +i\beta$ with real $\alpha$, $\beta$, we have $$\alpha=-1+\frac{\delta}{2}, \quad
\beta=\sqrt{\frac{1}{\lambda^{2}} +\delta-\frac{\delta^{2}}{4}}
\label{disk1}.$$ The contour $\Gamma$ is the piece of the covering of the imaginary axis in the upper sheet between $[-i,i]$, the function $G$ has the form $$G(K)=\frac{\sqrt{(K^{2}-\alpha)^{2}+\beta^{2}}+K^{2}+1}{
\sqrt{(K^{2}-\alpha)^{2}+\beta^{2}}-K^{2}-1}.
\label{disk2}$$ The physical parameters vary between $\delta=0$, the solution which was first given in [@NeuMei95], and $\delta_{s}=2(1+\sqrt{1+1/\lambda^{2}})$, the static limit in which $\beta=0$. In the latter case the Riemann surface degenerates, the resulting Ernst potential (\[ernst2\]) is real and be expressed in terms of objects corresponding to the surface $\mathcal{L}_{0}$ of genus 0 defined by the relation $\mu_{0}^{2}=(K-\xi)(K-\bar{\xi})$. The parameter $\lambda$ varies between $\lambda=0$, the so-called Newtonian limit where the branch points $E_{i}$, $F_{i}$ tend to infinity. Since $G$ is also of order $\lambda$ in this limit, the lowest order contributions are again real and defined on the surface $\mathcal{L}_{0}$. This case corresponds to the disk limit of the Maclaurin ellipsoids, see [@bitr]. The upper limit for $\lambda$ is infinity for $\delta\neq 0$ and $\lambda_{c}=4.629\ldots$ for $\delta=0$. The limiting situation is special in the second case since the resulting spacetime is no longer asymptotically flat and since the axis is singular. The invariant circumference of the disk is zero in this case which implies that the disk shrinks to a point for an observer in the exterior of the disk, see [@prd4].
For physical reasons the solution was discussed in [@prd4] in dependence of two other real parameters $\epsilon$ and $\gamma$. Here $\epsilon$ is related to the redshift of photons emitted at the center of the disk and detected at infinity. It varies between 0 in the Newtonian limit, and 1 in the ultra-relativistic limit, where photons cannot escape to infinity. Thus, $\epsilon$ is a measure of how relativistic the situation is. The parameter $\gamma$ is a measure of how static the solution is, it varies between 0, indicating the static limit and 1. For the functional relations between $\epsilon$, $\gamma$ and $\lambda$, $\delta$ see [@prd4]. The constant $\Omega$ (with respect to the physical coordinates) to appear in the following can be considered as a natural scale for the angular velocities in the disk, for a definition see [@prd4].
The coordinate $\rho$ can take all non-negative real values, the coordinate $\zeta$ all real values. The example we are studying here has an equatorial symmetry, $$\mathcal{E}(\rho,-\zeta)=\bar{\mathcal{E}}(\rho,\zeta)
\label{eq:eqsym}.$$ It is therefore sufficient to consider only non-negative values of $\zeta$. The case $\rho=0$ corresponds to the axis of symmetry where the branch cut $[\xi,\bar{\xi}]$ degenerates to a point. As was shown in [@prd2; @prd4], the Ernst potential can be written in this limit in terms of theta-functions on the elliptic surface $\mathcal{L}_{1}$ defined by $\mu_{1}^{2}= (K^{2}-\alpha)^{2}+\beta^{2}$, i.e. the surface $\mathcal{L}$ with the cut $[\xi,\bar{\xi}]$ removed. Near the axis the Ernst potential has the form (see [@fay; @prd2]) $$\mathcal{E}(\rho,\zeta)=\mathcal{E}_{0}(\zeta)+\rho^{2}
\mathcal{E}_{1}(\zeta)+\mathcal{O}(\rho^{4});
\label{eq:nearaxis}$$ here $\mathcal{E}_{0}$ and $\mathcal{E}_{1}$ are independent of $\rho$, $\mathcal{E}_{0}$ is the axis potential. This formula could be used to calculate the potential close to the axis. However we considered only values of $\rho$ greater than $10^{-5}$ and did not experience any numerical problems. Consequently we did not use formula (\[eq:nearaxis\]).
For large values of $r=|\xi|$, the Ernst potential has the asymptotic expansion $$\mathcal{E}=1-\frac{2m}{r}+\frac{2m^{2}}{r^{2}}
-\frac{2iJ\zeta}{r^{3}}+\mathcal{O}(1/r^{3});
\label{eq:ernstinfinity}$$ here the constants (with respect to $\xi$) $m$ and $J$ are the ADM-mass and, respectively, the angular momentum of the space-time. They can be calculated on the axis in terms of elliptic theta-functions, see [@prd4]. Formula (\[eq:ernstinfinity\]) is used for values of $r>10^{6}$.
In the limit $\xi=E_{2}$, the Ernst potential can be given on the surface $\Sigma_{0}$ of genus 0 obtained by removing the cuts $[\xi,\bar{\xi}]$ and $[E_{2},F_{2}]$ from the surface $\mathcal{L}$. The potential can thus be given in this case in terms of elementary functions, see [@prd4].
In the equatorial plane $\zeta=0$, the Riemann surface $\mathcal{L}$ has an additional involution $K\to-K$ as can be seen from (\[hyper1\]). This implies that the surface can be considered as a covering of an elliptic surface, see [@algebro; @prd2]. The theta-functions in (\[ernst2\]) can be written as sums of theta-functions on the covered surface and on the Prym variety which happens to be an elliptic surface as well in this case. We use this fact at the disk ($\zeta=0$, $\rho\leq 1$), where the moving branch points are situated on $\Gamma$. There all quantities can be expressed in terms of quantities defined on the Prym surface $\Sigma_{w}$ defined by $\mu_{w}^{2}=(K+\rho^{2})((K-\alpha)^{2}+\beta^{2})$, see [@prd4].
Numerical implementations {#sec:spectral}
=========================
The numerical task in this work is to approximate and evaluate analytically defined functions as accurately and efficiently as possible. To this end it is advantageous to use (pseudo-)spectral methods which are distinguished by their excellent approximation properties when applied to smooth functions. Here the functions are known to be analytic except for isolated points. In this section we explain the basic ideas behind the use of spectral methods and describe in detail how the theta-functions and the Ernst potential can be obtained to a high degree of accuracy.
Spectral approximation
----------------------
The basic idea of spectral methods is to approximate a given function $f$ globally on its domain of definition by a linear combination $$f \approx \sum_{k=0}^N a_k \phi_k,$$ where the function $\phi_k$ are taken from some class of functions which is chosen appropriately for the problem at hand.
The coefficients $a_k$ are determined by requiring that the linear combination should be ‘close’ to $f$. Thus, one could require that $||f - \sum_{k=0}^N a_k \phi_k||$ should be minimal for some norm. Another possibility is to require that $\left< f -\sum_{k=0}^N
a_k \phi_k, \chi_l\right> = 0$ for $l=0:N$ with an appropriate inner product and associated orthonormal basis $\chi_l$. This is called the Galerkin method. Finally, one can demand that $f(x_l) = \sum_{k=0}^N a_k
\phi_k(x_l)$ at selected points $(x_l)_{l=0:N}$. This is the so called collocation method which is the one we will use in this paper. In this case the function values $f_l=f(x_l)$ and the coefficients $a_k$ are related by the matrix $\Phi_{lk} =
\phi_k(x_l)$.
The choice of the expansion basis depends to a large extent on the specific problem. For periodic functions there is the obvious choice of trigonometric polynomials $\phi_k(x) = \exp(2\pi i k/N)$ while for functions defined on a finite interval the most used functions are orthogonal polynomials, in particular Chebyshev and Legendre polynomials. While the latter are important because of their relationship with the spherical harmonics on the sphere, the former are used because they have very good approximation properties and because one can use fast transform methods when computing the expansion coefficients from the function values provided one chooses the collocation points $x_l=\cos(\pi l/N)$ (see [@fornberg] and references therein). We will use here collocation with Chebyshev polynomials.
Let us briefly summarize here their basic properties. The Chebyshev polynomials $T_n(x)$ are defined on the interval $I=[-1,1]$ by the relation $$T_n(\cos(t)) = \cos(n t), \text{where } x = \cos(t),\qquad t\in[0,\pi].$$ They satisfy the differential equation $$\label{eq:diffeqcheb}
(1-x^2)\, \phi''(x) - x \phi'(x) + n^2 \phi(x) = 0.$$ The addition theorems for sine and cosine imply the recursion relations $$\label{eq:recurscheb}
T_{n+1}(x) - 2 x\, T_n(x) + T_{n-1}(x) = 0,$$ for the polynomials $T_n$ and $$\label{eq:recursderiv}
\frac{T'_{n+1}(x)}{n+1} - \frac{T'_{n-1}(x)}{n-1} = 2 T_n(x)$$ for their derivatives. The Chebyshev polynomials are orthogonal on $I$ with respect to the hermitian inner product $$\left< f, g \right> = \int_{-1}^1 f(x) \bar g(x) \,\frac{d x}{\sqrt{1-x^2}}.$$ We have $$\label{eq:ortho}
\left< T_m , T_n \right> = c_m \frac\pi2\, \delta_{mn}$$ where $c_0=2$ and $c_l=1$ otherwise.
Now suppose that a function $f$ on $I$ is sampled at the points $x_l=\cos(\pi l/N)$ and that $\sum_{n=0}^N a_n T_n$ is the interpolating polynomial. Defining $c_0=c_N=2$, $c_n=1$ for $0<n<N$ in the discrete case and the numbers $F_n = c_n a_n$ we have $$\begin{split}
f_l &= \sum_{n=0}^N a_n T_n(x_l) = \sum_{n=0}^N a_n T_n(\cos(\pi
l/N)) \\
&= \sum_{n=0}^N a_n \cos(\pi nl/N)
= \sum_{n=0}^N \frac{F_n}{c_n}\cos(\pi nl/N)
\end{split}.$$ This looks very much like a discrete cosine series and in fact one can show [@briggshenson] that the coefficients $F_n$ are related to the values $f_l$ of the function by an inverse discrete Fourier transform (DCT) $$F_n = \frac2N\sum_{l=0}^N \frac{f_l}{c_l}\cos(\pi nl/N).$$ Note, that up to a numerical factor the DCT is idempotent, i.e., it is its own inverse. This relationship between the Chebyshev polynomials and the DCT is the basis for the efficient computations because the DCT can be performed numerically by using the fast Fourier transform (FFT) and pre- and postprocessing of the coefficients [@fornberg]. The fast transform allows us to switch easily between the representations of the function in terms of its sampled values and in terms of the expansion coefficients $a_n$ (or $F_n$).
The fact that $f$ is approximated globally by a finite sum of polynomials allows us to express any operation applied to $f$ approximately in terms of the coefficients. Let us illustrate this in the case of integration. So we assume that $f = p_N =\sum_{n=0}^N a_n
T_n$ and we want to find an approximation of the integral for $p_N$, i.e., the function $$F(x) = \int_{-1}^x f(s)\, ds,$$ so that $F'(x)=f(x)$. We make the ansatz $F(x) = \sum_{n=0}^N b_n\,
T_n(x)$ and obtain the equation $$F' = \sum_{n=0}^N b_n\,T'_n = \sum_{n=0}^N a_n T_n = f.$$ Expressing $T_n$ in terms of the $T'_n$ using and comparing coefficients implies the equations $$b_1 = \frac{2a_{0} - a_{2}}{2}, \qquad b_n = \frac{a_{n-1} -
a_{n+1}}{2n}\quad \text{for }0< n < N,\qquad b_N = \frac{a_{N-1}}{2N}.$$ between the coefficients which determines all $b_l$ in terms of the $a_n$ except for $b_0$. This free constant is determined by the requirement that $F(-1)=0$ which implies (because $T_n(-1)=(-1)^n$) $$b_0 = - \sum_{n=1}^N (-1)^n b_n.$$ These coefficients $b_n$ determine a polynomial $q_N$ of degree $N$ which approximates the indefinite integral $F(x)$ of the $N$-th degree polynomial $f$. The exact function is a polynomial of degree $N+1$ whose highest coefficient is proportional to the highest coefficient $a_N$ of $f$. Thus, ignoring this term we make an error whose magnitude is of the order of $|a_N|$ so that the approximation will be the better the smaller $|a_N|$ is. The same is true when a smooth function $f$ is approximated by a polynomial $p_N$. Then, again, the indefinite integral will be approximated well by the polynomial $q_N$ whose coefficients are determined as above provided the highest coefficients in the approximating polynomial $p_N$ are small.
From the coefficients $b_n$ we can also find an approximation to the definite integral $\int_{-1}^1 f(s)\,ds = F(1)$ by evaluating $$q_N(1) = \sum_{n=0}^Nb_n = 2\sum_{l=0}^{\lfloor N/2\rfloor}b_{2l+1}.$$ Thus, to find an approximation of the integral of a function $f$ we proceed as described above, first computing the coefficients $a_n$ of $f$, computing the $b_n$ and then calculating the sum of the odd coefficients.
Implementation of the square-root
---------------------------------
The Riemann surface $\mathcal{L}$ is defined by an algebraic curve of the form $$\mu^{2}=(K-\xi)(K-\bar{\xi})\prod_{i=1}^{g}(K-E_{i})(K-\bar E_{i}),$$ where in our case we have $g=2$ throughout. In order to compute the periods and the theta-functions related to this Riemann surface it is necessary to evaluate the square-root $\sqrt{\mu^2(K)}$ for arbitrary complex numbers $K$. In order to make this a well defined problem we introduce the cut-system as indicated in Fig. \[fig:cut-system\].
On the cut surface the square-root $\mu(K)$ is defined as in [@heil] as the product of square-roots of monomials $$\mu=\sqrt{K-\xi\phantom{\bar\xi}} \sqrt{K-\bar{\xi}}
\prod_{i=1}^{g} \sqrt{K-E_{i}} \sqrt{K-\bar E_{i}}.
\label{eq:root}$$ The square-root routines such as the one available in MATLAB usually have their branch-cut along the negative real axis. The expression (\[eq:root\]) is holomorphic on the cut surface so that we cannot simply take the builtin square-root when computing $\sqrt{\mu^2(K)}$. Instead we need to use the information provided by the cut-system to define adapted square-roots.
Let $\arg(z)$ be the argument of a complex number $z$ with values in $]-\pi,\pi[$ and consider two factors in (\[eq:root\]) such as $$\sqrt{K-P_1}\sqrt{K-P_2}$$ where $P_1$ and $P_2$ are two branch-points connected by a branch-cut. Let $\alpha=\arg(P_2-P_1)$ be the argument at the line from $P_1$ to $P_2$. Now we define the square-root $\sqrt[(\alpha)]{\cdot}$ with branch-cut along the ray with argument $\alpha$ by computing for each $z\in\mathbb{C}$ the square-root $s:=\sqrt{z}$ with the available MATLAB routine and then putting $$\sqrt[(\alpha)]{z} = \left\{
\begin{array}{rl}
s & \alpha/2<\arg(s)<\alpha/2 + \pi\\
-s & \text{otherwise}
\end{array}
\right. .$$ With this square-root we compute the two factors $$\sqrt[(\alpha)]{K-P_1}\sqrt[(\alpha)]{K-P_2}.$$ It is easy to see that this expression changes sign exactly when the branch-cut between $P_1$ and $P_2$ is crossed. We compute the expression (\[eq:root\]) by multiplying the pairs of factors which correspond to the branch-cuts.
This procedure is not possible in the case of the non-linear transformations we are using to evaluate the periods in certain limiting cases. In these cases the root is chosen in a way that the integrand is a continuous function on the path of integration.
Numerical treatment of the periods
----------------------------------
The quantities entering formula (\[ernst2\]) for the Ernst potential are the periods of the Riemann surface and the line integrals $\mathbf{u}$ and $I$. The value of the theta-function is then approximated by a finite sum.
The periods of a hyperelliptic Riemann surface can be expressed as integrals between branch points. Since we need in our example the periods of the holomorphic differentials and the differential of the third kind with poles at $\infty^{\pm}$, we have to consider integrals of the form $$\int_{P_{i}}^{P_{j}}\frac{K^{n}dK}{\mu(K)}, \quad n=0,1,2
\label{period1},$$ where the $P_{i}$, $i,j=1,\ldots,6$ denote the branch points of $\mathcal{L}$.
In general position we use a linear transformation of the form $K
=ct+d$ to transform the integral (\[period1\]) to the normal form $$\label{eq:int_aperiod} \int_{-1}^1 \frac{\alpha_0 + \alpha_1 t +
\alpha_2 t^2}{\sqrt{1-t^2}} \;H(t) \,dt,$$ where the $\alpha_i$ are complex constants and where $H(t)$ is a continuous (in fact, analytic) complex valued function on the interval $[-1,1]$. This form of the integral suggests to express the powers $t^n$ in the numerator in terms of the first three Chebyshev polynomials $T_0(t)=1$, $T_1(t)=t$ and $T_2(t)= 2t^2-1$ and to approximate the function $H(t)$ by a linear combination of Chebyshev polynomials $$H(t) = \sum_{n\ge0} h_n T_n(t).$$ The integral is then calculated with the help of the orthogonality relation (\[eq:ortho\]) of the Chebyshev polynomials.
Since the Ernst potential has to be calculated for all $\rho,\zeta\in
\mathbb{R}^{+}_{0}$, it is convenient to use the cut-system (\[fig:cut-system\]). In this system the moving cut does not cross the immovable cut. In addition the system is adapted to the symmetries and reality properties of $\mathcal{L}$. Thus the periods $a_{2}$ and $b_{2}$ are related to $a_{1}$ and $b_{1}$ via complex conjugation. For the analytical calculations of the Ernst potential in the limit of collapsing cuts, we have chosen in [@prd2] cut systems adapted to the respective situation. In the limit $\xi\to
\bar{\xi}$ we were using for instance a system where $a_{2}$ is the cycle around the cut $[\xi,\bar{\xi}]$. This has the effect that only the $b$-period $b_{2}$ diverges logarithmically in this case whereas the remaining periods stay finite as $\rho$ tends to 0. In the cut systems \[fig:cut-system\], all periods diverge as $\ln \rho$. Since the divergence is only logarithmical this does not pose a problem for values of $\rho>10^{-5}$. In addition the integrals which have to be calculated in the evaluation of the periods are the same in both cut-system. Thus there is no advantage in using different cut systems for the numerical work.
To test the numerics we use the fact that the integral of any holomorphic differential along a contour surrounding the cut $[E_{1},F_{1}]$ in positive direction is equal to minus the sum of all $a$-periods of this integral. Since this condition is not implemented in the code it provides a strong test for the numerics. It can be seen in Fig. \[fig:test\_periods\] that 16 to 32 polynomials are sufficient in general position to achieve optimal accuracy. Since MATLAB works with 16 digits, machine precision is in general limited to 14 digits due to rounding errors. These rounding errors are also the reason why the accuracy drops slightly when a higher number of polynomials is used. The use of a low number of polynomials consequently does not only require less computational resources but has the additional benefit of reducing the rounding errors. It is therefore worthwhile to reformulate a problem if a high number of polynomials would be necessary to obtain optimal accuracy.
These situations occur in the calculation of the periods when the moving branch points almost coincide which happens on the axis of symmetry in the space-time or at spatial infinity. As can be seen from Fig. \[fig:test\_periods\], for $\rho=10^{-3}$ and $\zeta=10^{3}$ not even 2048 polynomials (this is the limit due to memory on the low end computers we were using) produce sufficient accuracy. The reason for these problems is that the function $H$ in (\[eq:int\_aperiod\]) behaves like $1/\sqrt{t+\rho}$ near $t=0$. For small $\rho$ this behavior is only satisfactorily approximated by a large number of polynomials. We therefore split the integral in two integrals between $F_{2}$ and $(F_{2}+\bar{\xi})/2$ and between $(F_{2}+\bar{\xi})/2$ and $\bar{\xi}$. The first integral is calculated with the Chebyshev integration routine after the substitution $t=\sqrt{K-F_{2}}$. This substitution leads to a regular integrand also at the branch point $F_{2}$. The second integral is calculated with the Chebyshev integration routine after the substitution $K-\zeta=\rho\sinh(t)$. This takes care of the almost collapsing cut $[\xi,\bar{\xi}]$. It can be seen in Fig. \[fig:test\_periods\] that 128 polynomials are sufficient to obtain machine precision even in almost degenerate situations.
The cut-system in Fig. \[fig:cut-system\] is adapted to the limit $\bar{\xi}\to F_{2}$ in what concerns the $a$-periods, since the cut which collapses in this limit is encircled by an $a$-cycle. However there will be similar problems as above in the determination of the $b$-periods. For $\bar{\xi}\sim F_{2}$ we split the integrals for the $b$-periods as above in two integrals between $F_{1}$ and $0$, and $0$ and $F_{2}$. For the first integral we use the integration variable $t
= \sqrt{K-F_1}$, for the second $K=\Re F_{2}-i\Im F_{2}\sinh t$. Since the Riemann matrix (the matrix of $b$-periods of the holomorphic differentials after normalization) is symmetric, the error in the numerical evaluation of the $b$-periods can be estimated via the asymmetry of the calculated Riemann matrix. We define the function $err(\rho,\zeta)$ as the maximum of the norm of the difference in the $a$-periods discussed above and the difference of the off-diagonal elements of the Riemann matrix. This error is presented for a whole space-time in Fig. \[fig:error\]. The values for $\rho$ and $\zeta$ vary between $10^{-4}$ and $10^{4}$. On the axis and at the disk we give the error for the elliptic integrals (only the error in the evaluation of the $a$-periods, since the Riemann matrix has just one component). For $\xi\to \infty$ the asymptotic formulas for the Ernst potential are used. The calculation is performed with 128 polynomials, and up to 256 for $|\xi|>10^{3}$. It can be seen that the error is in this case globally below $10^{-13}$.
Numerical treatment of the line integrals
-----------------------------------------
The line integrals $\mathbf{u}$ and $I$ in (\[ernst2\]) are linear combinations of integrals of the form $$\int_{-i}^{i}\frac{\ln G(K)K^{l}dK}{\mu(K)}, \qquad l=0,1,2
\label{eq:line1}.$$ In general position, i.e. not close to the disk and $\lambda$ small enough, the integrals can be directly calculated after the transformation $K=it$ with the Chebyshev integration routine. To test the numerics we consider the Newtonian limit ($\lambda\to0$) where the function $\ln G$ is proportional to $1+K^{2}$, i.e. we calculate the test integral $$\int_{-i}^{i}\frac{(1+K^{2})\;dK}{\sqrt{(K-\zeta)^{2}+\rho^{2}}}
\label{eq:testline}.$$ We compare the numerical with the analytical result in Fig. \[fig:line\]. In general position machine precision is reached with 32 polynomials.
When the moving cut approaches the path $\Gamma$, i.e., when the space-time point comes close to the disk, the integrand in (\[eq:testline\]) develops cusps near the points $\xi$ and $\bar{\xi}$. In this case a satisfactory approximation becomes difficult even with a large number of polynomials. Therefore we split the integration path in $[-i,-i\rho]$, $[-i\rho,i\rho]$ and $[i\rho,i]$. Using the reality properties of the integrands, we only calculate the integrals between $0$ and $i\rho$, and between $i\rho$ and $i$. In the first case we use the transformation $K=
\zeta+\rho\sinh t$ to evaluate the integral with the Chebyshev integration routine, in the second case we use the transformation $t = \sqrt{K-\bar{\xi}}$. It can be seen in figure \[fig:line\] that machine precision can be reached even at the disk with 64 to 128 polynomials. The values at the disk are, however, determined in terms of elliptic functions which is more efficient than the hyperelliptic formulae.
To treat the case where $\delta\lambda^{2}$ is not small, it is convenient to rewrite the function $G$ in (\[disk2\]) in the form $$\ln G(K) =2\ln \left(\sqrt{(K^{2}-\alpha)^{2}+\beta^{2}}+K^{2}
+1\right)-\ln \left(\frac{1}{\lambda^{2}}-\delta K^{2}\right)
\label{eq:log}.$$ In the limit $\delta \lambda^{2}\to \infty$ with $\delta$ finite, the second term in (\[eq:log\]) becomes singular for $K=0$. Even for $\delta\lambda^{2}$ large but finite, the approximation of the integrand by Chebyshev polynomials requires a huge number of coefficients as can be seen from Fig. \[fig:logreg\]. It is therefore sensible to ‘regularize’ the integrand near $K=0$. We consider instead of the function $\ln( \frac{1}{\lambda^{2}}-\delta
K^{2}) F(K)$ where $F(K)$ is a $C^{\infty}$ function near $K=0$, the function $$\ln \left(\frac{1}{\lambda^{2}}-\delta K^{2}\right)\left(
F(K)-F(0)-F'(0)K-\ldots-\frac{1}{n!}F^{(n)}(0)K^{n}\right)
\label{eq:logreg}.$$ The parameter $n$ is chosen such that the spectral coefficients of (\[eq:logreg\]) are of the order of $10^{-14}$ for a given number of polynomials, see Fig. \[fig:logreg\]. There we consider the integral $$\int_{-i}^{i}\frac{\ln
G(K)dK}{\sqrt{(K^{2}-\alpha)^{2}+\beta^{2}}}
\label{eq:axisreg},$$ which has to be calculated on the axis. We show the absolute values of the coefficients $a_{k}$ in an expansion of the integrand in Chebyshev polynomials, $\sum_{k=1}^{N}a_{k}T_{k}$. It can be seen that one has to include values of $n=6$ in (\[eq:logreg\]). The integral $\int_{\Gamma}^{}\ln G(K) F(K)$ is then calculated numerically as the integral of the function (\[eq:logreg\]), the subtracted terms are integrated analytically.
In this way one can ensure that the line integrals are calculated in the whole space-time with machine precision: close to the Newtonian limit, we use an analytically known test function to check the integration routine, for general situations we check the quality of the approximation of the integrand by Chebyshev polynomials via the spectral coefficients which have to become smaller than $10^{-14}$.
Theta-functions
---------------
The theta series (\[theta\]) for the Riemann theta-function (the theta function in (\[theta\]) with zero characteristic, theta functions with characteristic follow from (\[theta2\])) is approximated as the sum $$\Theta(\mathbf{x}|\mathbf{B})
=\sum_{n_{1}=-N}^{N}\sum_{n_{2}=-N}^{N}\exp\left\{
\frac{1}{2}n_{1}^{2}B_{11}+n_{1}n_{2}B_{12}+\frac{1}{2}B_{22}
+n_{1}x_{1}+n_{2}x_{2}\right\}.
\label{eq:thetasum}$$ The value of $N$ is determined by the condition that terms in the series (\[theta\]) for $n>N$ are strictly smaller than some threshold value $\epsilon$ which is taken to be of the order of $10^{-16}$. To this end we determine the eigenvalues of $\mathbf{B}$ and demand that $$N> -\frac{1}{B_{max}}\left(||\mathbf{x}||+\sqrt{||\mathbf{x}||^{2}
+2\ln \epsilon B_{max}}\right)
\label{eq:N},$$ where $B_{max}$ is the real part of the eigenvalue with maximal real part ($\mathbf{B}$ is negative definite). For a more sophisticated analysis of theta summations see [@deconinck03]. In general position we find values of $N$ between 4 and 8. For very large values of $\zeta$ close to the axis, $N$ can become larger that 40 which however did not lead to any computational problems. To treat more extreme cases it could be helpful to take care of the fact that the eigenvalues of $\mathbf{B}$ can differ by more than an order of magnitude in our example. In these cases a summation over an ellipse rather than over a sphere in the plane $(n_{1},n_{2})$, i.e.different limiting values for $n_{1}$ and $n_{2}$ as in [@deconinck03] will be more efficient.
In our case the computation of the integrals entering the theta-functions was however always the most time consuming such that an optimization of the summation of the theta-function would not have a noticeable effect. Due to the vectorization techniques in MATLAB, the theta summation always took less than 10 % of the calculation time for a value of the Ernst potential. Between 50 and 70 % of the processor time are used for the determination of the periods. On the used low-end PCs, the calculation time varied between 0.4 and 1.2s depending on the used number of polynomials. We show a plot of the real part of the Ernst potential for $\lambda=10$ and $\delta=1$ in Fig. \[fig:f\]. For $\rho,\zeta>1$, we use $1/\rho,1/\zeta$ as coordinates which makes it possible to plot the whole space-time in Weyl coordinates. The non-smoothness of the coordinates across $\rho=1=1/\rho$ and $\zeta=1=1/\zeta$ is noticeable in the plot. Asymptotically the potential is equal to 1. The disk is situated in the equatorial plane between $\rho=0$ and $\rho=1$. At the disk, the normal derivatives of $f$ are discontinuous.
The imaginary part of the Ernst potential in this case is given in Fig. \[fig:b\]. It vanishes at infinity and at the regular part of the equatorial plane. At the disk, the potential has a jump.
Integral identities {#sec:integrals}
===================
In the previous section we have tested the accuracy of the numerics locally, i.e. at single points in the space-time. Integral identities have the advantage that they provide some sort of global test of the numerical precision since they sum up the errors. In addition they require the calculation of the potentials in extended regions of the space-time which allows to explore the numerics for rather general values of the physical coordinates.
The identities we are considering in the following are the well known equivalence of a mass calculated at the disk (the Komar mass) and the ADM mass determined at infinity, see [@komar; @wald], and a generalization of the Newtonian virial identity, see [@virial] and the appendix. The derivatives of the Ernst potential occurring in the integrands can be related to derivatives of theta-functions, see [@prd2]. Since we are interested here in the numerical treatment of theta-functions with spectral methods, we determine the derivatives with spectral methods, too (see section 3). The integrals are again calculated with the Chebyshev integration routine. The main problem in this context is the singular behavior of the integrands e.g. at the disk which is a singularity for the space-time. As before this will lead to problems in the approximation of these terms via Chebyshev polynomials. This could lead to a drop in accuracy which is mainly due to numerical errors in the evaluation of the integrand and not of the potentials which we want to test. An important point is therefore the use of integration variables which are adapted to the possible singularities.
Mass equalities
---------------
The equality between the ADM mass and the Komar mass provides a test of the numerical treatment of the elliptic theta-functions at the disk by means of the elliptic theta-functions on the axis. Since this equality is not implemented in the code, it provides a strong test.
The Komar mass at the disk is given by formula (\[virial2\]) of the appendix. In the example we are considering here, the normal derivatives at the disk can be expressed via tangential derivatives (see [@prd4]) which makes a calculation of the derivatives solely within the disk possible. We implement the Komar mass in the form $$m_{K}= \int_{0}^{1}d\rho\frac{b_{\rho}}{4\Omega^{2}\sqrt{\rho^{2}-\delta
f^{2}+2f/\lambda}}
\left(f+\frac{\Omega^{2}}{f}(\rho^{2}-a^{2}f^{2})\right)
\label{eq:komar2}.$$
The integrand is known to vanish as $\sqrt{1-\rho^{2}}$ at the rim of the disk, which is the typical behavior for such disk solutions. Since $\sqrt{1-\rho^{2}}$ is not analytic in $\rho$, an expansion of the integrand (\[eq:komar2\]) in Chebyshev polynomials in $\rho$ would not be efficient. We will thus use $t= \sqrt{1-\rho^{2}}$ as the integration variable. This takes care of the behavior at the rim of the disk. Since in general the integrand in \[eq:komar2\] depends on $\rho^{2}$, this variable can be used in the whole disk. In the ultra-relativistic limit for $\delta\neq 0$, the function $f$ vanishes as $\rho$. In such cases it is convenient either to take two domains of integration or to use a different variable of integration. We chose the second approach with $\rho=\sin x$ (this corresponds to the disk coordinates (\[eq:diskcoor\])). Yet, strongly relativistic situations still lead to problems since $f$ vanishes in this case at the center of the disk as does $b_{\rho}$ which leads to a ‘0/0’ limit. In Fig. \[fig:mtest\] one can see that the masses are in general equal to the order of $10^{-14}$. In these calculations 128 up to 256 polynomials were used. We show the dependence for $\gamma=0.7$ and several values of $\epsilon$, as well as for $\epsilon=0.8$ and several values of $\gamma$. The accuracy drops in the strongly relativistic, almost static situations ($\epsilon$ close to 1, $\gamma$ close to zero) since the Riemann surface is almost degenerate in this case ($\beta\to 0$). In the ultra-relativistic limit for $\delta=0$, the situation is no longer asymptotically flat which implies that the masses formally diverge. For $\epsilon=0.95$, the masses are still equal to the order of $10^{-13}$. Not surprisingly the accuracy drops for $\epsilon=0.9996$ to the order of $10^{-4}$.
Virial-type identities
----------------------
Generalizations of the Newtonian virial theorem are used in numerics (see [@virial]) as a test of the quality of the numerical solution of the Einstein equations. Since they involve integrals over the whole space-time, they test the numerics globally and thus provide a valid criterion for the entire range of the physical coordinates.
The identity which is checked here is a variant of the one given in [@virial] which is adapted to possible problems at the zeros of the real part of the Ernst potential, the so-called ergosphere, see [@prd4] for the disk solutions discussed here. Eq. (\[virial20\]) relates integrals of the Ernst potential and its derivatives over the whole space-time to corresponding integrals at the disk. Since the numerics at the disk has been tested above, this provides a global test of the evaluation of the Ernst potential. As before, derivatives and integrals will be calculated via spectral methods.
The problem one faces when integrating over the whole space-time is the singular behavior of the fields on the disk which represents a discontinuity of the Ernst potential. The Weyl coordinates in which the solution is given are not optimal to describe the geometry near the disk. Hence a huge number of polynomials is necessary to approximate the integrands in (\[virial20\]). Even with $512$ polynomials for each coordinate, the coefficients of an expansion in Chebyshev polynomials did not drop below $10^{-6}$ in more relativistic situations. Though the computational limits are reached, the identity (\[virial20\]) is only satisfied to the order of $10^{-8}$ which is clearly related to the bad choice of coordinates.
We therefore use for this calculation so-called disk coordinates $\eta$, $\theta$ (see [@bitr]) which are related to the Weyl coordinates via $$\rho+i\zeta=\cosh(\eta+i\theta)
\label{eq:diskcoor}.$$ The coordinate $\eta$ varies between $\eta=0$, the disk, and infinity, the coordinate $\theta$ between $-\pi/2$ and $\pi/2$. The axis is given by $\pm \pi/2$, the equatorial plane in the exterior of the disk by $\theta=0$ and $\eta\neq0$. Because of the equatorial symmetry, we consider only positive values of $\theta$. The surfaces of constant $\eta$ are confocal ellipsoids which approach the disk for small $\eta$. For large $\eta$, the coordinates are close to spherical coordinates.
To evaluate the integrals in (\[virial20\]), we perform the $\eta$-integration up to a value $\eta_{0}$ as well as the $\theta$-integration with the Chebyshev integration routine. The parameter $\eta_{0}$ is chosen in a way that the deviation from spherical coordinates becomes negligible, typically $\eta_{0}=15$. The integral from $\eta_{0}$ to infinity is then carried out analytically with the asymptotic formula (\[eq:ernstinfinity\]). It turns out that an expansion in $64$ to $128$ polynomials for each coordinate is sufficient to provide a numerically optimal approximation within the used precision. This illustrates the convenience of the disk coordinates in this context. The virial identity is then satisfied to the order of $10^{-12}$. We plot the deviation of the sum of the integrals in (\[virial20\]) from zero for several values of $\lambda$ and $\gamma$ in Fig. \[fig:virialtest\].
The drop in accuracy for strongly relativistic almost static situations ($\gamma$ small and $\epsilon$ close to 1) is again due to the almost degenerate Riemann surface. The lower accuracy in the case of strongly relativistic situations for $\gamma=1$ reflects the fact that the disk is shrinking to a point in this limit. To maintain the needed resolution one would have to use more polynomials in the evaluation of the virial-type identity which was not possible on the used computers.
Testing `LORENE` {#sec:lorene}
================
One purpose of exact solutions of the Einstein equations is to provide test-beds for numerical codes to check the quality of the numerical approximation. In the previous sections we have established that the theta-functional solutions can be numerically evaluated to the order of machine precision which implies they can be used in this respect.
The code we are considering here is a C++-library called `LORENE` [@Lorene] which was constructed to treat problems from relativistic astrophysics such as rapidly rotating neutron stars. The main idea is to solve Poisson-type equations iteratively via spectral methods. To this end an equation as the Ernst equation (\[ernst1\]) is written in the form $$\Delta \mathcal{F} = \mathcal{G}(\mathcal{F},r,\theta,\phi)
\label{eq:poisson},$$ where spherical coordinates $r$, $\theta$, $\phi$ are used, and where $\mathcal{G}$ is some possibly non-linear functional of $\mathcal{F}$ and the coordinates. The system (\[eq:poisson\]) is to be solved for $\mathcal{F}$ which can be a vector. In an iterative approach, the equation is rewritten as $$\Delta \mathcal{F}_{n+1} =
\mathcal{G}(\mathcal{F}_{n},r,\theta,\phi),\quad n=1,2,\ldots
\label{eq:poisson2}.$$ Starting from some initial function $\mathcal{F}_{0}$, in each step of the iteration a Poisson equation is solved for a known right-hand side. For the stationary axisymmetric Einstein equations which we are considering here, it was shown in [@schaudt] that this iteration will converge exponentially for small enough boundary data if the initial values are close to the solution of the equation in some Banach space norm. It turns out that one can always start the iteration with Minkowski data, but it is necessary to use a relaxation: instead of the solution $\mathcal{F}_{n+1}$ of (\[eq:poisson2\]), it is better to take a combination $\tilde{\mathcal{F}}_{n+1}=\mathcal{F}_{n+1}+\kappa \mathcal{F}_{n}$ with $\kappa\in ]0,1[$ (typically $\kappa=0.5$) as a new value in the source $\mathcal{G}_{n+1}$ to provide numerical stability. The iteration is in general stopped if $||\mathcal{F}_{n+1}
-\mathcal{F}_{n}||<10^{-10}$.
The Ernst equation (\[ernst1\]) is already in the form (\[eq:poisson\]), but it has the disadvantage that the equation is no longer strongly elliptic at the ergo-sphere where $\Re(\mathcal{E})=0$. In physical terms, this apparent singularity is just a coordinate singularity, and the theta-functional solutions are analytic there. The Ernst equation in the form (\[eq:poisson\]) has a right-hand side of the form ‘$0/0$’ for $\Re \mathcal{E}=0$ which causes numerical problems especially in the iteration process since the zeros of the numerator and the denominator will only coincide for the exact solution. The disk solutions we are studying here have ergo-spheres in the shape of cusped toroids (see [@prd4]). Therefore it is difficult to take care of the limit $0/0$ by using adapted coordinates. Consequently the use of the Ernst picture is restricted to weakly relativistic situations without ergo-spheres in this framework.
To be able to treat strongly relativistic situations, we use a different form of the stationary axisymmetric vacuum Einstein equations which is derived from the standard $3+1$-decomposition, see [@eric1]. We introduce the functions $\nu$ and $N_{\phi}$ via $$e^{2\nu}=\frac{\rho^{2}f}{\rho^{2}-a^{2}f^{2}},\quad
N_{\phi}=\frac{\rho af^{2}}{\rho^{2}-a^{2}f^{2}},
\label{eq:nuN}$$ where $ae^{2U}$ is the $g_{t\phi}$ component of the metric leading to the Ernst potential, see (\[eq:wlp\]) in the appendix. Expressions for $a$ in terms of theta-functions are given in [@prd4]. The vacuum Einstein equations for the functions (\[eq:nuN\]) read $$\begin{aligned}
\Delta \nu & = & \frac{1}{2}\rho^{2}e^{-4\nu}(N_{\phi,\rho}^{2}+
N_{\phi,\zeta}^{2})
\label{eq:nu}, \\
\Delta N_{\phi} -\frac{1}{\rho^{2}}N_{\phi}& = &
4\rho(N_{\phi,\rho}(e^{2\nu})_{\rho}+
N_{\phi,\zeta}(e^{2\nu})_{\zeta}).
\label{eq:Npfi}\end{aligned}$$ By putting $V=N_{\phi}\cos\phi$ we obtain the flat 3-dimensional Laplacian acting on $V$ on the left-hand side, $$\Delta V = 4\rho(V_{\rho}(e^{2\nu})_{\rho}+
V_{\zeta}(e^{2\nu})_{\zeta}).
\label{eq:V}$$ Since the function $e^{2\nu}$ can only vanish at a horizon, it is globally non-zero in the examples we are considering here. Thus the system of equations (\[eq:nu\]) and (\[eq:V\]) is strongly elliptic, even at an ergo-sphere.
The disadvantage of this regular system is the non-linear dependence of the potentials $\nu$ and $N_{\phi}$ on the Ernst potential and $a$ via (\[eq:nuN\]). Thus we loose accuracy due to rounding errors of roughly an order of magnitude. Though we have shown in the previous sections that we can guarantee the numerical accuracy of the data for $f$ and $af$ to the order of $10^{-14}$, the values for $\nu$ and $V$ are only reliable to the order of $10^{-13}$.
To test the spectral methods implemented in `LORENE`, we provide boundary data for the disk solutions discussed above on a sphere around the disk. For these solutions it would have been more appropriate to prescribe data at the disk, but `LORENE` was developed to treat objects of spherical topology such as stars which suggests the use of spherical coordinates. It would be possible to include coordinates like the disk coordinates of the previous section in `LORENE`, but this is beyond the scope of this article. Instead we want to use the Poisson-Dirichlet routine which solves a Dirichlet boundary value problem for the Poisson equation for data prescribed at a sphere. We prescribe the data for $\nu$ and $N_{\phi}$ on a sphere of radius $R$ and solve the system (\[eq:nu\]) and (\[eq:V\]) iteratively in the exterior of the sphere. If the iteration converges, we compare the numerical solution in the exterior of the sphere with the exact solution.
Since spherical coordinates are not adapted to the disk geometry, a huge number of spherical harmonics would be necessary to approximate the potentials if $R$ is close to the disk radius. The limited memory on the used computers imposes an upper limit of 64 to 128 harmonics. We choose the radius $R$ and the number of harmonics in a way that the Fourier coefficients in $\theta$ drop below $10^{-14}$ to make sure that the provided boundary data contain the related information to the order of machine precision. The exterior of the sphere where the boundary data are prescribed is divided in two domains, one from $R$ to $2R$ and one from $2R$ to infinity. In the second domain $1/r$ is used as a coordinate. For the $\phi$ dependence which is needed only for the operator in (\[eq:V\]), 4 harmonics in $\phi$ are sufficient.
Since `LORENE` is adapted to the solution of the Poisson equation, it is to be expected that it reproduces the exact solution best for nearly static situations, since the static solutions solve the Laplace equation. The most significant deviations from the exact solution are therefore expected for $\delta=0$. For the case $\lambda=3$, we consider 32 harmonics in $\theta$ on a sphere of radius $R=1.5$. The iteration is stopped if $||\mathcal{F}_{n+1}-\mathcal{F}_{n}<5*10^{-10}$ which is the case in this example after 90 steps. The exact solution is reproduced to the order of $10^{-11}$. The absolute value of the difference between the exact and the numerical solution on a sphere of radius 3 is plotted in Fig. \[fig:maxdifftheta\] in dependence of $\theta$. There is no significant dependence of the error on $\theta$.
The maximal deviation is typically found on or near the axis. As can be seen from Fig. \[fig:maxdiffr\] which gives the dependence on $r$ on the axis, the error decreases almost linearly with $1/r$ except for some small oscillations near infinity.
We have plotted the maximal difference between the numerical and the exact solution for a range of the physical parameters $\lambda$ and $\delta$ in Fig. \[fig:gamma\]. As can be seen, the expectation is met that the deviation from the exact solution increases if the solution becomes more relativistic (larger $\epsilon$). As already mentioned, the solution can be considered as exactly reproduced if the deviation is below $10^{-13}$. Increasing the value of $\gamma$ for fixed $\epsilon$ leads to less significant effects though the solutions become less static with increasing $\gamma$.
,
For $\delta=0$, the ultra-relativistic limit $\lambda\to 4.629\ldots$ corresponds to a space-time with a singular axis which is not asymptotically flat, see [@prd4]. Since `LORENE` expands all functions in a Galerkin basis with regular axis in an asymptotically flat setting, solutions close to this singular limit cannot be approximated. Convergence gets much slower and can only be achieved with considerable relaxation. For $\lambda=4$ and $\delta=0$ we needed nearly 2000 iterations with a relaxation parameter of $\kappa=0.9$. The approximation is rather crude (in the order of one percent). For higher values of $\lambda$ no convergence could be obtained.
This is however due to the singular behavior of the solution in the ultra-relativistic limit. In all other cases, `LORENE` is able to reproduce the solution to the order of $10^{-11}$ and better, more static and less relativistic cases are reproduced with the provided accuracy.
Conclusion {#sec:concl}
==========
In this article we have presented a scheme based on spectral methods to treat hyperelliptic theta-functions numerically. It was shown that an accuracy of the order of machine precision could be obtained with an efficient code. As shown, spectral methods are very convenient if analytic functions are approximated. Close to singularities such as the degeneration of the Riemann surface, analytic techniques must be used to end up with analytic integrands in the discussed example.
The obtained numerical data were used to provide boundary values for the code `LORENE` which made possible a comparison of the numerical solution to the boundary value problem with the numerically evaluated theta-functions. For a large range of the physical parameters the numerical solution was of the same quality as the provided data. The main errors in `LORENE` are introduced by rounding errors in the iteration. This shows that spectral methods provide a reliable and efficient numerical treatment both for elliptic equations and for hyperelliptic Riemann surfaces. However, to maintain the global quality of the numerical approximation an analytical understanding of the solutions is necessary in order to treat the non-analyticities of the solutions.
Einstein equations and integral identities
==========================================
The Ernst equation has a geometric interpretation in terms of the stationary axisymmetric Einstein equations in vacuum. The metric can be written in this case in the Weyl-Lewis-Papapetrou form (see [@exac]) $$ds^{2}=g_{ab}dx^{a}dx^{b}=-f(dt+ad\phi)^{2}+(e^{2k}(d\rho^{2}+d\zeta^{2})
+\rho^{2}d\phi^{2})/f
\label{eq:wlp},$$ where $\rho$ and $\zeta$ are Weyl’s canonical coordinates and $\partial_{t}$ and $\partial_{\phi}$ are the commuting asymptotically timelike respectively spacelike Killing vectors.
In this case the vacuum field equations are equivalent to the Ernst equation (\[ernst1\]) for the complex potential $\mathcal{E}$. For a given Ernst potential, the metric (\[eq:wlp\]) can be constructed as follows: the metric function $f$ is equal to the real part of the Ernst potential. The functions $a$ and $k$ can be obtained via a line integration from the equations $$a_{\xi}=2\rho\frac{(\mathcal{E}-\bar{\mathcal{E}})_{\xi}}{
(\mathcal{E}+\bar{\mathcal{E}})^{2}}
\label{axi},$$ and $$k_{\xi}=(\xi-\bar{\xi})
\frac{\mathcal{E}_{\xi}\bar{\mathcal{E}}_{\xi}}{
(\mathcal{E}+\bar{\mathcal{E}})^{2}}\;.
\label{kxi}$$ This implies that $a$ is the dual of the imaginary part of the Ernst potential. The equation (\[kxi\]) for $k$ follows from the equations $$R_{\alpha\beta}=\frac{1}{2f^{2}}\Re(\mathcal{E}_{\alpha}
\bar{\mathcal{E}}_{\beta}),\quad \alpha,\beta=1,2,3
\label{eq:ricci},$$ where $R$ is the (three-dimensional) Ricci tensor corresponding to the spatial metric $\mathrm{h}=\mbox{diag}(e^{2k},e^{2k},\rho^{2})$. This reflects a general structure of the vacuum Einstein equations in the presence of a Killing vector. For the Ricci scalar one finds $$-\frac{1}{2}e^{2k}R = k_{\rho\rho}+k_{\zeta\zeta}
\label{virial17}.$$ We denote by $h$ the determinant of the metric $\mathrm{h}$.
The Komar integral [@komar; @wald] of the twist of the timelike Killing vector $\xi=\partial_{t}$ over the whole spacetime establishes the equivalence between the asymptotically defined ADM mass and the Komar mass $m_{K}$, $$2
\int_{disk}^{}dV\left(T_{ab}-\frac{1}{2}g_{ab}T^{c}_{c}\right)n^{a}
\xi^{b}
\label{virial2}=: m_{K},$$ where the integration is carried out over the disk, where $n_{a}$ is the normal at the disk, and where $T_{ab}$ is the energy momentum tensor of the disk given in [@prd4]. In other words the ADM mass can be calculated either asymptotically or locally at the disk.
To obtain an identity which does not involve only surface integrals, we consider as in [@virial] an integral over the trace of equation (\[eq:ricci\]) for the Ricci-tensor, $$R=\frac{h^{\alpha\beta}\mathcal{E}_{\alpha}
\bar{\mathcal{E}}_{\beta}}{2f^{2}}
\label{virial4}.$$ To avoid numerical problems at the set of zeros of $f$, the so-called ergo-sphere (see [@prd4] for the disk solutions studied here), we multiply both sides of equation (\[eq:ricci\]) by $f^{3}$. Integrating the resulting relation over the whole space-time, we find after partial integration $$-\int_{0}^{1}d\rho \rho f^{3}k_{\zeta}+
\int_{0}^{\infty}d\rho\int_{-\infty}^{\infty}d\zeta ((\rho
f^{3})_{\rho}k_{\rho}+(\rho f^{3})_{\zeta}k_{\zeta})=
\int_{0}^{\infty}d\rho\int_{-\infty}^{\infty}d\zeta \rho f(
\mathcal{E}_{\rho}\bar{\mathcal{E}}_{\rho}+
\mathcal{E}_{\zeta}\bar{\mathcal{E}}_{\zeta})
\label{eq:virial5};$$ here the only contributions of a surface integral arise at the disk, since $k\propto 1/r^{2}$ for $r\to\infty$ and since the axis is regular ($k$ vanishes on the axis). If we replace $k$ via (\[kxi\]), we end up with an identity for the Ernst potential and its derivatives, $$\begin{aligned}
&& -\int_{0}^{1}d\rho \rho^{2}f (\mathcal{E}_{\rho}
\bar{\mathcal{E}}_{\zeta}
+\mathcal{E}_{\zeta}\bar{\mathcal{E}}_{\rho})
+\frac{3}{2}\int_{0}^{\infty}\int_{0}^{\infty}d\rho d\zeta
\rho^{2}(\mathcal{E}_{\rho}(\bar{\mathcal{E}}_{\rho}^{2}
+\bar{\mathcal{E}}_{\zeta}^{2})+\bar{\mathcal{E}}_{\rho}
(\mathcal{E}_{\rho}^{2}+\mathcal{E}_{\zeta}^{2})) \nonumber\\ && =
2\int_{0}^{\infty}\int_{0}^{\infty}d\rho d\zeta \rho f
\mathcal{E}_{\zeta} \bar{\mathcal{E}}_{\zeta}
\label{virial20}.\end{aligned}$$ This identity (as the identity given in [@virial]) can be seen as a generalization of the Newtonian virial theorem. The relation (\[virial20\]) coincides with the corresponding relation of [@virial] only in the Newtonian limit. This reflects the fact that generalizations of a Newtonian result to a general relativistic setting are not unique. Our formulation is adapted to the Ernst picture and avoids problems at the ergo-spheres, thus it seems optimal to test the numerics for Ernst potentials in terms of theta-functions.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank A. Bobenko, D. Korotkin, E. Gourgoulhon and J. Novak for helpful discussions and hints. CK is grateful for financial support by the Marie-Curie program of the European Union and the Schloessmann foundation.
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www.lorene.obspm.fr
www-sfb288.math.tu-berlin.de/ jtem/
| ArXiv |
---
abstract: 'FLAMINGOS-2 (PI: S. Eikenberry) is a \$5M facility-class near-infrared (1-2.5 ) multi-object spectrometer and wide-field imager being built at the University of Florida for Gemini South. Here we highlight the capabilities of FLAMINGOS-2, as it will be an ideal instrument for surveying the accreting binary population in the Galactic Center.'
author:
- 'S. Nicholas Raines, Stephen S. Eikenberry, and Reba M. Bandyopadhyay'
title: 'FLAMINGOS-2: A Near-IR Multi-Object Spectrometer Ideal for Surveying the Galactic Center'
---
Introduction
============
[[*Chandra *]{}]{}observations have shown that there is a much larger population of accreting binaries in our Galaxy than was previously recognized, with a particular concentration in the Galactic Center (GC; Wang [[*et al.*]{}]{} 2002; Muno [[*et al.*]{}]{} 2003). IR spectroscopy is the only way to definitively identify the true stellar counterparts to these X-ray sources ([[*e.g. *]{}]{}Bandyopadhyay [[*et al.*]{}]{} 1999). Due to the extreme field crowding, to successfully find the true counterparts to the XRBs in the GC we need to obtain spectra of $\sim$1000-1500 IR stars - a nearly infeasible task to perform via traditional longslit single-object spectroscopy.
FLAMINGOS-2 (PI: S. Eikenberry; Eikenberry [[*et al.*]{}]{} 2006) is a facility-class near-IR (1-2.5 ) multi-object spectrometer (MOS) and wide-field imager being built at the University of Florida for Gemini South. Here we highlight the capabilities of FLAMINGOS-2, as it will be an ideal instrument for surveying the accreting binary population in the GC. Utilizing custom aperture masks in a 2$\times$6 arcminute$^{2}$ field-of-view (FOV), simultaneous multi-object spectroscopy of up to $\sim$90 targets will be possible at resolving power $\sim$1300 in the $H+K$ band. With this resolution and FOV, combined with the Gemini 8-m aperture, our team at UF will be able to efficiently perform the first spectroscopic survey of this GC population with high S/N to a limiting magnitude of $K\sim$17 during our guaranteed time (*cf.* Eikenberry [[*et al.*]{}]{}, Bandyopadhyay [[*et al.*]{}]{}, these proceedings).
FLAMINGOS-2 Optical Path & Components
=====================================
FLAMINGOS-2 has a fully cryogenic optical train, illustrated in Fig.1, with 9 spherical refractive elements and two front-surface gold flat mirrors. The window and 7 of the lenses are single-crystalline CaF$_2$; the other two lenses are made from Ohara SFTM-16. Progressing from the top to the bottom of the figure, light first passes through the window to a focus. The MOS wheel lies at the telescope focus and contains an imaging aperture and, most importantly, a selection of custom aperture masks (“mosplates”); it also caries 6 long slits. It is immediately followed by a selectable baffle, called the Decker wheel, and the field lens. The optical path then is folded by the flat mirrors to the other two elements of the collimator optics. At this point the beam is collimated. Two wheels carrying a selection of filters for imaging and spectroscopy are spaced on either side of the Lyot wheel. FLAMINGOS-2 was designed for operation with the telescope’s *f/16* beam but it also can accept the $\sim$*f/30* beam from the Gemini Multi-Conjugate Adaptive Optics (MCAO) system, and the Lyot wheel carries pupil stops for both modes of operation. The final mechanism is a wheel carrying a selection of grisms; it also includes a clear aperture for the imaging mode of operation. Light is then re-imaged onto the Hawaii-II array by a 6-element camera lens assembly.
Standard near-IR $J$, $H$, and $Ks$ filters are installed in one of the filter wheels; Gemini may additionally offer a $Y$-band filter (0.97-1.07 ). Two specialty spectroscopy filters are installed in the other filter wheel, one covering the $J+H$ bandpasses, the other covering $H+K$. Three grisms reside in the grism wheel, two with moderate resolving power, $R (= \lambda/\delta\lambda) \sim1300$, and one grism with high resolving power, $R \sim 3300$. The $R \sim
1300$ grisms are used in conjunction with the $J+H$ or $H+K$ bandpass filters, while the $R \sim 3300$ grism is used with the $J$, $H$, or $Ks$ standard near-IR filters for out-of-bandpass blocking.
FLAMINGOS-2 MOS Mode
=====================
Several features of FLAMINGOS-2 make it ideal for surveying the Galactic Center: *(a)* a wide imaging FOV of $\sim$6.2 arcminutes$^{2}$ ($\sim$3.1 arcminutes$^{2}$ with MCAO), *(b)* the ability to carry up to 9 custom mosplates at a time, *(c)* the mosplates’ large spectroscopic FOV of 2$\times$6.2 square arcminutes (1$\times$3.1 square arcminutes with MCAO), *(d)* the cooling of the masks to cryogenic temperatures which allow low internal instrument background for operation in the K-band, and *(e)* the ability to quickly exchange the set of mosplates.
The MOS wheel, shown in Fig. 2a, is 0.9 meters in diameter. It has three circular apertures positioned around the periphery of the wheel; one is left open for imaging and the other two usually contain pinhole masks for engineering. Equally spaced between the circular apertures are 9 rectangular slots for holding custom mosplates. A test mosplate is shown in Fig. 2b. The mosplate FOV (*f/16-mode*) has sufficient sky-coverage to design custom mosplates containing up to $\sim$90 slitlets.
Mosplates can be changed during the daytime without thermally cycling the entire instrument. Also shown in Fig. 1 is a *gate valve*, positioned between the field lens and the folding flat mirrors. At the end of a night of observing this valve is closed, the MOS dewar cooling is halted, and a warm-up heater is turned on. Several hours later, during the daytime, an engineer can open the access port on the side of the MOS dewar. Each mosplate is held in a frame which slides into the edge of the MOS wheel. The observed plates are removed, and a new set of mosplates, each one already mounted in a frame, are slid into place. The engineer then closes up the access port and begins the process of evacuating and cooling the dewar. Once it is cold enough the gate valve is then re-opened. By design, this should be completed in time for the observers who return that evening.
If each plate is observed for only 1 hour, $\sim$800 spectra could be obtained with a single night’s observation using all 9 mosplates. With only three nights of observing potentially up to $\sim$2400 spectra could be obtained. Thus FLAMINGOS-2’s MOS mode of operation is ideally suited for identifying the true stellar counterparts to the X-ray sources in the GC.
Bandyopadhyay, R.M., [[*et al.*]{}]{}, 1999, MNRAS 306, 417
Eikenberry, S.S., [[*et al.*]{}]{}, 2006, Proc. SPIE 6269, 39
Muno, M.P., [[*et al.*]{}]{}, 2003, ApJ 589, 225
Wang, Q.D., Gotthelf, E.V., & Lang, C.C., 2002, Nature 414, 148
| ArXiv |
---
author:
- 'Han Dong [^1], Ying-bin Wang [^2]'
- 'Xin-he Meng [^3]'
date: 'Received: date / Revised version: date'
title: 'Extended Birkhoff’s Theorem in the $f(T)$ Gravity'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction
============
Since the discovery of the accelerating expansion of the universe, people have made great efforts to investigate the hidden mechanism, which also provides us with great opportunities to deeply probe the fundamental theories of gravity. As one of modified gravitational theories, the $f(T)$ gravity is firstly invoked to drive inflation by Ferraro and Fiorini [@fT0]. Later, Bengochea and Ferraro [@fT], as well as Linder [@fT1], propose to use the $f(T)$ theory to drive the current accelerated expansion of our universe without invoking the mysterious dark energy. The framework is a generalization of the so-called *Teleparallel Equivalent of General Relativity* (TEGR) which is firstly propounded by Einstein in 1928 [@einstein] and maturates in the 1960s (For some reviews, see [@TEGR1; @TEGR2]). Contrary to the theory of general relativity which is based on Riemann geometry involving only curvature, the TEGR is based on the so named Weitzenböck geometry with the non-vanishing torsion. Owing to the definition of Weitzenböck connection rather than the Levi-Civita connection, the Riemann curvature is automatically vanishing in the TEGR framework, which brings the theory a new name, *Teleparallel Gravity*. For a specific choice of parameters, the TEGR behaves completely equivalent to the Einstein’s theory of general relativity. Furthermore, by using the torsion scalar $T$ as the Lagrangian density, the TEGR can give a field equation with second order only, instead of the fourth order as in the Einstein’s field equation, and avoids the instability problems caused by higher order derivatives as demonstrated in the metric framework $f(R)$ gravity models.
Similar to the generalization of Einstein’s theory of general relativity to the $f(R)$ theory (For some references, see [@fR0; @fRrev; @fR1; @fR11; @fRa0; @fRa1; @fRa2; @fRa3; @fRa4; @fRa5; @fRa6; @fRa7; @fRa8; @fRa9; @fR2; @fR3; @fR4]), the modified version of teleparallel gravity assumes a general function $f(T)$ as the model Lagrangian density. Also, the $f(T)$ theory can be directly reduced to the TEGR if we choose the simplest case, that is, $f(T)=T$. The Lorentz invariance and conformal invariance of the $f(T)$ theory is also investigated [@fT_Lorentz; @fT_conformal], with many interesting results presented. A class of $f(T)$ models with diagonal tetrad are proposed in succession to explain the late-time acceleration of the cosmic expansion without the mysteriously so-called dark energy, and are fitted the cosmological data-sets very well (e.g. [@fT; @fT1; @fT_w; @fT2; @fT3; @fT4; @fT5; @fT6; @fT7]). Most of the previous works consider the $f(T)$ gravity with diagonal tetrad field only. Noting that the tetrad field has sixteen components rather than ten as in the metric frame, there are more freedoms and more physical meaning from the extra uncertain six components. In our previous work [@fT_birkhoff], we have proved the validity of Birkhoff’s theorem in $f(T)$ gravity with a specific diagonal tetrad. In this letter, we study this issue more generally with also the off diagonal tetrad field, and discuss the physical meaning in a more extended context.
The Birkhoff’s theorem is also called Jebsen-Birkhoff theorem, for it was actually discovered by Jebsen two years before George D. Birkhoff in 1923 [@birkhoff; @bb]. The theorem states that the spherically symmetric gravitational field in vacuum must be static, with a metric uniquely given by the Schwarzschild solution form of Einstein equations [@weinberg]. It is well known that the Schwarzschild metric is found in 1918 as the external (vacuum) solution of a static and spherical symmetric star. The Birkhoff’s theorem means that any spherically symmetric object possesses the same static gravitational field, as if the mass of the object were concentrated at the center. Even if the central spherical symmetric object is dynamic motion, such as the case in the collapse and pulsation of stars, the external gravitational field is still static if only the radial motion is spherically symmetric. The same feature is held in the classical Newtonian gravity.
In this work we investigate the Birkhoff’s theorem in the $f(T)$ gravity model generally with both the diagonal and the off diagonal tetrad fields, analyze the extended meaning of this theorem, and study the equivalence between both Einstein frame and Jordan frame. First, in section two we briefly review the $f(T)$ theories, and in section three we prove the validity of Birkhoff’s theorem of the $f(T)$ gravity with both off diagonal tetrad and diagonal tetrad fields. In section four, we then discuss the validity of the Birkhoff’s theorem in the frame of $f(T)$ gravity via conformal transformation by regarding the Brans-Dicke-like scalar as effective matter. Both the Jordan and Einstein frames are discussed in this section. And some new conclusions and discussions are provided in the last section.
Elements of $f(T)$ Gravity
==========================
Instead of the metric tensor, the vierbein field $\mathbf{e}_{i}(x^{\mu})$ plays the role of the dynamical variable in the teleparallel gravity. It is defined as the orthonormal basis of the tangent space at each point $x^{\mu}$ in the manifold, namely, $\mathbf{e}_{i}\cdot \mathbf{e}_{j}=\eta_{ij}$, where $\eta_{ij}=diag(1,-1,-1,-1)$ is the Minkowski metric. The vierbein vector can be expanded in spacetime coordinate basis: $\mathbf{e}_{i}=e^{\mu}_{i} \partial_{\mu}$, $\mathbf{e}^i=e^i_\mu{\rm d}x^\mu$. According to the convention, Latin indices and Greek indices, both running from 0 to 3, label the tangent space coordinates and the spacetime coordinates respectively. The components of vierbein are related by $e_{\mu}^i
e^{\mu}_j=\delta^{~i}_{j}$, $e_{\mu}^i
e^{\nu}_i=\delta_{\mu}^{~\nu}$.
The metric tensor is determined uniquely by the vierbein as $$g_{\mu\nu}=\eta_{ij} e_{\mu}^i e_{\nu}^i,$$ which can be equivalently expressed as $\eta_{ij}=g_{\mu\nu}
e^i_{\mu} e^j_{\nu}$. The definition of torsion tensor is given then by $$T^{\rho}_{~\mu\nu}=\Gamma^{\rho}_{~\nu\mu}-\Gamma^{\rho}_{~\mu\nu},$$ where $\Gamma^{\rho}_{~\mu\nu}$ is the connection. Evidently, $T^{\rho}_{~\mu\nu}$ vanishes in the Riemann geometry since the Levi-Civita connection is symmetric with respect to the two covariant indices. Differing from that in Einstein’s theory of general relativity, the teleparallel gravity uses Weitzenböck connection defined directly from the vierbein: $$\Gamma^{\rho}_{~\mu\nu}=e_i^{\rho} \partial_{\nu} e^i_{\mu}.$$ Accordingly, the antisymmetric non-vanishing torsion is $$\label{torsion}
T^{\rho}_{~\mu\nu}=e_i^{\rho}(\partial_{\mu}e^i_{\nu} - \partial_{\nu}e^i_{\mu}).$$ It can be confirmed that the Riemann curvature in this framework is precisely vanishing: $$R^\rho_{~\theta\mu\nu}=\partial_\mu \Gamma^\rho_{~\theta\nu}-\partial_\nu \Gamma^\rho_{~\theta\mu}+\Gamma^\rho_
{~\sigma\mu}\Gamma^\sigma_{~\theta\nu}-\Gamma^\rho_{~\sigma\nu} \Gamma^\sigma_{\theta\mu}=0.$$
In order to get the action of the teleparallel gravity, it is convenient to define other two tensors: $$\label{contorsion}
K^{\mu\nu}_{~~\rho}=-\frac{1}{2}(T^{\mu\nu}_{~~\rho}-T^{\nu\mu}_{~~\rho}-T_{\rho}^{~\mu\nu}),$$ and $$\label{S}
S_\rho^{~\mu\nu}=\frac{1}{2}(K^{\mu\nu}_{~~\rho}+\delta_\rho^{~\mu}T^{\theta\nu}_{~~\theta}-\delta_\rho^{~\nu}T^
{\theta\mu}_{~~\theta}).$$ Then the torsion scalar as the teleparallel Lagrangian density is defined by $$\label{T}
T=S_{\rho}^{~\mu\nu} T^{\rho}_{~\mu\nu}.$$ The action of teleparallel gravity is then expressed as $$I=\frac{1}{16\pi G}\int {\rm d}^4 x~e\,T ,$$ where $e=$det$(e^i_{\mu})=\sqrt{-g}$. Performing variation of the action with respect to the vierbein, one can get the equations of motion which are equivalent to the results of Einstein’s theory of general relativity.
Just as in the $f(R)$ theory, the generalized version of teleparallel gravity could be obtained by extending the Lagrangian density directly to a general function of the scalar torsion $T$ : $$\label{action}
I=\frac{1}{16\pi G}\int {\rm d}^4x~e\,f(T).$$ This modification is expected possibly to provide a natural way to understand the cosmological observations, especially for the dark energy phenomena, as a motivation. Then the variation of the action with respect to vierbein leads to the following equations: $$\label{field eqn}
\begin{split}
{\big[}e^{-1}e^i_\mu\partial_\sigma(eS_i^{~\sigma\nu})-T^\rho_{~\sigma\mu}S_\rho^{~\nu\sigma}{\big]}f_T+
S_\mu^{~\rho\nu}\partial_\rho Tf_{TT}\\
-\frac{1}{4}\delta_\mu^{~\nu}f=4\pi GT_\mu^{~\nu} ,
\end{split}$$ where $f_T$ and $f_{TT}$ represent the first and second order derivatives with respect to $T$ respectively, and $S_i^{~\sigma\nu}=e_i^\rho S_\rho^{~\sigma\nu}$. $T_\mu^{~\nu}$ is the energy-momentum tensor of the particular matter, with assuming that matter couples to the metric in the standard form.
Extended Birkhoff’s theorem in $f(T)$ Gravity with both the off diagonal tetrad and the diagonal tetrad {#sec3}
=======================================================================================================
In our previous work [@fT_birkhoff] we have proved that the Birkhoff’s theorem is valid in $f(T)$ gravity with diagonal tetrad. But the $f(T)$ gravity with diagonal tetrad will give a strong constraint for a constant torsion scalar, which is shown in Ref. [@fT_relativistic; @stars], while another off diagonal tetrad field is able to construct interesting exact solutions to these field equations. The off diagonal tetrad field can provide bigger room to modify the gravity for its six more freedoms. Therefore, we think that the research of the Birkhoff’s theorem with off diagonal tetrad field is necessary and realistically meaningful.
We consider the external vacuum gravitational field solution of a spherically symmetric object. The spherically symmetric metric generally can always be written in the following form: $$\label{metric}
{\rm d} s^2={\rm e}^{a(t,r)}{\mathrm{d}}t^2-{\rm e}^{b(t,r)}{\mathrm{d}}r^2-r^2{\mathrm{d}}\theta^2-r^2 \sin^2\!\theta~{\mathrm{d}}\phi^2,$$ where $a(t,r)$, $b(t,r)$ are arbitrary functions of the coordinates $t$ and $r$. One possible corresponding off diagonal tetrad field can be written as $$e^i_\mu =
\left( \begin{array}{cccc}
e^{\frac{a(t,r)}{2}} & 0 & 0 & 0 \\
0 & {\rm e}^{\frac{b(t,r)}{2}}\!\sin\!\theta \cos\!\phi\, & r\!\cos\!\theta \cos\!\phi\, & -r\!\sin\!\theta \sin\!\phi \\
0 & {\rm e}^{\frac{b(t,r)}{2}}\!\sin\!\theta \sin\!\phi\, & r\!\cos\!\theta \sin\!\phi\, & -r\!\sin\!\theta \cos\!\phi \\
0 & {\rm e}^{\frac{b(t,r)}{2}}\!\cos\!\theta\, & -r\!\sin\!\theta\, & 0 \\
\end{array} \right)$$ The determinant of vierbein is $e={\rm e}^{\frac{a(t,r)+b(t,r)}{2}}r^2\sin\!\theta$. Then the tensors defined in Eqs. (\[torsion\],\[contorsion\],\[S\]) are determined, and the torsion scalar is given by $$\label{T2}
T=\frac{2{\big(}{\rm e}^{\frac{b(t,r)}{2}}-1{\big)}{\big(}{\rm e}^{\frac{b(t,r)}{2}}-a^\prime\!(t,r)\,r-1{\big)}}{{\rm e}^{b(t,r)}r^2},$$ where a prime denotes the derivative with respect to $r$, while a dot overhead denotes the derivative with respect to $t$. We will follow these conventions throughout this work.
For convenience, we introduce the tensor $E_\mu^{~\nu}$ to represent of the left hand side of Eq. (\[field eqn\]), and the field equation can be then re-expressed concisely as $$E_\mu^{~\nu}=4\pi GT_\mu^{~\nu}.$$ Then, we work out all the components of $E_\mu^{~\nu}$, and find most of them are not vanishing, including some quite complicated ones. But the two components we used, fortunately not very complex, are given by respectively $$\begin{aligned}
E_2^{~0}&=&\frac{1}{4}\cot\!\theta \, e^{-a(t,r)} \, \dot b(t,r) f_T,\label{E20}\\
E_2^{~1}&=&\frac{1}{4 r} \cot\!\theta \, e^{-b(t,r)} \, {\big(}2-2{\rm e}^{\frac{b(t,r)}{2}}+a^\prime(t,r) r{\big)}f_T.\label{E21}\end{aligned}$$ Since the non-diagonal elements of energy-momentum tensor for spherically symmetric gravitational source are naturally equal to zero, $E_2^{~0}$ always vanishes. And $f_T$ should not be trivially zero for the real universe observation, which is restricting $b(t,r)$ to be only the function of $r$, that is, $$\label{B}
b(t,r)=b(r).$$ For the same reason as to $E_2^{~0}$, that $E_2^{~1}$ is also equal to zero. After some manipulations, Eq. (\[E21\]) leads to $$\label{constrain}
1+\frac{a(t,r)^\prime r}{2}={\rm e}^{\frac{b(r)}{2}}.$$ For $b(r)$ is independent of $t$, the left of Eq. (\[constrain\]) should be also a function of $r$. After performing an integration with respect to $r$, the function $a(t,r)$ could be simply expressed as $$\label{A}
a(t,r)=\widetilde a(r)+c(t),$$ where $c(t)$ is an arbitrary function of $t$, and the $\widetilde a(r)$ is an integral function of the variable $r$. Therefore the function ${\rm e}^{a(t,r)}$ can be written as $${\rm e}^{a(t,r)}={\rm e}^{\widetilde a(t)}{\rm e}^{c(t)}.$$ The factor ${\rm e}^{c(t)}$ can always be absorbed in the metric through a coordinate transformation $t\to t^\prime$, where $t^\prime$ is the new time coordinate defined as: $${\rm d}t^\prime={\rm e}^{\frac{c(t)}{2}} {\rm d}t.$$ Therefore the metric presented in Eq. (\[metric\]) becomes $$\label{metric1}
{\rm d} s^2={\rm e}^{\widetilde a\!(r)}~{\mathrm{d}}t^2-{\rm e}^{b\!(r)}~{\mathrm{d}}r^2-r^2{\mathrm{d}}\theta^2-r^2 \sin^2\!\theta~{\mathrm{d}}\phi^2,$$ This is exactly a static metric, which is required by the Birkhoff’s theorem.
As a demonstration, we also perform the computation with the corresponding diagonal tetrad field, which can be written as $$e^i_\mu={\rm diag}{\big(}{\rm e}^{\frac{a(t,r)}{2}},~{\rm e}^{\frac{b(t,r)}{2}},~r,~r\sin\!\theta{\big)},$$ and the determinant of vierbein is $e={\rm e}^{\frac{a(t,r)+b(t,r)}{2}}r^2\sin\theta$. It is easy to find some of the non-vanishing components of $E_\mu^{~\nu}$ . The two components we used are given by respectively $$E_1^{~0} = -\frac{{\rm e}^{-a(t,r)}}{2 r} \dot b(t,r) f_T,\label{e10}$$ and $$\begin{split}
E_0^{~2} = \frac{2{\rm e}^{- b(t,r)} \cos\theta}{r^4 \sin\theta} \left(\dot{a}^\prime(t,r) r-\dot{b}(t,r)-r \dot{b}(t,r) a^\prime(t,r)\right)\\
\cdot f_{TT}.\label{e02}
\end{split}$$ Since the non-diagonal elements of energy-momentum tensor are naturally equal to zero, $E_1^{~0}$ and $E_0^{~2}$ always vanishes. Like the previous discussion, $\dot b(t,r) = 0$, we get $$\label{b}
b(t,r)=b(r).$$ Substituted into equation (\[e02\]), it leads to $$\label{constrain2}
\dot{a}^\prime(t,r) r=0.$$ Similar to Eqs.(\[B\]) and (\[constrain\]), the above two equations deduce the same conclusion as the case of the off diagonal tetrad field.
In the previous proof, we have not used any specific model form for $f(T)$ gravity. What we only require is that it should satisfy the necessary physical meaning, which means nontrivially $f_T \neq 0$.
Note that the integral performed on Eq.(\[constrain\]) is over the external region, and therefore the distribution and motion of the internal source matter cannot influence $\widetilde a(r)$ any way. We then come to the conclusion that the spherically symmetric vacuum solution of the $f(T)$ gravity must be static, and is independent of the radial distribution and motion of the source matter, implying that Birkhoff¡¯s theorem still holds generally.
The Birkhoff’s theorem in the frame of $f(T)$ Gravity via conformal transformation
==================================================================================
It is well known that the $f(R)$ gravity is dynamically equivalent to a particular class of scalar-tensor theories via conformal transformation, but the Birkhoff’s theorem generally does not hold in scalar-tensor gravity. The case of $f(T)$ gravity via conformal transformation is more complicated than that of $f(R)$ theories, which has been proved in the work [@fT_conformal].
We will explore the difference between $f(T)$ gravity and scaler-tensor theory, and compare the results obtained from the Jordan and Einstein frames via conformal transformation. Firstly, let us write the general action for a Brans-Dicke-like $f(T)$ theory, $$\label{SBD}
S_{BD} = \int {\rm d}^4 x~e\bigg[ \phi T - \frac{\omega}{\phi}g^{\mu\nu} \nabla_{\mu}\phi \nabla_{\nu}\phi
- V(\phi) + 2k^2\mathcal{L}_m(e_{\mu}^{~i}) \bigg],$$ where we have assumed $\omega$ to be constant. This action is written in the Jordan frame, which is related with the Einstein frame by the conformal transformation, $$\label{trans}
e_{\mu}^{~i}=\Omega^{-1} \tilde e_{\mu}^{~i}, \quad where \quad \phi=\Omega^{2}$$ under which the action (\[SBD\]) can be transformed to the Einstein frame as, $$\begin{aligned}
\label{SE1}
S_{E} &=& \int {\rm d}^4 x~\tilde{e}\bigg[\tilde{T} - 2\phi^{-1}\tilde \partial^{\mu}\phi \tilde T^{\rho}_{~\rho\mu}
- \frac{\omega-3/2}{\phi^2}\tilde \nabla_{\mu}\phi \tilde \nabla^{\mu}\phi \nonumber \\
&-& \frac{V(\phi)}{\phi^2} \bigg] + 2k^2 \int {\rm d}^4 x~\tilde{e}\,\mathcal{\tilde L}_m(\tilde e_{\mu}^{~i})\end{aligned}$$ By redefining the scalar field as $\phi=e^{\varphi/\sqrt{2\omega-3}}$ for the observation of the solar system $\omega\approx500$, and $U(\varphi)=\frac{V(\phi)}{\phi^2}$, we change the action (\[SE1\]) as, $$\begin{aligned}
\label{SE2}
S_{E}
&=& \int {\rm d}^4 x~\tilde{e}\bigg[\tilde{T}
- \frac{2}{\sqrt{2\omega-3}}\tilde \partial^{\mu}\varphi \tilde T^{\rho}_{~\rho\mu}
- \frac{1}{2}\tilde g^{\mu\nu}\tilde \nabla_{\mu}\varphi\tilde \nabla_{\nu}\varphi \nonumber \\
&-& U(\varphi)\bigg] + 2k^2 \int {\rm d}^4 x~\tilde{e}\,\mathcal{\tilde L}_m(\tilde e_{\mu}^{~i})\end{aligned}$$ Differing from the case in $f(R)$ gravity, an additional scalar-torsion coupling term presents in the action. Therefore, the $f(T)$ gravity is not simply dynamically equivalent to the TEGR action plus a scalar field via conformal transformation, and one cannot use the results of scalar-tensor theories directly to $f(T)$ gravity. Nonetheless, we can also obtain the field equations by varying the action (\[SE2\]) with respect to the tetrad field $e_{\alpha}^{~i}$ and the scalar field $\varphi$, which yields $$\begin{aligned}
\tilde e^{-1}\tilde G^{\alpha}_{~i}
&=& \frac{1}{2\sqrt{2\omega-3}}\tilde\partial_{\mu}\big[\tilde\partial^{\alpha}\varphi \tilde e^{\mu}_{~i}
- \tilde\partial^{\mu}\varphi \tilde e^{\alpha}_{~i}\big] \nonumber \\
&-& \frac{\tilde\partial^{\mu}\varphi}{2\sqrt{2\omega-3}}\tilde e^{\alpha}_{~i}\tilde T^{\rho}_{~\rho\mu}
+ \frac{\tilde\partial^{\mu}\varphi}{2\sqrt{2\omega-3}}\tilde e^{\rho}_{~i}\tilde T^{\alpha}_{~\rho\mu}\nonumber \\
&+& \frac{1}{4}\tilde e^{\nu}_{~i}\tilde \nabla^{\alpha}\varphi\tilde \nabla_{\nu}\varphi
- \frac{1}{8}\tilde e^{\alpha}_{~i}\tilde \nabla^{\sigma}\varphi\tilde \nabla_{\sigma}\varphi
\nonumber \\
&-& \frac{1}{4}e^{\alpha}_{~i}U(\varphi) + \frac{k^2}{2}e^{\rho}_{~i}\tilde T^{\alpha \,(m)}_{~\rho} , \label{field eqn0}
\end{aligned}$$ and $$\begin{aligned}
\label{scalar field eqn0}
-2k^2\frac{\delta(\tilde{e}\,\mathcal{\tilde L}_m)}{\tilde{e}\delta\varphi} &=& \tilde \Box \varphi
- \frac{{\rm d}U(\varphi)}{{\rm d}\varphi} \nonumber \\
&+& \frac{2}{\sqrt{2\omega-3}}\tilde e^{-1}\tilde\partial_{\mu}\big(\tilde{e}\tilde g^{\mu\nu}\tilde T^{\rho}_{~\rho\nu}\big),\end{aligned}$$ where the $\tilde G^{\alpha}_{~i}$ in Eq.(\[field eqn0\]) is defined by $$\tilde G^{\alpha}_{~i} = \tilde\partial_{\mu}(\tilde{e}\tilde e^{\rho}_{~i}\tilde S_{\rho}^{~\mu\alpha})
+ \tilde{e}\tilde e^{\nu}_{~i}\tilde T^{\rho}_{~\mu\nu}\tilde S_{\rho}^{~\mu\alpha}
- \frac{1}{4}\tilde{e}\tilde e^{\alpha}_{~i}\tilde T.$$ The field equation (\[field eqn0\]) seems very complicated, while the components that we need can be simplified. If the $\varphi=\varphi_{0}$, which is a constant, the field equation (\[field eqn0\]) degenerates to the teleparallel gravity with the cosmological constant $\Lambda = 2 U(\varphi)$. In this case the Birkhoff’s Theorem holds both in Einstein frame and Jordan frame with the diagonal and off diagonal tetrad fields.
Assuming $\varphi=\varphi(t,r)$, because of the relation $\phi=e^{\varphi/\sqrt{2\omega-3}}$, we can generally define $\phi=\phi(t,r)$. On one hand, the $E_2^{~0}$ and $E_2^{~1}$, which we use to prove the validity of Birkhoff’s Theorem with the off diagonal tetrad field, change as, $$\begin{aligned}
E_2^{~0}&=&\frac{1}{4}\cot\!\theta \, e^{-a(t,r)} \, \dot b(t,r) f_T
+ \frac{\tilde\partial^{\mu}\varphi}{2\sqrt{2\omega-3}}\tilde T^{t}_{~\theta\mu}, \\
E_2^{~1}&=&\frac{1}{4 r}\cot\!\theta \, e^{-b(t,r)} \, {\big(}2-2{\rm e}^{\frac{b(t,r)}{2}}+a^\prime(t,r) r{\big)}f_T \nonumber \\
&+& \frac{\tilde\partial^{\mu}\varphi}{2\sqrt{2\omega-3}}\tilde T^{r}_{~\theta\mu}.\end{aligned}$$ Then we consider the non-zero components of the antisymmetric torsion tensor $T^{\rho}_{~\mu\nu}$ for the two covariant indices with the off diagonal tetrad field, $$\begin{array}{cccc}
T^{\theta}_{~r\theta}\!\! &= \frac{1-e^{b(t,r)/2}}{r}, \quad &T^{t}_{~t r}\!\! &= -\frac{a'(t,r)}{2}, \\
T^{\psi}_{~r\psi}\!\! &= \frac{1-e^{b(t,r)/2}}{r}, \quad &T^{r}_{~t r}\!\! &= \frac{\dot b(t,r)}{2}.
\end{array}$$ According to the transformation relations (\[trans\]) of the tetrad fields, we can get $$\begin{aligned}
\tilde T^{\rho}_{~\mu\nu} &=& T^{\rho}_{~\mu\nu} + [\Omega^{-1}\delta^{\rho}_{\nu}\partial_{\mu}\Omega
- \Omega^{-1}\delta^{\rho}_{\mu}\partial_{\nu}\Omega] \nonumber \\
&=& T^{\rho}_{~\mu\nu} + \bigg[\frac{\delta^{\rho}_{\nu}\partial_{\mu}\phi(t,r)}{2\phi(t,r)}
- \frac{\delta^{\rho}_{\mu}\partial_{\nu}\phi(t,r)}{2\phi(t,r)}\bigg]\end{aligned}$$ Because $\partial_{\theta}\phi(t,r)=0$, the additional items $\tilde T^{t}_{~\theta\mu}$ and $\tilde T^{r}_{~\theta\mu}$ both disappear in the $E_2^{~0}$ and $E_2^{~1}$, and the result is not different from that we have proved. The Birkhoff’s Theorem in $f(T)$ gravity with off diagonal tetrad field in Einstein frame still holds.
Then we transform back the metric from Einstein frame to Jordan frame. According to the transformation relations (\[trans\]) of the tetrad fields, we can get the metric in Jordan frame $$\begin{aligned}
{\rm d} s^2
&=& \phi(t,r)^{-1}{\rm e}^{a\!(r)}~{\mathrm{d}}t^2-\phi(t,r)^{-1}{\rm e}^{b\!(r)}~{\mathrm{d}}r^2 \nonumber \\
&-& \phi(t,r)^{-1}r^2{\mathrm{d}}\theta^2-\phi(t,r)^{-1}r^2 \sin^2\!\theta~{\mathrm{d}}\phi^2.\end{aligned}$$ Obviously, the metric in the Jordan frame clearly depends on time, indicating that the Birkhoff’s theorem is not satisfied. This result suggests the non-physical equivalence between both frames.
On the other hand, the $E_1^{~0}$ and $E_0^{~2}$, which we introduce to prove the validity of Birkhoff’s Theorem with diagonal tetrad field, change as, $$\begin{aligned}
E_1^{~0} &=& -\frac{{\rm e}^{-a(t,r)}}{2 r} \dot b(t,r) f_T
+ \frac{\tilde\partial^{\mu}\varphi}{2\sqrt{2\omega-3}}\tilde T^{t}_{~r\mu},\label{e10T} \\
E_0^{~2} &=& \frac{2{\rm e}^{- b(t,r)} \cos\theta}{r^4 \sin\theta} \bigg(\dot{a}^\prime(t,r) r - \dot{b}(t,r) \nonumber \\
&-& r \dot{b}(t,r) a^\prime(t,r)\bigg)\cdot f_{TT} + \frac{\tilde\partial^{\mu}\varphi}{2\sqrt{2\omega-3}}\tilde T^{\theta}_{~t\mu}.\end{aligned}$$ Considering the diagonal tetrad field, the non-zero components of the antisymmetric torsion tensor $T^{\rho}_{~\mu\nu}$ for the two covariant indices, $$\begin{array}{cccc}
T^{\theta}_{~r\theta}\!\! &= \frac{1}{r}, \quad &T^{t}_{~t r}\!\! &= -\frac{a'(t,r)}{2}, \\
T^{\psi}_{~r\psi}\!\! &= \frac{1}{r}, \quad &T^{r}_{~t r}\!\! &= \frac{\dot b(t,r)}{2}, \\
T^{\psi}_{~\theta\psi}\!\! &= \cot\!\theta .
\end{array}$$ The additional item $\tilde\partial^{\theta}\varphi(t,r) \tilde T^{\theta}_{~t\theta}$ disappears in the $E_0^{~2}$ for $\tilde\partial^{\theta}\varphi(t,r)=0$, but the equation (\[e10T\]) changes as $$E_1^{~0} = -\frac{{\rm e}^{-a(t,r)}}{2 r} \dot b(t,r) f_T + \frac{\dot \varphi(t,r)}{2\sqrt{2\omega-3}}
\bigg(\frac{a'(t,r)}{2}+\frac{\phi'}{2\phi}\bigg)$$ So the result is different from that case we have proved with diagonal tetrad field. This means that $\dot \varphi(t,r)=0$ or $\dot \phi(t,r)=0$, because that the non-diagonal elements of energy-momentum tensor are naturally equal to zero. The Birkhoff’s Theorem in $f(T)$ gravity with diagonal tetrad field in Einstein frame still holds for $\phi=\phi(r)$. Consequently, the metric in the Jordan frame clearly does not depend on time, indicating that the Birkhoff’s theorem is still satisfied.
In the above analysis, we have studied the equivalence between both Einstein frame and Jordan frame. If we do not consider the ill-defined $\phi \rightarrow 0^+$, because of $\phi=e^{\varphi/\sqrt{2\omega-3}}$, the transformation relations (\[trans\]) of the tetrad fields only depend on the concrete form of the $\phi$ field. In other words, the equivalence between both frames depends on the constraint on the $\phi$ field by the theoretical model. We also can find the uncertainty of the $\phi$ field comes from the freedom of tetrad field, The extra six degrees of freedom in the off diagonal tetrad conceal the physical meaning of the $\phi$ field depending on time. The physical reasons for the above consequence of Birkhoff’s theorem in $f(T)$ gravity is that the concrete form of the tetrad field is not determined.
Discussions and Conclusions {#sec4}
===========================
In this letter we prove the validity of Birkhoff’s theorem in $f(T)$ gravity with both the diagonal and the off diagonal tetrad fields. In our previous work [@fT_birkhoff], we have detailedly discussed the physical meanings of the Birkhoff’s theorem in ordinary conditions, namely, the external vacuum gravitational field. More generally, we consider a spherically symmetric matter distribution as shown in Fig.\[fig\](a). The gravitational field of the interlining vacuum region is spherically symmetric, because of the symmetric distribution of source matter. Accordingly, the above analysis in this letter is applicable, and the gravitational field of the vacuum region is static. The only property of the source matter may appear in $\widetilde a(r)$ of equation (\[A\]) is the mass of internal source $M_I$ and mass of external source $M_E$. (A similar problem is discussed specifically in [@ZhangSH; @Penna] for black hole). The radial motion and distribution of the source matter cannot affect the gravitational field any way.
The second conclusion is that the main feature of the Birkhoff’s theorem is applicable in non-vacuum regions, such as the case shown in Fig.\[fig\](b). Note that we actually do not claim that the vanishing of density and pressure of matter is necessary to prove the validity of Birkhoff’s theorem. To obtain Eqs.(\[E20\], \[E21\]), what we really demand is that the non-diagonal elements of the energy-momentum tensor are zero, which is always satisfied for perfect fluid models. As a conclusion, the gravitational field is static inside the spherically symmetric matter, such as the region denoted by dashed line in Fig.\[fig\](b). This conclusion is correct only if there is no radial motion or convection across the sphere (the dashed line in Fig.\[fig\](b)).
As is known to all, Hawking’s theorem [@Hawking] states that a stationary space-time containing a black hole is a solution of the Brans-Dicke field equations with $V(\phi)=0$ if and only if it is a solution of the field equations, and therefore it must be axially symmetric or static. The proof of Hawking¡¯s theorem is performed in the Einstein frame, in order to obtain that the rescaled Brans-Dicke scalar has canonical kinetic energy density and obeys the weak and null energy conditions, and prove that the scalar field is static. But we can find that an additional scalar-torsion coupling term in the action (\[SE2\]) breaks previous conditions. And considering the scalar field equation (\[scalar field eqn0\]) with $U(\varphi)$=$\frac{V(\phi)}{\phi^2}$=$0$ in vacuum, which is different from $\tilde \Box \varphi=0$, so we cannot vanish the contribution from the additional term on the horizon. The Hawking’s theorem in $f(T)$ gravity needs more detailed study.
In this letter we have proved Birkhoff’s theorem validity in the $f(T)$ gravity with both the diagonal and the off diagonal tetrad fields, and discussed the significations of this theorem both in vacuum and non-vacuum conditions. We do not deal with perturbations in this present work, which will be left for our future work in preparation. The Birkhoff’s theorem generally does not hold in the frame of $f(R)$ gravity by using its scalar-tensor representation, and it is invalid at first linear order of perturbations in the Jordan frames [@invalid; @fR2; @fR_perturbation]. Nevertheless, the case of $f(T)$ gravity is more complicated for an additional scalar-torsion coupling term generated by conformal transformation [@fT_conformal]. The perturbations in $f(T)$ gravity have been studied in [@pt]. We can introduce some constraints on scalar field like what S. Capozziello, *et al*, have done [@fR_perturbation], by assuming a constant zero older scalar field as the background solution. Consequently, the zero-order solution in perturbations will give the Schwarzschild-(Anti)de Sitter solution, which will be proved in our next work [@fT_perturbation]. We must make certain whether the higher order of the additional scalar-torsion and scalar field would disappear like in the case of $f(R)$ gravity, or they still exist and have some effect, which may affect the validity of Birkhoff’s theorem in the perturbative approach. We will deal with this problem in detail in our next work.
Acknowledgement {#acknowledgement .unnumbered}
===============
We cordially thank Prof. Lewis H Ryder and Prof. Sergei D Odintsov for lots of interesting discussions on possible roles the torsion may play in gravity and cosmology physics during the project over years. This work is partly supported by Natural Science Foundation of China under Grant Nos.11075078 and 10675062 and by the project of knowledge Innovation Program (PKIP) of Chinese Academy of Sciences (CAS) under the grant No. KJCX2.YW.W10 through the KITPC where we have initiated this present work.
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[^1]:
[^2]:
[^3]: (corresponding author)
| ArXiv |
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abstract: 'We present new results regarding the features of high energy photon emission by an electron beam of 178GeV penetrating a 1.5cm thick single Si crystal aligned at the Strings-Of-Strings(SOS) orientation. This concerns a special case of coherent bremsstrahlung where the electron interacts with the strong fields of successive atomic strings in a plane and for which the largest enhancement of the highest energy photons is expected. The polarization of the resulting photon beam was measured by the asymmetry of e$^+$e$^-$ pair production in an aligned diamond crystal analyzer. By the selection of a single pair the energy and the polarization of individual photons could be measured in an the environment of multiple photons produced in the radiator crystal. Photons in the high energy region show less than 20% linear polarization at the 90% confidence level.'
bibliography:
- 'na59-sos.bib'
title: Measurement of Coherent Emission and Linear Polarization of Photons by Electrons in the Strong Fields of Aligned Crystals
---
\[sec:intro\]Introduction
=========================
Interest in the generation of intense, highly polarized high energy photon beams [@proposal; @e159] comes in part from the need to investigate the polarized photo-production mechanisms. For example, the so-called “spin crisis of the nucleon” and its connection to the gluon polarization has attracted much attention [@compass]. Future experiments will require intense photon beams with a high degree of polarization. The radiation emitted by electrons passing through oriented single crystals is important for these purposes. The coherent bremsstrahlung(CB) of high energy unpolarized electrons is a well established and widely applied technique for producing intense photon beams with a high degree of linear polarization. The coherence arises in this case due to crystal effects which become pronounced when the electron incidence angle with respect to a major plane is small. The resulting CB radiation differs from incoherent bremsstrahlung(ICB) in an amorphous target in that the cross section is substantially enhanced and relatively sharp coherent peaks appear in the photon spectrum. The position of these peaks can be tuned by adjusting the electron beam incidence angle with respect to the major plane of the lattice.
There is another less well known method of producing greater enhancement as well as a harder photon spectrum than the CB case. This is achieved by selecting a very specific electron incident angle with respect to the crystal. If the electron beam is incident very close to the plane (within the planar channelling critical angle) and also closely well aligned to a major axis (but beyond the axial channelling critical angle), then the electron interacts dominantly with successive atomic strings in the plane. This orientation had been aptly described by the term “String-Of-Strings"(SOS) by Lindhard, a pioneer of beam-crystal phenomena [@lindhard]. The NA43 Collaboration has studied the radiation emitted by electrons incident in the SOS orientation. The reference [@kirsebom01] and references therein are an account of this study, as well as many other related effects. There remained the issue of the polarisation of the SOS radiation. Polarisation measurements have been reported [@kirsebom99] which could be consistent with substantial polarisation of the hard component of SOS radiation. However the ability to distinguish clearly a single photon spectrum from the total radiated energy spectrum was not yet developed for that measurement. In this paper, a new study of SOS-produced high energy photon beams is reported in which we were able to study the beam on a photon-by-photon basis, and measure both the enhancement and the linear polarisation as a function of photon energy.
\[theory\]Theoretical Description
=================================
The CB mechanism produces linearly polarized photons in a selected energy region when the crystal type, its orientation with respect to the electron beam, and the electron energy are appropriately chosen. In the so-called point effect(PE) orientation of the crystal the direction of the electron beam has a small angle with respect to a chosen crystallographic plane and a relatively large angle with the crystallographic axes that are in that plane. For this PE orientation of the single crystal only one reciprocal lattice vector contributes to the CB cross section. The CB radiation from a crystal aligned in this configuration is more intense than the ICB radiation in amorphous media and a high degree of linear polarization can be achieved [@termisha]. The PE orientation of the crystal was used in a previous NA59 experiment, where a large linear polarization of high energy photons was measured. The photons had been produced by an unpolarized electron beam. The conversion of the linear polarisation to circular polarization induced by a birefringent effect in an aligned single crystal was also studied [@na59-1; @na59-2].
The character of the radiation, including its linear polarization, is changed when the direction of the electron (i) has a small angle with a crystallographic axis and (ii) is parallel with the plane that is formed by the atomic strings along the chosen axes. This is the so-called SOS orientation. It produces a harder photon spectrum than the CB case because the coherent radiation arises from successive scattering off the axial potential, which is deeper than the planar potential. The radiation phenomena in single crystals aligned in SOS mode have been under active theoretical investigation since the NA43 collaboration discovered, for the first time, two distinct photon peaks, one in the low energy region and one in the high energy region of the radiated energy spectrum for about 150GeV electrons traversing a diamond crystal [@new-effect]. It was established that the hard photon peak was a single photon peak. However, the radiated photons were generally emitted with significant multiplicity in such a way that a hard photon would be accompanied by a few low energy photons. It will be seen later that two different mechanisms are responsible for the soft and the hard photons. In the former case, it is planar channelling(PC) radiation, while in the latter case, it is SOS radiation.
The issue of the polarisation of SOS radiation also came into question. Early experiments with electron beams of up to 10GeV in single crystals showed a smaller linear polarization of the more intense radiation in the SOS orientation than in the PE orientation (see [@saenz] and references therein). The first measurements of linear polarization for high energy photons ($E_{\gamma} \approx 50-150$GeV) were consistent with a high degree of linear polarization of the radiated photons [@kirsebom99]. At this stage the theoretical prediction of the SOS hard photon polarisation was unresolved. However, it was clear that the photons emitted by the PC mechanism would be linearly polarised. This experiment therefore could not be considered conclusive, as the polarimeter recorded the integral polarisation for a given radiated energy, which was likely to have a multi-photon character. The extent to which pile-up from the low energy photons perturbed the high energy part of the total radiated energy spectrum was not resolved. These results therefore required more theoretical and experimental investigation.
A theory of photon emission by electrons along the SOS orientation of single crystals has since been developed. The theory takes into account the change of the effective electron mass in the fields due to the crystallographic planes and the crossing of the atomic strings [@bks]. The authors show that the SOS specific potential affects the high energy photon emission and also gives an additional contribution in the low energy region of the spectrum. In Refs. [@simon; @strakh] the linear polarization of the emitted photons was derived and analysed for different beam energies and crystal orientations. The predicted linear polarization of hard photons produced using the SOS orientation of the crystal is small compared to the comparable case using the PE orientation of the crystal. On the other hand, the additional soft photons produced with SOS orientation of the crystal are predicted to exhibit a high degree of polarization.
The emission mechanism of the high energy photons is CB connected to the periodic structure of the crystal [@termisha].
The peak energy of the CB photons, $E_\gamma$, is determined from the condition ( the system of units used here has $\hbar={\rm c}=1$ ), $$\frac{1}{|q_{\Vert}|} = 2 \lambda_c \gamma \frac{E_0-E_\gamma}{E_\gamma}~,$$ where $|q_{\Vert}|$ is the component of the momentum recoil, $\mathbf{q}$, parallel to the initial electron velocity and the other symbols have their usual meanings. Recall, in a crystal possible values of $\mathbf{q}$ are discrete: $\mathbf{q}=\mathbf{g}$ [@termisha], where $\mathbf{g}$ is a reciprocal lattice vector. The minimal reciprocal lattice vector giving rise to the main CB peak in both the PE and the SOS orientations is given by $$|g_{\Vert}|_{min} = \frac{2\pi}{d}\Theta.$$
For the PE orientation, $d$ is the interplanar distance and $\Theta=\psi$, the electron incident angle with respect to the plane. For the SOS orientation $d$ is the spacing between the axes (strings) forming the planes, and $\Theta=\theta$, the electron incident angle with respect to the axis. The position of the hard photon peak can be selected by simultaneous solution of the last two equations, $$\Theta =\frac{d}{4\pi\gamma\lambda_c}\frac{E_{\gamma}}{E_0-E_{\gamma}}.$$
With the appropriate choice of $\theta=\Theta$ the intensity of the SOS radiation may exceed the ICB radiation by an order of magnitude.
When a thin silicon crystal is used with an electron beam of energy $E_0
=178$GeV incident along the SOS orientation, within the $(110)$ plane and with an angle of $\theta=0.3$ mrad to the $<100>$ axis, the hard photon peak position is expected at $E_{\gamma}=129$GeV.
In the current experiment, a 1.5cm thick silicon crystal was used in the SOS orientation with the electron beam ($E_0 =178$GeV) incident within the $(110)$ plane with an angle of $\theta=0.3$ mrad to the $<100>$ axis. This gives the hard photon peak position at $x_{max}=0.725$. This corresponds to the photon energy $E_{\gamma}=129$GeV. Under this condition the radiation is expected to be enhanced by about a factor 30 with respect to the ICB for a randomly oriented crystalline Si target.
The coherence length determines the effective longitudinal dimension of the interaction region for the phase coherence of the radiation process: $$l_{coh} = \frac{1}{|q_{\Vert}|}.$$
The radiation spectrum with the crystal aligned in SOS orientation has in addition to the CB radiation a strong component at a low energy which is characteristic of PC radiation. As the electron direction lines up with a crystallographic plane in the SOS orientation, the planar channelling condition is fulfilled. For channelling radiation the coherence length is much longer than the interatomic distances and the long range motion, characteristic of planar channelled electrons, becomes dominant over short range variations with the emission of low energy photons. Theoretical calculations [@strakh; @armen] predict a more intense soft photon contribution with a high degree of linear polarization of up to 70%.
The simulation of the enhancements of both the low energy and the high energy components of the radiation emission for the SOS orientation under conditions applicable to this experiment are presented in Fig. \[F:Strak-1b\].
![\[F:Strak-1b\] Photon power yield per unit of thickness, $E_\gamma d^2N/dE_\gamma dl$, for a thin silicon crystal in the SOS orientation for a $E_0 =178$GeV electron beam incident within the $(110)$ plane and at an angle of $\theta=0.3$mrad to the $<100>$ axis. At low energy the PC radiation dominates and at high energies the SOS radiation peaks. The solid curve represents the total of the contributions from (green dash-dotted)ICB, (blue dotted)PC, and (red dashed) SOS radiation. vThe insert is a logarithmic representation and shows the flat incoherent contribution and the enhancement with a factor of about 30 for SOS radiation at 129GeV.](na59-sos-fig1)
\[setup\]Experimental Setup
===========================
The NA59 experiment was performed in the North Area of the CERN SPS, where unpolarized electron beams with energies above 100GeV are available. We used a beam of 178GeV electrons with angular divergence of $\sigma_{x'}=48\,\mu$rad and $\sigma_{y'}=35\,\mu$rad in the horizontal and vertical plane, respectively.
The experimental setup shown in Fig. \[F:setup\] was also used to investigate the linear polarization of CB and birefringence in aligned single crystals [@na59-1; @na59-2]. This setup is ideally suited for detailed studies of the photon radiation and pair production processes in aligned crystals.
The main components of the experimental setup are: two goniometers with crystals mounted inside vacuum chambers, a pair spectrometer, an electron tagging system, a segmented leadglass calorimeter, wire chambers, and plastic scintillators. In more detail a 1.5cm thick Si crystal can be rotated in the first goniometer with 2$\mu$rad precision and serves as radiator. A multi-tile synthetic diamond crystal on the first goniometer can be rotated with 20$\mu$rad precision and is used as the analyzer of the linear polarization of the photon beam.
The photon tagging system consists of a dipole magnet B8, wire chamber dwc0, and scintillators T1 and T2. Given the geometrical acceptances and the magnetic field, the system, tags the radiated energy between 10% and 90% of the electron beam energy. Drift chambers dch1up, dch2up, and delay wire chamber dwc3 define the incident and the exit angle of the electron at the radiator.
The e$^+$e$^-$ pair spectrometer consists of dipole magnet Trim 6 and of drift chambers dch05, dch1, dch2, and dch3. The drift chambers measure the horizontal and vertical positions of the passing charged particles with 100$\mu$m precision. Together with the magnetic field in the dipole this gives a momentum resolution of $\sigma_p/p^2=0.0012$ with $p$ in units ofGeV/c. The pair spectrometer enables the measurement of the energy of a high energy photon, $E_\gamma$, in a multi-photon environment. Signals from the plastic scintillators S1, S2, S3, T1, T2, S11 and veto detector ScVT provide several dedicated triggers.
The total radiated energy $E_{tot}$ is measured in a 12-segment array of leadglass calorimeter with a thickness of 24.6 radiation lengths and a resolution of $\sigma_E=0.115~\sqrt{E}$ with $E$ in units ofGeV. A central element of this leadglass array is used to map and to align the crystals with the electron beam.
A detailed description of the NA59 experimental apparatus can be found in reference [@na59-1].
Results and Discussion
======================
The experiment can be divided in two parts: (A) production of the photon beam by the photon radiation of the 178GeV electron beam in the Si radiator oriented in the SOS mode and (B) measurement of the linear polarization by using diamond crystals as analyzers. Prior to the experiment Monte Carlo(MC) simulations were used to estimate the photon yield, the radiated energy, and the linear polarization of the photon beam and we optimized the orientation of the crystal radiator. The MC calculations also included the crystal analyzer to estimate the asymmetry of the e$^+$e$^-$ pair production. The simulations further included the angular divergence of the electron beam, the spread of 1% in the beam energy, and the generation of the electromagnetic shower that develops in oriented crystals. To optimize the processing time of the MC simulation, energy cuts of 5GeV for electrons and of 500MeV for photons were applied.
Photon Beam
-----------
We used a beam angle of $\theta=0.3$mrad to the $\langle 100 \rangle$ axis in the $(110)$ plane of the 1.5cm thick Si crystal which is the optimal angle for a high energy SOS photon peak at 129GeV (see Fig. \[F:Strak-1b\]). As is mentioned above, the radiation probability with a thin radiator is expected to be 30 times larger at that energy than the Bethe-Heitler(ICB) prediction for randomly oriented crystalline Si.
![\[F:sps\] Photon power yield, $E_\gamma dN/dE_\gamma$, as a function of the energy $E_\gamma$ of individual photons radiated by an electron beam of 178GeV in the SOS-aligned 1.5cm Si crystal. The black crosses are the measurements with the pair spectrometer, the vertical lines represent the errors including the uncertainty in the acceptance of the spectrometer. The (red solid) histogram represent the MC prediction for our experimental conditions. The (green dotted) represent the small contribution due to incoherent interactions. For completeness, we also show the theoretical predictions if the experimental effects are ignored (blue dashed).](na59-sos-fig3)
However, there are several consequences for the photon spectrum due to the use of a 1.5 cm thick crystal For the chosen orientation of the Si crystal, the emission of mainly low energy photons from planar coherent bremsstrahlung (PC) results in a total average photon multiplicity above 15. And the most probable radiative energy loss of the 178 GeV electrons is expected to be 80%. The beam energy decreases significantly as the electrons traverse the crystal. The peak energy of both SOS and PC radiation also decreases with the decrease in electron energy. Consequently, the SOS radiation spectrum is not peaked at the energy for a thin radiator, but becomes a smooth energy distribution. Clearly, many electrons may pass through the crystal without emitting SOS radiation and still lose a large fraction of their energy due to PC and ICB. Hard photons emitted in the first part of the crystal that convert in the later part do not contribute anymore to the high energy part of the photon spectrum. A full Monte Carlo calculation is necessary to propagate the predicted photon yield with a thin crystal, as shown in Fig. \[F:Strak-1b\] for 178 GeV electrons, to the current case with a 1.5cm thick crystal.
This has been implemented for the measured photon spectrum shown in Fig. \[F:sps\]. We see that the measured SOS photon spectrum shows a smoothly decreasing distribution. The low energy region of the photon spectrum is especially saturated, due to the abundant production of low energy photons. Above 25GeV however, there is satisfactory agreement with the theoretical Monte Carlo prediction, which includes the effects mentioned above.
The enhancement of the emission probability compared to the ICB prediction is given in Fig. \[F:enh\] as a function of the total radiated energy as measured in the calorimeter. The maximal enhancement is about a factor of 18 at 150GeV and corresponds well with the predicted maximum of about 20 at 148GeV. This is a multi-photon spectrum measured with the photon calorimeter. The peak of radiated energy is situated at 150GeV, which means that each electron lost about 80% of its initial energy due to the large thickness of the radiator. This means that the effective radiation length of the oriented single crystal is several times shorter in comparison with the amorphous target. The low energy region is depleted due to the pile-up of several photons.
![\[F:enh\] Enhancement of the intensity with respect to the Bethe-Heitler(ICB) prediction for randomly oriented polycrystalline Si as a function of the total radiated energy $E_{tot}$ in the SOS-aligned Si crystal by 178GeV electrons. The black crosses are the measurements and the red histogram represent the MC prediction.](na59-sos-fig4)
The expected linear polarization is shown in Fig. \[F:SOS-pol\] as a function of photon energy. It is well known that channelling radiation in single crystals is linearly polarized [@Adishchev; @Vorobyov] and the low energy photons up to 70GeV are also predicted to be linearly polarized in the MC simulations. High energy photons are predicted with an insignificant polarization.
Asymmetry Measurement
---------------------
In this work, the photon polarization is always expressed using the Stoke’s parametrization with the Landau convention, where the total elliptical polarization is decomposed into two independent linear components and a circular component. In mathematical terms, one writes:
$$P_{\hbox {linear}}=\sqrt{\eta _{1}^{2}+\eta _{3}^{2}},
\quad \; P_{\hbox {circular}}=\sqrt{\eta _{2}^{2}},
\quad \; P_{\hbox {total}}=\sqrt{P_{\hbox {linear}}^{2}+P_{\hbox
{circular}}^{2}} \quad .
\label{eq:pol-def}$$
![\[F:SOS-pol\] Expected linear polarization as a function of the energy $E_\gamma$ of the photons produced in the SOS-aligned Si crystal by 178GeV electrons.](na59-sos-fig5)
The radiator angular settings were chosen to have the total linear polarization from the SOS radiation purely along $\eta _{3}$, that is $\eta _{1}=0$. The $\eta _{2}$ component is also zero because the electron beam is unpolarized. The expected $\eta _{3}$ component of the polarization shown is in Fig. \[F:SOS-pol\].
In order to determine the linear polarization of the photon beam the method proposed in reference [@barbiellini] with an oriented crystal was chosen. This method of measurement of the linear polarization of high energy photons is based on coherent e$^+$e$^-$ pair production(CPP) in single crystals which depends on the orientation of the reciprocal lattice vector and the linear polarization vector. Thus, the dependence of the CPP cross section on the linear polarization of the photon beam makes an oriented single crystal suitable as an efficient polarimeter for high energy photons.
The basic characteristic of the polarimeter is the analyzing power $R$ of the analyzer crystal [@barbiellini]. By choosing the appropriate crystal type and its orientation a maximal analyzing power can be obtained. The relevant experimental quantity is the asymmetry $A$ of the cross sections $\sigma (\gamma \rightarrow e^+e^-)$ for parallel and perpendicular polarization, where the polarization direction is defined with respect to a particular crystallographic plane of the [*analyzer*]{} crystal. This asymmetry is related to the linear polarization of the photon beam, $P_{\rm linear}$, through: $$A \equiv \frac{\sigma (\gamma _{\perp }\rightarrow e^{+}e^{-})-\sigma
(\gamma _{\parallel }\rightarrow e^{+}e^{-})}{\sigma (\gamma _{\perp }
\rightarrow e^{+}e^{-})+\sigma (\gamma _{\parallel }\rightarrow
e^{+}e^{-})}
=R \times P_{\rm linear}.
\label{eq:asym}$$ The analyzing power $R$ corresponds to the asymmetry expected for photons that are 100% linearly polarized perpendicular to the chosen crystallographic plane.
Denoting the number of e$^+$e$^-$ pairs produced in perpendicular and parallel cases by $p_{1}$ and $p_{2}$, and the number of the normalisation events in each case by $n_{1}$ and $n_{2}$, respectively, the measured asymmetry can be written as: $$A=\frac{p_{\perp }/n{\perp } - p_{\parallel }/n_{\parallel }}{p_{\perp
}/n_{\perp } + p_{\parallel }/n_{\parallel }},
\label{eq:asy-meas}$$ where $p$ and $n$ are acquired simultaneously and therefore correlated. Further details of this method, as well as refinements to enhance the analyzing power $R$ by using kinematic cuts on the pair spectra, may be found in reference [@na59-1].
The existence of a strong anisotropy for the channelling of the e$^+$e$^-$ pairs during their formation is the reason for the polarization dependent CPP cross section of photons passing through oriented crystals. This means that perfect alignment along a crystallographic axis is not an efficient analyzer orientation due to the approximate cylindrical symmetry of the crystal around atomic strings. However, for small angles of the photon beam with respect to the crystallographic symmetry directions the conditions for the formation of the e$^+$e$^-$ pairs prove to be very anisotropic. As it turns out, the orientations with the highest analyzing power are those where the e$^+$e$^-$ pair formation zone is not only highly anisotropic but also inhomogeneous with maximal fluctuations of the crystal potential along the electron path. At the crystallographic axes the potential is largest and so are the fluctuations. These conditions are related to the ones of the SOS orientation: (i) a small angle to a crystallographic axis to enhance the pair production (PP) process by the large fluctuations and (ii) a smaller angle to the crystallographic plane to have a long but still anisotropic formation zone for CPP.
In the NA59 experiment we used a multi-tile synthetic diamond crystal as an analyzer oriented with the photon beam at 6.2 mrad to the axis and at 465$\mu$rad from the $(110)$ plane. This configuration is predicted to have a maximal analyzing power for a photon energy of 125GeV as is shown in Fig. \[F:anpow\]. The predicted analyzing power in the high energy peak region is about 30%.
![\[F:anpow\] Analyzing power $R$ with the aligned diamond crystal as a function of the photon energy $E_\gamma$ (black curve) for an ideal photon beam without angular divergence and (red curve) for the Monte Carlo simulation of photons with the beam conditions in the NA59 experiment.](na59-sos-fig6)
The measured asymmetry and the predicted asymmetry are shown in Fig. \[F:asy\]. One can see that the measured asymmetry is consistent with zero over the whole photon energy range. For the photon energy range of 100-155GeV we find less than 5% polarization at 0.9 confidence level. The null result is expected to be reliable as the correct operation of the polarimeter had been confirmed in the same beam-time in measurements of the polarisation of CB radiation [@na59-1]. Note, that the expected asymmetry is small, especially in the high energy range of 120-140GeV, where the analyzing power is large, see Fig. (\[F:anpow\]). This corresponds to the expected small linear polarization in the high energy range, see Fig. (\[F:SOS-pol\]).
![\[F:asy\] Asymmetry of the e$^+$e$^-$ pair production in the aligned diamond crystal as a function of the photon energy $E_\gamma$ which is measured to determine the $P_1$ component of the photon polarization in the SOS-aligned Si crystal by 178GeV electrons. The black crosses are the measurements and the red histogram represent the MC prediction.](na59-sos-fig7)
In contrast to the result of a previous experiment [@kirsebom99], our results are consistent with calculations that predict negligible polarization in the high energy photon peak for the SOS orientation. The analyzing power of the diamond analyzer crystal in the previous experiment’s [@kirsebom99] setup peaked in the photon energy range of 20-40GeV where a high degree of linear polarization is expected. But in the high energy photon region we expect a small analyzing power of about 2-3%, also following recent calculation [@simon; @strakh]. The constant asymmetry measured in a previous experiment [@kirsebom99] over the whole range of total radiated energy may therefore not be due to the contribution of the high energy photons.
From Fig. \[F:SOS-pol\] one can expect a large linear polarization for photons in the low energy range of 20-50GeV. However, the analyzing power was optimized for an photon energy of 125GeV and is small in the region where we expect a large polarization. A different choice of orientation of the analyzer crystal can move the analyzing power peak to the low energy range and may be used to measure the linear polarization in the low energy range.
Conclusion
==========
We have performed an investigation of both enhancement and polarisation of photons emitted in the so called SOS radiation. This is a special case of coherent bremsstrahlung for multi-hundred GeV electrons incident on oriented crystalline targets, which provides some advantages comparing with other types of CB orientations. The experimental set-up had the capacity to deal with the relatively high photon multiplicity and single photon spectra were measured. This is very important in view of the fact that there are several production mechanisms for the multiphotons, which have different radiation characteristics.
We have confirmed the single photon nature of the hard photon peak produced in SOS radiation.
The issue of the polarisation of the SOS photons had previously not been conclusively settled. Earlier results in a previous experiment [@kirsebom99] had indicated that a large polarization might be obtained for the high energy SOS photons. Our experimental results show that the high energy photons emitted by electrons passing through the Si crystal radiator oriented in the SOS mode have a linear polarization smaller than 20% at a confidence level of 90%.
Since the previous experiments, the theoretical situation for the polarisation of hard SOS photons has also become clearer. Our results therefore also confirm recent calculations which predict that the linear polarization of high energy photons created in SOS orientation of the crystal is small compared to the polarization obtained with the PE orientation.
Photon emission by electrons traversing single crystals oriented in the SOS orientation has interesting peculiarities since three different radiation processes are involved: (1) incoherent bremsstrahlung, (2) channelling radiation, and (3) coherent bremsstrahlung induced the periodic structure of the atomic strings in the crystal that are crossed by the electron. The calculations presented here have taken these three processes into account, and predict around a 5% polarization for the high energy SOS photons. This prediction is consistent with our null polarization asymmetry measurement for the single photons with energies above 100GeV.
We dedicate this work to the memory of Friedel Sellschop. We express our gratitude to CNRS, Grenoble for the crystal alignment and Messers DeBeers Corporation for providing the high quality synthetic diamonds. We are grateful for the help and support of N. Doble, K. Elsener and H. Wahl. It is a pleasure to thank the technical staff of the participating laboratories and universities for their efforts in the construction and operation of the experiment.
This research was partially supported by the Illinois Consortium for Accelerator Research, agreement number 228-1001. UIU acknowledges support from the Danish Natural Science research council, STENO grant no J1-00-0568.
| ArXiv |
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abstract: 'Given a polynomial $\phi$ over a global function field $K/\mathbb{F}_q(t)$ and a wandering base point $b\in K$, we give a geometric condition on $\phi$ ensuring the existence of primitive prime divisors for almost all points in the orbit $\mathcal{O}_\phi(b):=\{\phi^n(b)\}_{n\geq0}$. As an application, we prove that the Galois groups (over $K$) of the iterates of many quadratic polynomials are large and use this to compute the density of prime divisors of $\mathcal{O}_\phi(b)$.'
author:
- Wade Hindes
title: Prime divisors in polynomial orbits over function fields
---
[^1] [^2]
[Introduction]{} From elliptic divisibility sequences to the Fibonacci numbers, it is an important problem in number theory to prove the existence of “new" prime divisors of an arithmetically defined sequence. For example, such ideas have applications ranging from the undecidability of Hilbert’s $10$th problem [@Poonen], to the classification of certain families of subgroups of finite linear groups [@Feit; @primcycl; @Linear; @trans; @alg]. In this paper, we study the set of prime divisors of polynomial recurrence sequences defined by iteration over global function fields.
To wit, let $K/\mathbb{F}_q(t)$ be a finite extension, let $V_K$ be a complete set of valuations on $K$, and let $\mathcal{B}=\{b_n\}\subseteq K$ be any sequence. We say that $v\in V_K$ is a *primitive prime divisor* of $b_n$ if $$v(b_n)>0\;\;\;\text{and}\;\;\; v(b_m)=0\;\; \text{for all}\; 1\leq m\leq n-1.$$ Likewise, we define the *Zsigmondy set of* $\mathcal{B}$ to be $$\mathcal{Z}(\mathcal{B}):=\big\{n\geq1\;\big\vert\; b_n\;\text{has no primitive prime divisors}\big\}.$$ Over number fields, there are numerous results regarding the finiteness (and size) of $\mathcal{Z}(\mathcal{B})$; for example, see [@PrimDiv; @Tucker; @Silv-Ing; @Krieger; @Silv-Vojta; @SilvPrimDiv].
In this paper, we are interested in studying the finiteness of $\mathcal{Z}(\phi,b):=\mathcal{Z}(\mathcal{O}_\phi(b))$, where $\mathcal{O}_\phi(b):=\{\phi^n(b)\}_{n\geq0}$ is the *orbit* of $b\in K$ for $\phi\in K[x]$; here the superscript $n$ denotes iteration (of $\phi$).
The key geometric notion, allowing us to use techniques in the theory of rational points on curves over $K$ to study $\mathcal{Z}(\phi,b)$, is the following:
Let $\phi\in K(x)$ and let $\ell\geq2$ be an integer. Then we say that $\phi$ is *dynamically $\ell$-power non-isotrivial* if there exists an integer $m\geq1$ such that, $${\label{Curve}} C_{m,\ell}(\phi):=\big\{(X,Y)\in \mathbb{A}^2(\bar{K})\;\big\vert\;Y^\ell=\phi^m(X)=(\underbracket{\phi\circ\phi\dots \circ\phi}_m)(X)\big\}$$ is a non-isotrivial curve [@isotrivial] of genus at least $2$.
Similarly, we have the following refined notions of primitive prime divisors and Zsigmondy sets:
Let $\phi\in K(x)$, let $b\in K$, and let $\ell$ be an integer. We say that a place $v\in V_K$ is an *$\ell$-primitive prime divisor* for $\phi^n(b)$ if all of the following conditions are satisfied:
1. $v(\phi^n(b))>0$,
2. $v(\phi^m(b))=0$ for all $1\leq m\leq n-1$ such that $\phi^m(b)\neq0$,
3. $v(\phi^n(b))\not\equiv0{\ (\textup{mod}\ \ell)}$.
Moreover, we call $${\label{Zig}} \mathcal{Z}(\phi,b,\ell):=\big\{n\;\big\vert\; \phi^n(b)\; \text{has no $\ell$-primitive prime divisors}\big\}$$ the *$\ell$-th Zsigmondy set* for $\phi$ and $b$.
Note that $\mathcal{Z}(\phi,b)\subseteq\mathcal{Z}(\phi,b,\ell)$ for all $\ell$. Hence, it suffices to show that $\mathcal{Z}(\phi,b,\ell)$ is finite for a single $\ell$ to ensure that all but finitely many elements of $\mathcal{O}_\phi(b)$ have primitive prime divisors. Moreover, we use the notions of height $h_K$ and canonical height $\hat{h}_\phi$ found in [@Baker].
[\[PrimDivThm\]]{} Suppose that $\phi\in K[x]$, $b\in K$, and $\ell\geq2$ satisfy the following conditions:
1. $\phi$ is dynamically $\ell$-power non-isotrivial,
2. $b$ is wandering (i.e. $\hat{h}_\phi(b)>0$).
Then $\mathcal{Z}(\phi,b,\ell)$ and $\mathcal{Z}(\phi,b)$ are finite. In particular, all but finitely many elements of $\mathcal{O}_\phi(b)$ have primitive prime divisors.
In addition to determining whether or not a sequence has primitive prime divisors, it is interesting to compute the “size" of its set of prime divisors (in terms of density) [@Hasse; @Lagarias], a problem which has applications to the dynamical Mordell-Lang conjecture [@Mordell-Lang] and to questions regarding the size of hyperbolic Mandelbrot sets [@RafeThesis].
To do this, let $\mathcal{O}_K$ be the integral closure of $\mathbb{F}_q[t]$ in $K$ and let $\mathfrak{q}\subseteq \mathcal{O}_K$ be a prime ideal, determining a valuation $v_\mathfrak{q}$ on $K$. For such $\mathfrak{q}$, define the *norm* of $\mathfrak{q}$ to be $N(\mathfrak{q}):=\#(\mathcal{O}_K/\mathfrak{q}\mathcal{O}_K)$, and let $\delta(\mathcal{P})$ be the *Dirchlet density* of a set of primes $\mathcal{P}$ of $K$: $$\delta(\mathcal{P}):=\lim_{s\rightarrow 1^+}\frac{\sum_{\mathfrak{q}\in\mathcal{P}}N(\mathfrak{q})^{-s}}{\sum_{\mathfrak{q}}N(\mathfrak{q})^{-s}}$$ We use Theorem \[PrimDivThm\] and ideas from the Galois theory of iterates to compute the density of $$\mathcal{P}_\phi(b):=\big\{\mathfrak{q}\in{\operatorname{Spec}}(\mathcal{O}_K)\;\big\vert\; v_\mathfrak{q}(\phi^n(b))>0\;\text{for some $n\geq0$}\big\},$$ the set of prime divisors of the orbit $\mathcal{O}_\phi(b)$. In particular, we establish a version of [@R.Jones Conj. 3.11]; see [@uniformity Theorem 1] for the corresponding statement in characteristic zero (with uniform bounds) and [@B-J; @R.Jones] for introductions to dynamical Galois theory.
[\[Galois\]]{} Let $K/\mathbb{F}_q(t)$ for some odd $q$ and let $\phi\in K[x]$ be a quadratic polynomial.\
Write $\phi(x)=(x-\gamma)^2+c$ and suppose that $\phi$ satisfies the following conditions:
1. $\phi$ is not post-critically finite (i.e. $\gamma$ is wandering),
2. the adjusted critical orbit $\widebar{\mathcal{O}}_\phi(\gamma)=\{-\phi(\gamma),\phi^n(\gamma)\}_{n\geq2}$ contains no squares in K,
3. the $j$-invariant of the elliptic cure $E_\phi: Y^2=(X-c)\cdot\phi(X)$ is non-constant.
Then all of the following statements hold:
1. $\mathcal{Z}(\phi,b,2)$ is finite for all wandering points $b\in K$,
2. $G_\infty(\phi)\leq{\operatorname{Aut}}(T(\phi))$ is a finite index subgroup,
3. $\delta(\mathcal{P}_\phi(b))=0$ for all $b\in K$.
One expects similar statements to hold for $\phi(x)=x^\ell+c$ and $\ell$ a prime. Namely, if one can show that $\phi$ is dynamically $\ell$-power non-isotrivial, then $\mathcal{Z}(\phi,0,\ell)$ is finite by Theorem \[PrimDivThm\] and $G_\infty(\phi)$ is a finite index subgroup of ${\operatorname{Aut}}(T(\phi))$ by [@Eventually Theorem 25].
In particular, we apply Theorem \[PrimDivThm\] and Corollary \[Galois\] to the explicit family $$\phi_{(f,g)}(X)=\big(X-f(t)\cdot g(t)^d\big)^2+g(t)\;\;\;\;\text{for}\;\;\; f\in\mathbb{F}_q(t),-g\notin(\mathbb{F}_q(t))^2,\;\text{and}\;d\geq1.$$ Note that by letting $f=0$ and $g=t$, we recover the main result of [@RafeThesis].
[\[eg\]]{} Let $K=\mathbb{F}_q(t)$ and let $\phi:=\phi_{(f,g)}$ be such that the $j$-invariant of the elliptic curve $$E_{\phi_{(f,g)}}: Y^2=(X-g(t))\cdot \phi_{(f,g)}(X)$$ is non-constant. Then $\mathcal{Z}(\phi,b,2)$ is finite for all wandering base points, $G_\infty(\phi)\leq{\operatorname{Aut}}(T(\phi))$ is a finite index subgroup, and $\delta(\mathcal{P}_\phi(b))=0$ for all $b\in K$.
[Primitive Prime Divisors and Superelliptic Curves]{} To prove Theorem \[PrimDivThm\], we build on our techniques from the characteristic zero setting [@uniformity]. There, among other things, we prove the uniform bound $$\#\mathcal{Z}(\phi,\gamma,2)\leq17$$ for all $\phi(x)=(x-\gamma(t))^2+c(t)$ satisfying $\deg(\gamma)\neq \deg(c)$; here $\gamma$ and $c$ are polynomials with coefficients in any field of characteristic zero. Additionally, we adapt ideas from [@Tucker] and use linear height bounds for rational points on non-isotrivial curves [@Kim] to prove our results.
Throughout the proof, it will be useful to consider only $\ell$-primitive prime divisors avoiding some finite subset $S\subseteq V_K$. Therefore, we make the following convention: $$\mathcal{Z}(\phi,b,S,\ell):=\big\{n\;\big\vert\; \phi^n(b)\; \text{has no $\ell$-primitive prime divisors}\; v\in V_K{\mathbin{\fgebackslash}}S\big\}.$$ Clearly $\mathcal{Z}(\phi,b,\ell)\subseteq\mathcal{Z}(\phi,b,\ell,S)$ for all $S$, and so it suffices to show that $\mathcal{Z}(\phi,b,\ell,S)$ is finite for some $S\subseteq V_K$. Note that there is no harm in enlarging $S$. In particular, we may assume that $$\text{(a)}.\;\;\;b\in\mathcal{O}_{K,S}\;\;\;\;\;\; \text{(b)}.\;\;\;\phi\in\mathcal{O}_{K,S}[x]\;\;\;\;\;\;\text{(c)}.\;\;\; v(a_d)=0\;\;\text{for all}\; v\notin S\;\;\;\;\;\; \text{(d)}.\;\;\; \mathcal{O}_{K,S}\;\text{is a UFD,}$$ where $a_d$ is the leading term of $\phi$. Note that condition (d) is made possible by [@Rosen Prop. 14.2 ]. Similarly, we see that $$\mathcal{Z}(\phi,b,S,\ell)\subseteq\mathcal{Z}(\phi,\phi^n(b),S,\ell)\cup\{t\in\mathbb{Z}\;\vert\; 1\leq t\leq n\}\;\;\,\text{for all}\;n\geq0.$$ Therefore, after replacing $b$ with $\phi^n(b)$ for some $n$, we may assume that $0\notin\mathcal{O}_\phi(b)$. By the assumptions on $S$ above, we see that $\phi^n(b)\in\mathcal{O}_{K,S}$ for all $n$, permitting us to write $${\label{decomp}} \phi^n(b)=u_n\cdot d_n\cdot y_n^\ell,\;\;\text{for some}\;\;\; d_n,y_n\in\mathcal{O}_{K,S}\;,\;u_n\in\mathcal{O}_{K,S}^*.$$ However, since $\mathcal{O}_{K,S}^*$ is a finitely generated group [@Rosen Prop. 14.2 ], we may write $u_n=\textbf{u}_1^{r_1}\cdot \textbf{u}_2^{r_2}\dots \textbf{u}_t^{r_t}$ for some basis $\{\textbf{u}_i\}$ of $\mathcal{O}_{K,S}^*$ and some integers $0\leq r_i\leq\ell-1$. In particular, the height $h_K(u_n)$ is bounded independently of $n\geq0$. Similarly, we may assume that $0\leq v(d_n)\leq\ell-1$ for all $v\notin S$. To see this, we use the correspondence $V_K{\mathbin{\fgebackslash}}S\longleftrightarrow{\operatorname{Spec}}(\mathcal{O}_{K,S})$ discussed in [@Rosen Ch. 14] and the fact that $\mathcal{O}_{K,S}$ is a UFD to write $$d_n=p_1^{e_1}\cdot p_2^{e_2}\cdots p_s^{e_s}\big(p_1^{q_1}\cdot p_2^{q_2}\cdots p_s^{q_s}\big)^\ell,\;\;\;\;\; p_i\in{\operatorname{Spec}}(\mathcal{O}_{K,S})$$ for some integers $e_i, q_i$ satisfying $v_{p_i}(d_n)=q_i\cdot\ell+e_i$ and $0\leq e_i<\ell$. In particular, by replacing $d_n$ with $\big(p_1^{e_1}\cdot p_2^{e_2}\cdots p_s^{e_s}\big)$ and $y_n$ with $\big(y_n\cdot p_1^{q_1}\cdot p_2^{q_2}\cdots p_s^{q_s}\big)$, we may assume that $0\leq v(d_n)\leq\ell-1$ for all $v\in V_K{\mathbin{\fgebackslash}}S$ as claimed.
Now suppose that $n\in Z(\phi,b,S,\ell)$. It is our goal to show that $n$ is bounded. To do this, first note that conditions (b) and (c) imply that $\phi$ has good reduction (see [@Silv-Dyn Thm. 2.15]) modulo the primes in ${\operatorname{Spec}}(\mathcal{O}_{K,S})$. In particular, if $p$ is such that $v_{p}(d_n)>0$ and $n\in Z(\phi,b,S,\ell)$, then $v_p(\phi^m(b))>0$ for some $1\leq m\leq n-1$. Moreover, $$\phi^{n-m}(0)\equiv\phi^{n-m}(\phi^{m}(b))\equiv\phi^n(b)\equiv0{\ (\textup{mod}\ p)};$$ see [@Silv-Dyn Thm. 2.18]. Therefore, we have the refinement $${\label{refinement}} d_n=\prod p_i^{e_i},\;\;\text{where}\;\; p_i\big\vert\phi^{t_i}(b)\;\text{or}\; p_i\big\vert\phi^{t_i}(0)\;\; \text{for some}\; 1\leq t_i\leq\Big\lfloor \frac{n}{2}\Big\rfloor.$$ Moreover, as noted above, we may assume that $0\leq e_i\leq\ell-1$. Hence $${\label{htestimate}} \boxed{h_K(d_n)\leq (\ell-1)\cdot\bigg(\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^i(b))+ \sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^j(0))\bigg)}$$ Now, choose (and fix) an integer $m\geq1$ such that the curve $$C_{m,\ell}(\phi): Y^\ell=\phi^m(X)$$ is nonsingular and of genus at least two - possible, since $\phi$ is dynamically $\ell$-power non-isotrivial. If $n\leq m$ for all $n\in\mathcal{Z}(\phi,b,\ell,S)$, then we are done. Otherwise, we may assume that $n>m$ so that (\[decomp\]) implies that $$P_n:=\big(\phi^{n-m}(b)\;,\;y_n\cdot\sqrt[\ell]{u_n\cdot d_n}\, \big)\in C_m(\phi)\big(K\big(\sqrt[\ell]{u_n\cdot d_n}\big)\big)\;\;.$$ It follows from any of the bounds (suitable to positive characteristic) discussed in the introductions of [@Kim] or [@htineq] that there exist constants $A_1, A_2>0$ such that $${\label{Szpiro}}
h_{\kappa(\phi,m)}(P_n)\leq A_1\cdot d(P_n)+A_2,$$ where $\kappa(\phi,m)$ is the canonical divisor of $C_{m,\ell}(\phi)$, $$d(P_n):=\frac{2\cdot {\operatorname{genus}}\big(K_n\big)-2}{\big[K_n:K\big]},\;\;\;\;\text{and}\;\;\;K_n:=K\big(\sqrt[\ell]{u_n\cdot d_n}\,\big).$$ Crucially, the bounds $A_i=A_i(\phi,\ell,m)$ are independent of both $b$ and $n$. We note that the bounds on (\[Szpiro\]) have been improved by Kim [@Kim Theorem 1], although we do not need them here.
On the other hand, it follows from [@ffields Prop. III.7.3 and Remark III.7.5] that $d(P_n)\leq B_1\cdot h_K(u_n\cdot d_n)+B_2$ for some positive constants $B_i=B_i(K)$. Likewise, $h_K(u_n\cdot d_n)\leq h_K(d_n)+B_3(K,S,\ell)$, since the height of $u_n$ is absolutely bounded. In particular, after combining these bounds with those on (\[Szpiro\]), we see that $${\label{combin1}}
h_{\kappa(\phi,m)}(P_n)\leq D_1\cdot h_K(d_n)+D_2, \;\;\;\;\;\;\; D_i=D_i(K,\phi,S,\ell,m)>0.$$ However, if $\mathcal{D}_1$ is an ample divisor on $C_{m,\ell}(\phi)$ and $\mathcal{D}_2$ is an arbitrary divisor, then $${\label{Divisor}} \lim_{h_{\mathcal{D}_1}(P)\rightarrow\infty}\frac{h_{\mathcal{D}_2}(P)}{h_{\mathcal{D}_1}(P)}=\frac{\deg\mathcal{D}_2}{\deg{\mathcal{D}_1}},\;\;\;\;\;\;P\in C_{m}(\phi);$$ see [@SilvA Thm III.10.2]. In particular, if $\pi:C_{m,\ell}(\phi)\rightarrow\mathbb{P}^1$ is the covering $\pi(X,Y)=X$, then a degree one divisor on $\mathbb{P}^1$ (giving the usual height $h_K$ on projective space) pulls back to a $\deg(\pi)$ divisor $\mathcal{D}_2$ on $C$ satisfying $h_{\mathcal{D}_2}(P)=h_K(\pi(P))$.
We deduce from (\[Divisor\]) that there exist constants $\beta$ and $E=E(\phi,m,\ell)$ such that $${\label{limit}} h_{\kappa(\phi,m)}(P)>\beta\;\;\;\;\;\text{implies}\;\;\;\;\; h_K(\pi(P))\leq E\cdot h_{\kappa(\phi,m)}(P)+1$$ for all $P\in C_m(\phi)(\bar{K})$. However, note that $${\label{finiteness}}
T:=\big\{P_n\;\big\vert\; h_{\kappa(\phi,m)}(P_n)\leq\beta\}\subseteq\big\{P\in C_m(\phi)(\bar{K})\;\big\vert\;\; h_{\kappa(\phi,m)}(P)\leq\beta\;\;\,\text{and}\,\;\; [K(P):K]\leq\ell\big\},$$ and the latter set is finite, since $\kappa(\phi,m)$ is ample; see [@SilvA Thm. 10.3]. Hence, (\[htestimate\]),(\[combin1\]), (\[limit\]), and (\[finiteness\]) together imply that $${\label{combin2}}
h_K(\phi^{n-m}(b))\leq F_1\cdot \bigg(\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^i(b))+ \sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^j(0))\bigg)+F_2$$ for all but finitely many $n\in Z(\phi,b,S,\ell)$ and some positive constants $F_i=F_i(K,\phi,S,\ell,m)$.
On the other hand, it is well known that the canonical height $\hat{h}_\phi$ satisfies: $$\text{(a).}\;\;\;\hat{h}_\phi=h_K+O(1)\;\;\;\;\;\;\;\;\;\;\;\; \text{(b).}\;\;\;\hat{h}_\phi(\phi^s(\alpha))=d^s\cdot \hat{h}_\phi(\alpha)$$ for all $\alpha\in K$ and all integers $s\geq0$; see [@Silv-Dyn Thm. 3.20]. In particular, (\[combin2\]) implies that $${\label{combin3}}
d^{n-m}\cdot\hat{h}_\phi(b)\leq G_1\cdot\bigg(\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}d^i\cdot \hat{h}_\phi(b)+ \sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}d^j\cdot\hat{h}_\phi(0) \bigg)+G_2\cdot n+G_3$$ for almost all $n\in Z(\phi,b,S,\ell)$ (those $n$ such that $P_n\notin T$) and some constants $G_i=G_i(K,\phi,S,\ell,m)$. In particular, for almost all $n\in Z(\phi,b,S,\ell)$, $$d^{n-m}\leq G\cdot \big(d^{\lfloor\frac{n}{2}\rfloor+1}+n+1\big),\;\;\;\;\text{where}\;\;G:=\max\bigg\{G_1,\frac{G_1\cdot \hat{h}_\phi(0)}{\hat{h}_\phi(b)},\frac{G_2}{\hat{h}_\phi(b)},\frac{G_2}{\hat{h}_\phi(b)}\bigg\}.$$ However, since $m$ is fixed, such $n$ are bounded.
[Dynamical Galois Groups]{} Our main interest in proving the finiteness of $\mathcal{Z}(\phi,b,\ell)$ comes from the Galois theory of iterates. In particular, if $\phi(x)=(x-\gamma)^2+c$ is a quadratic polynomial and $\mathcal{Z}(\phi,\gamma,2)$ is finite, then the Galois groups of $\phi^n$ are large [@RafeThesis Theorem 3.3], enabling us to compute the density of prime divisors $\mathcal{P}_\phi(b)$ (for all $b\in K$) via a suitable Chebotarev density theorem [@Eventually; @Jones].
To define the relevant dynamical Galois groups, let $\phi$ be a polynomial and assume that $\phi^n$ is separable for all $n\geq1$; hence, the set $T_n(\phi)$ of roots of $\phi, \phi^2,\dots ,\phi^n$ together with $0$, carries a natural $\deg(\phi)$-ary rooted tree structure: $\alpha,\beta\in T_n(\phi)$ share an edge if and only if $\phi(\alpha) =\beta$ or $\phi(\beta)=\alpha$. Furthermore, let $K_n:=K(T_n(\phi))$ and $G_n(\phi):={\operatorname{Gal}}(K_n/K)$. Finally, we set $${\label{Arboreal}}
T_\infty(\phi):=\bigcup _{n \geq 0} T_n(\phi)\;\;\text{and}\;\; G_\infty(\phi)=\varprojlim G_n(\phi).$$ Since $\phi$ is a polynomial with coefficients in $K$, it follows that $G_n(\phi)$ acts via graph automorphisms on $T_n(\phi)$. Hence, we have injections $G_n(\phi) \hookrightarrow {\operatorname{Aut}}(T_n(\phi))$ and $G_\infty(\phi) \hookrightarrow {\operatorname{Aut}}(T_\infty(\phi))$ called the *arboreal representations* associated to $\phi$.
A major problem in dynamical Galois theory, especially over global fields, is to understand the size of $G_\infty(\phi)$ in ${\operatorname{Aut}}(T_\infty(\phi))$; see [@B-J; @Me; @R.Jones; @Odoni; @Stoll]. We now use Theorem \[PrimDivThm\] to prove a finite index result for many quadratic polynomials (c.f. [@uniformity Theorem 1]), including the family defined in Corollary \[eg\], and provide an outline for further examples.
Let $\phi(x)=(x-\gamma)^2+c$ and let $m\geq2$. Then we have a map $${\label{map}} \Phi_m:C_{2,m}(\phi)\rightarrow E_\phi,\;\;\;\; \Phi(x,y)=\big(\phi^{m-1}(x),\;y\cdot(\phi^{m-2}(x)-\gamma)\big).$$ It follows from Proposition \[prop\] below that $C_{2,m}$ is non-isotrivial. On the other hand, since $\widebar{\mathcal{O}}_\phi(\gamma)$ contains no squares in $K$, [@Jones Proposition 4.2] implies that $\phi^n$ is an irreducible polynomial over $K$ for all $n$; hence, $C_{2,m}(\phi)$ and $E_\phi$ are non-singular; see [@Jones Lemma 2.6]. Therefore, we may choose $m$ so that $C_{2,m}(\phi)$ is a non-isotrivial curve of genus at least $2$. In particular, $\phi$ is dynamically 2-power non-isotrivial and Theorem \[PrimDivThm\] implies that $\mathcal{Z}(\phi,b,2)$ is finite for all wandering $b\in K$.
For the second claim, we apply this fact to $b=\gamma$ and use [@RafeThesis Theorem 3.3] to deduce that $G_\infty(\phi)\leq{\operatorname{Aut}}(T(\phi))$ is a finite index subgroup. Finally, [@R.Jones Theorem 4.2] an [@RafeThesis Theorem 1.3] imply that the density of $\mathcal{P}_\phi(b)$ is zero for all $b\in K$.
We now apply Corollary \[Galois\] to the family $\phi_{(f,g)}$ defined in Corollary \[eg\].
Let $K=\mathbb{F}_q(t)$ and let $\phi(x)=\phi_{(f,g)}=(x-f(t)\cdot g(t)^d)^2+g(t)$. It suffices to check conditions (a) and (b) of Corollary \[Galois\] hold to prove Corollary \[eg\]. We first show that the adjusted critical orbit of $\phi$, the set $\{-\phi(f\cdot g^d),\phi^2(f\cdot g^d),\phi^3(f\cdot g^d),\dots\}$, contains no perfect squares in $K$; in particular, $\phi^n$ is an irreducible polynomial over $K$ for all $n\geq1$; see [@Jones Proposition 4.2].
Note that $-\phi(f\cdot g^d)=-g$ is not a square in $K$ by assumption. On the other hand, we let $h:=g-f\cdot g^d$ and suppose that $${\label{iterate}} j^2=\phi^n(f\cdot g^d)=((((h^2+h)^2+h)^2+h)^2+\dots+h)^2+g$$ for some polynomial $j\in \mathbb{F}_q[t]$ and some $n\geq2$. Hence, $j^2=g^2\cdot k^2+g=g\cdot(g\cdot k^2+1)$ for some $k\in \mathbb{F}_q[t]$, since $g\vert h$. However, because $\mathbb{F}_q[t]$ is a UFD and $g$ and $g\cdot k^2+1$ are coprime, it follows that $g=l^2$ and $g\cdot h^2+1=m^2$ for some $l,m\in \mathbb{F}_q[t]$. In particular, $1=(m+l\cdot h)(m-l\cdot h)$ and both factors are constant. Hence, $2m=(m+l\cdot h)+(m-l\cdot h)$ is also constant. Finally, since $m^2-1=g\cdot h^2$ and $h=g-f\cdot g^d$, it follows that $g, h$, and $f$ are all constant. In particular, the $j$-invariant of $E_\phi$ is constant, a contradiction.
On the other hand, the right hand side of (\[iterate\]) implies that $$\deg(\phi^n(f\cdot g^d))\leq\max\big\{2^{n-1}\cdot\deg(h),\deg(g)\big\}=\max\big\{2^{n-1}\cdot[\deg(g)+\deg(1-f\cdot g^{d-1})],\deg(g)\big\},$$ with equality if the terms are unequal. In particular, if $\deg(h)\neq 0$, then we may choose $n$ large enough so that $\deg(\phi^n(f\cdot g^d))=2^{n-1}\cdot\deg(h)\geq 2^{n-1}$, and $\phi$ is post-critically infinite. Otherwise, we may assume that $h$ is constant. Consequently, since $h=g\cdot(1-f\cdot g^{d-1})$, we deduce that either $g$ is constant or $1-f\cdot g^{d-1}=0$. However, $g$ and $h$ constant implies that $f$ is constant, a contradiction. We deduce that $1-f\cdot g^{d-1}=0$. In particular, $g=f\cdot g^d$ and $\phi(x)=(x-g)^2+g$. In this case the elliptic curve $$E_\phi:\;Y^2=(X-g)\cdot\phi(X)=(X-g)\cdot\big((X-g)^2+g)$$ has $j$-invariant $1728$, contradicting our assumption that $E_\phi$ have non-constant $j$-invariant.
We expect that most $\phi\in K[x]$ are dynamically $\ell$-power non-isotrivial for some $\ell$, and we state a conjecture along these lines:
Suppose that $\phi\in K[x]{\mathbin{\fgebackslash}}\widebar{\mathbb{F}}_q[x]$ satisfies the following conditions:
1. $\deg(\phi)\geq2$,
2. $\phi$ is not conjugate to $x^d$ (as a function on $\mathbb{P}^1$)
3. $\gcd(\deg(\phi),q)=1$,
4. $0\notin \mathcal{O}_\phi(\gamma)$ for all critical points $\gamma\in\bar{K}$ of $\phi$.
Then there exists $\ell\geq2$ such that $\phi$ is dynamically $\ell$-power non-isotrivial and $\gcd(\ell,q)=1$.
Conditions (3) and (4) imply that the discriminant of $\phi^n$ is non-zero for all $n$; see [@Jones Lemma 2.6]. In particular, the curves $C_{m,\ell}(\phi)$ are non-singular and eventually of large genus, since $\deg(\phi^m)$ grows exponentially by condition (1).
We finish by proving an auxiliary result used in the proof of Corollary \[eg\]. The argument below was suggested by Bjorn Poonen.
[\[prop\]]{} Let $\Phi: C_1\rightarrow C_2$ be a non-constant morphism. If $C_1$ is isotrivial, then $C_2$ is isotrivial.
Suppose not. That is, suppose that $C_1$ is isotrivial but $C_2$ is not. By replacing $K$ by a finite extension, we may assume that $C_1$ is constant. We may also assume that $\Phi$ is separable (if not, then it factors as a power of Frobenius composed with a separable morphism, say $C_1 \rightarrow C_1' \rightarrow C_2$, and then (maybe after a finite extension) $C_1'$ is isomorphic to the curve obtained by taking the $p^n$-roots of all the coefficients of $C_1$, so $C_1'$ is constant too, and is separable over $C_2$; rename $C_1'$ as $C_1$).
We fix some notation. Let $g_i$ be the genus of $C_i$, and let $J_i$ be the Jacobian of $C_i$. Since $C_2$ is non-isotrivial, $g_2>0$.
Let $\Omega$ be an uncountable algebraically closed field containing $F_q$. Specializing (by choosing an $F_q$-homomorphism $\sigma: K \rightarrow \Omega$) gives separable maps over $\Omega$ from the same $C_1$ to uncountably many non-isomorphic curves $C_2^\sigma$. Each isogeny class of an elliptic curve over $\Omega$ consists of at most countably many elliptic curves, so if $g_2=1$, this would imply that in the decomposition of $J_i$ up to isogeny, uncountably many isogeny factors occur, which is impossible. If $g_2>1$, then the infinitely many maps from $C_1$ to various curves $C_2^\sigma$ contradict [@Samuel Theorem 2].
With some additional hypothesis, Proposition \[prop\] holds for projective varieties of arbitrary dimension [@isotrivial]. Such a result is necessary if one hopes to generalize Theorem \[PrimDivThm\] as in [@Silv-Vojta]
**Acknowledgements:** It is a pleasure to thank Joseph Silverman, Bjorn Poonen, Felipe Voloch, and Rafe Jones for the discussions related to the work in this paper.
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[^1]: 2010 *Mathematics Subject Classification*: Primary: 14G05, 37P55. Secondary: 11R32.
[^2]: *Key words and phrases*: Arithmetic Dynamics, Rational Points on Curves, Galois Theory.
| ArXiv |
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abstract: 'The spectra of two early B-type supergiant stars in the Sculptor spiral galaxy NGC 300 are analysed by means of non-LTE line blanketed unified model atmospheres, aimed at determining their chemical composition and the fundamental stellar and wind parameters. For the first time a detailed chemical abundance pattern (He, C, N, O, Mg and Si) is obtained for a B-type supergiant beyond the Local Group. The derived stellar properties are consistent with those of other Local Group B-type supergiants of similar types and metallicities. One of the stars shows a near solar metallicity while the other one resembles more a SMC B supergiant. The effects of the lower metallicity can be detected in the derived wind momentum.'
author:
- 'Miguel Alejandro Urbaneja, Artemio Herrero[^1] , Fabio Bresolin, Rolf-Peter Kudritzki[^2] , Wolfgang Gieren[^3] and Joachim Puls[^4]'
title: 'Quantitative spectral analysis of early B-type supergiants in the Sculptor galaxy NGC 300[^5]'
---
Introduction
============
The 8-10 meter class telescopes and their new generation instruments make it possible to extend the quantitative stellar spectroscopy beyond the Local Group. Early B-type supergiant stars are ideal targets for detailed spectroscopy even at low resolution (R$\sim$1000). Their blue spectra are rich in metal features which allows us the analysis of chemical species like C, N, O, Si and Mg. Although our knowledge of the evolution of massive stars still has open questions, most of the recent works indicate that the blue luminous supergiants do not show any contamination of their oxygen surface abundances during the early stages of their evolution, neither the O-types [@villamariz2002], nor the B-types [@smartt1997; @monteverde2000; @smartt2002], nor the A-types [@venn1995; @takeda1998; @przybilla2002], which enables a direct comparison between the stellar oxygen abundances and the ones derived from regions. This has become extremely important, especially in the extragalactic field where oxygen is used as the primary metallicity indicator, due to the fact that at high metallicity (larger than approx. 0.5 solar) strong line methods must be used, for which the choice of the calibration strongly influences the derived abundances [@kewley2002; @pilyugin2002]. In addition to chemical abundance studies, blue luminous stars have strong radiatively driven mass outflows which can provide us with information on extragalatic distances by means of the Wind Momentum - Luminosity Relationship, WLR [@kudritzki2000 and references therein].
Recently, and within a wide program aimed at the spectroscopy study of luminous blue stars beyond the Local Group, first steps have been done for A-type supergiants in NGC 3621 [6.7 Mpc away, @bresolin2001]. Quantitative spectroscopy has been shown to be possible for A-type supergiants [@bresolin2002a] and Wolf-Rayet stars [@bresolin2002b] in NGC 300, 2.02 Mpc away in the Sculptor group. Here we report the first quantitative analysis of B-type supergiants (hereafter B-Sg) out of the Local Group, presenting the detailed chemical pattern along with the stellar parameters and the wind properties. The technique will be applied in a forthcoming paper to a large set of early B-Sg located at several galactocentric distances in order to derive radial abundance gradients of the $\alpha$-elements. Combined with the results of a similar study of A-type supergiants it will provide a wealth of information on the chemical evolution of the host galaxy NGC 300.
Observations
============
The stars are part of a spectroscopic survey of photometrically selected blue luminous supergiants in the Sculptor galaxy NGC 300, obtained at the VLT with the FORS multiobject spectrograph, and described in detail by @bresolin2002a, which presents a spectral catalog of 70 luminous blue supergiants in the blue region ($\sim$ 4000 - 5000 Å). The selected stars are identified as B-12 and A-9 in that spectral catalog (see their Table 2 and finding charts). In September 2001 the spectra of the H$\alpha$ region were obtained in order to measure the mass-loss rates, which provide us with a complete coverage of the 3800 - 7200 Å wavelength range at R$\sim$1000 resolution. The reader is referred to @bresolin2002a for a detailed description of the observations and reduction process, as well as for the photometry and the spectral classification of the stars.
Spectral analysis
=================
The spectra of early B-Sg are dominated by the lines, followed by /, /, / and , in addition to H and lines. At high resolution it is possible to detect some other metal lines of , / and but, due to their intrinsic weakness, these lines do not have any influence in the analysis at low resolution and could hardly be used to fix the abundance of such elements. Fig. \[fig1\] shows the high resolution - high S/N ratio (R$\sim$15000, SNR$\sim$350) blue spectrum of the Galactic supergiant HD14956 (B1.5Ia), and the same spectrum degraded to the resolution of the NGC 300 data, R$\sim$1000 (labeled as [*\#d*]{} in the figure). We have also included the identification of the more important lines. As can be seen, only a few strong lines remain isolated at that low resolution, therefore the analysis must be based on the comparison of the observed spectra to a set of model atmospheres that include a vast number of lines in the calculation of the emergent fluxes. We have taken into account more than two hundreds metal lines in the 3800 - 6000 Å wavelength range. It is important to include extense metal line lists because of the fact that some spectral features are formed by the contribution of several chemical species (e.g. the strong blend of O, N and C at $\sim$ 4650 Å). We have excluded some strong isolated lines because our atomic models do not consider the levels involved in these transitions. Nevertheless, these lines are isolated and have no influence on the results.
Even considering the noise effects in the lower resolution FORS spectra (displayed also in Fig. \[fig1\]), strong metal features can still be detected and used for a detailed chemical abundance analysis. In particular a wealth of information can be extracted from the selected regions at 4070, 4320, 4420 (), 4550 - 4570 (), 4600 - 4660 (, , and ) and 5010 ( and ).
Atmosphere models
-----------------
We use the newest version of the FASTWIND code [first presented by @santolayarey1997] which solves the radiation transfer in a moving media by means of suitable approximations which simplify the numerical treatment of the problem but without affecting the physical significance of the results. The atmospheric structure is treated in a consistent way, assuming a $\beta$-velocity law in the wind, ensuring a smooth transition between the “photosphere” and the “wind”; the temperature structure is approximated by means of [*non-LTE Hopf functions*]{} carefully chosen to ensure the flux conservation better than 2 % at any depth point; rate equations are solved in the co-moving frame scheme, with the coupling between the radiation field and the rate equations solved using local ALOs [following @puls1991]. This new version includes the effects of the [*line blanketing*]{}. The reader is referred to Puls et al. (2003, in preparation) for a detailed description. We have analysed two Galactic stars, 10 Lac (O9V) and HD209975 (O9.5Ib) in order to compare our results with the ones obtained with other codes. In the case of 10 Lac, our results agree with the recent ones by @herrero2002 [see their comparison to the results by Hubeny et al. 1998]. The derived parameters for HD209975 are consistent with the results by @villamariz2002 which used plane-parallel model with line blocking.
A model is prescribed by the effective temperature $T_{eff}$, the surface gravity [*log g*]{}, the stellar radius $R_*$ (all these three quantities are defined at $\tau_{Ross.} = 2/3$), the mass-loss rate $\dot{M}$, the wind terminal velocity $v_\infty$, the $\beta$ exponent of the wind velocity law, the He abundance $Y_{He}$, the microturbulent velocity $v_{turb}$ and, in the case of B-type stars, the [*Si*]{} abundance. The $T_{eff}$ is well determined from the triplet and the blends of (with at 4090 Å and with / at 4120 Å), and the surface gravity from the Balmer hydrogen lines, provided that the mass-loss rate information is extracted from the H$\alpha$ profile. An important issue concerns the wind terminal velocity, that must be adopted from a spectral type - v$_\infty$ empirical calibration [@haser1995; @kudritzki2000]. The assumed terminal velocity affects the derived $\dot{M}$ and the [*log g*]{}. But, with the joined information from H$\alpha$ and H$\beta$, the mass-loss rate and v$_\infty$ can be constrained to yield reasonable uncertainties in [*log g*]{}. The stellar radius is derived interactively from the absolute magnitude, deduced from the apparent magnitude after adopting a distance modulus [$\mu =
26.53$, @freedman2001], and the model emergent flux [@kudritzki1999], which also provides the reddening by the comparison of the synthetic colors with the observed ones.
Results
-------
Best-fitting models are displayed in Fig. \[fig1\] and the results summarized in Tab. \[tabla1\]. The derived $\beta$ values are consistent with those obtained by @kudritzki1999 for Galactic B-Sg, with lower values excluded by the arise of emission wings in the synthetic H$\alpha$ profiles. We estimated an uncertainty of $\pm$0.25 in $\beta$. In the case of B-12 only the higher Balmer lines have been considered in the surface gravity determination, as the cores of H$\gamma$ and H$\beta$ are particularly affected by the sky substraction. As it has been quoted, the O and N abundances are very well constrained because of the large number of features from these species. The presence of a lot of weak metal lines in the 4600 - 4700 Å wavelength range makes the selection of the continuum level in this area difficult, good S/N ratio is also needed to disentangle between a real feature and the noise effects. Final abundance uncertainties are estimated to be $\pm\ 0.2$ [*dex*]{} from model comparisons (see Fig. \[fig4\]).
We define the mean metallicity as the sum of the $\alpha$-elements abundances, $X_{Si} +
X_{Mg} + X_{O}$ and refer it to the Sun abundances by @grevesse1998; at the early stages of massive star evolution, the O surface abundance is not affected by the CNO cycle, which means that the abundance of the $\alpha$-elements is a direct measurement of the ZAMS metallicity of the star. The results for B-12, located close to the galactic center, resembles the abundance patterns of the early B-Sg in the solar neighborhood, having a solar metallicity within the uncertainties of the analysis. On the other hand A-9, in the outskirts of the galaxy, has clearly a lower metallicity, around 0.3 $Z_\odot$. This is in agreement with the results for A-8, a B9-A0 supergiant close to A-9, by @bresolin2002a [see the Fig. 2]. These authors find a mean metallicity of 0.2 $Z_\odot$ for A-8. We must emphasize that both the model atmospheres and the metallicity indicators are different, but the results agree extremely well. The metallicity and the spatial location of both stars in NGC 300 points to a M33-like radial metallicity gradient. The CNO abundances indicate a different degree of chemical evolution, while B-12 displays a normal CNO spectrum, A-9 shows indications of strong N enrichment.
Synthetic magnitudes and colors (see Tab. \[tabla3\]) are consistent with almost no reddening for both stars, except the observed $(V-I)$ for B-12 that seems to be anomalous, probably reflecting the presence of the region. Fig. \[fig2\] shows the location of the stars on the Hertzprung-Russel diagram, along with theoretical stellar tracks without rotation at solar metallicity from @schaller1992. We have also added the location of the Galactic stars 10 Lac, HD209975 and HD14956 as a reference.
Comparing the wind momentum of both NGC 300 stars with the results for Galactic supergiants (Fig. \[fig5\]), B-12 agrees well with the results by Herrero et al. (2002) for O-type supergiants in the Galactic association Cyg OB2, as does HD14956. Note, however that the Herrero et al. (2002) stars are considerably hotter than the ones considered here. The wind momentum of A-9 is also compatible with the WLR of Galactic early B-Sg as derived by @kudritzki1999. With respect to this relationship, however, B-12 (being an early B-type supergiant as well) shows an enhanced wind momentum rate, which might be related to clumping effects in the wind that would produce an overestimation of the mass-loss rate. The failure of our models to reproduce the blue absorption of H$\alpha$ for B-12, in parallel with an H$\gamma$ core which is too strongly refilled might then be explained by this effect, at least in part, and not only by the rather problematic sky substraction outlined above. The location of A-9, compared to HD14956, reflects the lower metal content of the NGC 300 supergiant. It must be considered here that we have adopted the same $v_\infty$ for both stars, HD14956 and A-9, while a lower value for A-9 could be expected due to its lower metallicity [@kudritzki2000]. In any case the effect of the lower wind terminal velocity would reduce even more the wind momentum of A-9, reinforcing the difference with respect to the Galactic B1.5Ia.
Recently @kudritzki2003 have proposed a new extragalactic distance indicator, the “Flux-weighted - Luminosity Relationship (FGLR)”. The results for both NGC 300 B-type supergiants, B-12 and A-9, follow this relationship, within the observed scatter (see the Fig. 2 of the latter reference).
We are gratefull to L. J. Corral for making us available the spectrum of HD14956. MAU thanks F. Najarro for providing the routines for the computation of the synthetic magnitudes. AH and MAU thank the Spanish MCyT for a support under proyect PNAYA2001-0436, partially funded with FEDER funds from the EU. WG gratefully acknowledges financial support for this work from the Chilean Center for Astrophysics FONDAP 15010003.
Bresolin, F., Kudritzki, R.-P., Méndez, R. H., & Przybilla, N. 2001, , 548, 149
Bresolin, F., Gieren, W., Kudritzki, R.-P., Pietrzyńki, G., & Przybilla, N. 2002a, , 567, 277
Bresolin, F., Kudritzki, R.-P., Najarro, F., Gieren, W. & Pietrzyńki, G. 2002b, , 577, L107
Freedman, W. L., et al. 2001, , 553, 47
Grevesse, N. & Sauval, A. J. 1998, Space Sci. Rev., 85, 161
Haser, S. M. 1995, Ph.D. thesis, Ludwing-Maximillians Univ., Munich
Herrero, A., Puls, J., & Najarro, F. 2002, , in press
Hubeny, I., Heap, S. R., & Lanz, T. 1998, ASP Conf. Series Vol 131, 108
Kewley, L. J. & Dopita, M. A. 2002, , 142, 35
Kudritzki, R.-P., et al. 1999, , 350, 970
Kudritzki, R.-P. & Puls, J. 2000, , 38, 613
Kudritzki, R.-P., Bresolin, F., & Przybilla, N. 2003, , 582, 83
Monteverde, M. I., Herrero, A., & Lennon, D. J. 2000, , 545, 813
Pilyugin, L. S. 2002, preprint (astro-ph/0211319)
Przybilla, N. 2002, Ph.D. thesis, Ludwing-Maximillians Univ., Munich
Puls, J. 1991, , 248, 581
Santolaya-Rey, E., Puls, J., & Herrero, A. 1997, , 323, 488
Schaller, G., Schaerer, D., Meynet, G., & Maeder, G. 1992, , 96, 269
Smartt, S. J., Dufton, P. L., & Lennon, D. J. 1997, , 326, 763
Smartt, S. J., Lennon, D. J., Kudrityzki, R.-P., Rosales, F., Ryans, R. S. I., & Wright, N. 2002, , 979, 991
Takeda, Y. & Takada-Hidai, M. 1998, , 50, 629
Venn, K. A. 1995, , 449, 839
Villamariz, M. R., Herrero, A., Becker, S. R., & Butler, K. 2002, , 388, 940
Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2000, , 362, 295
[lccccccccccccccrc]{} B-12 & 24.0$\pm$1.0 & 2.60$\pm$0.15 & 43.5$\pm$1.5 & 0.10 & 20. & 1500. & 3.00$\pm$0.50 & 1.50 & 7.45 & 8.65 & 7.50 & 7.50 & 8.00 & 1.00 & 0.00 & 5.75$\pm$0.10\
A-9 & 21.0$\pm$1.0 & 2.50$\pm$0.15 & 32.0$\pm$1.0 & 0.10 & 15. & 800. & 0.25$\pm$0.07 & 2.00 & 7.10 & 8.30 & 7.20 & 8.00 & 7.60 & 0.30 & -0.50 & 5.24$\pm$0.11\
[lccccccccccc]{} B-12 & 19.30 & -0.18 & 0.00 & & -7.29 & -0.17 & -0.23 & -2.33 & & 0.00 & 0.23\
A-9 & 20.23 & -0.17 &…& & -6.36 & -0.16 & -0.20 & -1.97 & & 0.00 &…\
[^1]: Instituto de Astrofísica de Canarias, Vía Láctea S/N, E-38200 La Laguna, Canary Islands, Spain, [email protected], [email protected]; Dpto. de Astrofísica, Universidad de La Laguna, Avda. Astrofísico Francisco Sanchéz, E-38271 La Laguna, Canary Islands, Spain, [email protected]
[^2]: Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, Hawaii 96822, [email protected], [email protected]
[^3]: Universidad de Concepción, Departamento de Física, Casilla 160-C, Concepción, Chile, [email protected]
[^4]: Universitäts-Sternwarte München, Scheinerstr. 1, D-81679 München, Germany, [email protected]
[^5]: Based on observations obtained at the ESO Very Large Telescope
| ArXiv |
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abstract: 'We show how simple kinks and jumps of otherwise smooth integrands over ${\mathbb{R}}^d$ can be dealt with by a preliminary integration with respect to a single well chosen variable. It is assumed that this preintegration, or conditional sampling, can be carried out with negligible error, which is the case in particular for option pricing problems. It is proven that under appropriate conditions the preintegrated function of $d-1$ variables belongs to appropriate mixed Sobolev spaces, so potentially allowing high efficiency of Quasi Monte Carlo and Sparse Grid Methods applied to the preintegrated problem. The efficiency of applying Quasi Monte Carlo to the preintegrated function are demonstrated on a digital Asian option using the Principal Component Analysis factorisation of the covariance matrix.'
author:
- 'Andreas Griewank, Frances Y. Kuo, Hernan Leövey and Ian H. Sloan'
date: August 2017
title: |
High dimensional integration of kinks and jumps\
– smoothing by preintegration
---
Introduction {#sec:intro}
============
In the present paper we analyse a natural method for numerical integration over ${\mathbb{R}}^d$, where $d$ may be large, in the presence of “kinks” (i.e.discontinuities in the gradients) or “jumps” (i.e. discontinuities in the function). In this method one of the variables is integrated out in a “preintegration” step, with the aim of creating a smooth integrand over ${\mathbb{R}}^{d-1}$.
Integrands with kinks and jumps arise in option pricing, because an option is normally considered worthless if the value falls below a predetermined strike price. In the case of a continuous payoff function this introduces a kink, while in the case of a binary or other digital option it introduces a jump.
A simple strategy is to ignore the kinks and jumps, and apply directly a method for integration over ${\mathbb{R}}^d$. While there has been very significant recent progress in *Quasi Monte Carlo [(]{.nodecor}QMC[)]{.nodecor} methods* [@DKS13] and *Sparse Grid [(]{.nodecor}SG[)]{.nodecor} methods* [@BG04] for high dimensional integration when the integrand is somewhat smooth, there has been little progress in understanding their performance when the integrand has kinks or jumps.
The performance of QMC and SG methods is degraded in the presence of kinks and jumps, but perhaps not as much as might have been expected, given that in both cases the standard error analysis fails in general for kinks and jumps: the standard assumption in both cases is that the integrand has mixed first partial derivatives for all variables, or at least that it has bounded Hardy and Krause variation over the unit cube $[0,1]^d$, whereas even a straight non-aligned kink (one that is not orthogonal to one of the axes) lacks mixed first partial derivatives even for $d=2$, and generally exhibits unbounded Hardy and Krause variation on $[0,1]^d$ for $d\ge3$ [@Owen05].
A possible path towards understanding the performance of QMC and SG methods in the presence of kinks and jumps was developed in [@GKS13]. That paper studied the terms of the “ANOVA decomposition” of functions with kinks defined on $d$-dimensional Euclidean space ${{\mathbb{R}}}^d$, and showed that under suitable circumstances all but one of the $2^d$ ANOVA terms can be smooth, with the single exception of the highest order ANOVA term, the one depending on all $d$ of the variables. If the “effective dimension” of the function is small, as is commonly thought to be the case in applications, then that single non-smooth term can be expected to make a very small contribution to both supremum and ${{\mathcal{L}}}_2$ norms. In a subsequent paper [@GKSnote] the same authors showed, by strengthening the theorems and correcting a mis-statement in [@GKS13], that the smoothing of all but the highest order ANOVA term is a reality for the case of an arithmetic Asian option with the Brownian bridge construction.
More precisely, the papers [@GKS13] and [@GKSnote] showed, for a function of the form $f({{\boldsymbol{x}}})= \max(\phi({{\boldsymbol{x}}}),0)$ with $\phi$ smooth (so that $f$ generically has a kink along the manifold $\phi({{\boldsymbol{x}}})=0$), that if the $d$-dimensional function $\phi$ has a positive partial derivative with respect to $x_j$ for some $j\in\{1,\ldots,d\}$, and if certain growth conditions at infinity are satisfied, then all the ANOVA terms of $f$ that do not depend on the variable $x_j$ are smooth. The underlying reason, as explained in [@GKS13], is that integration of $f$ with respect to $x_j$, under the stated conditions, results in a $(d-1)$-dimensional function that no longer has a kink, and indeed is as often differentiable as the function $\phi$.
Going beyond kinks, we prove in this paper that Theorem 1 in [@GKSnote] can be extended from kinks to jumps – thus jumps are smoothed under almost the same conditions as kinks. The smoothing occurs even in situations (such as occur in option pricing) where the location of the kink or jump treated as a function of the other $d-1$ variables moves off to infinity for some values of the other variables.
In this paper we pay particular attention to proving that the presmoothed integrand belongs to an appropriate mixed-derivative function space.
The preintegration method studied in the present paper has appeared as a practical tool under other names in many other papers, including those related to “conditional sampling” (see [@GlaSta01]; the paragraph leading up to and including Lemma 7.2 in [@ACN13a]; the remark at the end of Section 3 in [@ACN13b]), and other root-finding strategies for identifying where the payoff is positive (see [@Hol11; @NuyWat12]), as well as those under the name “smoothing” (see [@BST17; @WWH17]). In contrast to the cited papers, the emphasis in this paper is on rigorous analysis. Also, we here prefer the description “preintegration” because to us “conditional sampling” suggests a probabilistic setting, which is not necessarily relevant here.
Even for the classical *Monte Carlo [(]{.nodecor}MC[)]{.nodecor} method* the preintegration step can be useful: to the extent that the preintegration can be considered exact, there is a reduction in the variance of the integrand, by the sum of the variances of all ANOVA terms that involve the preintegration variable $x_j$ (since the ANOVA terms are eliminated because their exact integrals with respect to $x_j$ are all zero). In our numerical experiments that reduction proves to be quite significant.
The problem class and the method are stated in Section \[sec:method\]. Immediately Section \[sec:app\] gives numerical examples in the context of an option pricing problem with 256 time steps, treated as a problem of integration in 256 dimensions. Section \[sec:var\] briefly discusses the variance reduction by preintegration for ${{\mathcal{L}}}_2$ functions. Section \[sec:smooth\] focuses on the smoothing effect of preintegration. It gives mathematical background on needed function spaces and states two new smoothing theorems, extended here in a non-trivial way from [@GKSnote Theorem 1]. Section \[sec:apply\] applies our theoretical results to the option pricing example. Technical proofs are given in Section \[sec:proof1\].
The problem and the method {#sec:method}
==========================
The problem is the approximate evaluation of $$\label{problem1}
I_d f
\,:=\, \int_{{\mathbb{R}}^d}f({{\boldsymbol{x}}})\rho_d({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}\,=\, \int_{-\infty}^\infty\ldots\int_{-\infty}^\infty
f(x_1,\ldots,x_d)\,\rho_d({{\boldsymbol{x}}})\,{{\mathrm{d}}}x_1 \cdots{{\mathrm{d}}}x_d,$$ with $$\rho_d({{\boldsymbol{x}}}) \,:=\, \prod_{k=1}^d \rho(x_k),$$ where $\rho$ is a continuous and strictly positive probability density function on ${\mathbb{R}}$ with some smoothness, and $f$ is a real-valued function integrable with respect to $\rho_d$.
To allow for both kinks and jumps we assume that the integrand is of the form $$\label{problem2}
f({{\boldsymbol{x}}}) \,=\, \theta({{\boldsymbol{x}}})\,{\mathop{\rm ind}}(\phi({{\boldsymbol{x}}})),$$ where $\theta$ and $\phi$ are somewhat smooth functions, and ${\mathop{\rm ind}}(\cdot)$ is the indicator function which gives the value $1$ if the input is positive and $0$ otherwise. When $\theta = \phi$ we have $f({{\boldsymbol{x}}}) =
\max(\phi({{\boldsymbol{x}}}),0)$ and thus we have the familiar kink seen in option pricing through the occurrence of a strike price. When $\theta$ and $\phi$ are different (for example, when $\theta({{\boldsymbol{x}}}) = 1$) we have a structure that includes binary digital options.
Our key assumption on $\phi({{\boldsymbol{x}}})$ is that it has a positive partial derivative (and so is an increasing function) with respect to some variable $x_j$, that is, we assume that for some $j\in\{1,\ldots,d\}$ we have $$\label{monotone}
\frac{\partial \phi}{\partial x_j}({{\boldsymbol{x}}}) > 0
\qquad \mbox{for all} \quad
{{\boldsymbol{x}}}\in {\mathbb{R}}^d.$$ In other words $\phi$ is monotone with respect to $x_j$.
We also make an assumption about the behavior as $x_j\to +\infty$. To state this it is convenient, given $j\in\{1,\ldots,d\}$, to write the general point ${{\boldsymbol{x}}}\in{\mathbb{R}}^d$ as ${{\boldsymbol{x}}}= (x_j,{{\boldsymbol{x}}}_{-j})$, where ${{\boldsymbol{x}}}_{-j}$ denotes the vector of length $d-1$ denoting all the variables other than $x_j$. With this notation, a second assumption is that $$\label{unbounded}
\lim_{x_j\to +\infty } \phi({{\boldsymbol{x}}})
\,=\, \lim_{x_j \to +\infty}\phi(x_j,{{\boldsymbol{x}}}_{-j})
\,=\, +\infty
\qquad\mbox{for fixed}\quad {{\boldsymbol{x}}}_{-j}\in {{\mathbb{R}}}^{d-1}.$$ The latter growth property follows automatically if we assume in addition to that $(\partial^2 \phi/ \partial x_j^2)({{\boldsymbol{x}}})\ge0$ for all ${{\boldsymbol{x}}}\in {{\mathbb{R}}}^d$. Additional properties at infinity will be assumed in Theorems \[thm:main1\] and \[thm:main2\].
Assuming that the properties and both hold for some $j\in\{1,\ldots,d\}$, the method is easily described: we write as the repeated integral using Fubini’s theorem $$I_d f
\,=\, \int_{{\mathbb{R}}^{d-1}}\left(\int_{-\infty}^\infty f(x_j,{{\boldsymbol{x}}}_{-j})\,\rho(x_j)\,{{\mathrm{d}}}x_j \right)
\rho_{d-1}({{\boldsymbol{x}}}_{-j})\,{{\mathrm{d}}}{{\boldsymbol{x}}}_{-j},$$ and first evaluate the inner integral for each needed value of ${{\boldsymbol{x}}}_{-j}$. This is the “*preintegration*” step. The essential point of the method is that the outer integral can then be evaluated by a standard QMC or SG method, in the knowledge that the integrand for this $(d-1)$-dimensional integral is smooth.
Looking more closely at the preintegration step, we write the operation of integration with respect to $x_j$ as $$\label{firstPj}
(P_j f)({{\boldsymbol{x}}}_{-j})
\,:=\, \int_{-\infty}^\infty f(x_j,{{\boldsymbol{x}}}_{-j})\,\rho(x_j)\,{{\mathrm{d}}}x_j.$$ It follows from and that the integrand in this integral has generically a jump at the (unique) point at which $\phi(x_j, {{\boldsymbol{x}}}_{-j})$ passes through zero. By the implicit function theorem (see Theorem \[thm:implicit\] below) for each ${{\boldsymbol{x}}}_{-j}$ there is a unique value $\psi({{\boldsymbol{x}}}_{-j})$ of $x_j$ at which $\phi(x_j,{{\boldsymbol{x}}}_{-j})$ passes from negative to positive values with increasing $x_j$. The preintegration step may then be written as $$(P_j f)({{\boldsymbol{x}}}_{-j})
\,=\,
\int_{\psi({{\boldsymbol{x}}}_{-j})}^\infty f(x_j,{{\boldsymbol{x}}}_{-j})\,\rho(x_j )\,{{\mathrm{d}}}x_j.$$ An essential ingredient in any implementation of the method is the accurate evaluation of $\psi({{\boldsymbol{x}}}_{-j})$, for each point ${{\boldsymbol{x}}}_{-j}$ of the outer integration rule. The semi-infinite integral $P_jf$ may then be evaluated, for each needed point ${{\boldsymbol{x}}}_{-j}$, by a standard method for 1-dimensional integrals, for example by a formula of Gauss type. On the other hand, in certain important applications such as option pricing, the integration can be performed in more or less closed form.
The monotonicity condition and the infinite growth condition imply that for fixed ${{\boldsymbol{x}}}_{-j}$ the function $\phi(x_j, {{\boldsymbol{x}}}_{-j})$ either has a simple root $x_j=\psi({{\boldsymbol{x}}}_{-j})$ or is positive for all $x_j\in{\mathbb{R}}$. The zero set of $\phi$, denoted by $$\phi^{-1}(0) \,:=\, \{{{\boldsymbol{x}}}\in{\mathbb{R}}^d:\phi({{\boldsymbol{x}}})=0\},$$ is then a hypersurface, i.e., a continuous manifold of dimension $d-1$. However, its projection onto ${\mathbb{R}}^{d-1}$ obtained by ignoring the component $x_j$ can be very complicated, even if $\phi$ is highly differentiable.
An example with $d=2$ and $j=1$ illustrating the complications that can arise is given by $$\begin{aligned}
\label{eq:example}
\phi(x_1,x_2) \,:=\, \begin{cases}
\exp(x_1) - x_2^m \sin(1/x_2) & \mbox{for }\, x_2>0,\\
\exp(x_1) & \mbox{for }\, x_2\le 0, \end{cases}\end{aligned}$$ for some large $m$. Since $\phi(x_1,x_2)$ is monotonically increasing in $x_1$, the explicit solution of $\phi(x_1,x_2)=0$ is $$x_1 \,= \, \psi(x_2) \,:=\, m \log(x_2)
+ \log((\sin(1/x_2))_+) \quad \mbox{for }\, x_2 \in U_1,$$ where $z_+ :=
\max(0,z)$, and $$\begin{aligned}
U_1 \,:=\,
\{x_2\in {\mathbb{R}}\;:\; \phi(x_1,x_2)=0 \;\mbox{ for some }\,x_1\in{\mathbb{R}}\}
\,=\, \left (\tfrac{1}{\pi}, \infty \right ) \,\cup\,
\bigcup_{k \in {\mathbb{N}}} \left ( \tfrac{1}{ (2 k+1)\pi}, \tfrac{1}{ 2 k\pi} \right ),\end{aligned}$$ while $\phi(x_1,x_2)=0$ has no solution for $x_2$ in the complicated complementary set $$U_1^+ \,:=\,
(-\infty, 0] \,\cup\, \bigcup_{k \in {\mathbb{N}}} \left [ \tfrac{1}{ 2 k\pi}, \tfrac{1}{(2 k- 1)\pi} \right ].$$
![The zero set $\phi^{-1}(0)$ for the example in with $m=2$.[]{data-label="figure:example"}](Graph_Psi_color){width="12cm" height="14cm"}
The graph of the zero set $\phi^{-1}(0)$ for $m=2$ is shown in Figure \[figure:example\]. The solid lines represent the zero set, while the broken lines parallel to the horizontal axis define the boundaries on the $x_2$ axis between subsets of $U_1^+$ for which there is no solution of $\phi(x_1,x_2) = 0$ and subsets of $U_1$ for which there is a solution $x_1 = \psi(x_2)$. The preintegrated version of $f$ given by for any smooth $\theta$ will rather clearly be differentiable on both $U_1$ and $U_1^+$, but it is not obvious that this is the case on the complicated boundary between the two sets. To ensure the necessary differentiability properties it turns out in Section \[sec:smooth\] to be necessary to assume that the functions $\theta$ and $\phi$ and their derivatives, when multiplied by the appropriate weight functions, decay sufficiently rapidly as $x_1= \psi(x_2)$ runs to $-\infty$.
Numerical experiments: application to option pricing {#sec:app}
====================================================
In this section we apply the preintegration method to an option pricing example, for which the payoff function is discontinuous.
An important aspect of the method presented in this paper is that the user needs to choose a variable $x_j$ such that the condition is satisfied. In the paper [@GKS13] it is shown that for the standard and Brownian bridge constructions for path simulation of Brownian motions every choice of the variable $x_j$ will be suitable. More interesting for the present paper is the popular Principal Component Analysis (or PCA) method of constructing the Brownian motion [@Glasserman]: for this case the only result known to us, from [@GKS10 Section 5], is that the property is guaranteed if $x_j$ is the variable that corresponds to the largest eigenvalue of the Brownian motion covariance matrix. For this reason it is of particular interest to apply the present theory to the PCA case, as we do below.
For our tests, we consider now the example of an arithmetic average digital Asian option. We assume that the underlying asset $S_t$ follows the *geometric Brownian motion* model based on the stochastic differential equation $$\label{eq:brown:sode}
{{\mathrm{d}}}S_t \,=\, r S_t \,{{\mathrm{d}}}t + \sigma S_t \,{{\mathrm{d}}}W_t\,,$$ where $r$ is the risk-free interest rate, $\sigma$ is the (constant) volatility and $W_t$ is the standard Brownian Motion. The solution of this stochastic equation can be given as $$\label{eq:brown:solution}
S_t \,=\, S_0 \exp \left ( \left ( r - \tfrac{\sigma^2}{2}\right ) t + \sigma W_t \right).$$ The problem of simulating asset prices can be reduced to the problem of simulating discretized Brownian motion paths taking values , where $d$ is the number of time steps taken in the disctretization of the continuous time period $[0, T]$. In our tests, the asset prices are assumed to be sampled at equally spaced times $t_\ell:=\ell \Delta t$, $\ell=1,\dots,d$, where $\Delta t := T/d $. The Brownian motion is a Gaussian process, therefore the vector $(W_{t_1},\dots,W_{t_d})$ is normally distributed, and in this particular case is a vector with mean zero and covariance matrix $C$ given by $$C \,=\, (\min( t_\ell,t_k ))_{\ell,k=1}^d .$$
The value of an arithmetic average digital Asian call option is $$V \,=\, \frac{e^{-rT}}{(2\pi)^{d/2} \sqrt{\det(C)}}
\int_{{\mathbb{R}}^d}
{\rm ind}\bigg( \frac{1}{d}\,\sum_{\ell=1}^{d} S_{t_\ell}(w_\ell) - K\bigg)
e^{- \frac{1}{2} {{\boldsymbol{w}}}^\top C^{-1} {{\boldsymbol{w}}}} \,{{\mathrm{d}}}{{\boldsymbol{w}}}\,,$$ with ${{\boldsymbol{w}}}=(W_{t_1},\dots,W_{t_d})^\top$. After a factorization $C=AA^\top$ of the covariance matrix is chosen (for the choice of $A$ is not unique), we can rewrite the integration problem using the substitution ${{\boldsymbol{w}}}=A{{\boldsymbol{x}}}$ as $$V \,=\, \frac{e^{-rT}}{(2\pi)^{d/2}}
\int_{{\mathbb{R}}^d}
{\rm ind}\bigg( \frac{1}{d}\,\sum_{\ell=1}^{d} S_{t_\ell}((A{{\boldsymbol{x}}})_\ell) - K\bigg)
e^{- \frac{1}{2} {{\boldsymbol{x}}}^\top {{\boldsymbol{x}}}} \,{{\mathrm{d}}}{{\boldsymbol{x}}}\,.$$ The new variable vector ${{\boldsymbol{x}}}=(x_1,\ldots,x_d)^{\top}$ can be assumed to consist of independent standard normally distributed random variables. Then the identity ${{\boldsymbol{w}}}=A{{\boldsymbol{x}}}$ defines a construction method for Brownian paths. We therefore have an integral of the form – with $\rho(x) = e^{-x^2}/\sqrt{2\pi}$, $\theta({{\boldsymbol{x}}}) = e^{-rT}$, and $$\label{phi_explicit}
\phi({{\boldsymbol{x}}}) \,=\,
\frac{S_0}{d} \sum_{\ell=1}^d
\exp\bigg( \left(r-\tfrac{\sigma^2}{2}\right) \ell\Delta t
+\sigma \sum_{k=1}^d A_{\ell k}\, x_k\bigg) -K.$$
We use in our experiments the PCA factorization of $C$, which is based on the orthogonal factorization $$C \,=\, ({{\boldsymbol{u}}}_{1};\ldots;{{\boldsymbol{u}}}_{d})\,
{\rm diag}(\lambda_{1},\ldots,\lambda_{d})\,({{\boldsymbol{u}}}_{1};\ldots;{{\boldsymbol{u}}}_{d})^{\top},$$ where the eigenvalues $\lambda_{1},\ldots,\lambda_{d}$ (all positive) are given in non-increasing order, with corresponding unit-length column eigenvectors ${{\boldsymbol{u}}}_1\ldots,{{\boldsymbol{u}}}_d$, and as a result $$A \,=\, (\sqrt{\lambda_{1}}{{\boldsymbol{u}}}_{1};\ldots;\sqrt{\lambda_{d}}{{\boldsymbol{u}}}_{d}).$$ Note that we have $A_{\ell 1}>0$ for $1\le \ell \le d$ because the elements of the eigenvector ${{\boldsymbol{u}}}_1$ are all positive.
For approximate integration with quadrature, we generate randomized QMC or MC samples ${{\boldsymbol{x}}}^{(1)},\dots,{{\boldsymbol{x}}}^{(N)}$ over ${\mathbb{R}}^d$ by first generating classical randomized QMC or MC samples over unit cube $(0,1)^d$, and then transforming them to ${\mathbb{R}}^d$ using in each coordinate the univariate inverse normal cumulative distribution function $\Phi^{-1}({\cdot})$. The randomized QMC points over $(0,1)^d$ are obtained by first generating Sobol$'$ points over $[0,1]^d$ with direction numbers taken from [@JoeKuo08], and then applying the random linear-affine scrambling method as proposed by Matousek [@Matousek] (as implemented in the statistics toolbox of MATLAB). Note that taking randomly scrambled Sobol$'$ points not only allows us to generate statistically independent QMC samples, but also allows us to avoid in practice having points lying on the boundary of $(0,1)^d$ (which is usually the case for non-randomized QMC points), since the boundary is sampled with zero probability. The MC points were taken from the Mersenne Twister PRNG. For the function $\Phi^{-1}({\cdot})$, we have used Moro’s algorithm [@Glasserman]. The matrix $A$ can be given explicitly [@GKS10], but more importantly, each matrix-vector multiplication $A{{\boldsymbol{x}}}^{(i)}, 1\le i \le N$ can be done with $O(d\,\log d)$ cost by means of the fast-sine transform [@Sche07] (as long as time steps for discretization are taken of equal size).
For the preintegration approach, we generate randomized QMC or MC points over $[0,1]^{d-1}$, following the procedure for the $d$-dimensional case, and so obtain $N$ sample points over ${\mathbb{R}}^{d-1}$. We then evaluate the paths without using the first variable $x_1$, i.e., we sample over the coordinates $x_2,\dots,x_d$. Once a sample point on these coordinates is fixed, the resulting problem is a one-dimensional integral on the variable $x_1$. We take then the approximation $$V \,\approx\, Q_{N,d-1} \left( P_1(f)\right)
\,=\, P_1\left( Q_{N,d-1}(f) \right)
\,=\, \frac{1}{N} \sum_{i=1}^N \int_{-\infty}^\infty f(\xi,{{\boldsymbol{x}}}^{(i)})\,\rho(\xi) \,{{\mathrm{d}}}\xi,$$ where the quadrature with respect to $\xi$ is to be carried out for each of the $N$ sample points ${{\boldsymbol{x}}}^{(i)}$ in ${\mathbb{R}}^{d-1}$. In the PCA case in this problem, the resulting $N$ univariate integrals can be calculated in terms of the normal cumulative distribution function by completing squares and identifying the points $\xi^{(i)}_\star, 1\le i \le N$ (if they exist), where we have $\frac{1}{d}\,\sum_{\ell=1}^{d} S_{t_\ell}((A
(\xi^{(i)}_\star,{{\boldsymbol{x}}}^{(i)})^{\top})_\ell)=K$. Finding the points $\xi^{(i)}_\star, 1\le i \le N$, is not a difficult numerical task since each of them can be obtained as the root of an equation defined by a univariate convex function, for which Newton’s method converges in few steps to a satisfactorily accurate solution.
![Root mean square errors for (from left to right) Monte Carlo, Quasi Monte Carlo, preintegrated Monte Carlo and preintegrated Quasi Monte Carlo[]{data-label="fig:PCA"}](Real_Binary_new3){width="16cm" height="8cm"}
The parameters in our tests were fixed to $K=100,S_0=100, r=0.1 ,
\sigma=0.1, T=1$. We summarize our numerical experiments in Figure \[fig:PCA\]. In the figure we show the box-plots of the $\log_{10}$ of relative root mean square error (RMSE), each obtained from 10 independent random replications, with PCA factorization of covariance matrix for the arithmetic average digital Asian option. Results are shown in four groups containing three box-plots each. Each group corresponds to one of the following method: in order, MC, QMC, MC with preintegration and QMC with preintegration. In each group we have three box-plots to characterize the error convergence, each box-plot containing RMSE sampled with a given sample size. For all integration methods we chose the sample sizes $N=2^{12},2^{14},2^{16}$. Note that for MC and QMC we generate samples over ${\mathbb{R}}^d$, while for the preintegration MC and QMC we generate samples only over ${\mathbb{R}}^{d-1}$.
The results show that randomized QMC exhibits higher convergence than MC, but the convergence rate is still not optimal ($\sim N^{-0.6}$). When we combine the preintegration method with MC, we observe an improvement in the implied error constant, as predicted, but the convergence rate remains the same as MC ($= N^{-0.5}$), as of course it should. Combining the preintegration method with randomized QMC reduces the error satisfactorily, and improves the convergence rate to close to the best possible rate $N^{-1.0}$.
Variance reduction by preintegration {#sec:var}
====================================
In this section we consider the space ${{\mathcal{L}}}_{2,\rho_d}$ of square-integrable functions on ${{\mathbb{R}}}^d$, with $\rho_d$-weighted ${{\mathcal{L}}}_2$ inner product and norm.
The preintegration step can be viewed more generally as a projection, which is the key operation underlying the well-known ANOVA decomposition. For a general function $g\in {{\mathcal{L}}}_{2,\rho_d}$ the ANOVA decomposition takes the form [@Sobol90] $$\label{anova}
g \,=\, \sum_{{{\mathfrak{u}}}\subseteq {{\mathfrak{D}}}} g_{{\mathfrak{u}}},$$ where the sum is over all the $2^d$ subsets of ${{\mathfrak{D}}}:=\{1,\ldots,d\}$, and each term $g_{{\mathfrak{u}}}$ depends only on the variables $x_k$ with $k\in{{\mathfrak{u}}}$, and with the additional property that the projection operator $P_k$ defined by (as in ) $$(P_k g)({{\boldsymbol{x}}}_{-k}) \,:=\, \int_{-\infty}^\infty g(x_k, {{\boldsymbol{x}}}_{-k})\,\rho(x_k)\,{{\mathrm{d}}}x_k$$ annihilates all ANOVA terms $g_{{\mathfrak{u}}}$ with $k\in{{\mathfrak{u}}}$: $$\label{Pkprop}
P_k g_{{\mathfrak{u}}}=0 \quad\mbox{for}\quad k\in{{\mathfrak{u}}}, \quad \mbox{whereas}\quad
P_k g_{{\mathfrak{u}}}=g_{{\mathfrak{u}}}\quad\mbox{for}\quad k\notin{{\mathfrak{u}}}.$$ The ANOVA terms can be written explicitly as [@KSWW10] $$g_{{\mathfrak{u}}}\,=\, \sum_{{{\mathfrak{v}}}\subseteq{{\mathfrak{u}}}} (-1)^{|{{\mathfrak{u}}}|-|{{\mathfrak{v}}}|} \bigg(\prod_{k\notin{{\mathfrak{v}}}} P_k\bigg) g.$$
It follows from , since $I_d$ involves integration with respect to every variable $x_k$ for $k\in{{\mathfrak{D}}}$, that $$I_d g \,=\, g_\emptyset.$$ Another consequence is that the ANOVA terms are orthogonal in ${{\mathcal{L}}}_{2,
\rho_d}$, $$\int_{{\mathbb{R}}^d} g_{{\mathfrak{u}}}({{\boldsymbol{x}}})\,g_{{\mathfrak{v}}}({{\boldsymbol{x}}})\,\rho_d({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}= 0
\quad \mbox{for} \quad {{\mathfrak{u}}}\ne{{\mathfrak{v}}}.$$ As a result, the variance of $g$ has the well known property that it is a sum of the variances of the separate ANOVA terms, $$\label{eq:var}
\sigma^2(g)
\,=\, \sum_{\emptyset\ne {{\mathfrak{u}}}\subseteq{{{\mathfrak{D}}}}} \sigma^2(g_{{\mathfrak{u}}}),$$ where $$\sigma^2(g) \,:=\,
\int_{{\mathbb{R}}^d} g^2({{\boldsymbol{x}}})\,\rho_d({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}- g_\emptyset^2
\qquad\mbox{and}\qquad
\sigma^2(g_{{\mathfrak{u}}}) \,=\, \int_{{\mathbb{R}}^d} g_{{\mathfrak{u}}}^2({{\boldsymbol{x}}})\, \rho_d({{\boldsymbol{x}}}) \,{{\mathrm{d}}}{{\boldsymbol{x}}}\quad \mbox{for}\quad {{\mathfrak{u}}}\ne\emptyset.$$
With these preparations, we are now ready to make a simple observation that preintegration is a variance reduction strategy for any general ${{\mathcal{L}}}_2$ function, not specific to our functions with kinks or jumps. This explains why the preintegration strategy improves the performance of MC methods.
\[lem:var\] The projection $P_k$ reduces the variance of $g$ for all $g \in
{{\mathcal{L}}}_{2.\rho_d}$ and all $k\in{{\mathfrak{D}}}$.
For any $g\in{{\mathcal{L}}}_{2,\rho_d}$ and any $k\in{{\mathfrak{D}}}$, it follows from and that $$ P_k g \,=\, \sum_{k\notin{{\mathfrak{u}}}\subseteq{{\mathfrak{D}}}} g_{{\mathfrak{u}}},$$ that is, the operation $P_k$ applied to $g$ has the effect of annihilating those ANOVA terms $g_{{\mathfrak{u}}}$ of $g$ with $k\in{{\mathfrak{u}}}$. As a result, the ANOVA terms of the resulting function $P_kg$ are precisely the ANOVA terms $g_{{\mathfrak{u}}}$ of $g$ for which $k\notin{{\mathfrak{u}}}$. Hence we have $$\label{eq:var-Pk}
\sigma^2(P_k g)
\,=\, \sum_{k\notin{{\mathfrak{u}}}\subseteq{{{\mathfrak{D}}}},\,{{\mathfrak{u}}}\ne\emptyset} \sigma^2(g_{{\mathfrak{u}}}).$$ The result follows by comparing with .
Smoothing by preintegration {#sec:smooth}
===========================
In this section we first slightly generalize the mathematical setting from [@GKS13], providing some details on Sobolev spaces and weak derivatives which are needed for the formulation of our main smoothing theorems. Then we establish two new smoothing theorems for these Sobolev spaces, extending [@GKSnote Theorem 1] from kink to jumps.
Sobolev spaces with generalized weight functions {#sec:sob}
------------------------------------------------
Following [@GKS13 Section 2.2], for $j\in{{\mathfrak{D}}}$ let $D_j$ denote the partial derivative operator $$(D_jg)({{\boldsymbol{x}}}) \,=\, \frac{\partial g}{\partial x_j}({{\boldsymbol{x}}}).$$ Throughout this paper, the term *multi-index* refers to a vector ${{\boldsymbol{\alpha}}}=(\alpha_1,\ldots,\alpha_d)$ whose components are nonnegative integers, and we use the notation $|{{\boldsymbol{\alpha}}}| = \alpha_1+\cdots+\alpha_d$ to denote the sum of its components. For any multi-index ${{\boldsymbol{\alpha}}}=(\alpha_1,\ldots,\alpha_d)$, we define $$\label{Dalpha}
D^{{\boldsymbol{\alpha}}}\,=\, \prod_{j=1}^d D_j^{\alpha_j}
\,=\, \prod_{j=1}^d \left(\frac{\partial}{\partial x_j}\right)^{\alpha_j}
\,=\, \frac{\partial^{|{{\boldsymbol{\alpha}}}|}}{\prod_{j=1}^d \partial
x_j^{\alpha_j}},$$ and we say that the derivative $D^{{\boldsymbol{\alpha}}}f$ is of order $|{{\boldsymbol{\alpha}}}|$.
Let ${{\mathcal{C}}}({{\mathbb{R}}}^d) = {{\mathcal{C}}}^0({{\mathbb{R}}}^d)$ denote the linear space of continuous functions defined on ${{\mathbb{R}}}^d$. For a nonnegative integer $r\ge
0$, we define ${{\mathcal{C}}}^r({{\mathbb{R}}}^d)$ to be the space of functions whose classical derivatives of order $\le r$ are all continuous at every point in ${{\mathbb{R}}}^d$, with no limitation on their behavior at infinity. For example, the function $g({{\boldsymbol{x}}}) = \exp(\sum_{j=1}^d x_j^2)$ belongs to ${{\mathcal{C}}}^r({{\mathbb{R}}}^d)$ for all values of $r$. For convenience we write ${{\mathcal{C}}}^\infty({{\mathbb{R}}}^d) = \cap_{r\ge 0} {{\mathcal{C}}}^r({{\mathbb{R}}}^d)$.
In addition to classical derivatives, we shall need also *weak* derivatives. By definition, the weak derivative $D^{{\boldsymbol{\alpha}}}g$ is a measurable function on ${{\mathbb{R}}}^d$ which satisfies $$\label{weak}
\int_{{{\mathbb{R}}}^d} (D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{x}}})\,v({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}\,=\, (-1)^{|{{\boldsymbol{\alpha}}}|} \int_{{{\mathbb{R}}}^d} g({{\boldsymbol{x}}})\,(D^{{\boldsymbol{\alpha}}}v)({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}\quad\mbox{for all}\quad
v\in {{\mathcal{C}}}^\infty_0({{\mathbb{R}}}^d),$$ where ${{\mathcal{C}}}^\infty_0({{\mathbb{R}}}^d)$ denotes the space of infinitely differentiable functions with compact support in ${{\mathbb{R}}}^d$, and where the derivatives on the right-hand side of are classical partial derivatives. It can be shown, using the definition , that $D_jD_k = D_kD_j$ for all $j,k\in{{\mathfrak{D}}}$, thus the ordering of the weak first derivatives that make up $D^{{\boldsymbol{\alpha}}}$ in is irrelevant.
If $g$ has classical continuous derivatives up to order $|{{\boldsymbol{\alpha}}}|$, then they satisfy , which in the classical sense is just the integration by parts formula on ${{\mathbb{R}}}^d$. Unless stated otherwise, the derivatives in this paper are weak derivatives, which in principle allows the possibility that they are defined only “almost everywhere”. However, a recurring theme is that our weak derivatives are shown to be continuous (or strictly, can be represented by continuous functions), in which case the weak derivatives are at the same time classical derivatives.
We now turn to the definition of the function spaces. For $p \in
[1,\infty]$, we first define weighted ${{\mathcal{L}}}_p$ norms: $$ \|g\|_{{{\mathcal{L}}}_{p,\widetilde\rho_d}} \,=\,
\begin{cases}
\left(\int_{{{\mathbb{R}}}^d} |g({{\boldsymbol{x}}})|^p\,\widetilde\rho_d({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}\right)^{1/p}
& \mbox{if } p\in [1,\infty), \\
{\operatornamewithlimits{ess\,sup}}_{{{\boldsymbol{x}}}\in {{\mathbb{R}}}^d} |g({{\boldsymbol{x}}})| & \mbox{if } p = \infty
\end{cases}$$ where $\widetilde\rho_d$ is a positive integrable function on ${\mathbb{R}}^d$.
When dealing with function spaces of derivatives of a function $g$, it turns out to be convenient to allow flexibility in the choice of weight function $\widetilde\rho_d$. We therefore generalize the setting in [@GKS13] and introduce a family $\zeta_{d,{{\boldsymbol{\alpha}}}}$ of such weight functions, one for each derivative $D^{{{\boldsymbol{\alpha}}}}$, given by $$\label{zetaprod}
\zeta_{d,{{\boldsymbol{\alpha}}}}({{\boldsymbol{x}}})
\,:=\, \prod_{k=1}^d \zeta_{\alpha_k}(x_k),$$ where $\{\zeta_i\}_{i\ge 0}$ is a sequence of continuous integrable functions on ${{\mathbb{R}}}$, satisfying $$\label{zbiggerrho}
\rho(x) \,=\, \zeta_0(x) \,\le\, \zeta_1(x) \,\le\, \zeta_2(x) \,\le\, \cdots \qquad\mbox{for all}\quad x\in{\mathbb{R}}.$$ The intuitive idea is that higher derivatives with respect to every coordinate need to be limited in their growth towards infinity by making $\zeta_{i}$ decay more slowly for larger order of derivatives $i$.
With these generalized weight functions $\zeta_{d,{{\boldsymbol{\alpha}}}}$, denoted collectively by ${{\boldsymbol{\zeta}}}$, we consider two kinds of Sobolev space: the *isotropic Sobolev space* with smoothness parameter $r\ge 0$, for $r$ a nonnegative integer, $$ {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r \,=\,
\left\{ g \,:\, D^{{\boldsymbol{\alpha}}}g \in {{\mathcal{L}}}_{p,\zeta_{d,{{\boldsymbol{\alpha}}}}}
\quad\mbox{for all}\quad |{{\boldsymbol{\alpha}}}| \le r \right\},$$ and the *Sobolev space of dominating mixed smoothness* with smoothness multi-index ${{\boldsymbol{r}}}=(r_1,\ldots,r_d)$, $$ {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}},{\mathrm{mix}}}^{{\boldsymbol{r}}}\,=\,
\left\{ g \,:\, D^{{\boldsymbol{\alpha}}}g \in {{\mathcal{L}}}_{p,\zeta_{d,{{\boldsymbol{\alpha}}}}}
\quad\mbox{for all}\quad {{\boldsymbol{\alpha}}}\le{{\boldsymbol{r}}}\right\},$$ where ${{\boldsymbol{\alpha}}}\le{{\boldsymbol{r}}}$ is to be understood componentwise, and the derivatives are weak derivatives. For convenience we also write ${{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^0 = {{\mathcal{L}}}_{p,\rho_d}$ and ${{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^\infty
= \cap_{r\ge 0} {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r$. Analogously, we define ${{\mathcal{C}}}^{{{\boldsymbol{r}}}}_{{\mathrm{mix}}}({{\mathbb{R}}}^d)$ to be the space of functions $g$ whose classical derivatives $D^{{\boldsymbol{\alpha}}}g$ with ${{\boldsymbol{\alpha}}}\le {{\boldsymbol{r}}}$ are all continuous at every point in ${{\mathbb{R}}}^d$, with no limitation on their behavior at infinity.
The norms corresponding to the two kinds of Sobolev space can be defined, for example, as in the classical sense, by $$ \|g\|_{{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r} =
\Bigg(\sum_{|{{\boldsymbol{\alpha}}}|\le r}
\|D^{{\boldsymbol{\alpha}}}g\|_{{{\mathcal{L}}}_{p,\zeta_{d,{{\boldsymbol{\alpha}}}}}}^2\Bigg)^{1/2}
\quad\mbox{and}\quad
\|g\|_{{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}},{\mathrm{mix}}}^{{\boldsymbol{r}}}} =
\Bigg(\sum_{{{\boldsymbol{\alpha}}}\le {{\boldsymbol{r}}}}
\|D^{{\boldsymbol{\alpha}}}g\|_{{{\mathcal{L}}}_{p,\zeta_{d,{{\boldsymbol{\alpha}}}}}}^2\Bigg)^{1/2}.$$ We have the following relationships between the spaces:
- ${{\mathcal{W}}}_{d,p',{{\boldsymbol{\zeta}}}}^r \subseteq {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r$ and ${{\mathcal{W}}}_{d,p',{{\boldsymbol{\zeta}}},{\mathrm{mix}}}^{{\boldsymbol{r}}}\subseteq
{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}},{\mathrm{mix}}}^{{\boldsymbol{r}}}$ for $1\le p\le p'\le \infty$.
- ${{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}},{\mathrm{mix}}}^{{\boldsymbol{r}}}\subseteq
{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r \iff \min_{j \in {{\mathfrak{D}}}} r_j \ge r \quad$ and $\quad {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r = \cap_{|{{\boldsymbol{r}}}|=r}
{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}},{\mathrm{mix}}}^{{\boldsymbol{r}}}$.
- ${{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}},{\mathrm{mix}}}^{s,\ldots,s} \subseteq
{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r \iff s\ge r \quad $ and $\quad {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r
\subseteq {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}},{\mathrm{mix}}}^{s,\ldots,s} \iff r\ge s\,d$.
- ${{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r \subseteq {{\mathcal{C}}}^k({{\mathbb{R}}}^d)$ if $r>k+d/p$ (Sobolev embedding theorem).
- For $p\in [1,\infty)$ and $r\ge 1$, if $g \in
{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r$ then $D^{{\boldsymbol{\alpha}}}g\in
{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^{r-|{{\boldsymbol{\alpha}}}|}$ for all $|{{\boldsymbol{\alpha}}}| \le r$.
- For $p\in [1,\infty)$ and ${{\boldsymbol{r}}}\ge {{\boldsymbol{1}}}$, if $g \in
{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^{{{\boldsymbol{r}}}}$ then $D^{{\boldsymbol{\alpha}}}g\in
{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^{{{\boldsymbol{r}}}-{{\boldsymbol{\alpha}}}}$ for all ${{\boldsymbol{\alpha}}}\le {{\boldsymbol{r}}}$.
Properties (i)–(iv) are straightforward. Properties (v) and (vi) are a bit more involved due to the varying generalized weight functions considered here. Indeed, when $\overline{{\boldsymbol{\alpha}}}$ is a multi-index satisfying $0\le |\overline{{\boldsymbol{\alpha}}}|\le r-|{{\boldsymbol{\alpha}}}|$, we have $$\begin{aligned}
\|D^{\overline{{\boldsymbol{\alpha}}}} (D^{{\boldsymbol{\alpha}}}g)\|_{{{\mathcal{L}}}_{p,\zeta_{d,\overline{{\boldsymbol{\alpha}}}}}}
&\,=\, \left(\int_{{{\mathbb{R}}}^d} |(D^{\overline{{\boldsymbol{\alpha}}}} (D^{{\boldsymbol{\alpha}}}g))({{\boldsymbol{x}}})|^p\,
\zeta_{d,\overline{{\boldsymbol{\alpha}}}}({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}\right)^{1/p} \\
&\,\le\, \left(\int_{{{\mathbb{R}}}^d} |(D^{\widehat{{\boldsymbol{\alpha}}}} g)({{\boldsymbol{x}}})|^p\,
\zeta_{d,\widehat{{\boldsymbol{\alpha}}}}({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}\right)^{1/p}
\,=\, \|D^{\widehat{{\boldsymbol{\alpha}}}}g\|_{{{\mathcal{L}}}_{p,\zeta_{d,\widehat{{\boldsymbol{\alpha}}}}}}
\,<\,\infty,\end{aligned}$$ where we introduced $\widehat{{\boldsymbol{\alpha}}}:= \overline{{\boldsymbol{\alpha}}}+ {{\boldsymbol{\alpha}}}$ and we used $\zeta_{d,\overline{{\boldsymbol{\alpha}}}}({{\boldsymbol{x}}})\le\zeta_{d,\widehat{{\boldsymbol{\alpha}}}}({{\boldsymbol{x}}})$ since $\overline{{\boldsymbol{\alpha}}}\le\widehat{{\boldsymbol{\alpha}}}$. The finiteness in the final step follows from $g\in {{\mathcal{W}}}^r_{d,p,{{\boldsymbol{\zeta}}}}$ and $|\widehat{{\boldsymbol{\alpha}}}|\le
r$. This justifies (v). The argument can easily be modified to justify (vi).
New smoothing theorems {#sec:newthms}
----------------------
In this subsection we establish two smoothing theorems: one for the isotropic Sobolev space, the other for the mixed Sobolev space. The proofs are modeled on the proof of [@GKSnote Theorem 1], but are extended here to cover discontinuous integrands.
\[thm:main1\] Let $d\ge 2$, $r\ge 1$, $p\in [1,\infty)$, and let $\rho\in{{\mathcal{C}}}^{r-1}({{\mathbb{R}}})$ be a strictly positive probability density function. Let $$ f({{\boldsymbol{x}}}) \,:=\, \theta({{\boldsymbol{x}}})\,{\rm ind}(\phi({{\boldsymbol{x}}})),
\quad\mbox{where}\quad
\theta,\phi\in{{\mathcal{W}}}^r_{d,p,{{\boldsymbol{\zeta}}}}\cap{{\mathcal{C}}}^r({{\mathbb{R}}}^d),$$ with generalized weight functions $\zeta_{d,{{\boldsymbol{\alpha}}}}$ satisfying and , and with ${\rm ind}(\cdot)$ denoting the indicator function. Let $j\in{{\mathfrak{D}}}:= \{1,\ldots,d\}$ be fixed, and suppose that $$\label{phi}
(D_j\phi)({{\boldsymbol{x}}}) > 0
\quad\forall{{\boldsymbol{x}}}\in {{\mathbb{R}}}^d,
\qquad\mbox{and}\qquad
\phi({{\boldsymbol{x}}}) \to \infty \quad\mbox{as}\quad x_j\to \infty \quad\forall {{\boldsymbol{x}}}_{-j} \in {{\mathbb{R}}}^{d-1}.$$ Writing ${{\boldsymbol{y}}}:= {{\boldsymbol{x}}}_{-j}$ so that ${{\boldsymbol{x}}}= (x_j,{{\boldsymbol{y}}})$, let $$ U_j \,:=\,
\{
{{\boldsymbol{y}}}\in{{\mathbb{R}}}^{d-1}
\,:\,
\phi(x_j,{{\boldsymbol{y}}})=0
\mbox{ for some }x_j\in{{\mathbb{R}}}\}
\qquad\mbox{and}\qquad
U_j^+ \,:=\, {{\mathbb{R}}}^{d-1}\setminus U_j.$$ If $U_j$ is empty, then $f = \theta$. If $U_j$ is not empty, then $U_j$ is open, and there exists a unique function $\psi \equiv \psi_j\in
{{\mathcal{C}}}^r(U_j)$ such that $\phi(x_j,{{\boldsymbol{y}}})=0$ if and only if $x_j =
\psi({{\boldsymbol{y}}})$ for ${{\boldsymbol{y}}}\in U_j$. In the latter case we assume that every function of the form $$\begin{aligned}
\label{h}
\begin{cases}
h({{\boldsymbol{y}}}) \,=\,
\displaystyle\frac{(D^{{{\boldsymbol{\eta}}}}\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,\prod_{i=1}^a
[(D^{{{\boldsymbol{\gamma}}}^{(i)}}\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]}{[(D_j\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]^b}\,
\rho^{(c)}(\psi({{\boldsymbol{y}}})),
\quad {{\boldsymbol{y}}}\in U_j,
\vspace{0.2cm} \\
\mbox{where $a,b,c$ are integers and ${{\boldsymbol{\gamma}}}^{(i)}$, ${{\boldsymbol{\eta}}}$ are
multi-indices with the constraints}
\vspace{0.1cm} \\
1\le a,b\le 2r-1,\quad
1\le |{{\boldsymbol{\gamma}}}^{(i)}|\le r,\quad
0\le |{{\boldsymbol{\eta}}}|,c\le r-1, \quad
1\le |{{\boldsymbol{\gamma}}}^{(i)}| + |{{\boldsymbol{\eta}}}| + c \le r,
\end{cases}\end{aligned}$$ satisfies both $$\label{h1}
h({{\boldsymbol{y}}})\to 0 \qquad
\mbox{as}\quad \mbox{${{\boldsymbol{y}}}$ approaches a boundary point of $U_j$ lying in $U_j^+$},$$ and $$\label{h2}
\int_{U_j} |h({{\boldsymbol{y}}})|^p\, \zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{y}}})\,{{\mathrm{d}}}{{\boldsymbol{y}}}\,<\, \infty,
\quad \text{ for all } |{{\boldsymbol{\alpha}}}|\le r \text{ with } \alpha_j=0,$$ where ${{\boldsymbol{\alpha}}}_{-j}$ denotes the multi-index with $d-1$ components obtained from ${{\boldsymbol{\alpha}}}$ by leaving out $\alpha_j$. Then $$P_j f \in {{\mathcal{W}}}^r_{d-1,p,{{\boldsymbol{\zeta}}}} \cap{{\mathcal{C}}}^r({{\mathbb{R}}}^{d-1}).$$
We defer the proof of this theorem to Section \[sec:proof1\].
In effect, under the conditions in the theorem, the single integration with respect to $x_j$ is sufficient to ensure that $P_jf$ inherits the full smoothness of $\theta$ and $\phi$.
We remark that when $\theta = \phi$ we are back at the same function $f({{\boldsymbol{x}}}) = \max(\phi({{\boldsymbol{x}}},0))$ as considered in [@GKSnote Theorem 1]. However, for this case we see that the new result is not as sharp as the old one in the sense that the upper bounds on the values of $a$, $b$, $c$, $|{{\boldsymbol{\gamma}}}^{(i)}|$ in the condition are larger than those in [@GKSnote Theorem 1]. This is because the explicit prior knowledge of $\theta = \phi$ means that we know a certain term vanishes (precisely, the second term on the right-hand side of [@GKSnote Formula (11)]). This observation also indirectly explains how the new result for jumps require stronger conditions on the functions $\theta$, $\phi$ and $\rho$ than the corresponding result for kinks.
The conditions and in the theorem are difficult to verify directly because the function $h$ depends explicitly on the inverse function $\psi({{\boldsymbol{y}}})$. Fortunately, a sufficient condition for to hold is that $$\begin{aligned}
\label{suff_new}
&\left|\frac{(D^{{{\boldsymbol{\eta}}}}\theta)(x_j,{{\boldsymbol{y}}})\,\prod_{i=1}^a
[(D^{{{\boldsymbol{\gamma}}}^{(i)}}\phi)(x_j,{{\boldsymbol{y}}})]}{[(D_j\phi)(x_j,{{\boldsymbol{y}}})]^b}\,
\rho^{(c)}(x_j)\right|
\,\le\, E_1(x_j) E_2({{\boldsymbol{y}}}),\end{aligned}$$ where $E_1,E_2$ are positive functions satisfying
- $E_1$ is bounded and $E_1(x_j)\to 0$ as $x_j\to - \infty$,
- $E_2$ is locally bounded (bounded over compact sets) and $\int_{{{\mathbb{R}}}^{d-1}} \left|E_2({{\boldsymbol{y}}}) \right|^p
\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{y}}})\,{{\mathrm{d}}}{{\boldsymbol{y}}}\,<\, \infty$ for all $|{{\boldsymbol{\alpha}}}|\le r$.
Considering a point ${{\boldsymbol{y}}}^\star$ on the boundary $\varGamma(U_j) \subset
U_j^+$, and a sequence $({{\boldsymbol{y}}}_n)_{n \in \mathbb{N}} \subset U_j $ such that ${{\boldsymbol{y}}}_n \to {{\boldsymbol{y}}}^\star$, we see that $E_2({{\boldsymbol{y}}}_n)$ is bounded and $E_1(\psi({{\boldsymbol{y}}}_n)) \to 0$ since $\psi({{\boldsymbol{y}}}_n) \to -\infty$. Therefore is sufficient for . Moreover, for $|{{\boldsymbol{\alpha}}}|\le r$ we have $$\begin{aligned}
&\int_{U_j}
\left | \displaystyle\frac{(D^{{{\boldsymbol{\eta}}}}\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,\prod_{i=1}^a
[(D^{{{\boldsymbol{\gamma}}}^{(i)}}\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]}{[(D_j\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]^b}\,
\rho^{(c)}(\psi({{\boldsymbol{y}}})) \right |^p \zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{y}}})\,{{\mathrm{d}}}{{\boldsymbol{y}}}\\
&\,\le\, \int_{U_j} \left|E_1(\psi({{\boldsymbol{y}}})) \right|^p \left|E_2({{\boldsymbol{y}}}) \right|^p
\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{y}}})\,{{\mathrm{d}}}{{\boldsymbol{y}}}\,\le\, B \int_{U_j} \left|E_2({{\boldsymbol{y}}}) \right|^p
\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{y}}})\,{{\mathrm{d}}}{{\boldsymbol{y}}}\,<\, \infty,
\end{aligned}$$ for some positive constant $B$. Therefore is also sufficient for .
We can also deduce a result for Sobolev spaces of dominating mixed smoothness.
\[thm:main2\] Let $d\ge 2$, $p\in [1,\infty)$, $j\in {{\mathfrak{D}}}$, and let $\rho\in{{\mathcal{C}}}^{r-1}({{\mathbb{R}}})$ be a strictly positive probability density function. Let ${{\boldsymbol{r}}}= (r_1,\ldots,r_d)$ be a multi-index satisfying $$r_j \,\ge\, \textstyle\sum_{1\le k\le d,\,k\ne j} r_k \,\ge\, 1.$$ If we replace the conditions on $\theta$, $\phi$ in Theorem \[thm:main1\] by $$\theta,\phi\in {{\mathcal{W}}}^{{{\boldsymbol{r}}}}_{d,p,{{\boldsymbol{\zeta}}},{\mathrm{mix}}}\cap{{\mathcal{C}}}^{{\boldsymbol{r}}}_{{\mathrm{mix}}}({{\mathbb{R}}}^d),$$ and further restrict – to functions $h$ with multi-indices ${{\boldsymbol{\gamma}}}^{(i)} \le{{\boldsymbol{r}}}$, ${{\boldsymbol{\eta}}}< {{\boldsymbol{r}}}$, and ${{\boldsymbol{\alpha}}}\le{{\boldsymbol{r}}}$, then the conclusion becomes: $$P_j f \in
{{\mathcal{W}}}^{{{\boldsymbol{r}}}_{-j}}_{d-1,p,{{\boldsymbol{\zeta}}}}\cap{{\mathcal{C}}}^{{{\boldsymbol{r}}}_{-j}}_{{{\mathrm{mix}}}}({{\mathbb{R}}}^{d-1}).$$
The proof is obtained from minor modifications of the proof of Theorem \[thm:main1\] in Section \[sec:proof1\]. In particular, the requirement that $r_j$ is greater than or equal to the sum of the remaining $r_k$ for $k\ne j$ is needed because, for any multi-index ${{\boldsymbol{\alpha}}}\le {{\boldsymbol{r}}}$ with $\alpha_j = 0$, it is clear from and a generalization of that the expression for $D^{{\boldsymbol{\alpha}}}P_jf$ includes some terms that depend on $D_j^{|{{\boldsymbol{\alpha}}}|}\phi$ and some terms that depend on $D_j^{|{{\boldsymbol{\alpha}}}|-1}\theta$.
Applying the theory to option pricing {#sec:apply}
=====================================
We now apply our results to the option pricing example.
Recall from Section \[sec:app\] that after PCA factorization the function $f$ from the digital option pricing example takes the form , with $\theta$ a constant function and $\phi$ given by . It follows that $$(D_j \phi)({{\boldsymbol{x}}}) \,=\,
\frac{\sigma\,S_0}{d} \sum_{\ell=1}^d
\exp\Bigg( \left(\mu-\tfrac{\sigma^2}{2}\right) \ell\Delta t
+\sigma \sum_{k=1}^d A_{\ell k}\, x_k\Bigg) A_{\ell j}.$$ In particular, we see that $(D_1 \phi)({{\boldsymbol{x}}})>0$ because, as explained in Section \[sec:app\], $A_{\ell 1}>0$ for all $\ell$, thus in this case it is appropriate to take $j=1$ in Theorem \[thm:main1\]. It is also clear that $\phi$ is in ${{\mathcal{C}}}^r({\mathbb{R}}^d)$ for all $r\in\mathbb{Z}^+$. Additionally, we may take all the weight functions $\zeta_i$ in equal to the standard normal density $\rho$. It is then clear that the sufficient condition is satisfied, and moreover that all the integrability and decay conditions in Theorem \[thm:main1\] are satisfied, because all derivatives of $\phi$ are “killed” at infinity by the Gaussian weight functions and their derivatives. It then follows from Theorem \[thm:main1\] that $$P_1f\in {{\mathcal{W}}}_{d-1,p,\boldsymbol{\rho}}^r\cap {{\mathcal{C}}}^r({\mathbb{R}}^{d-1})\quad
\forall \; r\in\mathbb{Z}^+, \quad \forall \; p\in[1,\infty).$$
Proof of the main smoothing theorem {#sec:proof1}
===================================
Before we proceed to prove Theorem \[thm:main1\], we quote three theorems from [@GKS13 Section 2.4], but state them with respect to the Sobolev spaces defined with generalized weight functions $\zeta_{d,{{\boldsymbol{\alpha}}}}$. We outline the subtle additional steps needed in the proofs of [@GKS13] to allow for this generalization.
The classical Leibniz theorem allows us to swap the order of differentiation and integration. In this paper we need a more general form of the Leibniz theorem as given below.
\[thm:swap\] Let $p\in [1,\infty)$. For $g \in {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^1$ with generalized weight functions $\zeta_{d,{{\boldsymbol{\alpha}}}}$ satisfying and , we have $$D_k P_j g \,=\, P_j D_k g
\qquad\mbox{for all}\quad j,k \in {{\mathfrak{D}}}\quad\mbox{with}\quad j \ne k.$$
We follow the proof of [@GKS13 Theorem 2.1] to the last paragraph where Fubini’s theorem was applied a second time. This application of Fubini’s Theorem is valid because $$\begin{aligned}
&\left|\int_{{{\mathbb{R}}}^d} \int_{-\infty}^\infty (D_kg)(t_j,{{\boldsymbol{x}}}_{-j})\,\rho(t_j)\,{{\mathrm{d}}}t_j\, v({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}\right| \\
&\,\le\, \int_{{{\mathbb{R}}}^{d-1}} \int_{-\infty}^\infty
|(D_kg)(t_j,{{\boldsymbol{x}}}_{-j})|\,\zeta_1(t_j)\,{{\mathrm{d}}}t_j\,\rho_{d-1}({{\boldsymbol{x}}}_{-j})
\frac{\int_{-\infty}^\infty|v(x_j,{{\boldsymbol{x}}}_{-j})|\,{{\mathrm{d}}}x_j}{\rho_{d-1}({{\boldsymbol{x}}}_{-j})}\, \,{{\mathrm{d}}}{{\boldsymbol{x}}}_{-j} \\
&\,\le\, \| D_kg \|_{{{\mathcal{L}}}_{1,\zeta_{d,{{\boldsymbol{e}}}_k}}}
\frac{\sup_{{{\boldsymbol{x}}}_{-j}\in V}\int_{-\infty}^\infty|v(x_j,{{\boldsymbol{x}}}_{-j})|\,{{\mathrm{d}}}x_j}
{\inf_{{{\boldsymbol{x}}}_{-j}\in V}\rho_{d-1}({{\boldsymbol{x}}}_{-j})}
\,<\, \infty,\end{aligned}$$ where ${{\boldsymbol{e}}}_k$ is the multi-index consisting of $1$ in the position $k$, and $0$ elsewhere, and where we made use of $\rho(t_j)\le \zeta_1(t_j)$ and $g\in {{\mathcal{W}}}^1_{d,1,{{\boldsymbol{\zeta}}}}$, and that the set $V$ defined in the proof of [@GKS13 Theorem 2.1] is a compact set because of the compactness of ${\rm supp}(v)$. The remainder of that proof then stands.
The next theorem is an application of the Leibniz theorem; it establishes that $P_jf$ inherits the smoothness of $g$.
\[thm:inher\] Let $r\ge 0$ and $p\in [1,\infty)$. For $g \in {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r$ with generalized weight functions $\zeta_{d,{{\boldsymbol{\alpha}}}}$ satisfying and , we have $$P_j g \,\in\, {{\mathcal{W}}}_{{d-1},p,{{\boldsymbol{\zeta}}}}^r
\qquad\mbox{for all}\quad j \in {{\mathfrak{D}}}.$$
For the case $r=0$ the proof is exactly the same as the proof of [@GKS13 Theorem 2.2]. Consider now $r\ge 1$. Let $j\in{{\mathfrak{D}}}$ and let ${{\boldsymbol{\alpha}}}$ be any multi-index with $|{{\boldsymbol{\alpha}}}|\le r$ and $\alpha_j = 0$. Since now $g\in {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^r$ with generalized weight functions, we have $\|D^{{\boldsymbol{\alpha}}}g\|_{{{\mathcal{L}}}_{p,\zeta_{d,{{\boldsymbol{\alpha}}}}}}<\infty$. To show that $P_jg \in {{\mathcal{W}}}_{d-1,p,{{\boldsymbol{\zeta}}}}^r$ we need to show that $\|D^{{\boldsymbol{\alpha}}}P_j g\|_{{{\mathcal{L}}}_{p,\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}}}<\infty$. Mimiking the proof of [@GKS13 Theorem 2.2], we write successively $$\begin{aligned}
\label{eq:step}
D^{{\boldsymbol{\alpha}}}P_j g
&\,=\, \left( \textstyle\prod_{i=1}^{|{{\boldsymbol{\alpha}}}|} D_{k_i}\right) P_j g
\,=\, \left( \textstyle\prod_{i=2}^{|{{\boldsymbol{\alpha}}}|} D_{k_i}\right) P_j D_{k_1} g \nonumber \\
&\,=\, \cdots
\,=\, D_{k_{|{{\boldsymbol{\alpha}}}}|} P_j
\left( \textstyle\prod_{i=1}^{|{{\boldsymbol{\alpha}}}|-1} D_{k_i} \right) g
\,=\, P_j \left( \textstyle\prod_{i=1}^{|{{\boldsymbol{\alpha}}}|} D_{k_i} \right) g
\,=\, P_j D^{{\boldsymbol{\alpha}}}g,\end{aligned}$$ where $k_i\in{{\mathfrak{D}}}\setminus\{j\}$ and $k_1,\ldots,k_{|{{\boldsymbol{\alpha}}}|}$ need not be distinct. Each step in involves a single differentiation under the integral sign, and is justified by the Leibniz theorem (Theorem \[thm:swap\]) because we know from the property (v) in Subsection \[sec:sob\] that $(\prod_{i=1}^{\ell} D_{k_i})
g\in{{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^{\,r-\ell} \subseteq {{\mathcal{W}}}_{d,p,{{\boldsymbol{\zeta}}}}^1$ for all $\ell\le |{{\boldsymbol{\alpha}}}|-1\le r-1$. We have therefore $$\begin{aligned}
&\|D^{{\boldsymbol{\alpha}}}P_j g\|_{{{\mathcal{L}}}_{p,\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}}}
\,=\, \|P_j D^{{\boldsymbol{\alpha}}}g\|_{{{\mathcal{L}}}_{p,\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}}} \\
&\,=\, \Bigg(\int_{{{\mathbb{R}}}^{{{\mathfrak{D}}}\setminus\{j\}}}
\left|
\int_{-\infty}^\infty (D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{x}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j
\right|^p
\,\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{x}}}_{-j})\,{{\mathrm{d}}}{{\boldsymbol{x}}}_{-j}
\Bigg)^{1/p} \\
&\,\le\, \Bigg(\int_{{{\mathbb{R}}}^{{{\mathfrak{D}}}\setminus\{j\}}}
\left(
\int_{-\infty}^\infty |(D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{x}}})|^p\,\rho(x_j)\,{{\mathrm{d}}}x_j
\right)
\,\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{x}}}_{-j})\,{{\mathrm{d}}}{{\boldsymbol{x}}}_{-j}
\Bigg)^{1/p}
\,=\, \|D^{{\boldsymbol{\alpha}}}g\|_{{{\mathcal{L}}}_{p,\zeta_{d,{{\boldsymbol{\alpha}}}}}} \,<\, \infty,\end{aligned}$$ where we applied Hölder’s inequality to the inner integral as in [@GKS13 Equation (2.11)] and used $ \zeta_{\alpha_j}(x_j) = \zeta_0(x_j)=\rho(x_j)$. This completes the proof.
The implicit function theorem stated below is crucial for the main results of this paper. In the rest of the paper, for any $r\ge 0$, $k\ge 1$, and an open set $U\subset{{\mathbb{R}}}^k$, we define ${{\mathcal{C}}}^r(U)$ to be the space of functions whose classical derivatives of order $\le r$ are all continuous at every point in $U$.
\[thm:implicit\] Let $j\in{{\mathfrak{D}}}$. Suppose $\phi \in {{\mathcal{C}}}^1({{\mathbb{R}}}^d)$ satisfies $$\label{djnot}
(D_j \phi) ({{\boldsymbol{x}}}) \,>\, 0 \qquad\mbox{for all}\quad {{\boldsymbol{x}}}\in {{\mathbb{R}}}^d.$$ Let $$U_j \,:=\, \{ {{\boldsymbol{x}}}_{-j}\in{{\mathbb{R}}}^{d-1} \,:\,
\phi(x_j,{{\boldsymbol{x}}}_{-j})=0 \mbox{ for some $($unique$)$ } x_j\in{{\mathbb{R}}}\}.$$ If $U_j$ is not empty then there exists a unique function $\psi_j \in
{{\mathcal{C}}}^1 (U_j)$ such that $$\phi (\,\psi_j ({{\boldsymbol{x}}}_{-j}),{{\boldsymbol{x}}}_{-j}) \,=\, 0
\qquad\mbox{for all}\quad {{\boldsymbol{x}}}_{-j}
\in U_j,$$ and for all $k \ne j$ we have $$\label{dkpsi-first}
(D_k\psi_j) ({{\boldsymbol{x}}}_{{{\mathfrak{D}}}\setminus\{j\}})
\,=\, - \frac{(D_k\phi)({{\boldsymbol{x}}})}{(D_j \phi)({{\boldsymbol{x}}})}\;
\bigg|_{\;x_j \,=\, \psi_j({{\boldsymbol{x}}}_{{{\mathfrak{D}}}\setminus\{j\}})}
\quad\mbox{for all}\quad {{\boldsymbol{x}}}_{{{\mathfrak{D}}}\setminus\{j\}}
\in U_j.$$ If in addition $\phi \in {{\mathcal{C}}}^r({{\mathbb{R}}}^d)$ for some $r\ge 2$, then $\psi_j
\in {{\mathcal{C}}}^r (U_j)$.
Note that the derivatives in the implicit function theorem are classical derivatives, and the condition needs to hold for *all* ${{\boldsymbol{x}}}\in{{\mathbb{R}}}^d$.
We are almost ready to prove Theorem \[thm:main1\]. But first we give a remark and a couple of auxiliary results.
It is easily seen that $U_j$ and $U_j^+$ in Theorem \[thm:main1\] can also be defined by $$\begin{aligned}
U_j
&\,=\, \left\{{{\boldsymbol{y}}}\in{{\mathbb{R}}}^{d-1}:\lim_{x_j\to -\infty}\phi(x_j,{{\boldsymbol{y}}})<0\right\},
\\
U_j^+
&\,=\, \left\{{{\boldsymbol{y}}}\in{{\mathbb{R}}}^{d-1}:\lim_{x_j\to -\infty}\phi(x_j,{{\boldsymbol{y}}})\ge 0\right\}.\end{aligned}$$
In the proof of the theorem we make essential use of the following lemma. This result is needed to ensure that all the derivatives we encounter are continuous across the boundary between $U_j$ and $U_j^+$.
\[lem:toinf\] Under the condition , the function $\psi_j:{\mathbb{R}}\to{\mathbb{R}}$ has the following property $$\label{psitomininfty}
\psi_j({{\boldsymbol{y}}})\to -\infty$$ as ${{\boldsymbol{y}}}$ approaches a point on the boundary of $U_j$.
Consider a point ${{\boldsymbol{y}}}^\star$ a point on the boundary of $U_j$, and hence (because $U_j$ is open) lying in $U_j^+$. Consider also a sequence $({{\boldsymbol{y}}}_n)_{n \in \mathbb{N}}\subset U_j$ with ${{\boldsymbol{y}}}_n \rightarrow
{{\boldsymbol{y}}}^\star$ as $n \rightarrow \infty$. We assert that the sequence $(\psi({{\boldsymbol{y}}}_n))_{n \in \mathbb{N}} $ has no accumulation points in ${\mathbb{R}}$. This is true because if we assume otherwise then there would exist a convergent subsequence $(\psi({{\boldsymbol{y}}}_{n_k}))_{k \in \mathbb{N}} $, with $\psi({{\boldsymbol{y}}}_{n_k}) \rightarrow x_j^\star$ as $k \rightarrow \infty$ for $x_j^\star \in {\mathbb{R}}$. But because of the continuity of $\phi$ we must have $$\phi(x_j^\star,{{\boldsymbol{y}}}^\star)=\lim_{k\rightarrow \infty}\phi(\psi({{\boldsymbol{y}}}_{n_k}), {{\boldsymbol{y}}}_{n_k}) = 0,$$ since by definition $\phi(\psi({{\boldsymbol{y}}}_{n_k}), {{\boldsymbol{y}}}_{n_k}) = 0, \forall k \in \mathbb{N}$. This implies that ${{\boldsymbol{y}}}^\star
\in U_j$, which is a contradiction. Therefore the sequence $(\psi({{\boldsymbol{y}}}_n))_{n \in \mathbb{N}} $ has no accumulation points in ${\mathbb{R}}$. This implies (due to the Bolzano-Weierstrass Theorem) that $(\psi({{\boldsymbol{y}}}_n))_{n \in \mathbb{N}}\cap [a,b]$ is a finite set, for each interval $[a,b], a,b \in {\mathbb{R}}$. Thus, $$\lim_{n \rightarrow \infty} \left| \psi({{\boldsymbol{y}}}_n) \right| = \infty.$$
To eliminate the possibility that $+\infty$ is an accumulation point of $\psi({{\boldsymbol{y}}}_n)$ we observe that, due to condition , and with ${{\boldsymbol{y}}}^\star$ as above, there exists an $x_j^\star$ such that $\phi(x_j^\star,{{\boldsymbol{y}}}^\star)>0$. Because of the continuity of $\phi$, there is a ball around the point $(x_j^\star,{{\boldsymbol{y}}}^\star)$, denoted by $B(x_j^\star,{{\boldsymbol{y}}}^\star)$ such that $\phi$ is positive for each point in $B(x_j^\star,{{\boldsymbol{y}}}^\star)$. Assume now that we have a subsequence $
(\psi({{\boldsymbol{y}}}_{n_m}))_{m \in \mathbb{N}}$ such that $\lim_{m \rightarrow
\infty} \psi({{\boldsymbol{y}}}_{n_m}) = +\infty$. Because ${{\boldsymbol{y}}}_{n_m}$ converges to ${{\boldsymbol{y}}}^\star$ as $m \rightarrow \infty$, it follows that $(x_j^\star,{{\boldsymbol{y}}}_{n_m}) \in B(x_j^\star,{{\boldsymbol{y}}}^\star) $ for all $m$ sufficiently large. But assumption $\psi({{\boldsymbol{y}}}_{n_m}) \to +\infty$, and the monotonicity condition in implies that for all $m$ sufficiently large we have $\psi({{\boldsymbol{y}}}_{n_m})>x_j^\star$, and therefore $0 <
\phi(x_j^\star,{{\boldsymbol{y}}}_{n_m})<\phi(\psi({{\boldsymbol{y}}}_{n_m}),{{\boldsymbol{y}}}_{n_m})=0$, which is clearly a contradiction. Therefore we conclude that $$\lim_{n \rightarrow \infty} \psi({{\boldsymbol{y}}}_n) = -\infty.$$
Another auxiliary result is needed to show that the assumption implies continuous differentiability of $P_j f$ at boundary points of $U_j$.
\[lem:Crproperty\] Let $r\ge 0$ and $k\ge 1$. Suppose $g \in {{\mathcal{C}}}^r(U)$ for some open domain $U \subset {\mathbb{R}}^k$ and $g({{\boldsymbol{y}}})=0$ for all ${{\boldsymbol{y}}}\in U^c$. Suppose that for any multi-index ${{\boldsymbol{\alpha}}}$ with $|{{\boldsymbol{\alpha}}}| \leq r$ and any sequence $({{\boldsymbol{y}}}_n)_{n \in \mathbb{N}}\subset U$ we have $$\label{Crprop:Assumption}
\lim_{n \rightarrow \infty} {{\boldsymbol{y}}}_n = {{\boldsymbol{y}}}^\star \mbox{ with } {{\boldsymbol{y}}}^\star \in U^c \quad \Rightarrow \quad
\lim_{n \rightarrow \infty} (D^{{{\boldsymbol{\alpha}}}} g)({{\boldsymbol{y}}}_n) = 0 \; .$$ Then we have $g \in {{\mathcal{C}}}^r({\mathbb{R}}^k)$, with $(D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{y}}}) = 0$ for all ${{\boldsymbol{y}}}\in U^c$.
The statement is obviously true for $r=0$ where mere continuity of $g$ is asserted. Now suppose it holds for a natural number $r_0 \geq 0$ and consider any multi-index ${{\boldsymbol{\alpha}}}= {{\boldsymbol{\alpha}}}_0 + {{\boldsymbol{e}}}_i$, with ${{\boldsymbol{e}}}_i$ denoting a canonical basis vector and $|{{\boldsymbol{\alpha}}}_0|= r_0$, and hence $|{{\boldsymbol{\alpha}}}| = r_0 +1 \le r$. Then we have to show that $(D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{y}}})$ exists at all ${{\boldsymbol{y}}}\in {\mathbb{R}}^k$ and is continuous at every point of ${\mathbb{R}}^k$. For points in $U$ the derivative $D^{{\boldsymbol{\alpha}}}g$ exists and is continuous by assumption, and for points in the interior of $U^c$ the derivative $D^{{\boldsymbol{\alpha}}}g$ exists and is continuous because $g$ is zero there. So it remains to consider the existence and continuity of $D^{{\boldsymbol{\alpha}}}g$ at any boundary point ${{\boldsymbol{y}}}^\star$, i.e., at any limit point ${{\boldsymbol{y}}}^\star$ of a sequence $({{\boldsymbol{y}}}_n)_{n \in \mathbb{N}} \subset U$.
To show the existence, consider the scalar valued function $D^{{{\boldsymbol{\alpha}}}_0}
g$ which is by the induction hypothesis continuous on all of ${\mathbb{R}}^k$ and vanishes at any boundary point ${{\boldsymbol{y}}}^\star$. Consider first the case $h>0$ and a point ${{\boldsymbol{y}}}^\star + h {{\boldsymbol{e}}}_i$. If ${{\boldsymbol{y}}}^\star + h {{\boldsymbol{e}}}_i \in U$, then because $U$ is open and ${{\boldsymbol{y}}}^\star \in U^c$, we have for $$\bar{h} \,:=\,
\sup\{h'\;:\; 0\le h' < h ,\; {{\boldsymbol{y}}}^* + h' {{\boldsymbol{e}}}_i \in U^c\}$$ that ${{\boldsymbol{y}}}^\star + h' {{\boldsymbol{e}}}_i \in U$ for all $\bar{h}<h'\le h$. Furthermore, because $U^c$ is closed it follows that $(D^{{{\boldsymbol{\alpha}}}_0}g)({{\boldsymbol{y}}}^\star+\bar{h}{{\boldsymbol{e}}}_i) = 0 =
(D^{{{\boldsymbol{\alpha}}}_0}g)({{\boldsymbol{y}}}^\star)$, and thus we conclude from the mean value theorem that $$(D^{{{\boldsymbol{\alpha}}}_0}g)({{\boldsymbol{y}}}^\star+h{{\boldsymbol{e}}}_i) - (D^{{{\boldsymbol{\alpha}}}_0}g)({{\boldsymbol{y}}}^\star)
= (D^{{{\boldsymbol{\alpha}}}_0}g)({{\boldsymbol{y}}}^\star+h{{\boldsymbol{e}}}_i) - (D^{{{\boldsymbol{\alpha}}}_0}g)(y^\star+\bar{h}{{\boldsymbol{e}}}_i)
= (h-\bar{h})\, (D_i D^{{{\boldsymbol{\alpha}}}_0}g)({{\boldsymbol{y}}}^*+h^\star{{\boldsymbol{e}}}_i)$$ for some $h^\star$ satisfying $\bar{h}< h^\star <h$. Hence we have for the quotient $$\frac{(D^{{{\boldsymbol{\alpha}}}_0} g)({{\boldsymbol{y}}}^\star + h {{\boldsymbol{e}}}_i) - (D^{{{\boldsymbol{\alpha}}}_0} g)({{\boldsymbol{y}}}^\star) }{h}
\,=\,
\begin{cases}
\frac{h-\bar{h}}{h}\, (D_iD^{{{\boldsymbol{\alpha}}}_0} g)({{\boldsymbol{y}}}^\star + h^\star {{\boldsymbol{e}}}_i)
, \, \; \bar{h}< h^\star <h,
& \text{ if } {{\boldsymbol{y}}}^* + h {{\boldsymbol{e}}}_i \in U,\\
0 & \text{ if } {{\boldsymbol{y}}}^\star + h {{\boldsymbol{e}}}_i \in U^c.
\end{cases}$$ Then using the assumption , letting $h$ be arbitrarily small, using that $|\bar{h}|\le |h|$, and considering the analogous situation for $h<0$, we obtain the existence of $D_iD^{{{\boldsymbol{\alpha}}}_0} g$ at $ {{\boldsymbol{y}}}^\star$, with $(D_iD^{{{\boldsymbol{\alpha}}}_0}
g)({{\boldsymbol{y}}}^\star)=0$.
To show the derivative continuity at a boundary point ${{\boldsymbol{y}}}^\star$, consider any sequence $({{\boldsymbol{y}}}_n)_{n \in \mathbb{N}}\subset {\mathbb{R}}^k$ with $\lim_{n \rightarrow \infty} {{\boldsymbol{y}}}_n = {{\boldsymbol{y}}}^\star$. For a given $n$ either ${{\boldsymbol{y}}}_n\in U$, in which case applies, or ${{\boldsymbol{y}}}_n\in U^c$, in which case $(D_i D^{{{\boldsymbol{\alpha}}}_0}g)({{\boldsymbol{y}}}_n)=0$, as above, so that both subsequences converge to $0=(D_iD^{{{\boldsymbol{\alpha}}}_0}
g)({{\boldsymbol{y}}}^\star)$. Finally, since all partial derivatives of order $r_0+1$ are now proved continuous in ${{\mathbb{R}}}^k$, those mixed partial derivatives are symmetric, and we can write $D_i D^{{{\boldsymbol{\alpha}}}_0}g = D^{{\boldsymbol{\alpha}}}g$.
Hence $D^{{\boldsymbol{\alpha}}}g$ exists and is continuous on all of ${\mathbb{R}}^k$, i.e., the induction step is proved. It follows then that $g\in {{\mathcal{C}}}^r({\mathbb{R}}^k)$.
We are now ready to prove Theorem \[thm:main1\].
[Theorem \[thm:main1\]]{} We focus on the non-trivial case when $U_j$ is not empty. Given that $\phi\in{{\mathcal{C}}}^r({{\mathbb{R}}}^d)$, that $(D_j\phi)({{\boldsymbol{x}}})\ne 0$ for all ${{\boldsymbol{x}}}\in{{\mathbb{R}}}^d$, and that $U_j$ is not empty, it follows from the implicit function theorem (see Theorem \[thm:implicit\]) that the set $U_j$ is open, and that there exists a unique function $\psi\equiv\psi_j\in{{\mathcal{C}}}^r(U_j)$ for which $$\label{psi}
\phi(x_j,{{\boldsymbol{y}}}) \,=\,0
\;\iff\;
\psi({{\boldsymbol{y}}}) \,=\, x_j
\quad\mbox{for all }\;
{{\boldsymbol{y}}}\in U_j.$$ This justifies the existence of the function $\psi$ as stated in the theorem.
For the function $f({{\boldsymbol{x}}}) = \theta(x_j,{{\boldsymbol{y}}})\,{\rm ind}(\phi(x_j,{{\boldsymbol{y}}}))$ we can write $P_jf$ defined by as $$\label{lim}
(P_jf)({{\boldsymbol{y}}})
\,=\, \int_{x_j\in {{\mathbb{R}}}\,:\, \phi(x_j,{{\boldsymbol{y}}})\ge 0}
\theta(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j.$$ It follows from the condition $(D_j\phi)({{\boldsymbol{x}}})> 0$ for all ${{\boldsymbol{x}}}\in{{\mathbb{R}}}^d$ and the continuity of $D_j\phi$ that, for fixed ${{\boldsymbol{y}}}$, $\phi(x_j,{{\boldsymbol{y}}})$ is a strictly increasing function of $x_j$.
We now determine the limits of integration in . If ${{\boldsymbol{y}}}\in
U_j^+$, then $\phi(x_j,{{\boldsymbol{y}}}) \ne 0$ for all $x_j\in{{\mathbb{R}}}$. Since $\phi$ is continuous, strictly increasing in $x_j$, and tends to $+\infty$ as $x_j\to +\infty$, we conclude that $\phi(x_j,{{\boldsymbol{y}}})
> 0$ for all $x_j\in{{\mathbb{R}}}$, and thus we integrate $x_j$ from $-\infty$ to $\infty$. On the other hand, if ${{\boldsymbol{y}}}\in U_j$, in which case $\phi(x_j,{{\boldsymbol{y}}})$ changes sign once as $x_j$ goes from $-\infty$ to $\infty$, then there exists a unique $x_j^* = \psi({{\boldsymbol{y}}})\in{{\mathbb{R}}}$ for which $\phi(x_j^*,{{\boldsymbol{y}}})=0$, and in this case we integrate $x_j$ from $\psi({{\boldsymbol{y}}})$ to $\infty$. Hence we can write as $$\begin{aligned}
(P_jf)({{\boldsymbol{y}}})
&\,=\,
\begin{cases}
\displaystyle\int_{-\infty}^\infty \theta(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j
& \mbox{if } {{\boldsymbol{y}}}\in U_j^+, \vspace{0.1cm}\\
\displaystyle\int_{\psi({{\boldsymbol{y}}})}^\infty \theta(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j
& \mbox{if } {{\boldsymbol{y}}}\in U_j.
\end{cases}\end{aligned}$$ Note that $P_jf$ is continuous across the boundary between $U_j$ and $U_j^+$, since from Lemma \[lem:toinf\] it follows that $\psi({{\boldsymbol{y}}})\to
-\infty$ as ${{\boldsymbol{y}}}\in U_j$ approaches a boundary point of $U_j$.
By the Leibniz Theorem and the Inheritance Theorem, we know that the function $(P_j\theta)({{\boldsymbol{y}}}) = \int_{-\infty}^\infty
\theta(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j$ for ${{\boldsymbol{y}}}\in {{\mathbb{R}}}^{d-1}$ is as smooth as $\theta$, i.e., $P_j\theta \in {{\mathcal{W}}}^r_{d-1,p,{{\boldsymbol{\zeta}}}}\cap
{{\mathcal{C}}}^r({{\mathbb{R}}}^{d-1})$. Therefore, to obtain the same smoothness property for $P_jf$ it suffices that we consider in the remainder of this proof the difference $$g({{\boldsymbol{y}}})
\,:=\, (P_jf)({{\boldsymbol{y}}}) - (P_j\theta)({{\boldsymbol{y}}})
\,=\, \begin{cases}
- \displaystyle\int_{-\infty}^{\psi({{\boldsymbol{y}}})} \theta(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j & \mbox{if } {{\boldsymbol{y}}}\in U_j,\\
0 & \mbox{if } {{\boldsymbol{y}}}\in U^+_j.
\end{cases}$$ First we differentiate $g$ with respect to the $k$th coordinate for any $k\ne j$. For ${{\boldsymbol{y}}}\in U_j$ we obtain, using the fundamental theorem of calculus, $$\begin{aligned}
\label{dkpjf}
(D_k g)({{\boldsymbol{y}}})
&\,=\,
-\int_{-\infty}^{\psi({{\boldsymbol{y}}})}
(D_k\theta)(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j
-\, \theta(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}) \,
\rho(\psi({{\boldsymbol{y}}}))\,
(D_k\psi)({{\boldsymbol{y}}}) \nonumber \\
&\,=\,
-\int_{-\infty}^{\psi({{\boldsymbol{y}}})}
(D_k\theta)(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j
+\, \theta(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,
\frac{(D_k \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})}{(D_j \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})}
\,\rho(\psi({{\boldsymbol{y}}})),\end{aligned}$$ where we substituted using $$\label{dkpsi}
(D_k \psi)({{\boldsymbol{y}}})
\,=\, - \frac
{(D_k \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})}
{(D_j \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})}.$$ It follows from and Lemma \[lem:toinf\] that both terms in go to $0$ as ${{\boldsymbol{y}}}\in U_j$ approaches a boundary point ${{\boldsymbol{y}}}^\star$ of $U_j$ lying in $U_j^+$. Hence the condition in Lemma \[lem:Crproperty\] holds with $r=1$, and we conclude that $g\in{{\mathcal{C}}}^1({{\mathbb{R}}}^{d-1})$. Next we differentiate with respect to the $\ell$th coordinate for any $\ell\ne j$ (allowing the possibility that $\ell=k$). For ${{\boldsymbol{y}}}\in U_j$ it is useful to note that for any sufficiently smooth $d$-variate function $\xi$ the rule for partial differentiation and the chain rule gives $$\label{chainrule}
D_\ell (\xi(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}))
\,=\, (D_\ell \xi )(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}) + (D_j \xi )(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\, (D_\ell \psi)({{\boldsymbol{y}}}).$$ Thus we find for ${{\boldsymbol{y}}}\in U_j$ $$\begin{aligned}
\label{dldkpjf}
&(D_\ell D_k g)({{\boldsymbol{y}}}) \,=\,
-\int_{-\infty}^{\psi({{\boldsymbol{y}}})}
(D_\ell D_k\theta)(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j
\,-\, (D_k\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\, \rho(\psi({{\boldsymbol{y}}})) \, (D_\ell\psi)({{\boldsymbol{y}}}) \nonumber \\
&\,+\, [(D_\ell\, \theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}) + (D_j\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,(D_\ell\psi)({{\boldsymbol{y}}})]\,
\frac{(D_k \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})}{(D_j \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})}\,\rho(\psi({{\boldsymbol{y}}})) \nonumber \\
&\,+\, \theta(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,
\frac{[(D_\ell D_k \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}) + (D_j D_k \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,(D_\ell\psi)({{\boldsymbol{y}}})]}
{(D_j \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})}\,\rho(\psi({{\boldsymbol{y}}})) \nonumber \\
&\,-\, \theta(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,
\frac{(D_k \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,
[(D_\ell D_j \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}) + (D_j D_j \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,(D_\ell\psi)({{\boldsymbol{y}}})]}
{[(D_j \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]^2}\,\rho(\psi({{\boldsymbol{y}}}))
\nonumber \\
&\,+\, \theta(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,
\frac{(D_k \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})}{(D_j \phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})}\,\rho'(\psi({{\boldsymbol{y}}}))\,
(D_\ell\psi)({{\boldsymbol{y}}}),\end{aligned}$$ where we used again . We have from and Lemma \[lem:toinf\] that all terms in go to $0$ as ${{\boldsymbol{y}}}\in U_j$ approaches a boundary point ${{\boldsymbol{y}}}^\star$ of $U_j$ lying in $U_j^+$. Hence the condition in Lemma \[lem:Crproperty\] holds with $r=2$, and we conclude that $g\in{{\mathcal{C}}}^2({{\mathbb{R}}}^{d-1})$. In general, for every non-zero multi-index ${{\boldsymbol{\alpha}}}=(\alpha_1,\ldots,\alpha_d)$ with $|{{\boldsymbol{\alpha}}}|\le r$ and $\alpha_j
= 0$, we claim that for ${{\boldsymbol{y}}}\in U_j$ $$\label{nice}
(D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{y}}}) \,=\,
-\int_{-\infty}^{\psi({{\boldsymbol{y}}})}
(D^{{\boldsymbol{\alpha}}}\,\theta)(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j
\,+\, \sum_{m=1}^{M_{|{{\boldsymbol{\alpha}}}|}} h_{{{\boldsymbol{\alpha}}},m}({{\boldsymbol{y}}}),$$ where $M_{|{{\boldsymbol{\alpha}}}|}$ is a nonnegative integer, and each function $h_{{{\boldsymbol{\alpha}}},m}$ is of the form , with integers $\beta,a,b,c$ and multi-indices ${{\boldsymbol{\gamma}}}^{(i)}$ and ${{\boldsymbol{\eta}}}$ satisfying $$\label{bounds}
1\le a,b\le 2|{{\boldsymbol{\alpha}}}|-1,\quad
1\le |{{\boldsymbol{\gamma}}}^{(i)}|\le |{{\boldsymbol{\alpha}}}|, \quad
0\le |{{\boldsymbol{\eta}}}|,c\le |{{\boldsymbol{\alpha}}}|-1, \quad
1\le |{{\boldsymbol{\gamma}}}^{(i)}|+|{{\boldsymbol{\eta}}}|+c \le |{{\boldsymbol{\alpha}}}|.$$ We have from and Lemma \[lem:toinf\] that all terms in go to $0$ as ${{\boldsymbol{y}}}\in U_j$ approaches a boundary point ${{\boldsymbol{y}}}^\star$ of $U_j$ lying in $U_j^+$. Hence the condition in Lemma \[lem:Crproperty\] holds for a general $r$, and we conclude that $g\in{{\mathcal{C}}}^r({{\mathbb{R}}}^{d-1})$. We will prove – by induction on $|{{\boldsymbol{\alpha}}}|$. The case $|{{\boldsymbol{\alpha}}}|=1$ is shown in ; there we have $M_1= 1$, $a=1$, $b=1$, $c=0$, $\beta=1$, $|{{\boldsymbol{\gamma}}}^{(1)}|=1$, $|{{\boldsymbol{\eta}}}| = 0$, and $|{{\boldsymbol{\gamma}}}^{(i)}|+|{{\boldsymbol{\eta}}}|+c =1$. The case $|{{\boldsymbol{\alpha}}}|=2$ is shown in ; there we have $M_2= 8$, $1\le a,b\le 3$, $0\le c\le 1$, $\beta = \pm 1$, $1\le |{{\boldsymbol{\gamma}}}^{(i)}|\le 2$, $0\le |{{\boldsymbol{\eta}}}|\le 1$, and $1\le |{{\boldsymbol{\gamma}}}^{(i)}|+|{{\boldsymbol{\eta}}}|+c \le 2$. To establish the inductive step we now differentiate $D^{{\boldsymbol{\alpha}}}g$ once more: for $\ell\ne j$ we have from $$\begin{aligned}
\label{term}
(D_\ell D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{y}}})
&\,=\, -\int_{-\infty}^{\psi({{\boldsymbol{y}}})}
(D_\ell D^{{\boldsymbol{\alpha}}}\theta)(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\,{{\mathrm{d}}}x_j
-\, (D^{{\boldsymbol{\alpha}}}\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,
\rho(\psi({{\boldsymbol{y}}})) \, (D_\ell \psi)({{\boldsymbol{y}}}) \nonumber \\
&\qquad +\, \sum_{m=1}^{M_{|{{\boldsymbol{\alpha}}}|}} (D_\ell\, h_{{{\boldsymbol{\alpha}}},m})({{\boldsymbol{y}}}).\end{aligned}$$ For a typical term in , we have from $$\begin{aligned}
&(D_\ell\, h)({{\boldsymbol{y}}}) \\
&\,=\, \beta\,
\frac{[(D_\ell D^{{\boldsymbol{\eta}}}\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}) + (D_jD^{{\boldsymbol{\eta}}}\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,(D_\ell\psi)({{\boldsymbol{y}}})]
\prod_{i=1}^a [(D^{{{\boldsymbol{\gamma}}}^{(i)}}\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]}
{[(D_j\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]^b}\, \rho^{(c)}(\psi({{\boldsymbol{y}}})) \\
&\qquad + \beta\,
\frac{(D^{{\boldsymbol{\eta}}}\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,D_\ell \big(\prod_{i=1}^a [(D^{{{\boldsymbol{\gamma}}}^{(i)}}\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]\big)}
{[(D_j\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]^b}\, \rho^{(c)}(\psi({{\boldsymbol{y}}})) \\
&\qquad + \beta\,
\frac{(D^{{\boldsymbol{\eta}}}\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,\prod_{i=1}^a [(D^{{{\boldsymbol{\gamma}}}^{(i)}}\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]}
{[(D_j\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]^b}\, \rho^{(c+1)}(\psi({{\boldsymbol{y}}}))\,
(D_\ell \psi)({{\boldsymbol{y}}}) \nonumber\\
&\qquad - \beta b\,
\frac{(D^{{\boldsymbol{\eta}}}\theta)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,\prod_{i=1}^a [(D^{{{\boldsymbol{\gamma}}}^{(i)}}\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]}
{[(D_j\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})]^{b+1}} \,\rho^{(c)}(\psi({{\boldsymbol{y}}})) \nonumber\\
&\qquad\qquad \Big[
(D_\ell D_j\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}) + (D_jD_j\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})\,
(D_\ell \psi)({{\boldsymbol{y}}}) \Big], \nonumber\end{aligned}$$ where $$\begin{aligned}
&D_\ell \bigg(\prod_{i=1}^a [(D^{{{\boldsymbol{\gamma}}}^{(i)}}\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})] \bigg) \\
&\,=\, \sum_{t=1}^a \Bigg(
\Big[(D_\ell D^{{{\boldsymbol{\gamma}}}^{(t)}}\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}) + (D_j D^{{{\boldsymbol{\gamma}}}^{(t)}}\phi)
(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}}) \, (D_\ell \psi)({{\boldsymbol{y}}}) \Big] \prod_{{\stackrel{\scriptstyle{i=1}}{\scriptstyle{i\ne t}}}}^a (D^{{{\boldsymbol{\gamma}}}^{(i)}}\phi)(\psi({{\boldsymbol{y}}}),{{\boldsymbol{y}}})
\Bigg).\end{aligned}$$ Thus we conclude that $D_\ell\, h$ is a sum of functions of the form , but with $a$ and $b$ increased by at most $2$, $c$ increased by at most $1$, $|\beta|$ multiplied by a factor of at most $b$, $|{{\boldsymbol{\gamma}}}^{(i)}|$ and $|{{\boldsymbol{\eta}}}|$ increased by at most $1$, and with $|{{\boldsymbol{\gamma}}}^{(i)}| + |{{\boldsymbol{\eta}}}| + c$ increased by at most $1$. Hence, $D_\ell
D^{{\boldsymbol{\alpha}}}g$ consists of a sum of functions of the form satisfying the constraints in . This completes the induction proof for –.
We now turn to the task of showing that $D^{{\boldsymbol{\alpha}}}g\in
{{\mathcal{L}}}_{p,\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}}$ for $p\in[1,\infty)$ and all ${{\boldsymbol{\alpha}}}$ satisfying $|{{\boldsymbol{\alpha}}}|\le r$ and $\alpha_j = 0$. We need to consider $$\begin{aligned}
\int_{{{\mathbb{R}}}^{d-1}}
|(D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{y}}})|^p\,
\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{y}}})\,
{{\mathrm{d}}}{{\boldsymbol{y}}}&\,=\,
\int_{U_j} |(D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{y}}})|^p\,
\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{y}}})\,
{{\mathrm{d}}}{{\boldsymbol{y}}},\end{aligned}$$ where we have split the integral noting that $U_j$ is open and its complement $U_j^+$ is closed, as they are both Borel measurable, and that $D^{{\boldsymbol{\alpha}}}g$ is zero on $U_j^+$.
For ${{\boldsymbol{y}}}\in U_j$, it follows from Hölder’s inequality and the special form of $D^{{\boldsymbol{\alpha}}}g$ in that $$\begin{aligned}
|(D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{y}}})|^p
&\,=\, \left| - \int_{-\infty}^{\psi({{\boldsymbol{y}}})}
(D^{{\boldsymbol{\alpha}}}\theta)(x_j,{{\boldsymbol{y}}})\,\rho(x_j)\, {{\mathrm{d}}}x_j
+ \sum_{m=1}^{M_{|{{\boldsymbol{\alpha}}}|}} h_{{{\boldsymbol{\alpha}}},m}({{\boldsymbol{y}}})\right|^p \\
&\,\le\, \left( \int_{-\infty}^{\psi({{\boldsymbol{y}}})}
|(D^{{\boldsymbol{\alpha}}}\theta)(x_j,{{\boldsymbol{y}}})|\, \rho(x_j)\, {{\mathrm{d}}}x_j
+ \sum_{m=1}^{M_{|{{\boldsymbol{\alpha}}}|}} |h_{{{\boldsymbol{\alpha}}},m}({{\boldsymbol{y}}})|\right)^p \\
&\,\le\, (M_{|{{\boldsymbol{\alpha}}}|}+1)^{p-1} \left( \left(\int_{-\infty}^{\psi({{\boldsymbol{y}}})}
|D^{{\boldsymbol{\alpha}}}\theta(x_j,{{\boldsymbol{y}}})|\, \rho(x_j)\, {{\mathrm{d}}}x_j\right)^p
+ \sum_{m=1}^{M_{|{{\boldsymbol{\alpha}}}|}} |h_{{{\boldsymbol{\alpha}}},m}({{\boldsymbol{y}}})|^p\right) \\
&\,\le\, (M_{|{{\boldsymbol{\alpha}}}|}+1)^{p-1} \left(\int_{-\infty}^{\psi({{\boldsymbol{y}}})}
|(D^{{\boldsymbol{\alpha}}}\theta)(x_j,{{\boldsymbol{y}}})|^p\, \rho(x_j)\, {{\mathrm{d}}}x_j
+ \sum_{m=1}^{M_{|{{\boldsymbol{\alpha}}}|}} |h_{{{\boldsymbol{\alpha}}},m}({{\boldsymbol{y}}})|^p\right),\end{aligned}$$ and thus $$\begin{aligned}
& \int_{U_j} |(D^{{\boldsymbol{\alpha}}}g)({{\boldsymbol{y}}})|^p\,
\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{y}}}) \,{{\mathrm{d}}}{{\boldsymbol{y}}}\\
&\,\le\,
(M_{|{{\boldsymbol{\alpha}}}|}+1)^{p-1} \Bigg(\int_{{{\mathbb{R}}}^d}
|(D^{{\boldsymbol{\alpha}}}\theta)({{\boldsymbol{x}}})|^p\,\zeta_{d,{{\boldsymbol{\alpha}}}}({{\boldsymbol{x}}})\,{{\mathrm{d}}}{{\boldsymbol{x}}}+ \sum_{m=1}^{M_{|{{\boldsymbol{\alpha}}}|}}
\int_{U_j} |h_{{{\boldsymbol{\alpha}}},m}({{\boldsymbol{y}}})|^p \,\zeta_{d-1,{{\boldsymbol{\alpha}}}_{-j}}({{\boldsymbol{y}}})\,{{\mathrm{d}}}{{\boldsymbol{y}}}\Bigg)
\,<\, \infty,\end{aligned}$$ with the finiteness coming because $\rho(x_j) = \zeta_0(x_j) =
\zeta_{\alpha_j}(x_j)$ and $\theta\in{{\mathcal{W}}}^r_{d,p,{{\boldsymbol{\zeta}}}}$, and because each integral involving $h_{{{\boldsymbol{\alpha}}},m}$ is finite due to the condition . This proves that ${{\color{blue}{g}}} \in {{\mathcal{W}}}^r_{d-1,p,{{\boldsymbol{\zeta}}}}$ as claimed.
**Acknowledgements** The authors acknowledge the support of the Australian Research Council under the projects FT130100655 and DP150101770.
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- [Andreas Griewank]{}\
School of Mathematical Sciences and Information Technology\
Yachay Tech\
Urcuqui 100119, Imbabura\
Ecuador\
email: [email protected]
- [Frances Y. Kuo]{}\
School of Mathematics and Statistics\
The University of New South Wales\
Sydney NSW 2052, Australia\
email: [email protected]
- [Hernan Leövey]{}\
Structured Energy Management Team\
Axpo AG\
Baden, Switzerland\
Post: Parkstrasse 23 CH-5401 Baden, Switzerland\
email: [email protected]
- [Ian H. Sloan]{}\
School of Mathematics and Statistics\
The University of New South Wales\
Sydney NSW 2052, Australia\
email: [email protected]
| ArXiv |
---
abstract: 'We investigate the thermopower due to the orbital Kondo effect in a single quantum dot system by means of the noncrossing approximation. It is elucidated how the asymmetry of tunneling resonance due to the orbital Kondo effect affects the thermopower under gate-voltage and magnetic-field control.'
address:
- 'Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan'
- 'Department of Physics, Kyoto University, Kyoto 606-8502, Japan'
author:
- 'R. Sakano'
- 'T. Kita'
- 'N. Kawakami'
title: Thermopower of Kondo Effect in Single Quantum Dot Systems with Orbital at Finite Temperatures
---
,
and
quantum dot ,Kondo effect ,transport 73.23.-b ,73.63.Kv ,71.27.+a ,75.30.Mb
Introduction
============
The Kondo effect due to magnetic impurity scattering in metals is a well known and widely studied phenomenon [@book:hewson]. The effect has recently received much renewed attention since it was found that the Kondo effect significantly influences the conductance in quantum dot (QD) systems [@pap:D.GG]. A lot of tunable parameters in QD systems have made it possible to systematically investigate electron correlations. In particular, high symmetry in shape of QDs gives rise to the orbital properties, which has stimulated extensive studies on the conductance due to the orbital Kondo effect [@pap:sasaki_st; @pap:st_Eto; @pap:Sasaki2; @pap:pjh; @pap:Choi; @pap:sakano]. The thermopower we study in this paper is another important transport quantity, which gives complementary information on the density of states to the conductance measurement: the thermopower can sensitively probe the asymmetric nature of the tunneling resonance around the Fermi level. So far, a few theoretical studies have been done on the thermopower in QD systems [@pap:Beenakker; @pap:boese; @pap:Turek; @pap:tskim; @pap:Matveev; @pap:BDong; @pap:Krawiec; @pap:Donabidowicz]. A recent observation of the thermopower due to the spin Kondo effect in a lateral QD system [@pap:Scheibner] naturally motivates us to theoretically explore this transport quantity in more detail. Here, we discuss how the asymmetry of tunneling resonance due to the orbital Kondo effect affects the thermopower under gate-voltage and magnetic-field control. By employing the noncrossing approximation (NCA) for the Anderson model with finite Coulomb repulsion, we especially investigate the Kondo effect of QD for several electron-charge regions.
Model and Calculation
=====================
Let us consider a single QD system with $N$-degenerate orbitals in equilibrium, as shown in Fig. \[fug:sch\].
![ Energy-level scheme of a single QD system with three orbitals coupled to two leads. []{data-label="fug:sch"}](pic/sch.eps){width="0.5\linewidth"}
The energy levels of the QD are assumed to be $$\begin{aligned}
&&\varepsilon_{\sigma l} = \varepsilon_d + l \Delta_{orb}, \\
&&l=-(N_{orb}-1)/2,-(N_{orb}-3)/2, \cdots, (N_{orb}-1)/2 \nonumber\end{aligned}$$ where $\varepsilon_d$ denotes the center of the energy levels and $\sigma$ ($l$) represents spin (orbital) index and $N_{orb}$ represents the degree of the orbital degeneracy. The energy-level splitting between the orbitals $\Delta_{orb}$ is induced in the presence of magnetic field $B$; $\Delta_{orb} \propto B$. In addition, the Zeeman splitting is assumed to be much smaller than the orbital splitting, so that we can ignore the Zeeman effect. In practice, this type of orbital splitting has been experimentally realized as Fock-Darwin states in vertical QD systems or clockwise and counterclockwise states in carbon nanotube QD systems. Our QD system is described by the multiorbital Anderson impurity model, $$\begin{aligned}
{\cal H} &=& {\cal H}_l + {\cal H}_d + {\cal H}_{t} \label{eq:hamiltonian} \\
%%%%%%%%
{\cal H}_l &=& \sum_{k \sigma l} \varepsilon_{k \sigma l} c^{\dagger}_{k \sigma l} c_{k \sigma l}, \\
%%%%%%%%%%
{\cal H}_d &=& \sum_{k \sigma l} \varepsilon_{\sigma l} d^{\dagger}_{\sigma l} d_{\sigma l}
+ U \sum_{\sigma l \neq \sigma' l'} n_{\sigma l} n_{\sigma' l'} \nonumber \\
&& \qquad -J \sum_{l \neq l'} \mbox{\boldmath$S$}_{dl} \cdot \mbox{\boldmath$S$}_{dl'} , \\
%%%%%%%%%
{\cal H}_{t} &=& V \sum_{k \sigma } \left( c^{\dagger}_{k \sigma l} d_{\sigma l} + \mbox{H. c.} \right),\end{aligned}$$ where $U$ is the Coulomb repulsion and $J(>0)$ represents the Hund coupling in the QD.
The non-equilibrium Green’s function technique allows us to study general transport properties, which gives the expression for the T-linear thermopower as [@pap:BDong], $$\begin{aligned}
S=-(1/eT) ({\cal L}_{12}/{\cal L}_{11}),\end{aligned}$$ with the linear response coefficients, $$\begin{aligned}
{\cal L}_{11} &=& \frac{\pi T}{h} \Gamma \sum_{\sigma l} \int d\varepsilon \, \rho_{\sigma l}(\varepsilon) \left( - \frac{\partial f(\varepsilon)}{\partial \varepsilon} \right), \\
{\cal L}_{12} &=& \frac{\pi T}{h} \Gamma \sum_{\sigma l} \int d\varepsilon \, \varepsilon \rho_{\sigma l} (\varepsilon) \left( - \frac{\partial f(\varepsilon)}{\partial \varepsilon} \right),\end{aligned}$$ where $\rho_{\sigma l}(\varepsilon)$ is the density of states for the electrons with spin $\sigma$ and orbital $l$ in the QD and $f(\varepsilon)$ is the Fermi distribution function. In order to obtain the thermopower it is necessary to evaluate $\rho_{\sigma l}(\varepsilon)$.
We exploit the NCA method to treat the Hamiltonian (\[eq:hamiltonian\]) [@pap:Bickers; @pap:Pruschke]. The NCA is a self-consistent perturbation theory, which summarizes a specific series of expansions in the hybridization $V$. This method is known to give physically sensible results at temperatures around or higher than the Kondo temperature. The NCA basic equations can be obtained in terms of coupled equations for the self-energies $\Sigma_m(z)$ of the resolvents $R_m(z)=1/[z-\varepsilon_m - \Sigma_m(z)]$, $$\begin{aligned}
\Sigma_m(z) &=& \frac{\Gamma}{\pi} \sum_{m'} \sum_{\sigma l} \left[ \left( M^{\sigma l}_{m' m} \right)^2 + \left( M^{\sigma l}_{m m'} \right)^2 \right] \nonumber \\
&& \qquad \times \int d\varepsilon R_{m'}(z+\varepsilon)f(\varepsilon),\end{aligned}$$ where the index $m$ specifies the eigenstates of ${\cal H}_d$ and the mixing width is $\Gamma=\pi \rho_c V^2$. The coefficients $M_{mm'}^{\sigma l}$ are determined by the expansion coefficients of the Fermion operator $d_{\sigma l}^{\dagger}=\sum_{mm'} M_{mm'}^{\sigma l} | m \rangle \langle m' |$. We compute the density of states by this method to investigate the thermopower.
Results
=======
Gate voltage control
--------------------
The thermopower for two orbitals is shown in Fig. \[fig:double\_vS\] as a function of the energy level $\varepsilon_d$ (gate-voltage control).
![ The thermopower for the two orbital QD system with finite Coulomb repulsion $U=8\Gamma$ as a function of the energy level of the QD. (a) The temperature dependence for $J=0$. The inset shows the conductance as a function of the dot level at $k_BT=0.20\Gamma$ (Coulomb resonance peaks). (b) The Hund-coupling dependence for $k_BT=0.04\Gamma$. []{data-label="fig:double_vS"}](pic/double_v.eps){width="6cm"}
There are four Coulomb peaks around $-\varepsilon_d/U \sim 0,1,2,3$ at high temperatures (see the inset of Fig. \[fig:double\_vS\](a)). As the temperature decreases, the thermopower in the region of $-1<\varepsilon_d/U < 0 (-3<\varepsilon_d/U < -2)$ with $n_d \sim 1 (3)$ is dominated by the [*SU*]{}(4) Kondo effect. The thermopower has negative values in the region $-1<\varepsilon_d/U < 0$, implying that the effective tunneling resonance, such as the Kondo resonance, is located above the Fermi level. At low enough temperatures, the [*SU*]{}(4) Kondo effect is enhanced with decrease of energy level down to $\varepsilon_d/U=-1/2$, which results in the enhancement of the thermopower. However, if the temperature of the system is larger than the [*SU*]{}(4) Kondo temperature, the Kondo effect is suppressed and the thermopower has a minimum in the regime $-1/2 <\varepsilon_d/U <0$. As the energy level further decreases, the [*SU*]{}(4) Kondo effect and the resulting thermopower are both suppressed. Note that the Hund coupling hardly affects the thermopower because of $n_d \sim 1$ in this regime, as shown in Fig. \[fig:double\_vS\] (b). Since the region of $-3<\varepsilon_d/U < -2$ can be related to $-1<\varepsilon_d/U < 0$ via an electron-hole transformation, we can directly apply the above discussions on the [*SU*]{}(4) Kondo effect to the former region by changing the sign of the thermopower.
Let us now turn to the region of $-2<\varepsilon_d/U<-1$, where $n_d \sim 2$. At $J=0$, the Kondo effect due to six-fold degenerate states occurs. Although the resulting Kondo effect is strongly enhanced around $\varepsilon_d/U=-3/2$ in this case, the thermopower is almost zero because the Kondo resonance is located just at the Fermi level. Therefore, when the dot level is changed, the position of the Kondo resonance is shifted across the Fermi level, which causes the sign change of the thermopower. Around $\varepsilon_d/U=-3/2$, even small perturbations could easily change the sign of the thermopower at low temperatures. Note that these properties are quite similar to those for the ordinary spin Kondo effect shown in Fig. \[fig:single\_vS\], because the filling is near half in both cases.
![The thermopower due to the ordinary spin Kondo effect as a function of the dot level. We set $U=6\Gamma$.[]{data-label="fig:single_vS"}](pic/single_vS.eps){width="5.5cm"}
For large Hund couplings $J$, the triplet Kondo effect is realized and the resulting Kondo temperature is very small, so that the thermopower shown in Fig. \[fig:double\_vS\](b) is dramatically suppressed.
Magnetic field control
----------------------
Let us now analyze the effects of orbital-splitting caused by magnetic fields. The computed thermopower for $\varepsilon_d/U=-1/2$ is shown in Fig. \[fig:double\_kS\] as a function of the orbital splitting $\Delta_{orb}$.
![ The thermopower for the two orbital QD system, in case of $\varepsilon_d=-U/2$, as a function of orbital splitting $\Delta_{orb}$. We set $U=8\Gamma$. []{data-label="fig:double_kS"}](pic/double_kS.eps){width="6cm"}
It is seen that magnetic fields dramatically suppress the thermopower, which is caused by the following mechanism. In the presence of magnetic fields, the Kondo effect changes from the [*SU*]{}(4) orbital type to the [*SU*]{}(2) spin type because the orbital degeneracy is lifted. As a consequence, the resonance peak approaches the Fermi level and the effective Kondo temperature is reduced, so that the thermopower at finite temperatures is reduced in the presence of magnetic fields.
Note that, in our model, magnetic fields change the lowest energy level $\varepsilon_{\sigma -\frac{1}{2}}$ from $-U/2$ to $-(U+\Delta_{orb})/2$. Accordingly, the peak position of the renormalized resonance shifts downward across the Fermi level (down to a little below the Fermi level). Thus, the large negative thermopower changes to a small positive one as the magnetic field increases at low temperatures. In strong fields, the effective Kondo resonance is located around the Fermi level with symmetric shape, so that even small perturbations could give rise to a large value of thermopower with either negative or positive sign.
Finally a brief comment is in order for other choices of the parameters. The thermopower for $\varepsilon_d/U=-5/2$ shows similar magnetic-field dependence to the $\varepsilon_d/U=-1/2$ case except that its sign is changed. For $\varepsilon_d/U=-3/2$, the thermopower is almost zero and independent of magnetic fields, because the Kondo resonance is pinned at the Fermi level and gradually disappears with increase of magnetic fields.
Summary
=======
We have studied the thermopower for the two-orbital QD system under gate-voltage and magnetic-field control. In particular, making use of the NCA method for the Anderson model with finite Coulomb repulsion, we have systematically investigated the low-temperature properties for several electron-charge regions. It has been elucidated how the asymmetric nature of the resonance due to the orbital Kondo effect controls the magnitude and the sign of the thermopower at low temperatures.
For $\varepsilon_d/U\sim-1/2$ ($\varepsilon_d/U\sim -5/2$), where $n_d \sim1 (3)$, the [*SU*]{}(4) Kondo effect is dominant and the corresponding thermopower is enhanced. These two regions are related to each other via an electron-hole transformation, which gives rise to an opposite sign of the thermopower. In addition, magnetic fields change the Kondo effect to an [*SU*]{}(2) type, resulting in two major effects: the effective resonance position approaches the Fermi level and the Kondo temperature is decreased. Therefore, the reduction of the thermopower occurs in the presence of magnetic fields.
For $\varepsilon_d/U\sim-3/2$, where $n_d \sim 2$, the Kondo effect due to six-fold degenerate states occurs for $J=0$. However, the thermopower is strongly reduced because the resonance peak is located near the Fermi level. When the Hund coupling is large, the triplet Kondo effect is dominant. The resulting small Kondo temperature suppresses the thermopower around $\varepsilon_d/U \sim -3/2$ at finite temperatures. In this region, magnetic fields do not affect the asymmetry of the resonance peak and the resulting thermopower remains almost zero because the filling is fixed.
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank S. Tarucha, A. C. Hewson, A. Oguri and S. Amaha for valuable discussions. RS was supported by the Japan Society for the Promotion of Science.
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| ArXiv |
---
author:
- |
Charlie Beil$^\dagger$, David Berenstein$^\ddagger$\
$^\dagger$ Department of Mathematics, UCSB, Santa Barbara, CA 93106\
$^\ddagger$ Department of Physics, UCSB, Santa Barbara, CA 93106\
$^\ddagger$ Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, China
title: Geometric aspects of dibaryon operators
---
Introduction and conclusion
===========================
In recent years, it has been understood that the low energy physics of D-branes near Calabi-Yau singularities gives rise to supersymmetric field theories where the gravitational degrees of freedom can be decoupled. The field theory dynamics and the geometry of the Calabi-Yau singularity are very closely related. This relation is most striking in the AdS/CFT correspondence setup [@Malda], where one ends up with a quantum field theory being exactly dual to a gravitational description of the near horizon geometry of the branes, meaning that they both describe the same dynamical system. The evidence for this identification is overwhelming and it suggests that for any object on the field theory side one can find a corresponding object in the gravitational side.
Given that the theories are supersymmetric, one can exploit holomorphy and the natural geometry of branes to obtain identifications between field theories and geometries to provide long lists of conjectured dual pairs (for the case of toric singularities see [@FHMSVW]). If we have a local Calabi-Yau singularity $V$, we expect that it can be described by a three dimensional non-compact algebraic variety. One usually believes that this local problem will be characterized by some affine space: a commutative ring with polynomial relations, rather than a more formal algebraic geometric space that is built by patching these classes of objects into one bigger geometric object. A similar holomorphic polynomial object is the chiral ring of a supersymmetric quantum field theory, so the two holomorphic structures should be related. One could ask how the commutative ring describing the variety $V$ is encoded in the supersymmetric quantum field theory. If moreover the field theory is conformal, the two rings will be graded and $V$ will be a complex cone.
The conformal field theory is usually considered to be a theory of $N$ point-like D-branes on $V$ and we usually expect that the moduli space of the field theory is given by $N$ points of $V$. However, when branes reach the singularity they fractionate: the singularity in moduli space is a physical singularity of the moduli space in the sense that new massless degrees of freedom should be present. This fractionation implies that the gauge symmetry of a single brane at the origin is not $U(1)$ (what one would expect for a single brane), but instead it is a product of gauge fields $\prod_i U(N_i)$, where the $N_i$ indicates the number of fractional branes of type $i$ that are present. One of the problems of establishing the $AdS/CFT$ duality is to determine these $N_i$ and the possible structures of the field theory given a singularity. In the toric case for a brane in the bulk, all the $N_i=1$, so the problem becomes simpler. However, for more general cases there is no uniform answer.
If we blow-up the singularity, the fractional branes are expected to be some branes wrapping cycles in the exceptional locus with some bundles on them and one would call this construction a geometric engineering of a field theory [@Vafa]. There are also situations where one is not allowed to blow-up the singularity because one has discrete torsion degrees of freedom that prevent it [@VW]. This means that there can be more than one class of quantum field theories that gives rise to the same algebraic variety $V$. It would be nice if we could find a way to read $V$ directly from an abstract field theory even when we don’t know how many branes of each type are needed to build a brane in the bulk (the $N_i$ are unknown). Such a program should also predict that the general form of the moduli space is some symmetric product of $N$ copies of $V$ (the branes are indistinguishable). This program were one begins with a field theory and produces the Calabi-Yau geometry where the branes live is the opposite process of geometric engineering, so one can label it reverse geometric engineering. An understanding of how $V$ appears in a given field theory has been described in [@Brev] (for previous work see [@BJL]), but one is not guaranteed to get a Calabi-Yau geometry from such a field theory and all the details of when this works are not understood in general.
The basic idea in [@Brev] is that the field theories that appear are of quiver form and that one can associate to such a quiver theory with superpotential a non-commutative associative algebra (ring) over the complex numbers. The ring is made of chiral gauge variant observables where one interprets the chiral matter fields of the quantum field theory as matrices that can be multiplied and added to make composite fields. One can then obtain gauge invariant observables by taking traces of these matrices. The advantage of this formulation in terms of algebra is that it produces a natural candidate for the variety $V$. The affine ring describing $V$ should be the set of elements of the algebra that belong to the center (in ordinary Lie algebra language these would be called the Cassimir-operators). This is a natural commutative subalgebra of any algebra. One also finds that a vacuum of the field theory is a representation of the algebra (independently of the rank of the gauge groups), so one can use this information to build general representations from the simplest ones and give a simple construction of the moduli space of vacua for arbitrary rank gauge groups.
The traces of elements of the algebra being gauge invariant give natural elements of the chiral ring of the given field theory. The usual notion of the moduli space is that it is a representation of the chiral ring for a collection of fixed gauge groups [@Seiberg], which can be quantum corrected by non-perturbative effects. The non-commutative algebra point of view is that the classical moduli space is a complete family of representations, which is given by direct sums of irreducibles ( a generalized notion of symmetric product), where the rank of the gauge groups is fixed once we fix a particular representative from the family.
One should notice that when one goes to the conformal limit, the $U(1)$ parts of the gauge field decouple, either because they become massive by mixing with closed string fields (this can happen if one cancels an anomaly via the Green-Schwarz mechanism), or because their coupling constant goes to zero. So in the end the gauge group is reduced to $\prod_iSU(N_i)$ and the non-anomalous gauge $U(1)$ fields at zero coupling are left over as global symmetries. These act as generalized baryonic symmetries. This slight change of gauge group has implications for the field theory dynamics, in that we can expect that this decoupling of extra degrees of freedom might modify the moduli space of vacua and make it bigger, because one gauged symmetry becomes just a global symmetry, so the coordinates on this bigger symmetry space become physical.
This enlargement of the moduli space does not destroy the symmetric product structure: all the previous gauge invariant variables and the relations between them are the same. Thus the true moduli space must have a projection map to $Sym^N(V)$ and it gives it an extra lift. The coordinates describing the bigger moduli space should carry the extra $U(1)$ charges. These can be described by elements of the chiral ring that carry baryon charges. In general the possible baryons are not given by traces, but rather by determinant operators. Thus in order to understand this technicality one needs to improve the description of the moduli space. The purely non-commutative algebraic point of view does not know about baryon charges, because those depend on knowing the $N_i$ ahead of time.
One can also show that under the AdS/CFT dictionary, the traces that can be used to generate the chiral ring should be mapped to particular graviton states in the dual AdS theory [@WGKP]. If the dual theory is described by $AdS_5\times X$, for a Freund-Rubin [@FR] ansatz where $V$ is a real cone over $X$ and $X$ is a Sasaki-Einstein manifold [@MP], it is expected that the baryonic operators are mapped to D-branes wrapping some non-contractible cycles in $X$ [@Wittenbar] (see also [@GK]).
Mikhailov [@Mikhailov] has shown that the condition for the D-brane to be BPS can be related to studying holomorphic submanifolds of $V$ (this was the generalization described by Beasley in [@Beasley]). Where the holomorphic submanifold intersects the Sasaki-Einstein manifold, we obtain a supersymmetric cycle that describes the instantaneous position of the brane at some time $t$. The basic conjecture is that dibaryon and multi-baryon operators are given by some cycle inside $X$, but there is no obvious rule to follow that tells us which particular dibaryon or multi-baryon operator goes to which particular cycle of $X$.
In a certain sense, natural conjectures as to how one should describe these objects are available in the literature [@IW], but they are not explicit enough to provide a complete solution of the problem in the most general case. Also, they need to be corrected because it turns out that not all the matter fields joining two nodes correspond to the same anomalous dimension [@BBC] (this results from the a-maximization procedure to calculate anomalous dimensions of fields [@IW1]).
We seem to be in a situation where the matching between operators and geometry is not obvious. The evidence for the identification is done by matching operator dimensions with the energies of branes in the gravity theory (one measures their volumes) plus any other quantum numbers that one has at ones disposal. So in situations where there is a lot of symmetry, one can in principle make a match of sufficiently symmetric objects. However, brane configurations can have degeneracies. This is because they come in families, and so we would like to know how to count these degeneracies as well as how to match them with the field theory. Moreover, we would like to have a precise map between field theory objects and geometric configurations, rather than a patchwork of matching of dimensions and degeneracies.
The purpose of this paper is to explain how to tie the conjectures between branes and dibaryons to a more precise formulation in terms of the (algebraic) geometry of the Calabi-Yau singularity that one wishes to study. In this way one can have a precise description that ties these two classes of objects (dibaryon operators and cycles) in a canonical way, rather than an ad-hoc prescription for matching dimensions and volumes of cycles. Our idea is to give an additional interpretation to the arrows of a quiver diagram as particular global sections of a non trivial holomorphic vector bundle over $V$. In two of the examples we study we have that the arrows can be considered as holomorhic global sections of line bundles over $V$, but we also study one more general example where this is not the case. Our techniques to show this depend on interpreting the quiver theory as a noncommutative geometric space as described above. Our end result is similar to the prescription [@BFZ] that was done for the toric case. However, our techniques in the end do not depend on having this type of constraint on the quiver theory and they can be applied more generally.
Under these conditions, the zero locus of a section of a line bundle is a geometric object that can be identified with the holomorphic objects that Mikhailov’s description requires. Moreover, we will see that the corresponding line bundles are determined by the coordinate ring of $V$: the locus where the vev of an arrow vanishes is a submanifold in the geometry determined by the coordinates of $V$. These submanifolds are (Cartier) divisors corresponding to line bundles on $V$. If $V$ is a holomorphic cone with base $B$ it admits a ${{\mathbb C}}^*$ action of rescalings. If the subvariety determined by the divisor of $V$ is invariant under the rescalings, then it also is a holomorphic submanifold of $B$.
We will show how this works in detail for three different Calabi-Yau geometries. We first consider the Klebanov-Witten setup of the conifold [@KW], following the previous work of the second author [@BHK]. The simplicity of this setup will provide intuition in how we approach our second more involved example, the Calabi-Yau complex cone over the first del Pezzo surface, which is the vev moduli space of the $Y^{2,1}$ gauge theory. In this example we will be able to show that once we match the cycles in the Sasaki-Einstein manifold via the prescription we conjecture, the $R$-charges determined by $a$-maximization will be precisely the volumes of the corresponding zero-vev loci. These nontrivial computations provide strong evidence for our conjecture. Finally, we will study a Seiberg dual realization of the orbifold geometry ${{\mathbb C}}^3/{{\mathbb Z}}_3$. In this geometry we show how to extend the results to more involved multi-baryon operators. The main difference with the previous two examples is that the arrows have to be interpreted as sections of some vector bundle of higher rank. If one assembles the various arrows with care, one finds that one can generate a vector bundle map that is locally described by a $k\times k$ matrix. The baryonic operators end up being related to a locus where the map is degenerate: the kernel and cokernel jump rank. This is captured by the matrix. If we take the determinant of the matrix, one can also be think about this more general object as the zero locus of a global section of the determinant line bundle associated to the map. However, in this case, the individual arrows will not have to vanish at the degenerate locus.
Our conclusion is that the procedure for identifying the geometry of dibaryons, although somewhat involved computationally in the examples, can be made canonical from the point of view of algebraic geometry. It is also clear that this works in non-toric geometries and that we have a general setup for performing these computations. It should be interesting to apply these techniques to other conformal field theories and see the match between the geometry of cycles and the counting of generalized dibaryons.
Dibaryon operators in the conifold theory
=========================================
The simplest example to begin with is the conifold geometry. The conformal field theory was described by Klebanov and Witten [@KW] and derived by Morrison and Plesser [@MP], and it consists of two gauge groups $SU(N_1)\times SU(N_2)$ with $N_1=N_2$ and a set of four bifundamental chiral fields $A_1, A_2$ transforming in the $(N_1, \bar N_2)$ and $B_1, B_2$ transforming in the $(\bar N_1, N_2)$. The superpotential of the theory is given by $$W\sim \epsilon_{ij} \epsilon_{lm}{\hbox{tr}}(A_i B_l A_j B_m)$$ where we use the natural contraction of indices as matrices to describe the superpotential. The $R$-charge of the fields $A,B$ is one half at the conformal point (the dimension of the scalar fields in the multiplet is $3/4$ rather than one).
The superpotential is invariant under an $SU(2)\times SU(2)$ global symmetry where the $A$ transform as a $(1/2,0)$ and the $B$ transform as a $(0,1/2)$. The superpotential is also invariant classically under a non-anomalous $U(1)_B$ baryonic symmetry where $A,B$ have equal and opposite charges. This charge is called the baryon charge. Indeed, one can think of the theories as being given by a $U(N)\times U(N)$ symmetry where only the $SU(N)$ part has been gauged. The extra non-gauged $U(1)$ symmetry is the baryonic number (there is a diagonal $U(1)$ which is decoupled). The fields $A,B$ have a non-zero anomalous dimension when the theory is conformal. The field theory at the conifold should be considered strongly coupled.
One can show that the F-term equations lead to vacuum configurations where the $A_iB_j$ product matrices commute with each other. Generically, the $AB$ matrices do not have degeneracies in the eigenvalues and they can be diagonalized simultaenously by a $GL(N,{{\mathbb C}})$ transformation. In the superfield formulation, the gauge symmetry can be understood as acting by conjugation by $GL(N,{{\mathbb C}})$ or $SL(N,{{\mathbb C}})$ transformations, so this should be understood as an allowed gauge transformation [@WBagger]. The gauge invariant variables are the eigenvalues of the $AB$ matrices.
If we use the notation $U=A_1B_1$, $V=A_2B_2$, $W=A_1B_2$, $Z=A_2B_1$, we find that the eigenvalues $u_i, v_i, w_i, z_i$ satisfy the equations $$u_iv_i= w_i z_i\label{eq:conifold}$$ so that each joint eigenvalue represents a point in the conifold, which is described by the equation $uv=wz$. Thus, we find that the moduli space can be thought of a describing $N$ points in the conifold, one for each eigenvalue (see [@Bcon] for more details).
The eigenvalues of the $AB$ products can be recovered by taking traces of products ${\hbox{tr}}(U^{n_1} V^{n_2} W^{n_3} Z^{n_4})$. The order inside the trace does not matter because the matrices commute with each other. These traces are invariant under joint permutations of the eigenvalues. This permutation symmetry is a residual gauge symmetry of the system.
These operators are elements of the chiral ring, which is defined as the set of chiral gauge invariant scalar operators modulo the F-term relations. It is the F-term relations that guarantee that the order of the matrices does not matter within the chiral ring.
However, we have missed one important detail. We did the diagonalization at the level of the products $AB$, but we did not work the details of $A,B$ individually to see if there is more information we are missing. Indeed, there is such additional information that is not immediately apparent from the F-term relations. The key to understanding this information is that one can be more precise as to how one solves the moduli space of vacua problem. This is where one can see that there is a difference between gauging by $U(N)$ and by $SU(N)$ transformations in the gauge theory, $GL(N,{{\mathbb C}})$ and $SL(N,{{\mathbb C}})$. The missing generators of the chiral ring are going to be baryonic objects that are charged under the additional $U(1)_B$ symmetry. The traces defined above are neutral with respect to the $U(1)_B$ symmetry.
For example, one can consider the objects given by $$\det(B_1)(0) \sim \frac 1 {N!}\epsilon_{i_1\dots i_N} \epsilon^{j_1\dots j_N} (B_1)^{i_1}_{j_1}\dots (B_1)^{i_N}_{j_N}(0)$$ This operator is invariant under $SL(N,{{\mathbb C}})\times SL(N,{{\mathbb C}})$ gauge transformations (the $\epsilon$ tensors are invariant tensors) and it is a baryonic object with charge $N$ (if each of the $B$ carry charge one). If we replace various $B_1$ by $B_2$, we find that this collection of operators transform as an $(0,N/2)$ representation of $SU(2)\times SU(2)$ [@BHK]. The symmetry is obtained by noticing that the $B$ are bosonic, and that the operator is totally symmetric in exchanges of the different $B$ objects. There are $N+1$ such operators. They are classified by how many $B_1$ and $B_2$ they have.
There are similar objects with $A$ fields rather than the $B$ fields. We can also consider a product of two or more of these operators. One important question is to count how many independent such products one can have. This counting can serve as a test of the $AdS/CFT$ correspondence if we can count states in the gravity side (one can sometimes do this via an index theorem for a quantization of a compact space of configurations). We should always ask this question in the interacting theory. Another important question is to understand how to describe these objects in the dual gravitational theory in terms of the AdS/CFT correspondence. These issues have been studied in detail for the case of the conifold in [@BHK], but a complete answer was not found for the multibaryon operators, nor for the correct counting of excited states of a given dibaryon. The problem of counting all chiral ring operators has been systematically started in the works [@FHH].
This same type of reasoning can be applied to other conformal field theories in four dimensions. In general, one would like to know the $AdS$ geometry corresponding to a given baryonic operator.
In the following subsection we outline the mathematical formalism that makes our prescription apparent. We describe the algebraic tools developed in [@BJL; @Brev] for the three examples we consider in this paper.
Quiver gauge theories as representations of algebras
-----------------------------------------------------
This section reviews the points of view developed in [@BJL; @Brev] regarding quiver theories as an algebra with an attached family of representations of the algebra. This mathematical point of view rewrites the problem of finding the solution to the moduli space of vacua into a problem of calculating the irreducible representation theory of an algebra. One can then assemble this information together into a full description of the moduli space of vacua that is easy to understand from the point of view of branes in some geometry. This is a generalization of the problem described above for a particular theory, where instead of commuting matrices one introduces a more general set of relations that need to be solved. In this description various aspects of the moduli space of vacua of a given field theory become obvious. In particular, the fact that the field theory moduli space is some generalized notion of symmetric product. Also, other aspects of brane fractionation can be characterized in a simple manner.
The idea is to begin with a quiver theory with some superpotential. A quiver is a graph with oriented arrows. The nodes of the graph will represent gauge groups and the arrows will represent matter fields.
The gauge groups will be either $U(N)$ or $SU(N)$. The case of $U(N)$ is simpler and it does not have baryonic operators. We will be interested in the problem for the gauge group $SU(N)$. This will lead to a generalized version of symmetric product that takes into account baryonic operators. Once we understand how this works, we will be able to attack the problem of how to relate these operators to geometry in the dual gravity theory. The new development in this section is on how to get this additional information encoded into the algebraic setup.
In the superfield formulation of supersymmetric theories, the full quiver gauge group is really the complexification of $U(n)$, which is $GL(n)$. The extended gauge invariance resulting from the complexification is reduced to the usual non-complexified gauge invariance in the Wess-Zumino gauge. At the level of calculating the moduli space of vacua, the extra constraint is realized by noticing that the D-terms of the gauge theory must vanish. These D-terms are interpreted as moment maps in the full theory, so one can describe the problem of finding the moduli space of vacua as a symplectic quotient construction, which becomes equivalent to a geometric Invariant theory quotient. This is a fundamental result that will be used throughout this paper implicitly [@LT].
The arrows in the quiver, joining nodes $i,j$ are in bifundamental representations $(N_i, \bar N_j)$ of the gauge groups associated to the end-points of the arrows. The directionality of the arrow indicates for which gauge group the matter is in the fundamental representation, and for which gauge group it is in the antifundamental representation.
The natural two index structure of these objects makes it possible to think of these chiral fields as matrices that connect two auxiliary vector spaces of dimensions $N_i$ and $N_j$ over ${{\mathbb C}}$. Let us call these auxiliary spaces $V_i$ and $V_j$. Thus, we can write in a formal way that $$\phi_{ij} \in Hom(V_j, V_i)$$ so that each arrow gives us a morphism between vector spaces. [We take the convention where $\phi_{ij}$ acts to the left of $V_j$, so for $v \in V_j$ we write $\phi_{ij}(v) \in V_i$.]{} The arrows should be drawn according to this convention (the direction in which the ).
Obviously, we cannot multiply an arrow ending at node $j$ with an arrow beginning at node $i$ in any non-trivial way that makes sense from the point of view of matrix multiplication, and so their product is defined to be zero. To do this in a slightly formal way, $$1 = \sum_i e_i$$
The statement that arrows begin and end on given nodes can be formalized in terms of algebraic matrix equations of the following form $$\phi_{ij} e_k = \delta_{jk} \phi_{ij}, \quad e_k\phi_{ij} = \delta_{ik} \phi_{ij}$$ while the ’gauge’ transformations by the $U(1)_i$ would be given by commutators with $e_i$. It is easy to convince oneself that these rules make sense.
Now, we can ask what is the role of $GL(N_i,{{\mathbb C}})$ and $SL(N_i,{{\mathbb C}})$ transformations from this more formal matrix point of view. Well, elements of $GL(N_i,{{\mathbb C}})$ or $SL(N_i,{{\mathbb C}})$ act on a natural way on the vector spaces $V_i$. They do specific changes in the basis of $V_i$. When we change the basis of $V_i, V_j$ we can think of the matrices $\phi_{ij}$ as being invariant objects that do not depend on a basis, but their specific components do transform with the change of basis. These will be identical to the gauge transformations of the fields $\phi_{ij}$ if we are careful. Thus, we realize that the fields $\phi_{ij}$ transform covariantly with respect to this auxiliary structure.
The other thing we notice is that composition of matrices is associative, so these multiplications of fields to make composite fields can be encoded naturally into a framework of having an associative product for the fields. The idea is that now we can abstract these concepts to state that the fields $\phi$ have a natural multiplication on their own, even in the absence of the vector spaces $V_i$ or the labels $N_i$. This is, the fields $\phi$ in this situation give rise to a natural algebra structure on their own. The generators of the algebra are the arrows of the quiver and the $e_i$, and all composite paths of arrows define abstractly an element of the algebra. We will allow to take general finite linear combinations of these objects with complex coefficients. The end result we get is a path algebra of the associated graph.
We are just showing that the quiver structure associated to gauge theories of a particular type naturally lead to a notion of an associative algebra. Let us name this algebra ${\cal A_Q}$, where ${\cal Q}$ is the quiver. So what happens when we substitute arrows for specific matrices? In the algebraic setup, this means that we have a map $\mu:\phi_{ij}\to Hom(V_j,V_i)$ where we have prescribed some particular vector spaces $V_i$, and such that the abstract definition of the algebra of the $\phi$ with it’s tautological (standard) multiplication is reflected into having matrices $M_{ij}$ that satisfy the same multiplication table. At this level, this is essentially trivial, because we have essentially no non-trivial relations between the generators, but from a formal point of view what we realize is that we have a representation of the algebra ${\cal A_Q}$ realized by matrices. This is, given the $V_i$, any collection of matrices will do, where the only ones that are fixed are $e_i$. They are such that $e_i v_j = \delta_{ij} v_j$ for any vector $v_j \in V_j$.
So far, the algebra we have is very easy to understand. We have relations from incidence into the different nodes of the quiver and that is all. The idea is that now we can consider a superpotential for the field theory.
Within perturbative string theory one usually generates a superpotential of the general single trace form (a disc diagram on the worldsheet). These are the natural objects that can be associated to geometry. The superpotential will have the general form $$W= {\hbox{tr}}( X )$$ where $X$ is any element of the path algebra and ${\hbox{tr}}$ stands for an ordinary matrix trace (this is invariant under cyclic permutations and also under similarity trasnformations). One can show easily that $X$ can only depend on [oriented cycles (closed paths) in the quiver]{}. This is because $$W= {\hbox{tr}}(1^2 X)={\hbox{tr}}( (\sum e_i)^2 X)= \sum{\hbox{tr}}(e_i^2 X)= \sum {\hbox{tr}}( e_i X e_i)$$ Notice that $e_i Xe_i$ are paths that begin and end on node $i$. This condition tells us that the superpotential ends up being gauge invariant, because on each such **cycle** the $SL(N_i,{{\mathbb C}})$ change of basis acts by conjugation and the trace is invariant due to the cyclic property.
If one considers the F-term equations associated to this superpotential, it is clear that they can also be written as algebraic relations involving the generators in the associative algebra [@BJL; @Brev]. Thus, what we have is a path algebra with relations and the relations are derived from a superpotential.
For example, in the case of the conifold, the F-term relations read $$\begin{aligned}
B_1 A_i B_2&=& B_2 A_i B_1\\
A_1B_i A_2&=& A_2 B_i A_1\end{aligned}$$ This means that the problem of solving the F-term constraints reduces to the problem of finding matrices that satisfy these relations, modulo gauge transformations (which at the level we have described are a change of basis of the $V_i$). This identification under change of basis can be thought of as equivalence classes of representations, so they are objects in the category of modules of the algebra ${{\cal A_Q}}$. It is easy to manipulate the above equations to show that $$\begin{aligned}
W_{ij} W_{lm} = W_{lm} W_{ij}\end{aligned}$$ where $W_{ij} = A_i B_j$. Similarly, we can consider the $\tilde W_{ij} = B_j A_i$ and we can show that these also commute with each other.
One can also use the more formal objects of the algebra $Z_{ij} = W_{ij}+ \tilde W_{ij}$. It is easy to show that $Z_{ij}$ commutes with all of the elements of the algebra. We do this by showing that it commutes with the $e_i$ and $A_i, B_i$, the generators of the algebra. This is a simple exercise [@Bcon].
One can also recover the original $W, \tilde W$ matrices by projecting with the $e_i$ as follows $W_{ij} = e_1 Z_{ij} e_1$ and $\tilde W_{ij} = e_2 Z_{ij} e_2$.
Finally, one also gets the relations $Z_{11} Z_{22} = Z_{12} Z_{21}$ which is a matrix version of the conifold geometry (\[eq:conifold\]).
Now, let us assume that we have two representations of the algebra and let us call them $R_1$, $R_2$. It is easy to show that $R_1\oplus R_2$ is also a representation. This is the standard direct sum of modules for the algebra. Another standard result of representation theory is that if one has a module map between representations of an associative algebra $$\mu: R_1\to R_2$$ then the kernel and the coset $R_2/\mu(R_1)$ are also representations. This means that we can build more general solutions of the relations by using smaller representations.
A representation $R$ is called irreducible if it has no subrepresentation inside it. This is, if one has any map $\mu: R_1\to R$, and $\mu(R_1)\neq 0$, then $\mu(R_1)=R$.
For irreducible representations one can use Schur’s lemma. This states that any element of the center is proportional to the identity. We have already seen that the $Z_{ij}$ all belong to the center. This means that in an irreducible representation they should be proportional to the identity. Notice that the $Z_{ij}$ define a commutative algebra over the complex numbers, and that they recover the conifold geometry as the algebraic variety defined by the center algebra.
Also, since the $e_i$ form a complete set of projectors that commute with each other, we can choose representations where the $e_i$ are diagonal. These are of the form $$e_1 \sim \begin{pmatrix}1&0\\0&0\end{pmatrix}, e_2\sim \begin{pmatrix}0&0\\0&1\end{pmatrix}\label{eq:gen}$$ where the block diagonal decomposition can be of arbitrary dimension.
If we also write the algebra carefully, we find that as a left module over the center, the algebra is generated by $A_i, B_i, e_1, e_2$. This is, a general algebra is finite dimensional over its center. Thus the full algebra of the quiver theory can be interpreted as a particular coherent sheaf of finite dimension over the usual algebraic variety (this turns out to be a holomorphic vector bundle away from the singularity).
Working a little bit harder, one finds that the irreducibles can be described with $2\times 2$ matrices where the $e_i$ are given as in (\[eq:gen\]), and the $A,B$ matrices are strictly off-diagonal. There are no relations between them. So, a naive guess is that the moduli space is ${{\mathbb C}}^4$. However, if we gauge the $GL(1,{{\mathbb C}})$ symmetry, we get that the moduli space is properly ${{\mathbb C}}^4/{{\mathbb C}}^*$ and the conifold is a quotient variety. The extra dimension we get in the moduli space should be thought of as a baryonic direction (this has been called the Master space [@Master]). Also, given any representation, the dimensions of $V_1, V_2$ can be calculated by taking traces of $e_1, e_2$ and in this case this gives us one for each.
Under these conditions, the $A$ become just numbers, as well as the $B$: they act as homomorphisms between one dimensional vector spaces.
The exception to this happens only if none of the $Z_{ij}$ are invertible, where one finds smaller representations, with $e_1=1, e_2=0$, or $e_2=1, e_1=0$ and all arrows given by zero. Let us call these $S_1, S_2$. These smaller representations at the singular locus can be called fractional brane representations. One can show that if we take the general representation $R$ of $2\times 2$ matrices and we let $A\to 0, B\to 0$, then we have $$\lim_{A,B\to 0} R \to S_1\oplus S_2$$ while one has a non-trivial short exact sequence $$0\to S_1\to \lim_{A\to 0} R \to S_2\to 0$$ that is parametrized by the values $b_2,b_1$. This makes the identification of arrows in the quiver with $Ext$ groups easy to understand. The $Ext^1(S_2,S_1)$ groups can be characterized by the equivalence classes of such extensions (this is the counting of massless modes between D-branes [@Douglas]). In this case, the dimension space of the extension space is two. This is the same as the number of arrows in the quiver diagram going from node one to node two. One can do a similar analysis with the other arrows, by exchanging the roles of $S_1, S_2$.
So far we have been cavalier with the role played by $GL(N,{{\mathbb C}})$ or $SL(N,{{\mathbb C}})$. At this level we have not encountered an obvious difference yet. The difference is at the level of which changes of basis are allowed. If we allow general changes of basis, then we are working with $GL(N,{{\mathbb C}})$. However, to work with $SL(N,{{\mathbb C}})$ we end up with a restriction on the changes of basis: we are only allowed to make a change of basis that preserves a volume form for the $V_i$. This is, we have to choose a preferred element of $\omega_1\in \Lambda^N V_1\simeq {{\mathbb C}}$ and $\omega_2\in\Lambda^N V_2\simeq {{\mathbb C}}$, which are one dimensional vector spaces over ${{\mathbb C}}$. This is not something that fits easily within a purely algebraic problem.
However, we can work around this by using a compensator. We can allow general changes of variables, so long as we compensate by rescalings of the volumes as we change variables. This means that for theories with $SL(N,{{\mathbb C}})$ groups we can still use the general changes of variables, but we have to add volume forms on the vertices. These volume forms also transform. If we ignore the volume forms, we get back the same problem as with $GL(N,{{\mathbb C}})$ group, and that has been solved in terms of representation theory already.
If we put many of these representations together, and we gauge the $SL(N,{{\mathbb C}})$ symmetry, we will get that the total moduli space is a line bundle over $N$ copies of the conifold. This is because the action of the $SL(N,{{\mathbb C}})$ gauges the $GL(1,{{\mathbb C}})^{N-1}$ diagonal subgroup. In this manner we find that $${\cal M}_{GL} = {\cal M}_{SL} //GL(1,{{\mathbb C}})$$
The fact that the representation theory still works means that the full moduli space in the case of the $SL(N,{{\mathbb C}})$ theory can be mapped onto the moduli space of the $GL(N,{{\mathbb C}})$ theory in a canonical way, and we have an algebraic fibration structure $$GL(1,{{\mathbb C}})\to {\cal M}_{SL} \to {\cal M}_{GL}$$
The fibration is parametrized by the (complex) size of the volume forms, so it should be thought of as a $({{\mathbb C}}^*)^2$ fibration, this is the same as having $GL(1,{{\mathbb C}})^2$ orbits. Now, a map like $A_1$, from $V_1$ to $V_2$ also acts on the volume forms in an obvious way (the matrix $A_1$ can act by products on tensors). We can call the quantity $$\det(A_1) \sim A_1( \omega_1)/ \omega_2$$ and this gives our notion of dibaryon operators. This is a number on a given representation with choices of volume forms. These are the new coordinates of the chiral ring.
Since the moduli space for the $SL(N,{{\mathbb C}})$ theory is naturally fibered over the moduli space for the $GL(N,{{\mathbb C}})$ theory and the fiber is finite dimensional, while the base can have arbitrarily large dimension, it makes sense to try to think geometrically in terms of the natural geometry of the base. The base is the moduli space of vacua of the $U(N)\times U(N)$ theory. It is a symmetric product $Sym^N(V)$, where $V$ is the conifold variety, the locus $uv=zw$ in ${{\mathbb C}}^4$. Each of the points of $V$ that is selected is described by an irreducible representation.
What we would like to have now is a geometric interpretation of the fields $A, B$ in terms of the geometry of the conifold, even if it is on an irreducible representation. So far, this is not apparent. This is where having the algebraic description given above will make a difference.
Consider the algebra of the conifold quiver ${\cal A_Q}$. The algebra can be considered as a left module over itself: we multiply by elements of the algebra on the left. The algebra can be split as a left module in the obvious form $${{{\cal A_Q}}} = {{\cal A_Q}}e_1 \oplus {{\cal A_Q}}e_2$$ This is because every arrow in the quiver ends in one of the two nodes. Indeed, if we multiply by elements of ${{\cal A_Q}}$ on the right, this commutes with left multiplication, so it provides a natural way to build module maps. If we multiply by $1= e_1+e_2$ on the right we recover the splitting above. Each of the modules ${{\cal A_Q}}e_1$ and ${{\cal A_Q}}e_2$ are direct summands of a free module (${{\cal A_Q}}$ itself) and therefore they are projective. Projective modules are the natural generalization of vector bundles in this context.
Each of these is also finitely generated over the center of the algebra (the algebra itself has that property). Thus, one should be able to think of ${{\cal A_Q}}e_1$ and ${{\cal A_Q}}e_2$ as vector bundles over the conifold geometry. Since the conifold geometry is singular, one can localize this property away from the singular locus $u=v=w=z=0$ by taking inverses for some of these variables.
For example, we can consider the complex submanifold described by $u$ being invertible. We can consider a different patch where $w$ is invertible. If $u$ is invertible, the quotient $w/u$ makes sense. Similarly, if $w$ is invertible, the quotient $u/w$ makes sense. With these two patches one can construct the blow-up of the conifold at the origin. The coordinates $\xi= w/u$ and $\xi'=u/w$ are patched as $\xi'= 1/\xi$. These describe a $\mathbb{CP}^1$ geometry. The blow up of the conifold is the total space of ${{\cal O}}(-1)\oplus{{\cal O}}(-1)$ over $\mathbb{CP}^1$. The coordinates $\xi, \xi'$ describe the coordinates of the base of this fibration.
Notice that on an irreducible representation we also have that $$u/w = Z_{11}/Z_{12} = b_1/b_2$$ but in the full algebra one should write instead $$Z_{11}/Z_{22} = e_1 B_1 B_2^{-1} e_1 + e_2 B_2^{-1} B_1 e_2$$ because the $B$ are not invertible in the full algebra, but they are as maps between the $V_i$.
For an irreducible representation the quotient of the arrows can be thought of as a set of coordinates on $\mathbb{CP}^1$. The non-normalized coordinates $b_1, b_2$ can be interpreted as two distinct holomorphic sections of the hyperplane bundle on this $\mathbb{CP}^1$. Notice also that the locus where $b_1$ vanishes does not depend on the normalization of the arrows and it defines a holomorphic submanifold of the blow-up. We can project this submanifold to the blowdown and we can therefore identify some geometric locus on the conifold $V$ associated to an arrow of the quiver itself. This is the locus where $b_1$ vanishes. We can do the same for the $A$ arrows.
In essence, from the point of view of geometry, we can interpret the noncommutative variables that extend the commutative geometry of the center to include non-diagonal matrices as holomorphic sections of particular (line) bundles on the complement of the singular locus. These also belong to $$Hom({{\cal A_Q}}e_1, {{\cal A_Q}}e_2)$$ because of our interpretation of arrows in the quiver as maps between these modules. These $Hom$ functors are also modules over the center and define for us a coherent sheaf on the conifold, whose global sections are the paths in the quiver starting on one node and ending on another one. A particular choice of an off-diagonal element corresponds to a particular choice of global section.
For the case of toric quiver diagrams the elements of $Hom({{\cal A_Q}}e_1, {{\cal A_Q}}e_2)$ describe a line bundle. This is because one expects that the set $e_i {{\cal A_Q}}e_i$ is isomorphic to the center of the algebra. Thus, if we take the two non-zero elements $v_1, v_2\in Hom({{\cal A_Q}}e_1, {{\cal A_Q}}e_2)$, we can compare them by taking quotients (the ratio $v_1/v_2$ makes sense at a generic point) and gives a rational function of elements of $e_1 {{\cal A_Q}}e_1$: the algebraic variety describing the singularity. This is the same type of comparison that tells us that we have a line bundle: we produce rational functions by taking ratios of global sections. The divisor characterizing the line bundle is the polar locus of the quotient.
We can also blow up the $A$ variables. This gives us a different $\mathbb{CP}^1$. Between the $A$ and the $B$ variables we can see that there is a ${\mathbb{CP}^1}\times {\mathbb{CP}^1}$ space appearing. The Sasaki-Einstein space $T^{11}$ is a regular circle fibration over $\mathbb{CP}^1\times \mathbb{CP}^1$. This $\mathbb{CP}^1\times \mathbb{CP}^1$ is the complex base of the Calabi-Yau cone.
The dibaryon operators in $AdS_5\times T^{11}$ are branes that are located at one point of either of the $\mathbb{CP}^1$. They wrap the other $\mathbb{CP}^1$ and the circle fiber. This is the intersection of the locus $a_1=0$ or $b_1=0$ with the Sasaki-Einstein base. The locus $a_1=0$ is the same locus as $\{(u,v,w,z) \in \mathbb{C}^4 \ | \ uv-wz=0 \text{ and } u=0 \cap w=0\}$. Remember that $u=a_1b_1$, $w=a_1b_2$, so that $a_1$ vanishing implies that both $u,w$ have to vanish.
Notice that in the conifold, even though this is a space of codimension one, it can not be described as the zero locus of a single holomorphic function (like setting $u=0$ alone). This is how we know algebraically that one needs a section of a nontrivial line bundle over the conifold in order to describe it.
Conjectures
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To summarize, our algebraic point of view has upgraded the chiral fields in a quiver with a superpotential to be related to global sections of some holomorphic vector bundles (coherent sheafs) on the geometry of the center algebra, which we will take to be the Calabi-yau geometry where D-branes are moving (see [@Brev] for conjectures regarding how these ideas fit together into one framework and what conditions are required for this point of view to be useful).
[The zero locus of a given global section defines a holomorphic submanifold of the Calabi-Yau geometry. Mikhailov has shown that in conformal field theories, holomorphic submanifolds can be used to define dibaryons and giant gravitons: they are the locus where the holomorphic submanifold intersects the Sasaki-Einstein base of the Calabi-Yau geometry, which is at a fixed distance from the origin. The Sasaki-Einstein manifold is itself a circle bundle whose base is a projective variety.]{}
For the case of the conifold, the base is a $\mathbb{CP}^1\times \mathbb{CP}^1$ geometry, and the Sasaki-Einstein manifold is a circle bundle over the $\mathbb{CP}^1\times \mathbb{CP}^1$ base. If we take a homogeneous global section (with fixed R-charge), the zero locus will be invariant under the $U(1)_R$ symmetry of the conformal field theory and it will wrap the circle direction of the Sasaki-Einstein space. This locus can be projected down to the $\mathbb{CP}^1\times \mathbb{CP}^1$ base and we get a holomorphic object on the base of the cone.
The idea is that we will associate exactly such a geometric locus to any element of the algebra that begins in one node and ends at another (possibly the same) node. This conjecture can be made more convincing with the ideas of emergent geometry as espoused in [@BHart]. One understands that the chiral ring is a holomorphic quantization of the moduli space of vacua. A holomorphic quantization would require a line bundle ${\cal L}$ on the moduli space of vacua, and the wave functions would be holomorphic sections of such a line bundle. If we just look at the symmetric product base, one usually takes the structure sheaf bundle for ${\cal L}$, but this is not required. Since the moduli space is a fibration over the symmetric product base, a dibaryon wave function is sensitive to the details of this fibration, and this would result in holomorphic sections over the base that are due to different line bundles. These would measure the twisting of the fibre by the dibaryon charge. This means that each dibaryon (or multi-baryon) charge picks a line bundle and a specific dibaryon-like operator would pick a global section of the associated line bundle. A similar prescription can be found in [@BFZ] which specializes to the case of toric geometries and the line bundles are constructed from the toric data.
As we have seen, for the case of the conifold, this a straightforward procedure and it matches the previous known geometric results exactly. Now we want to apply this to the case of the Calabi-Yau manifold called $Y^{2,1}$. This is a non regular Sasaki-Einstein manifold and its metric was constructed in [@GMSW].
An example: the $Y^{2,1}$ quiver gauge theory
=============================================
The $Y^{2,1}$ quiver gauge theory has quiver $$\xy (0,0)*+{2}="1";(0,25)*+{3}="2";(25,25)*+{4}="3";(25,0)*+{1}="4";
{\ar@/^/|-{a_2}"1";"2"};{\ar@/_/|-{b_2}"1";"2"};{\ar@/^/|-{a_4}"3";"4"};{\ar@/_/|-{b_4}"3";"4"};
{\ar@{->}|-{d}"2";"3"};{\ar@/^1pc/|-{a_1}"4";"1"};{\ar@/_1pc/|-{b_1}"4";"1"};
{\ar@{->}^{c_3}"1";"3"};{\ar@{->}_{c_2}"2";"4"}; {\ar@{->}|-{c_1}"4";"1"};
\endxy$$ and superpotential $$\label{hi} W = \text{tr}\left(c_3 \left( b_2a_1 - a_2b_1 \right) + c_1 \left( b_4da_2 - a_4b_2 \right) + c_2 \left( b_1a_4 - a_1b_4 \right)\right).$$ The theory was constructed in [@FHHtoric] (see also [@HW] for a different viewpoint), and has been analyzed extensively in many papers. We will follow some of the algebraic geometric details described in [@BHOP] to make everything as explicit as possible.
It has been proven in [@B1] that a single brane at a generic point of the vacuum moduli space (that is, an irreducible representation of the corresponding algebra) will have a 1 dimensional vector space at each node and so the arrows will be represented by complex numbers; this is expected since the $Y^{2,1}$ geometry is toric. The gauge group will then be $\prod_{1 \leq i \leq 4}GL(1,{{\mathbb C}})$, and consequently only the traces of fields (representations of cycles) count as gauge invariant observables. These can be characterized by the set of cycles centered at any of the nodes. It is a nontrivial task to show that if one solves the F-term constraints, then the set of cycles centered at one node are related to the set of cycles centered at any other node, and their representations give identical numerical values (this has also been proven in [@B1]).
As we will show below, the vacuum moduli space arising from the quiver (the moduli space of irreducible representations of the algebra) is a complex cone whose base is the first del Pezzo surface $dP_1$, which is $\mathbb{CP}^2$ blown-up at one point. It is conjectured that the moduli space is also a real cone over the $Y^{2,1}$ non-spherical horizon.
We will first describe the algebraic geometry of the $dP_1$ surface in detail. Then we will see how we can read the $dP_1$ geometry from studying the quiver algebra. The idea is that $dP_1$ can be embedded in a simple way into the algebraic variety $\mathbb{CP}^2\times \mathbb{CP}^1$. Using the projective coordinates $\left[x_1: y_1: z_1; x_2:y_2 \right]$ for the $\mathbb{CP}^2\times \mathbb{CP}^1$ space [^1] $dP_1$ may be described as the locus $\{ x_2y_1 = y_2x_1 \} \subset \mathbb{CP}^2 \times \mathbb{CP}^1$. The $dP^1$ has a clear projection to $\mathbb{CP}^2$ which is done by using the forgetful map that sends $f: [x_1:y_1:z_1; x_2:y_2] \to [x_1:y_1;z_1]$. This map is one to one away from $f^{-1}[0,0,1]$. The exceptional divisor of $dP^1$ is the inverse image of $[0,0,1]$, and we call it $E= f^{-1}[0,0,1]$. This inverse image is a copy of $\mathbb{CP}^1$ (the equation that the $dP^1$ variety satisfies is solved for arbitrary values of $[x_2:y_2]$).
If we consider an hyperplane $H$ on the $\mathbb{CP}^2$ projection that does not pass through $[0,0,1]$, then the inverse image $f^{-1}(H)$ will be a subvariety of $dP^1$ which we will also call the hyperplane locus. It is easy to establish the following intersection numbers $H\cdot E=0, H\cdot H=1, E\cdot E=-1$. The subvarieties $E,H$ generate the second homology group of $dP_1$. It is also convenient to introduce a cycle which we will call a line $L$. The immersion on $\mathbb{CP}^2\times \mathbb{CP}^1$ has a projection $\pi$ onto the $\mathbb{CP}^1$ (again defined by a forgetful map onto the second set of coordinates). If we take any point $p \in \mathbb{CP}^1$, we will define the class of $L$ as $\pi^{-1}(p)$.
It is easy to establish that $L$ projected onto $\mathbb{CP}^2$ gives a line passing through the point that is the blow-down of the exceptional divisor (we will call this point the origin in $\mathbb{CP}^2$). One easily shows then that $L\cdot E=1$, $L\cdot H=1$, $L\cdot L=0$. But the class of $L$ should be a class in the second homology of $dP^1$, which is generated by $H,E$. One can determine that $[L]=[H]- [E]$. Thus, it will be better for us to think of $H= L+E$ in homology. Also, if one has a Kahler metric on $dP_1$, we will find that $Vol(H)= Vol(L)+Vol(E)$.
Now, let us consider the complete linear system associated to the divisor $3H-E$ (this is a line bundle such that the zero locus of any of it’s global sections is in the class $3H-E$ ). The complete linear system is a linear space made of all the holomorphic sections of a given line bundle. If the dimension of this linear system is $d$ (the number of linearly independent holomorphic sections), this complete linear system can be used to define a Veronese map from $dP^1$ to $\mathbb{CP}^{d-1}$, where the embedding coordinates are the values of the global sections at each point and they are mapped to the homogeneous coordinates of $\mathbb{CP}^{d-1}$. The multiplicative changes due to how one patches a line bundle together cancel because the homogeneous coordinates of $\mathbb{CP}^{d-1}$ are well defined only up to multiplications by a common factor.
If we project the cycle $3H-E$ onto $\mathbb{CP}^2$, the fact that $[3H-E]\cdot E=1$ implies that the projection must pass through the origin. Moreover $[3H-E]\cdot H= 3$, so the projection gives a degree $3$ curve in $\mathbb{CP}^2$. This means that the curve is characterized by the zero locus of a cubic equation in $x_1, y_1, z_1$. Let us call $$S= \sum_{i+j+k=3, i,j,k\geq 0} a_{ijk} x_1^iy_1^jz_1^k$$ the associated equation. Since $S$ passes through the origin, it is such that $S[0,0,1]=0$, and this tells us that $a_{003}=0$.
Any such curve can be lifted to $dP_1$ uniquely. Since $dP_1$ and $\mathbb{CP}^2 $ are birationally equivalent, if we take $\mathbb{CP}^2$ and remove the origin, we can describe the Veronese embedding due to this linear system on $\mathbb{CP}^2 \setminus \{0\}$. It’s completion in $dP_1$ will be the associated map. The global sections are the monomials of $S$ that are allowed. This gives us a system of dimension $8$. This means that there is a natural embedding of $dP^1$ into $\mathbb{CP}^8$ and that the embedding is of degree $3$ for the hyperplane bundle. This is a standard construction in algebraic geometry [@GH] and it will make an appearance in the $Y^{2,1}$ field theory. The embedding is one to one.
According to the conjectures we have described, the locus where a bifundamental field vanishes is a submanifold of the base. The $R$-charge of the field is then the volume of this submanifold with respect to a Kahler-Einstein metric [@Wittenbar; @GK] (see also [@BHK; @IW]). To test the conjecture for a given field we first find the zero locus of its vev and show that the volume of this locus is exactly the fields $R$-charge as computed by $a$-maximization.
For concreteness we will consider the minimal cycles at node 1 (again, the results are independent of what node is chosen), where a minimal cycle is one which has no proper cyclic subpaths. Using the constraints $\partial_a W = 0$, it was shown in [@BHOP] that every minimal cycle at node 1 is equal to one of the following minimal cycles:
$$\begin{array}{ccc}
\varphi_{00} &= & a_4c_2a_1\\
\varphi_{01} &= & b_4c_2a_1\\
\varphi_{02} &= & b_4c_2b_1
\end{array} \ \ \
\begin{array}{ccc}
\varphi_{10} &= & a_4da_2a_1\\
\varphi_{11} &= & b_4da_2a_1\\
\varphi_{12} &= & b_4db_2a_1\\
\varphi_{13} & =& b_4db_2b_1
\end{array} \ \ \
\begin{array}{ccc}
\varphi_{20} & =&c_3a_2c_1\\
\varphi_{21} &= &c_3b_2c_1
\end{array}$$
Given a generic point in the vev moduli space (irreducible representation), it has been shown in [@B1] that there exists a choice of gauge (isoclass representative) for which $$\left\langle a_2 \right\rangle = \left\langle a_4 \right\rangle, \ \ \ \left\langle b_2 \right\rangle = \left\langle b_4 \right\rangle, \ \ \
\left\langle c_1 \right\rangle = \left\langle c_2 \right\rangle = \left\langle c_3 \right\rangle, \ \ \ \left\langle d \right\rangle = \frac{\left\langle a_1 \right\rangle}{\left\langle a_2 \right\rangle} = \frac{\left\langle b_1 \right\rangle}{\left\langle b_2 \right\rangle}.$$ Let us call the variables as follows $$\left\langle a_{2i+1} \right\rangle =: x_1, \ \ \ \left\langle b_{2i+1} \right\rangle =: y_1, \ \ \ \left\langle c_i \right\rangle =: z,$$
$$\left\langle a_{2i} \right\rangle =: x_2, \ \ \ \left\langle b_{2i} \right\rangle =: y_2, \ \ \ \left\langle d \right\rangle = \frac{x_1}{x_2}=\frac{y_1}{y_2}.$$
Using this notation and setting $\phi_{jk}:= \left\langle \varphi_{jk} \right\rangle$, the vev’s of the above minimal cycles are as follows: $$\begin{array}{ccc}
\phi_{00} &= & x_2 z x_1\\
\phi_{01} &= & y_2zx_1\\
\phi_{02} &= & y_2zy_1
\end{array} \ \ \
\begin{array}{ccc}
\phi_{10} &= & x_2^2x_1\\
\phi_{11} &= & y_2x_2x_1\\
\phi_{12} &= & y_2^2x_1\\
\phi_{13} & =& y_2^2y_1
\end{array} \ \ \
\begin{array}{ccc}
\phi_{20} & =&x_2z^2\\
\phi_{21} &= &y_2z^2
\end{array}$$
At the apex of the cone the vev of each $\phi_{jk}$ is zero, and using the $F$-constraints we find that if we are away from the apex, not all $x_1,y_1,z$ can be zero and not both $x_2,y_2$ can be zero. The base of the cone “vanishes” at the apex, so if we want to consider the base we must require not all $x_1,y_1,z$ be zero and not both $x_2,y_2$ be zero. There is one $F$-constraint between these coordinates, namely $0=\partial_{c_3}W = a_2b_1-b_2a_1$. The resulting locus is then $\{x_2y_1=y_2x_1 \} \subset \mathbb{CP}^2 \times \mathbb{CP}^1$, which is $dP_1$. Thus from the quiver we obtain homogeneous coordinates for $\mathbb{CP}^2\times \mathbb{CP}^1$ that describe the $dP_1$ locus. The notation we have used makes the identification obvious.
Notice that in the quiver theory, the parameterizing variables $x_1,y_1,z,x_2,y_2$ are not homogeneous coordinates, so their rescalings are physical: they change the location of the branes. This is what gives us the cone structure for the total space. Notice also that the gauge invariants $\phi_{ij}$ are cubic in the parameterizing variables, so the moduli space of gauge invariants gives us an embedding into $\mathbb{C}^9$. If we projectivize the embedding, we get the embedding into $\mathbb{CP}^8$ described above. The relations between the monomials $\phi_{ij}$ are exactly the relations of this embedding[^2] since they are determined by the parameterizing variables.
To compute the volumes of interest, consider the divisor $$H := f^{-1}\{ z =0 \},$$ the exceptional divisor $$E := \left[ 0:0:1; x_2: y_2 \right] = \{ \text{a point} \} \times \mathbb{CP}^1,$$ and their difference $$L:= H - E = \left[ 0:y_1:z_1;0:1 \right] \subset \mathbb{CP}^2 \times \{ \text{a point} \}.$$ The following table describes the zero loci of the various bifundamental fields; the results in the fourth column are derived below and we note that the last column follows from the fourth upon substituting $H =L+E$.
$$\begin{array}{|c||c|c|c|c|c|}
\hline
0 = & \begin{array}{c}\text{\scriptsize{so the only possible}}\\ \text{\scriptsize{nonzero coordinates are}} \end{array}& \text{\scriptsize{with constraints}} & \begin{array}{c}\text{\scriptsize{and thus the}}\\\text{\scriptsize{zero locus is}} \end{array} & \text{\scriptsize{or alternatively}}\\
\hline
\hline
\left\langle d \right\rangle & \left[ \phi_{20}: \phi_{21} \right] &\text{\scriptsize{none}} & E & E\\
\hline
z & \left[ \phi_{10}: \phi_{11}: \phi_{12}: \phi_{13} \right] & \phi_{10}\phi_{13} = \phi_{11}\phi_{12}, & H & L+E\\
&& \phi_{11}^2 = \phi_{12}\phi_{10}, &&\\
&& \phi_{12}^2 = \phi_{11}\phi_{13}.&&\\
\hline
x_2 & \left[ \phi_{02}:\phi_{13}: \phi_{21}\right] & \phi_{02}^2 = \phi_{13}\phi_{21} & 2(H-E) & 2L\\
\hline
y_2 & \left[ \phi_{00}: \phi_{10}: \phi_{20} \right] & \phi_{00}^2 =\phi_{20}\phi_{10} & 2(H-E) & 2L\\
\hline
x_1 & \left[ \phi_{02}: \phi_{13}: \phi_{20}: \phi_{21} \right] & \phi_{02}^2 = \phi_{13}\phi_{21} & 2(H-E)+E & 2L+E\\
&&& = 2H -E &
\\
\hline
y_1 & \left[ \phi_{00}: \phi_{10}: \phi_{20}: \phi_{21} \right]& \phi_{00}^2 = \phi_{10}\phi_{20} & 2(H-E)+E & 2L+E\\
&&&= 2H-E&\\
\hline
\end{array}$$ By “and thus the zero locus is” we specifically mean inside $dP_1$ and not in the ambient space $\mathbb{CP}^8$. The multiplicities are computed by calculating the degrees of the appropriate locus in $\mathbb{CP}^8$. Notice that the condition that an arrow in the quiver vanishes implies that one of the variables above vanishes because of the gauge choices that are made. Thus the locus $a_2=0$ coincides with the locus $a_4=0$ for example.
Here we justify these computations.
Define the divisors $D_{x_i} := \{x_i=0\}$, $D_{y_i} := \{y_i = 0\}$, and $D_{z} := \{z=0\}$. In order to write the classes of these divisors in terms of the basis $\{ \left[ L \right], \left[ E \right] \}$ of $H^1(dP_1)$, we compute their intersection numbers.
- $D_{\left\langle d \right\rangle} = E$ since $\left\langle d \right\rangle = 0$ implies $x_1 = y_1 = 0$.
- $D_{z} \cdot E = 0$ since $z = 0$ implies $x_1 \not = 0$ or $y_1 \not = 0$, so $D_z$ does not intersect the exceptional divisor. $D_z \cdot H = 1$ in $dP_1$ so $D_z =H$, though we note that in $\mathbb{CP}^8$ the zero locus $z=0$ is a twisted cubic and so $\deg(D_z) =3$.
- $x_2 =0$ implies $y_2 \not =0$ and since $y_1x_2 =x_1y_2$, it must be that $x_1 =0$. But then the resulting coordinates $\left[ 0: y_1:x_1; 0:1 \right]$ are those of $L$. Thus $D_{x_2} \cdot H = 2$ since $D_{x_2}$ is given by the degree 2 hypersurface $\phi_{02}^2 - \phi_{13}\phi_{21}=0$. $D_{x_2} \cdot E = 2$ since again $D_{x_2}$ is a given by a degree 2 hypersurface and intersects $E$ transversely at the single point $\left[ 0:0:1;0:1\right]$. Thus $D_{x_2} = 2(H -E)=2L$. Similarly, $D_{y_2}=2(H-E)=2L$.
- The only difference between $D_{x_1}$ and $D_{x_2}$ is that $D_{x_1}$ is given by both coordinates $\left[ \phi_{20}:\phi_{21} \right]$ of the exceptional divisor whereas $D_{x_1}$ is only given by the single coordinate $\phi_{21}$. Thus $$\{x_1=0\} = \{ x_2 =0\} \cup E,$$ and hence $D_{x_1} = D_{x_2}+E = 2L -E$. Similarly, $D_{y_1}= 2L-E$.
We now consider the $R$-charges of the bifundamental fields. These are determined by $a$-maximization [@IW] and were computed in [@BBC] (a general case was done in [@BHK2] for the toric phases of the $Y^{p,q}$ quivers for the case we need and we use their notation). They are as follows: $$\begin{array}{rcl}
r(a_1) = r(b_1) & = & (3q-2p+\sqrt{4p^2-3q^2})/3q\\
&=& \frac 13 \left( -1 + \sqrt{13} \right)\\
r(a_{2i}) =r(b_{2i})& = & 2p(2p-\sqrt{4p^2-3q^2})/3q^2\\
& = & \frac 43 \left( 4 - \sqrt{13} \right)\\
r(c_i) & = & (-4p^2+3q^2+2pq+(2p-q)\sqrt{4p^2-3q^2})/3q^2\\
& = & -3 + \sqrt{13}\\
r(d) &= &(-4p^2+3q^2-2pq+(2p+q)\sqrt{4p^2-3q^2})/3q^2\\
& = & \frac 13 \left( -17+ 5 \sqrt{13} \right)
\end{array}$$ Setting $$r(c_i) := \int H \ \ \text{ and } \ \ \ r(d) := \int E,$$ we find $$\begin{array}{ccl}
r(a_{2i}) & = & \frac 43 \left( 4 - \sqrt{13} \right)\\
&= & \frac 13 \left( 6\left( -3+\sqrt{13} \right) + -2 \left( -17 + 5 \sqrt{13} \right)\right)\\
& =& 6 \left( \frac 13 r(c_i) \right) + -2 r(d) ,\\
&&\\
r(a_1) & = & \frac 13 \left( -1 + \sqrt{13} \right)\\
& = & \frac 13 \left( 6 \left( -3 + \sqrt{13} \right) -1 \left( -17 + 5 \sqrt{13} \right) \right)\\
& = & 6 \left( \frac 13 r(c_i) \right) - r(d),
\end{array}$$ and similarly for $b_{2i}$ and $b_1$. These are exactly the same relations we get using the divisors described above. Moreover, as computed in [@BBC], the volumes of the corresponding cycles in the Sasaki-Einstein manifold give the same dimensions. Here we see that the geometry we found matches exactly what is found in the AdS dual setup. Notice that our procedure was systematic and did not require any guesswork to make the match. Moreover all the multiplicities are accounted for by the algebraic geometry calculation.
It should also be noticed that a general analysis of D-branes in the $Y^{p,q}$ and $L^{a,b,c}$ geometries has been performed in [@Edelstein], so it should be interesting to check the match between algebraic geometry and the geometry on the dual $AdS$ side for these examples. A recent analysis of how to get some handle on the geometry and some other studies of these operators can be found in [@EK; @FGU].
Also notice that although the examples we analyzed were given by toric geometries, in principle the techniques we have described do not depend on having so much symmetry. Instead, they only depend on the study of the details of the algebraic geometric space that one is considering. One should also notice that the property of a quiver field theory being toric is not preserved by Seiberg dualities [@DualC], but the non-commutative algebraic geometry setup is preserved [@BD]. This implies that any arrow in a quiver diagram can be interpreted as some element of $Hom(S_1, S_2)$ between two projective modules as described by us. These are always sections of a global line bundle over the center of the algebras, which is common between them. One can think of the center variety as the moduli space of a point-like brane in the bulk, and remembering that the moduli spaces should be invariant under Seiberg dualities (see[@Brev] and also [@Asp] for more details on the geometry of points).
A Seiberg dual of the ${{\mathbb C}}^3/{{\mathbb Z}}_3$ orbifold
================================================================
The ${{\mathbb C}}^3/{{\mathbb Z}}_3$ quiver theory is usually represented by three nodes $e_1,e_2,e_3$, and between pairs of consecutive nodes one usually has three superfields $x_i,y_i,z_i$, giving a total of 9 arrows in the quiver. Setting $x = \sum x_i$, $y= \sum y_i$, and $z = \sum z_i$ (the summation is formal, as described above), the F-terms can be deduced from the superpotential $$W={\hbox{tr}}([x,y]z)$$ which is the same superpotential as ${\cal N}=4 $ SYM, properly projected to account for the three node quiver structure of the orbifold.
Since the variables $x,y,z$ commute with each other, any polynomial in these variables will also commute. However, one can show that only cubic polynomials in the $x_i,y_i,z_i$ and their products can commute with the $e_i$. The center is then generated by $$\alpha_{ijk} = x^i y^j z^k$$ where $i+j+k=3$. The three variables $x,y,z$ have an associated $SU(3)$ symmetry of rotations between them. The cubic polynomials in $x,y,z$ transform as a ten dimensional totally symmetric representation of $SU(3)$. For irreducible representations, the vector space on each node is one dimensional, and so each arrow is represented by a scalar. This is a special orbifold that has a toric description.
There is a one-to-one correspondence between the center of the algebra and the cycles at any given node. Thus again each node represents a line bundle on the Calabi-Yau cone. The cone is a complex cone over $\mathbb{CP}^2$.
The $\alpha_{ijk}$ give us a Veronese embedding of $\mathbb{CP}^2$ into $\mathbb{CP}^9$ very similar to the case of the first del Pezzo surface. The basic dibaryon operators are objects like $\det( x_i)$. If $x_i$ vanishes, we find that all the $\alpha_{ijk}=0$ if $i>0$, and the locus $x=0$ corresponds to a particular hyperplane section of the base of the cone. Also, the three non-gauge invariant objects $x_i,y_i,z_i$ at the node $i$ can be understood as a set of homogeneous coordinates for $\mathbb{CP}^2$; they are global sections of the ${{\cal O}}(1)$ line bundle on $\mathbb{CP}^2$.
The dibaryon operators for this orbifold theory have been analyzed by [@GRW]. The fact that there are three elementary dibaryons associated to the same hyperplane section on the $\mathbb{CP}^2$ base and that the lift to the $S^5/{{\mathbb Z}}_3$ Sasaki-Einstein space is not simply connected suggested that each of these corresponds to a D-brane with different Wilson lines along the non-simply connected cycle.
Our idea is to show how one can redo the analysis of the basic dibaryons in a setting where the total space is not described as a toric variety from the field theory point of view. Our problem is to show how one can recover the geometric description of the baryonic operators above from a Seiberg dual theory [@Sdual]. According to the original arguments of Seiberg reproducing the counting of baryonic operators was given as evidence for the duality. Thus, we know it will work at this level. The question we will address is how to see the geometry from this more involved baryonic setup.
If we take the quiver diagram and do a Seiberg duality on any of the nodes, we get a new quiver diagram with three nodes. We find that there are six arrows between two of the nodes, say $e_3$ and $e_2$. These should be thought of as transforming in the $6$ of $SU(3)$. We shall call them $\phi^{[ab]}$. The associated fields have dimension two. Moreover, there are also three arrows from $e_3$ to $e_1$ and three arrows from $e_1$ to $e_2$. These transform in the $\bar 3$ representation of $SU(3)$ and have dimension $1/2$ (these computations can be found in [@DualC]). The reason they transform in opposite representations of $SU(3)$ is that the $SU(3)$ is part of the ’flavor symmetry’ with respect to the gauge group node that was dualized. Remember that in a Seiberg duality one changes the direction of the arrows that begin and end on the node that is dualized (one trades fundamentals by antifundamentals plus mesons of the global symmetry).
The superpotential is given by $$W\sim {\hbox{tr}}(\chi^{[\alpha\beta]} \phi_\alpha\xi_\beta$$ where the upper indices are fundamentals of $SU(3)$, and the lower indices are antifundamentals (see [@Brev] for notation).
For the irreducible representations of the algebra, we expect that the Seiberg duality corresponds to a derived equivalence between two different algebras [@BD], and that the points on the Calabi-Yau cone are skyscraper sheafs over the center of the original algebra that go to skyscarper sheafs over the center of the dualized algebra.
The arrows transforming in the $6$ of $SU(6)$ can be thought of as global sections of the ${{\cal O}}(2)$ bundle on $\mathbb{CP}^2$, so one would expect a similar strategy as in the example over $Y^{2,1}$ and the conifold to deal with the corresponding dibaryons: one would obtain curves of degrees two on $\mathbb{CP}^2$ by studying general linear combinations of the bundles.
For the arrows transforming as a $\bar 3$ of $SU(3)$, these cannot be thought of as global sections for any line bundle on $\mathbb{CP}^2$. Line bundles on $\mathbb{CP}^2$ are described by the degree $d$. These generate a set of global sections that are in the $d$-symmetric product of $SU(3)$ (the d-symmetric product of global sections of ${{\cal O}}(1)$).
Thus we find that we cannot interpret the arrows between nodes one and two, and between nodes two and three in terms of line bundles over $\mathbb{CP}^2$. This should not be so surprising. When one considers the irreducible representations of the algebra away from the singular locus, the node $e_1$ gets an associated vector space of dimension two over the complex numbers. In physical setups one can argue this value from anomaly cancellation of the field theory, whereas at the other nodes one would have spaces of dimension one. Thus, not all the vector spaces are the same dimension and one can have composition maps of arrows that give zero even though none of the arrows is zero itself. This would not be possible for sections of line bundles. What this means is that our prescription of choosing a zero locus of a global section of a line bundle needs to be generalized to a different construction that depends on having sections of general bundles rather than line bundles. After all, the spaces $Hom({{\cal A_Q}}e_i, {{\cal A_Q}}e_j)$ will always be modules over the center and can always be interpreted in terms of coherent sheafs over the complement of the singular locus.
If one considers cycles based at node $e_2$, one finds that all the minimal cycles that can be non-vanishing belong to a ten dimensional representation of $SU(3)$ [@Brev], and that they all commute with each other in a trivial way on all irreducible representations of the algebra (since they are all $1\times 1$ matrices, this is, complex numbers). As a vector space one realizes that this is the same as the vector space of the $\alpha_{ijk}$ that we described before. Thus one sees indirectly that the center of the algebra is invariant under the derived equivalence.
So how do we build invariants if we have volume forms attached to each node, and they have different dimension? Remember that in the conformal field theory the rank of the gauge group at $e_1$ will be double the rank of the gauge group at $e_2$. Let $V_1 = \mathbb{C}^{2N}$ be the vector space on which the gauge group at $e_1$ acts, and similarly let $V_2 = \mathbb{C}^{N}$ be the vector space on which the gauge group at $e_2$ acts. We first consider linear maps $V_2 \rightarrow V_1$, that is, representations of the arrows from node $e_2$ to node $e_1$. Since the dimension of $V_2$ will be half that of $V_1$, by considering two such maps $\gamma,\delta: V_2 \rightarrow V_1$, we can generate in the general case two subvector spaces at node $e_1$ of dimension $N$. If we pushforward the volume form of $e_2$ via each of these maps, we get two elements of the vector space $\omega_{\gamma}, \omega_{\delta} \in \Lambda^N V_1$. Via the wedge product of these two, we can find a unique number associated to these two maps, namely $\omega_{\gamma}\wedge \omega_{\delta}\in \Lambda^{2N} V_1$ which can now be compared with the volume form at node $e_1$. Thus we can produce a single complex number given this information. This is the evaluation map of a dibaryon operator on the given configuration.
On the moduli space, the maps split into irreducible representations, so these objects factorize between irreducibles, and for each irreducible we find a similar object with $N=1$. If we use two maps in the $\bar 3$ representation, because of the wedge product, we find an antisymmetric object in the two entries with specific transformations under the $SU(3)$ symmetry. These transform as a $3$ of $SU(3)$ and can again be interpreted as a section of a line bundle (namely ${{\cal O}}(1)$ on $\mathbb{CP}^2$). The locus associated to the vanishing of this object is when the pair of maps $\gamma, \delta$ are degenerate (they give the same one dimensional subspace). This can also be written as follows.
Consider the map from $V_2\oplus V_2\to V_1$ given by $\mu(v, u) = \gamma(v)+\delta(u)$. The object we have described above is what one would ordinarily write as $\det(\mu)$. This vanishes if the map is not invertible, and is exactly when the map $\mu$ has a jump in the dimension of the kernel (or one could phrase it in terms of a jump of the dimension of the co-kernel as well). Notice also that this is a larger locus than the vanishing of any individual arrow.
The corresponding determinant can be written as the following gauge invariant combination of fields $$\det(\mu) = \epsilon_{i_1, \dots i_N} \epsilon_{j_1,\dots, j_N} X^{i_1}_{k_1}\dots X^{i_N}_{k_N}
Y^{j_1}_{k_{N+1}} \dots Y^{j_N}_{k_{2N}} \epsilon^{k_1\dots k_{2N}}$$ and as we argued, this factorizes on the moduli space to a product of sections of line bundles for each individual irreducible. However, the arrows themselves have to be interpreted here as global sections of general bundles that can degenerate. In this case, one can argue that the pair $(X,Y)$ belongs to $Hom(V_2\oplus V_2 , V_1)$ and that since we have spaces of dimension two on the irreducibles, the $(X,Y)$ are naturally global sections of a sheaf of matrix-valued $2\times 2$ matrices over the center. The volume of the singular locus determined by these maps gives a holomorphic submanifold of the base of the cone that we identify with the $R$-charge of the multibaryon operator.
Notice that in the Seiberg dual the dimension of what we called $X,Y$ are one half, and that the object we have described has the same dimension as the dibaryons of the original orbifold quiver theory. Moreover, we have argued that the same locus on the geometry of the base is associated to the two theories.
One can similarly construct the other baryons of the right dimension. If one considers composite maps from $V_2 \to V_3$, the F-terms also tell us that on an irreducible these maps transform as a $3$ of $SU(3)$. Similarly, one can consider general maps $V_1\to V_3\oplus V_3$ to define the third class of elementary baryon of dimension $N= 2N * 1/2$.
[**Acknowledgements:**]{} We would like to thank S. Franco and D. Morrison for many discussions related to this work. Work supported in part by DOE under grant DE-FG02-91ER40618. D. B. would like to thank the hospitality of the Kavli Institute of Theoretical Physics China for their support.
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[^1]: In this notation, $x_1, y_1,z_1$ cannot all be zero and $x_2, y_2$ cannot both be zero. Also for any $\lambda \in \mathbb{C}^*$, $$\left[ \lambda x_1: \lambda y_1: \lambda z_1; x_2:y_2 \right] \sim \left[ x_1:y_1:z_1; x_2:y_2 \right],$$ $$\left[ x_1: y_1:z_1; \lambda x_2: \lambda y_2 \right] \sim \left[ x_1:y_1:z_1; x_2:y_2 \right].$$
[^2]: In general the $k$th del Pezzo surface, $dP_k$, which is $\mathbb{CP}^2$ blown up in $k$ points, admits an embedding $dP_k \hookrightarrow \mathbb{CP}^{9-k}$ for $k \leq 6$ (for $k >6$ we run out of variables since $\mathbb{CP}^2$ has only three variables). See [@GH] for more details.
| ArXiv |
---
abstract: 'We extend a result of Ahlgren and Ono [@ao] on congruences for traces of singular moduli of level $1$ to traces defined in terms of Hauptmodul associated to certain groups of genus 0 of higher levels.'
address: 'Department of Mathematics $\&$ Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6'
author:
- Robert Osburn
title: Congruences for Traces of Singular moduli
---
Introduction
============
Let $j(z)$ denote the usual elliptic modular function on $\operatorname{SL}_2(\mathbb Z)$ with $q$-expansion ($q:=e^{2\pi i
z}$)
$j(z)=q^{-1}+744+196884q+21493760q^2+\cdots.$
The values of $j(z)$ at imaginary quadratic arguments in the upper half of the complex plane are known as singular moduli. Singular moduli are important algebraic integers which generate ring class field extensions of imaginary quadratic fields (Theorem 11.1 in [@cox]), are related to supersingular elliptic curves ([@ao]), and to Borcherds products of modular forms ([@b1], [@b2]).
Let $d$ denote a positive integer congruent to 0 or 3 modulo 4 so that $-d$ is the discriminant of an order in an imaginary quadratic field. Denote by $\mathcal{Q}_d$ the set of positive definite integral binary quadratic forms $$Q(x,y)=ax^2+bxy+cy^2$$ with discriminant $-d=b^2-4ac$. To each $Q \in \mathcal{Q}_d$, let $\alpha_Q$ be the unique complex number in the upper half plane which is a root of $Q(x,1)$; the singular modulus $j(\alpha_Q)$ depends only on the equivalence class of $Q$ under the action of $\Gamma:= PSL_2(\mathbb Z)$. Define $\omega_Q\in \{1, 2, 3\}$ by $$\omega_Q:
=\left \{ \begin{array}{l}
2 \quad \mbox{if $Q \sim_{\Gamma} [a,0,a]$},\\
3 \quad \mbox{if $Q \sim_{\Gamma} [a,a,a]$},\\
1 \quad \mbox{otherwise.}
\end{array}
\right. \\$$ Let $J(z)$ be the Hauptmodul $$J(z):= j(z)-744=q^{-1}+196884q+21493760q^2+\cdots.$$ Zagier [@z] defined the trace of the singular moduli of discriminant $-d$ as $$t(d):=\sum_{Q\in \mathcal{Q}_d/\Gamma}
\frac{J(\alpha_Q)}{\omega_Q}=\sum_{Q\in \mathcal{Q}_d/\Gamma}
\frac{j(\alpha_Q)-744}{\omega_Q} \in \mathbb{Z}.$$
Zagier has shown that $t(d)$ has some interesting properties. Namely, the following result (see Theorem 1 in [@z]) shows that the $t(d)$’s are Fourier coefficients of a half-integral weight modular form.
Let $\theta_1(z)$ and $E_4(z)$ be defined by $$\begin{aligned}
&E_4(z):=1+240\sum_{n=1}^{\infty}\frac{n^3q^n}{1-q^n},\\
&\theta_1(z):=
\frac{\eta^2(z)}{\eta(2z)}
=\sum_{n=-\infty}^{\infty}(-1)^nq^{n^2}=1-2q+2q^4-2q^9+\cdots.\\
\end{aligned}$$ and let $g(z)$ be defined by $$\begin{aligned}
g(z): &=-q^{-1}+2+\sum_{0<
d\equiv 0, 3\pmod 4}t(d)q^d \\
\end{aligned}$$
Then
$$\begin{aligned}
g(z) &=-\frac{\theta_1(z)E_4(4z)}{\eta^6(4z)} \\
&=-q^{-1}+2-248q^3+492q^4-4119q^7\cdots \\
\end{aligned}$$
i.e., $g(z)$ is a modular form of weight $\frac{3}{2}$ on $\Gamma_{0}(4)$, holomorphic on the upper half plane and meromorphic at the cusps.
Now what about divisibility properties of $t(d)$ as $d$ varies? In this direction, Ahlgren and Ono [@ao] recently proved the following result which shows that these traces $t(d)$ satisfy congruences based on the factorization of primes in certain imaginary quadratic fields.
If $d$ is a positive integer for which an odd prime $l$ splits in $\mathbb Q(\sqrt{-d})$, then $$t(l^2d)\equiv 0\pmod l.$$
Recently, Kim [@kim2] and Zagier [@z] defined an analogous trace of singular moduli by replacing the $j$-function by a modular function of higher level, in particular by the Hauptmodul associated to other groups of genus 0. Let $\Gamma_0(N)^{*}$ be the group generated by $\Gamma_0(N)$ and all Atkin-Lehner involutions $W_e$ for $e||N$, i.e., $e$ is a positive divisor of $N$ for which gcd$(e, N/e)=1$. There are only finitely many values of $N$ for which $\Gamma_0(N)^{*}$ is of genus 0 (see [@fricke1], [@fricke2], or [@ogg]). In particular, there are only finitely many prime values of $N$. For such a prime $p$, let $j_{p}^{*}$ be the corresponding Hauptmodul. For these primes $p$, Kim and Zagier define a trace $t^{(p)}(d)$ (see Section 3 below) in terms of singular values of $j_{p}^{*}$. The goal of this paper is to prove that the same type of congruence holds for $t^{(p)}(d)$, namely
Let $p$ be a prime for which $\Gamma_0(p)^{*}$ is of genus 0. If $d$ is a positive integer such that $-d$ is congruent to a square modulo $4p$ and for which an odd prime $l \neq p$ splits in $\mathbb
Q(\sqrt{-d})$, then $$t^{(p)}(l^2d)\equiv 0\bmod l.$$
Preliminaries on Modular and Jacobi forms
=========================================
We first recall some facts about half-integral weight modular forms (see [@kob], [@koh]). If $f(z)$ is a function of the upper half-plane, $\lambda \in \frac{1}{2}\mathbb Z$, and $\left(\begin{matrix} a & b \\
c & d \\ \end{matrix} \right) \in GL_{2}^{+}(\mathbb R)$, then we define the slash operator by
$\displaystyle f(z)|_{\lambda}\left(\begin{matrix} a & b \\
c & d \\ \end{matrix} \right) :=
(ad-bc)^{\frac{\lambda}{2}}(cz+d)^{-\lambda}f\Big(\frac{az+b}{cz+d}\Big)$
Here we take the branch of the square root having non-negative real part. If $\gamma=\left(\begin{matrix} a & b \\
c & d \\ \end{matrix} \right) \in \Gamma_0(4)$, then define
$\displaystyle j(\gamma, z):=\Big(\frac{c}{d}\Big)\epsilon_d^{-1}\sqrt{cz+d}$,
where
$$\epsilon:=
\left \{ \begin{array}{l}
1 \quad \mbox{if $d \equiv 1 \pmod 4$},\\
i \quad \mbox{if $d \equiv -1 \pmod 4$}.
\end{array}
\right. \\$$
If $k$ is an integer and $N$ is an odd positive integer, then let $\mathcal{M}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$ denote the infinite dimensional vector space of nearly holomorphic modular forms of weight $k+\frac{1}{2}$ on $\Gamma_0(4N)$. These are functions $f(z)$ which are holomorphic on the upper half-plane, meromorphic at the cusps, and which satisfy
$$f({\gamma}z)=j(\gamma, z)^{2k+1}f(z)$$
for all $\gamma \in \Gamma_0(4N)$. Denote by $\mathcal{M}^{+}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$ the “Kohnen plus-spaces” (see [@koh]) of nearly holomorphic forms which transform according to (1) and which have a Fourier expansion of the form
$\displaystyle \sum_{(-1)^{k}n \equiv 0,1 \pmod 4} a(n)q^n$.
We recall some properties of Hecke operators on $\mathcal{M}^{+}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$. If $l$ is a prime such that $l \nmid N$, then the Hecke operator $T_{k+\frac{1}{2}, 4N}(l^2)$ on a modular form
$f(z):= \displaystyle \sum_{(-1)^{k}n \equiv 0,1 \pmod 4} a(n)q^n
\in \mathcal{M}^{+}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$
is given by
$f(z)|T_{k+\frac{1}{2}, 4N}(l^2):= \displaystyle \sum_{(-1)^{k}n
\equiv 0,1 \pmod 4} \Bigl ( a(l^{2}n) +
\Biggl(\frac{(-1)^{k}n}{l}\Biggr)l^{k-1}a(n) + l^{2k-1}a(n/l^2)
\Bigr ) q^n$
where $\Bigl(\frac{*}{l}\Bigr)$ is a Legendre symbol. Let us now recall some facts about Jacobi forms (see [@ez]). A Jacobi form on $\operatorname{SL}_2(\mathbb Z)$ is a holomorphic function
$\phi: \frak{H} \times \mathbb C \to \mathbb C$
satisfying
$\displaystyle \phi\Big(\frac{a{\tau}+b}{c{\tau}+d}, \frac{z}{c{\tau} + d}\Big)=
(c{\tau}+d)^{k} e^{2{\pi}iN\frac{cz^2}{c{\tau}+ d}} \phi(\tau, z)$
$\displaystyle \phi({\tau}, z+{\lambda}{\tau}+ {\mu})=
e^{-2{\pi}iN({\lambda}^2{\tau}+2{\lambda}z)} \phi(\tau, z)$
for all $\left( \begin{matrix} a & b \\
c & d \\ \end{matrix} \right) \in \operatorname{SL}_2(\mathbb Z)$ and $(\lambda,\mu) \in {\mathbb Z}^2$, and having a Fourier expansion of the form ($q=e^{2{\pi}i{\tau}}$, $\zeta=e^{2{\pi}iz}$)
$\phi(\tau, z) = \displaystyle \sum_{n=0}^{\infty}
\sum_{\substack{r\in \mathbb Z \\ r^2 \leq 4Nn}}
c(n,r)q^{n}{\zeta}^{r}$.
Here $k$ and $N$ are the weight and index of $\phi$, respectively. Let $J_{k,N}$ denote the space of Jacobi forms of weight $k$ and index $N$ on $\operatorname{SL}_2(\mathbb Z)$. By Theorem 2.2 in [@ez], the coefficient $c(n,r)$ depends only on $4Nn - r^2$ and $r \bmod 2N$. By definition $c(n,r)=0$ unless $4Nn
- r^2 \geq 0$. If we drop the condition $4Nn - r^2 \geq 0$, we obtain a nearly holomorphic Jacobi form. Let $J^{!}_{k,N}$ be the space of nearly holomorphic Jacobi forms of weight $k$ and index $N$.
Traces
======
Let $\Gamma_0(N)^{*}$ be the group generated by $\Gamma_0(N)$ and all Atkin-Lehner involutions $W_e$ for $e||N$, that is, $e$ is a positive divisor of $N$ for which gcd$(e, N/e)=1$. $W_e$ can be represented by a matrix of the form $\frac{1}{\sqrt{e}}\left( \begin{matrix} ex & y \\
Nz & ew \\ \end{matrix} \right)$ with $x$, $y$, $z$, $w \in
\mathbb Z$ and $xwe-yzN/e=1$. There are only finitely many values of $N$ for which $\Gamma_0(N)^{*}$ is of genus 0 (see [@fricke1], [@fricke2], or [@ogg]). In particular, if we let $\frak{S}$ denote the set of prime values for such $N$, then
$\frak{S}=\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71
\}.$
For $p \in \mathfrak{S}$, let $j_{p}^{*}$ be the corresponding Hauptmodul with Fourier expansion
$q^{-1} + 0 + a_{1}q + a_{2}q^2 + \dots$
Let us now define a trace in terms of the $j_{p}^{*}$’s. Let $d$ be a positive integer such that $-d$ is congruent to a square modulo $4p$. Choose an integer $\beta \bmod 2p$ such that ${\beta}^2 \equiv -d \bmod 4p$ and consider the set
$\mathcal{Q}_{d, p, \beta}= \{[a,b,c] \in \mathcal{Q}_d : a \equiv
0 \bmod p$, $b \equiv \beta \bmod 2p \}$.
Note that $\Gamma_0(p)$ acts on $\mathcal{Q}_{d, p, \beta}$. Assume that $d$ is not divisible as a discriminant by the square of any prime dividing $p$, i.e. not divisible by $p^2$. Then we have a bijection via the natural map between
$\mathcal{Q}_{d, p, \beta} \diagup \Gamma_0(p)$
and
$\mathcal{Q}_d \diagup \Gamma$
as the image of the root $\alpha_{Q}$, $Q \in
\mathcal{Q}_{d, p, \beta}$, in $\Gamma_0(p) \diagup \mathfrak{H}$ corresponds to a Heegner point. We could then define a trace $t^{(p, \beta)}(d)$ as the sum of the values of $j_{p}^{*}$ with $Q$ running over a set of representatives for $\mathcal{Q}_{d, p,
\beta} \diagup \Gamma_0(p)$. As $t^{(p, \beta)}(d)$ is independent of $\beta$, we define the trace $t^{(p)}(d)$ (see Section 8 of [@z] or Section 1 of [@kim2])
$t^{(p)}(d) = \displaystyle \sum_{Q}
\frac{j_{p}^{*}(\alpha_{Q})}{\omega_{Q}} \in \mathbb{Z}$
where the sum is over $\Gamma_0(p)^{*}$ representatives of forms $Q=[a,b,c]$ satisfying $a \equiv 0 \bmod p$.
For $p=2$, we have $t^{(2)}(4)=\frac{1}{2}
j_{2}^{*}(\frac{1+i}{2})=-52$, $t^{(2)}(7)=
j_{2}^{*}(\frac{1+\sqrt{-7}}{4})=-23$, $t^{(2)}(8)=
j_{2}^{*}(\frac{\sqrt{-2}}{2})=152$. For $p=3$, we have $t^{(3)}(3)=\frac{1}{3} j_{3}^{*}(\frac{-3+\sqrt{-3}}{6})=-14$, $t^{(3)}(11)= j_{3}^{*}(\frac{1+\sqrt{-11}}{6})=22$. Moreover by the table in Section 8 of [@z], we have:
[c|ccc]{}\
$d$ & $t^{(2)}(d)$ & $t^{(3)}(d)$ & $t^{(5)}(d)$\
3 & & $-14$ &\
4 & $-52$ & & $-8$\
7 & $-23$ & &\
8 & 152 & $-34$ &\
11 & & 22 & $-12$\
12 & $-496$ & 52 &\
15 & $-1$ & $-138$ & $-38$\
16 & 1036 & & $-6$\
19 & & & 20\
20 & $-2256$ & $-116$ & 12\
23 & $-94$ & 115 &\
24 & 4400 & 348 & $-44$\
27 & & $-482$ &\
28 & $-8192$ & &\
The empty entries correspond to $-d$ which are not congruent to squares modulo $4p$.
By the discussion in Section 8 of [@z] or Section 2.2 in [@kim2], there exist forms $\phi_{p} \in J^{!}_{2,p}$ uniquely characterized by the condition that their Fourier coefficients $c(n,r)=B(4pn-r^2)$ depend only on $r^2-4pn$ and where $B(-1)=1$, $B(d)=0$ if $d=4pn-r^2<0$, $\neq -1$ and $B(0)=-2$. Define $g_p(z)$ as
$g_p(z):= q^{-1} + \displaystyle \sum_{d \geq 0} B(d)q^d$.
By the correspondence between Jacobi forms and half-integral weight forms (Theorem 5.6 in [@ez]), $g_p(z) \in
\mathcal{M}^{+}_{\frac{3}{2}}(\Gamma_0(4p))$. As the dimension of $J_{2,p}$ is zero, we have that for every integer $d \geq 0$ such that $-d$ is congruent to a square modulo $4p$, there exists a unique $f_{d,p} \in \mathcal{M}^{+}_{\frac{1}{2}}(\Gamma_0(4p))$ with Fourier expansion
$f_{d,p}(z)= q^{-d} + \displaystyle \sum_{0<D\equiv 0,1 \pmod 4}
A(D,d) q^D$.
An explicit construction of $f_{d,p}$ can be found in the appendix of [@kim1] and the uniqueness of $f_{d,p}$ follows from the discussion at the end of Section 2 in [@kim1]. The following result relates the Fourier coefficients $A(1,d)$ and $B(d)$ and shows that the traces $t^{(p)}(d)$ are Fourier coefficients of a nearly holomorphic Jacobi form of weight 2 and index $p$ (see Theorem 8 in [@z] or Lemma 3.5 and Corollary 3.6 in [@kim2]).
Let $p$ be a prime for which $\Gamma_0(p)^{*}$ is of genus 0.\
(i) Let $d=4pn-r^2$ for some integers $n$ and $r$. Let $A(1,d)$ be the coefficient of $q$ in $f_{d,p}$ and $B(d)$ be the coefficient of ${q^n}{\zeta^{r}}$ in $\phi_{p}$. Then $A(1,d)=-B(d)$.\
(ii) For each natural number $d$ which is congruent to a square modulo $4p$, let $t^{(p)}(d)$ be defined as above. We also put $t^{(p)}(-1)=-1, t^{(p)}(d)=0$ for $d<-1$. Then $t^{(p)}(d)=-B(d)$.
Proof of Theorem 1.3
====================
The proof requires the study of Hecke operators $T_{k+\frac{1}{2},
4p}(l^2)$ on the forms $g_p(z)$ and $f_{d,p}(z)$. Define integers $A_{l}(d)$ and $B_{l}(d)$ by
$A_{l}(d):=$ the coefficient of $q$ in $f_{d,p}|T_{\frac{1}{2},
4p}(l^2)$,
$B_{l}(d):=$ the coefficient of $q^{d}$ in $g_p(z)|T_{\frac{3}{2},
4p}(l^2)$.
From equation (19) of [@z], we have
$A_{l}(d)=A(1,d) + lA(l^2,d)$.
Also note that we have
$g_p(z)|T_{\frac{3}{2}, 4p}(l^2)= q^{-1} + lq^{-{l^2}} +
\displaystyle \sum_{0<d \equiv 0, 3 \pmod 4} \Bigl ( B(l^{2}d) +
\Biggl(\frac{-d}{l}\Biggr)B(d) + lB(d/l^2) \Bigr ) q^d$
and so $B_{l}(d) = B(l^{2}d) +
\Bigl(\frac{-d}{l}\Bigr)B(d) + lB(d/l^2)$. Now suppose $p$ is in $\frak{S}$ and $d$ is a positive integer such that $-d$ is a square modulo $4p$ and for which an odd prime $l\neq p$ splits in $\mathbb Q(\sqrt{-d})$. Then $\Bigl(\frac{-d}{l}\Bigr)=1$. By Theorem 3.2 and the above calculations, we have $$\begin{aligned}
t^{(p)}(l^{2}d)&= -B(l^{2}d) \\
&= -B_{l}(d) + \Bigl(\frac{-d}{l}\Bigr)B(d) + lB(d/l^{2}) \\
&\equiv -B_{l}(d) + B(d) \bmod l \\
&\equiv A_{l}(d) + B(d) \bmod l \\
&\equiv A(1,d) + lA(l^{2},d) + B(d) \bmod l \\
&\equiv -B(d) + B(d) \bmod l \\
&\equiv 0 \bmod l.\\
\end{aligned}$$
We now illustrate Theorem 1.3. If $p=2$ and $l=3$, then for every non-negative integer $s$, we have
$t^{(2)}(3^2(24s+23)) \equiv 0 \bmod 3$.
In particular, if we want to compute $t^{(2)}(207)$, then we are interested in $\phi_{2} \in J^{!}_{2,2}$. By Theorem 9.3 in [@ez] and the discussion preceding Table 8 in [@z], $J^{!}_{2,2}$ is the free polynomial algebra over
$\mathbb C[E_4(\tau), E_6(\tau),
{\Delta}^{-1}]\diagup({E_4(\tau)}^{3} - {E_6(\tau)}^{2})$
on two generators $a$ and $b$ where $\displaystyle \Delta=\frac{{E_4(\tau)}^{3} - {E_6(\tau)}^{2}}{1728}$. The Fourier expansions of $a$ and $b$ begin
$$\begin{aligned}
a&=(\zeta - 2 + {\zeta}^{-1}) + (-2{\zeta}^{2} + 8{\zeta} - 12 +
8{\zeta}^{-1} - 2{\zeta}^{-2})q + ({\zeta}^{3} -12{\zeta}^2 +
39{\zeta} - 56 + \cdots)q^2 \\ &+ (8{\zeta}^3 - 56{\zeta}^2 +
152{\zeta} - 208 + \cdots)q^3 + \cdots.
\end{aligned}$$
$$\begin{aligned}
b&=(\zeta + 10 + {\zeta}^-1) + (10{\zeta}^2 -64{\zeta} + 108 -
64{\zeta} + 10{\zeta}^2)q + ({\zeta}^3 + 108{\zeta}^2 - 513{\zeta}
\\ &+ 808 - \cdots)q^2 + (-64{\zeta}^3 + 808{\zeta}^2 - 2752\zeta +
4016 - \cdots)q^3 + \cdots.
\end{aligned}$$
The coefficients for $a$ and $b$ can be obtained using Table 1 or the recursion formulas on page 39 of [@ez]. The representation of $\phi_{2}$ in terms of $a$ and $b$ is:
$\displaystyle \phi_{2}=\frac{1}{12}a(E_4(\tau)b-E_6(\tau)a)$.
By Theorem 3.2, we have $t^{(2)}(207)=-B(207)$. As $8n-r^2=207$ has a solution $n=29$ and $r=5$, then $B(207)$ is the coefficient of $q^{29}{\zeta}^5$ which is $-113643$. Thus
$t^{(2)}(207)=113643 \equiv 0 \bmod 3$.
\(1) Zagier actually defined $t^{(N)}(d)$ and proved part (ii) of Theorem 3.2 for all $N$ such that $\Gamma_0(N)^{*}$ is of genus 0 (see Section 8 in [@z]). One might be able to prove part (i) of Theorem 3.2 in the case $N$ is squarefree. If so, then a congruence, similar to Theorem 1.3, should hold for $t^{(N)}(d)$, $N$ squarefree. If $N$ is not squarefree, then C. Kim has kindly pointed out part (i) of Theorem 3.2 does not hold. For example, if $N=4$ and $d=7$, one can construct $f_{7,4}$ (see the appendix in [@kim2]) and compute that
$f_{7,4}=q^{-7} -55q + 0q^{4} + 220q^{9} + \cdots.$
Thus $A(1,7)=-55$. But $B(7)=23$ (see Remark 3.1).
\(2) We should note that Theorem 1.3 is an extension of the simplest case of Theorem 1 in [@ao]. Ono and Ahlgren have also proven congruences for $t(d)$ which involve ramified or inert primes in quadratic fields. In fact, they prove that a positive proportion of primes yield congruences for $t(d)$ (see parts (2) and (3) of Theorem 1 in [@ao]). It would be interesting to see if such congruences hold for $t^{(p)}(d)$ or $t^{(N)}(d)$.
\(3) The Monster $\mathbb{M}$ is the largest of the sporadic simple groups of order
$2^{46}3^{20}5^{9}7^{6}11^{2}13^{3}17\cdot19\cdot23\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71$
Ogg [@ogg] noticed that the primes dividing the order of $\mathbb{M}$ are exactly those in the set $\frak{S}$. The monster $\mathbb{M}$ acts on a graded vector algebra $V=V_{-1}
\bigoplus_{n\geq 1} V_n$ (see Frenkel, Lepowsky, and Meurman [@flm] for the construction). For any element $g \in \mathbb{M}$, let $Tr(g|V_n)$ denote the trace of $g$ acting on $V_n$ for each $n$. Then $Tr(g|V_{-1})=1$ and $Tr(g|V_n) \in \mathbb Z$ for every $n \geq 1$. The Thompson series is defined by:
$T_{g}(z)= q^{-1} + \displaystyle \sum_{n \geq 1} Tr(g|V_n)q^n$.
The authors in [@cy] study Thompson series evaluated at imaginary quadratic arguments, i.e. “singular moduli” of Thompson series. It is possible to define a trace of singular moduli of Thompson series. A natural question is “do we have congruences for these traces?”
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank Imin Chen, Chang Heon Kim, Ken Ono, and Noriko Yui for their valuable comments.
[10]{}
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| ArXiv |
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abstract: 'The theory of magnetized induced scattering off relativistic gyrating particles is developed. It is directly applicable to the magnetosphere of a pulsar, in which case the particles acquire gyration energies as a result of resonant absorption of radio emission. In the course of the radio beam scattering into background the scattered radiation concentrates along the ambient magnetic field. The scattering from different harmonics of the particle gyrofrequency takes place at different characteristic altitudes in the magnetosphere and, because of the rotational effect, gives rise to different components in the pulse profile. It is demonstrated that the induced scattering from the first harmonic into the state under the resonance can account for the so-called low-frequency component in the radio profile of the Crab pulsar. The precursor component is believed to result from the induced scattering between the two states well below the resonance. It is shown that these ideas are strongly supported by the polarization data observed. Based on an analysis of the fluctuation behaviour of the scattering efficiencies, the transient components of a similar nature are predicted for other pulsars.'
author:
- |
S. A. Petrova [^1]\
Institute of Radio Astronomy, NAS of Ukraine, 4, Chervonopraporna Str., 61002 Kharkov, Ukraine
date: 'Received…'
title: 'Interpretation of the Low-Frequency Peculiarities in the Radio Profile Structure of the Crab Pulsar'
---
\[firstpage\]
pulsars: general – pulsars: individual (the Crab pulsar) – radiation mechanisms: non-thermal – scattering
Introduction
============
Radio emission components outside of the main pulse
---------------------------------------------------
The Crab pulsar is known for its complex radio profile [e.g. @mh96]. It is built of a total of seven components, which are spread out over the whole pulse period and exhibit substantially distinct spectral and polarization properties. At the lowest frequencies, $\la 600$ MHz, the profile consists of the three components: the main pulse (MP), the precursor (PR) $\sim
15^\circ$ ahead of the MP and the interpulse (IP), which lags the MP by $\sim 150^\circ$ and is connected to it by a weak emission bridge [@r70; @mht72; @v73]. The PR component is distinguished by its complete linear polarization and extremely steep spectrum. At frequencies $\ga 1$ GHz, where the PR has already vanished, there appears another component $\sim 36^\circ$ in advance of the MP [@mh96]. This so-called low-frequency component (LFC) is broader and weaker than the PR. Moreover, the percentage of linear polarization in the LFC is less, $\sim 40\%$ [@mh98], though still markedly exceeds that in the MP and IP ($\sim 25\%$ and $15\%$, respectively).
The IP and LFC become invisible at $\sim 3$ and 5 GHz, and at still higher frequencies the profile structure changes drastically [@mh96; @mh98]. In addition to the MP, there is the interpulse (IP’), which re-appears some $10^\circ$ earlier in phase, and two high-frequency components (HFC1 and HFC2), $\sim 70^\circ$ and $130^\circ$ behind the IP’. All these new components are characterized by high linear polarization and relatively flat spectra, so that at frequencies $\ga 8$ GHz the MP disappears. The fluctuation properties of these components are also worth noting. According to the recent high-frequency studies, all the components show occasional giant pulses [@h05; @slow05], the temporal and frequency structure of the giant MPs and IP’s being essentially different [@hc07].
The components outside of the MP are present in other pulsars as well. About $40\%$ of the millisecond pulsars and $2\%$ of the normal ones are known to exhibit IPs [@k99]. Besides that, a handful of pulsars have firmly established PRs. Note that the PRs are met in the profiles with IPs. Thus, the structure of the Crab profile at the lowest frequencies is similar to that in some other pulsars (e.g., PSR B1055-52, @m76 and PSR B1822-09, @f81). However, in these cases the profile structure is preserved over a wide frequency range, though the component spectra are also somewhat different. Such components of the Crab profile as the LFC, HFC1 and HFC2 as well as the high-frequency shift of the IP to earlier pulse longitudes are unique. It should be noted, however, that some millisecond pulsars have even more complex structure (e.g., PSR J0437-4715, @mj95), but it is not well studied and classified.
The mechanisms of the IP and PR emissions are still a matter of debate, while the nature of other components of the Crab profile is completely obscure. The IP components are usually interpreted in terms of geometrical models. It is assumed that the IP emission originates in a distinct region (e.g. in the outer magnetosphere or at the opposite magnetic pole) and can only be observable due to a specific geometry of the pulsar (in cases of approximate alignment or approximate orthogonality of the rotational and magnetic axes). Recently @d05 have developed a generalized geometrical model for the PSR B1822-09, which includes the formation of the PR as well. It has been suggested that the PR component originates well above the MP and the backward emission from this region forms the IP. Such a pattern can be observable if the pulsar is a nearly orthogonal rotator.
It should be noted that the geometrical models are insufficient to account for the bulk of observational facts. Firstly, the components outside of the MP show peculiar polarization and spectral properties. Secondly, the fluctuation behaviour of the components strongly testifies to their physical connection with the MP. In particular, the subpulse modulation in the MP and IP of the PSR B1702-19 has been found to be intrinsically in phase [@welt07]. All this calls for a physical interpretation.
Recently we have proposed a physical mechanism of the PR and IP components based on propagation effects in pulsar magnetosphere [@p07a; @p07b]. These components are suggested to result from induced scattering of the MP emission into the background. In case of efficient scattering, the scattered radiation grows substantially and concentrates in the direction corresponding to the maximum scattering probability. In the regime of a superstrong magnetic field, the scattered component is directed along the ambient field and can be identified with the PR. In a moderately strong magnetic field, the radiation is predominantly scattered in the opposite direction, giving rise to the IP. Within the framework of this model, the basic features of the components as well as their connection to the MP are explained naturally. Our theory can be elaborated further to explain the complicated radio emission pattern of the Crab pulsar. The present paper is devoted to the generalized mechanism of the two components, the PR and LFC, which precede the MP and develop at relatively low frequencies. The formation of the high-frequency components of the Crab will be addressed in our forthcoming paper. It will be argued that the IP’ results from the backward scattering of the PR, while the HFC1 and HFC2 present the backscattered emission of the LFC.
Statement of the problem
------------------------
The magnetosphere of a pulsar contains the ultrarelativistic electron-positron plasma, which streams outwards along the open magnetic field lines and leaves the magnetosphere as a pulsar wind. The pulsar radio emission is generally believed to originate deep in the open field line tube, and on its way in the magnetosphere it should propagate through the plasma flow. As the brightness temperatures of the pulsar radio emission are extremely high, one can expect that induced scattering off the plasma particles is significant. Deep in the magnetosphere the magnetic field is strong enough to affect the scattering process considerably by modifying both the scattering cross-section and the particle recoil. This happens for the waves below the cyclotron resonance, as long as the frequency in the particle rest frame remains much less than the electron gyrofrequency, $\omega^\prime\ll\omega_G\equiv eB/mc$. The magnetized induced scattering in pulsars is known to be efficient [@bs76] and is suggested to have a number of observational consequences [@lp96; @p04a; @p04b; @p07a; @p07b]. As the magnetic field strength decreases with distance from the neutron star, in the outer magnetosphere the radio waves pass through the resonance. The scattering by the pulsar wind holds in the non-magnetic regime and can also be efficient in pulsars [@wr78; @lp96]. Note that in the resonance region itself the radio waves are subject to resonant absorption rather than scattering [@bs76; @lp98; @p02; @p03; @melr_a; @melr_b].
Close to the neutron star surface, the magnetic field is so strong that any perpendicular momentum of the particles is almost immediately lost via synchrotron re-emission. Hence, the particles are confined to the magnetic field lines, and it is usually assumed that they perform ultrarelativistic rectilinear motion throughout the open field line tube. However, in the outer magnetosphere, where synchrotron re-emission is already inefficient, the particles can easily gain relativistic gyration energies as a result of resonant absorption of the radio emission [@lp98; @p02; @p03]. As has been shown in @p02 [@p03], the absorbing particles reach relativistic gyration at the very bottom of the resonance region for radio waves, in the course of absorption of the highest-frequency waves, $\nu\ga 10$ GHz. Then the lower-frequency waves, $\nu\ll 10$ GHz, which are still below the resonance, $\omega^\prime\ll \omega_G$, are scattered by the particles performing relativistic helical motion.
The spontaneous scattering off a gyrating electron in application to pulsars has recently been studied in @p07c. The process has been found to differ substantially from the scattering by the particle at rest. The total cross-section of the former process appears much larger, and the scattered waves predominantly concentrate at high harmonics of the gyrofrequency, close to the maximum of the synchrotron emission of the scattering electron, whereas the scattering by the particle at rest leaves the wave frequency unchanged. Although the spontaneous scattering by gyrating particles appears weak enough to markedly suppress pulsar radiation, except for the lowest frequencies, $\nu <100$ MHz, it supplements the synchrotron re-emission in reprocessing the radio photons into the high-energy emission of the optical or soft X-ray band.
All the previous studies of induced scattering in pulsars have assumed rectilinear motion of the scattering particles. It has been demonstrated that even in the regime of a superstrong magnetic field the induced scattering is most efficient well above the emission region, at distances roughly comparable with the radius of cyclotron resonance [@p07a]. Therefore the gyration of the scattering particles needs to be taken into account. In the present paper, we examine the induced scattering off the spiraling particles. In contrast to the spontaneous scattering, the induced scattering between the states corresponding to close harmonics of the gyrofrequency is more efficient. We consider the induced scattering between the zeroth-harmonic states, when the incident and final frequencies are both below the resonance, study the scattering from several first harmonics to the zeroth one and conclude that these processes can be efficient in pulsars. The kinetic equations derived are applied to the problem of the radio beam scattering into the background. It is found that the radiation is predominantly scattered along the ambient magnetic field. Since the scattering regions lie at different altitudes above the emission region, the rotational aberration places the scattered components at different pulse longitudes ahead of the MP. Proceeding from the numerical estimate of the radius of cyclotron resonance in the Crab pulsar we show that the PR component should result from the scattering under the resonance and the LFC from the first-harmonic scattering.
The plan of the paper is as follows. Basic formalism of induced scattering off the spiraling particles is developed in Sect. 2. The problem on the radio beam scattering into the background is examined in Sect. 3. Applications to the the Crab pulsar are presented in Sect. 4. Our results are discussed in Sect. 5 and briefly summarized in Sect. 6.
Basic formalism of induced scattering
=====================================
Let us consider the induced scattering of transverse electromagnetic waves by the system of particles in the presence of an external magnetic field. The scattering particles are assumed to perform relativistic helical motion. The rate of change of the photon occupation number $N$ as a result of the scattering is given by $$\frac{\partial N}{\partial t}=n_e\int wNN_1\frac{{\rm d}^3{\bmath
k_1 }}{(2\pi)^3}\frac{\partial f}{\partial{\bmath p}}\Delta{\bmath
p}{\rm d^3}{\bmath p},$$ where $N=N(\bmath k)$, $N_1=N_1(\bmath k_1)$, $\bmath k$ and $\bmath k_1$ are the photon wavevectors in the initial and final states, $w=w({\bmath k},{\bmath k_1},{\bmath p})$ is the scattering probability, $n_e$ is the number density of the scattering particles, $f(\bmath p)$ is the particle distribution function in momenta normalized as $\int f(\bmath p){\rm
d}^3{\bmath p}=1$, $\Delta\bmath p$ is the momentum increment in the scattering act. In case of the scattering by the particles at rest, the recoil is extremely small, $\Delta k/k\sim \hbar k/mc<<<
1$, so that the photon frequency is approximately unaltered in the scattering act, $\omega_1^{(\rm r)}\approx \omega^{(\rm r)}$. In case of the scattering by the spiraling particles, the situation is essentially distinct: The frequencies of the scattered photons present a discrete set. In the guiding centre frame, $$\omega_1^{(\rm c)}=\omega^{(\rm c)}+s\omega_H/\gamma_0,\quad
s=0,\pm 1,\pm 2,\dots ,$$ where $\gamma_0$ is the Lorentz-factor of the circular motion of the particle, $\gamma_0\equiv (1-\beta_0^2)^{-1/2}$, $\beta_0$ is the particle velocity in units of $c$. In the laboratory frame, equation (2) is written as $$\omega_1\gamma_\Vert\eta_1=\omega\gamma_\Vert\eta+s\omega_H/\gamma_0,\quad
s=0,\pm 1,\pm 2,\dots ,$$ where $\eta=1-\beta_\Vert\cos\theta$, $\eta_1=1-\beta_\Vert\cos\theta_1$, $\beta_\Vert$ is the longitudinal component of the particle velocity normalized to $c$, $\theta$ and $\theta_1$ are the tilts of the initial and final wavevectors to the magnetic field, $\gamma_\Vert\equiv(1-\beta_\Vert^2)^{-1/2}$.
For a given harmonic of the gyrofrequency, $s$, the scattering probability is related to the differential cross-section per elementary solid angle ${\rm d}O_1$ as $$w_s=(2\pi)^3c^4\frac{\omega}{\omega_1^3}\frac{{\rm
d}\sigma_s}{{\rm
d}O_1}\delta(\omega_1-\omega\eta/\eta_1-s\omega_H/\gamma\eta_1)$$ and the relativistic transformation of the cross-section from the guiding-centre frame reads $$\frac{{\rm d}\sigma}{{\rm d}O_1}=\left (\frac{{\rm d}\sigma}{{\rm
d }O_1}\right )^{({\rm c})}\frac{\eta^2}{\gamma_\Vert^2\eta_1^3}$$ [for more details see @p07c]. In equation (4) above, $\gamma$ stands for the total Lorentz-factor of the particle, and using the invariance of the transverse momentum, $p_\perp=\beta_\perp\gamma mc=\beta_0\gamma_0mc$, it is easy to see that $\gamma=\gamma_0\gamma_\Vert$. A general form of the scattering cross-section in the guiding centre frame is given by eqs. (11)-(12) in @p07c. It actually includes four cross-sections, which correspond to one of the two possible polarizations of the photons in the initial and final states. The photons are polarized either in the plane of the wavevector and the ambient magnetic field (A-polarization) or perpendicularly to this plane (B-polarization).
Given that $\beta_\Vert=0$, $s=0$ and $\omega=\omega_1\ll
\omega_H/\gamma_0$, the cross-sections are reduced to [see also eq. (17) in @p07c] $$\frac{{\rm d}\sigma_0^{AA}}{{\rm
d}O_1}=\frac{r_e^2\sin^2\theta\sin^2\theta_1}{\gamma_0^2},$$ $$\frac{{\rm d}\sigma_0^{AB}}{{\rm
d}O_1}=\frac{r_e^2\omega^2}{\omega_H^2}\left
[\cos\theta\cos\Delta\phi -\frac{\beta_0^2\sin\theta
\sin\theta_1\cos\theta_1}{2}\right ]^2,$$ $$\frac{{\rm d}\sigma_0^{BA}}{{\rm
d}O_1}=\frac{r_e^2\omega^2}{\omega_H^2}\left
[\cos\theta_1\cos\Delta\phi -\frac{\beta_0^2\sin\theta
\sin\theta_1\cos\theta}{2}\right ]^2,$$ $$\frac{{\rm d}\sigma_0^{BB}}{{\rm
d}O_1}=\frac{r_e^2\omega^2}{\omega_H^2}\sin^2\Delta\phi,$$ where $\Delta\phi$ is the difference of the azimuthal coordinates of the wavevectors in the initial and final states. Besides that, we are interested in the induced scattering between the states, one of which still corresponds to the low frequency, $\omega_1\ll\omega_H/\gamma_0$, while in another one $\omega\approx s\omega_H/\gamma_0$. The induced scattering is believed to be efficient for small enough $s$, $s\sim 1$, and the scattering between the states with $\omega\sim n\omega_H/\gamma_0$ and $\omega_1\sim l\omega_H/\gamma_0$, where $n,l\gg 1$, should be much weaker. In the case of interest, the cross-sections take the form $$\frac{{\rm d}\sigma_s^{AA}}{{\rm
d}O_1}=\frac{r_e^2}{\gamma_0^2}J_s^2(s\beta_0\sin\theta)
\sin^2\theta\sin^2\theta_1\cot^4\theta,$$ $$\frac{{\rm d}\sigma_s^{AB}}{{\rm
d}O_1}=\frac{r_e^2\omega_1^2}{\omega_H^2}J_s^2(s\beta\sin\theta)\cot^4\theta$$ $$\times\left\{\sin\theta\cos\Delta\phi+\sin\theta_1\left [
1-\beta_0^2(1-\cos\theta\cos\theta_1)/2\right ]\right\}^2,$$ $$\frac{{\rm d}\sigma_s^{BA}}{{\rm
d}O_1}=\frac{r_e^2}{\gamma_0^2}J_s^{\prime^2}(s\beta_0\sin\theta)
\beta_0^2\cos^2\theta\sin^2\theta_1,$$ $$\frac{{\rm d}\sigma_s^{BB}}{{\rm
d}O_1}=\frac{r_e^2\beta_0^2\omega_1^2}{\omega_H^2}J_s^{\prime^2}(s\beta\sin\theta)$$ $$\times\left\{\sin\theta\cos\Delta\phi+\sin\theta_1\left [
1-\beta_0^2(1-\cos\theta\cos\theta_1)/2\right ]\right\}^2.$$ For the inverse scattering, $\omega\ll\omega_H/\gamma_0$, $\omega_1\approx s\omega_H/\gamma_0$, the cross-sections are given by eq. (18) in @p07c. Comparing those results with equation (7) above, one can see that the scattering probability (4) is symmetrical with respect to the initial and final photon states. This is known to be the fundamental property of this quantity.
The momentum increment $\Delta\bmath p$ can be found from the conservation laws in the elementary scattering act. In the presence of an external magnetic field, the energy and the momentum component parallel to the field are conserved, $$\Delta\gamma mc^2=\hbar (\omega-\omega_1),$$ $$\Delta p_\Vert=\hbar (k\cos\theta-k_1\cos\theta_1).$$ Using equation (3) and differentiating the relation $\gamma^2m^2c^4\equiv m^2c^4+(p_\perp^2+p_\Vert^2)c^2$, one can find that $$\Delta p_\perp =-s\hbar\omega_H\frac{m}{p_\perp}.$$ Thus, the perpendicular momentum is changed if $s\neq 0$.
Induced scattering of pulsar radio beam into background
=======================================================
Now let us concentrate on the problem of induced scattering into background based on the general theory presented above. Pulsar radiation is known to be highly directional: At any point of the emission cone it is concentrated into a beam of the opening angle $\la 1/\gamma$, which is typically much less than the angular width of the emission cone. Therefore at any point of the scattering region the incident radiation can be approximately presented by a single wavevector $\bmath k$. Far enough from the emission region, the radiation propagates quasi-transversely with respect to the ambient magnetic field, $1/\gamma\ll\theta\la 1$.
Since the rate of induced scattering depends on the photon occupation number in the final state, the scattering out of the beam is impossible unless there are some background photons which trigger the scattering process. Such photons should indeed be present. In particular, they may result from the spontaneous scattering from the radio beam. Although the occupation numbers of the background are extremely small, they may still stimulate efficient induced scattering from the beam. Then the beam photons are predominantly scattered into the state $\bmath k_1$ corresponding to the maximum scattering probability, and the photon occupation number in this state may become comparable with the original occupation number of the beam photons. Thus, we come to the problem on induced scattering between the two photon states, $\bmath k$ and $\bmath k_1$; the parameters of the background state corresponding to the maximum scattering probability will be specified below based on an analysis of the concrete kinetic equations.
The case $s=0$
--------------
We start from the case when the frequency in the guiding centre frame remains unchanged in the scattering act, i.e. $s=0$ and $\omega\eta=\omega_1\eta_1$. Then $\Delta p_\perp =0$ and the scattering is analogous to that off the rectilinearly moving particles. The latter has been examined in @p07a, and now we are interested if there are any quantitative differences between the two processes. It is convenient to replace the photon occupation numbers $N$ and $N_1$ with the intensities $i_\nu$ and $i_{\nu_1}$ defined as $$i_\nu (\nu,\theta,\phi){\rm d}\nu{\rm d}O\equiv
2\hbar\omega cN({\bmath k})\frac{{\rm d}^3{\bmath k}}{(2\pi)^3},$$ $$i_{\nu_1} (\nu_1,\theta_1,\phi_1){\rm d}\nu_1{\rm d}O_1\equiv
2\hbar\omega_1 cN({\bmath k_1})\frac{{\rm d}^3{\bmath
k_1}}{(2\pi)^3},$$ and to perform trivial integration over the solid angle, introducing the spectral intensities $I_\nu\equiv\int i_\nu{\rm
d}O$ and $I_{\nu_1}\equiv\int i_{\nu_1}{\rm d}O_1$. The scattering is assumed to be stationary: The intensity varies only on account of the beam propagation through the scattering region, whereas the parameters of the scattering process do not depend on time, and therefore $\partial /\partial t=c{\rm d}/{\rm d}r$.
Recall that the scattering cross-sections for gyrating particles given by equation (6) are obtained in the approximation $\omega^{(\rm c)}\ll\omega_H/\gamma_0$. One can see that $\left
({\rm d }\sigma^{AB}_0/{\rm d}O_1,{\rm d }\sigma^{BA}_0/{\rm
d}O_1,{\rm d }\sigma^{BB}_0/{\rm d}O_1\right )/\left ({\rm d
}\sigma^{AA}_0/{\rm d}O_1\right )
\sim\omega^{(c)^2}\gamma_0^2/\omega_H^2\ll 1$, i.e. the scattering in the polarization channel $A\to A$ strongly dominates those in the other channels. Note that the same polarization signature is also characteristic of the scattering by the particle at rest.
The cross-section ${\rm d }\sigma^{AA}_0/{\rm d}O_1$ can be rewritten in terms of the quantities of the laboratory frame using the relativistic transformations $\sin\theta^{(\rm
c)}=\sin\theta/\gamma_\Vert\eta$ and $\sin\theta_1^{(\rm
c)}=\sin\theta_1/\gamma_\Vert\eta_1$. Then, substituting equations (4) and (5) into equation (1) and performing integration over the wavenumber with the help of delta-function, one can obtain $$\frac{{\rm d}I_\nu}{{\rm d}r}=I_\nu
I_{\nu_1}r_e^2n_ec\int\frac{\cos\theta
-\cos\theta_1}{2\nu_1^2\eta_1^2}\frac{\sin^2\theta\sin^2\theta_1}
{\eta^2\eta_1^2}$$ $$\times\frac{m^2c^2p_\perp^4}{p_\Vert^6}\frac{\partial f }{\partial
p_\Vert}{\rm d}p_\Vert p_\perp{\rm d}p_\perp .$$ Here it is taken into account that $\gamma_\Vert=\gamma/\gamma_0$, $p_\Vert =\beta_\Vert\gamma mc\approx \gamma mc$ and $p_\perp
=\beta_0\gamma_0mc\approx\gamma_0mc$. For the inverse scattering we find analogously $$\frac{{\rm d}I_{\nu_1}}{{\rm d}r}=I_\nu
I_{\nu_1}r_e^2n_ec\int\frac{\cos\theta_1
-\cos\theta}{2\nu^2\eta^2}\frac{\sin^2\theta\sin^2\theta_1}
{\eta^2\eta_1^2}$$ $$\times\frac{m^2c^2p_\perp^4}{p_\Vert^6}\frac{\partial f }{\partial
p_\Vert}{\rm d}p_\Vert p_\perp{\rm d}p_\perp .$$ Keeping in mind that $\nu\eta=\nu_1\eta_1$, one can see that the right-hand sides of equations (11) and (12) are equal in absolute value and have opposite signs. Performing integration by parts and noticing that the scattering probability peaks at $\theta_1^{\rm
max }\sim 1/\gamma_\Vert$, we come to the system $$\frac{{\rm d}I_\nu}{{\rm d}r}=-a_0I_\nu
I_{\nu_1},$$ $$\frac{{\rm d}I_{\nu_1}}{{\rm d}r}=a_0I_\nu I_{\nu_1},$$ where $$a_0\sim\frac{24n_er_e^2\gamma_0^2}{m\nu^2\theta^4\gamma^5},$$ $\gamma$ and $\gamma_0$ stand for the characteristic values of these quantities for a given particle distribution. It should be noted that the system (13) along with equation (14) are literally the same as those for the scattering off the rectilinearly moving particles, except for the factor $\gamma_0^2$ entering equation (14) [cf. eqs. (8)-(9) in @p07a]. Generally speaking, one can write that $\gamma_0^2/\gamma^5=1/\gamma_\Vert^5\gamma_0^3$ and conclude that the scattering by spiraling particles with the longitudinal Lorentz-factor $\gamma_\Vert$ is $\gamma_0^3$ times less efficient than that by the particles streaming along the magnetic field with the same Lorentz-factor $\gamma_\Vert$. However, in pulsar case the situation is somewhat different. As has been shown in @lp98, the synchrotron absorption, which determines the evolution of the particle distribution function, acts to slow down the longitudinal motion and increase the transverse momenta in such a way that $\gamma=\gamma_0\gamma_\Vert$ keeps constant as long as the particle pitch-angle is small enough, $\gamma_0/\gamma <\theta$. Thus, it is reasonable to compare the scatterings by the original and evolved distributions of particles, in which case $\gamma$ is the same. Then the spiraling particles scatter $\gamma_0^2$ times more efficiently than the streaming particles. Note also that in the two processes the directions of predominant scattering of the radio beam photons differ, $\theta_1^{\rm max}\sim
1/\gamma_\Vert=\gamma_0/\gamma$ and $\theta_1^{\rm max}\sim
1/\gamma$, respectively, implying different relations between the interacting frequencies, $\nu_1\sim\nu\theta^2\gamma^2/\gamma_0^2$ and $\nu_1\sim\nu\theta^2\gamma^2$.
The system (13) has the first integral, $$I\equiv I_\nu+I_{\nu_1}={\rm const},$$ and the solution is written as $$I_\nu=\frac{I/x}{1+1/x},$$ $$I_{\nu_1}=\frac{I}{1+1/x},$$ where $x\equiv [I_{\nu_1}^{(0)}/I_\nu^{(0)}]\exp (\Gamma_0)$ and $\Gamma_0\equiv Ia_0r$. The latter quantity characterizes the scattering efficiency, whereas the former one the extent of intensity transfer from the radio beam to the background. As long as $x\ll 1$, $I_{\nu_1}\sim I_{\nu_1}^{(0)}\exp (\Gamma_0)$ and $I_\nu\approx I_\nu^{(0)}$. At $x\sim 1$ $I_{\nu_1}$ becomes comparable with $I_\nu^{(0)}$ and enters the stage of saturation. Given that $x\gg 1$, $I_{\nu_1}\approx I_\nu^{(0)}$ and $I_\nu\sim
I_\nu^{(0)}/x$. Since initially the intensity ratio of the background and the beam is extremely small, $I_{\nu_1}^{(0)}/I_\nu^{(0)}<<<1 $, $\Gamma_0\sim n\times 10$ is necessary to provide $x\sim 1$, in which case a substantial part of the beam intensity is transferred to the background. As has been shown in @p07a, in case of the scattering by the streaming particles this condition may well be satisfied in pulsars. For the scattering by the spiraling particles $\Gamma_0$ is $\gamma_0^2$ larger and, correspondingly, the growth of the scattered component should be even more significant.
The case $\omega^{(\rm c)}\sim s\omega_H/\gamma_0$, $\omega_1^{(\rm c)}\ll \omega_H/\gamma_0$
---------------------------------------------------------------------------------------------
Now we turn to induced scattering between such two states that one of the frequencies is close to the resonance, $\omega\gamma\eta\sim s\omega_H$, and another one is well below the resonance, $\omega_1\gamma\eta_1\equiv\omega\gamma\eta
-s\omega_H\ll\omega_H$. The intensity evolution in the two states is given by $$\frac{{\rm d}I_\nu^i}{{\rm d}r}=I_\nu^iI_{\nu_1}^j\frac{n_ec}{2}
\int\frac{\eta^2\nu}{\eta_1^3\gamma_\Vert^2\nu_1^4}\left
(\frac{{\rm d}\sigma_{\nu\to\nu_1}^{ij}}{{\rm d}O_1}\right )^{(\rm
c )}\frac{s\omega_H}{2\pi}$$ $$\times\left [\frac{\partial f}{\partial p_\perp}\frac{mc}{p_\perp}+\frac{\partial f}
{\partial p_\Vert}\frac{mc\cos\theta}{p_\Vert\eta}\right ]p_\perp
{\rm d}p_\perp{\rm d}p_\Vert,$$ $$\frac{{\rm d}I_{\nu_1}^j}{{\rm
d}r}=-I_\nu^iI_{\nu_1}^j\frac{n_ec}{2}
\int\frac{\eta_1^2\nu_1}{\eta^3\gamma_\Vert^2\nu^4}\left
(\frac{{\rm d}\sigma_{\nu_1\to\nu}^{ji}}{{\rm d}O}\right )^{(\rm c
)}\frac{s\omega_H}{2\pi}$$ $$\times\left [\frac{\partial f}{\partial
p_\perp}\frac{mc}{p_\perp}+\frac{\partial f} {\partial
p_\Vert}\frac{mc\cos\theta}{p_\Vert\eta}\right ]p_\perp {\rm
d}p_\perp{\rm d}p_\Vert,$$ where the subscripts $i,j$ denote the polarization states of the initial and final photons, $({\rm d}\sigma_{\nu\to\nu_1}^{ij}/{\rm
d}O_1)^{(\rm c )}$ is the cross-section of the scattering from $\nu$ to $\nu_1$ in the guiding centre frame (it is given by eq. (7) and should be expressed via the quantities of the laboratory system), the components of the momentum increment are given by equations (8) and (9) and it is taken into account that $\omega_1\gamma\eta_1\ll\omega_H,\omega\gamma\eta$.
The symmetry of the scattering probability (4) with respect to direct and inverse scatterings implies that $({\rm
d}\sigma_{\nu_1\to\nu}^{ji}/{\rm d}O)^{(\rm c )}=(\nu^4/\nu_1^4)
({\rm d}\sigma_{\nu\to\nu_1}^{ij}/{\rm d}O_1)^{(\rm c )}$, and one can see that $\vert{\rm d}I_\nu^i/{\rm d}r\vert=\alpha\vert{\rm
d}I_{\nu_1}^j/{\rm d}r\vert$, where $\alpha\equiv\nu\eta/\nu_1\eta_1\gg 1$. Comparing the increments of the transverse and longitudinal momenta we find that $\Delta
p_\perp\partial f/\partial
p_\perp\sim(\theta^2p_\Vert^2/p_\perp^2)\Delta p_\Vert\partial
f/\partial p_\Vert$. In our consideration $p_\perp/p_\Vert\ll\theta$, i.e. the particle pitch-angle is much less than the photon propagation angle, and hence the contribution of the longitudinal increment can be neglected. Note that $p_\perp$ enters equation (17) via $f$ as well as via the argument $s\xi$ of the Bessel function and its derivative in the cross-sections (7), $\xi\equiv\beta_0\sin\theta^{(c)}=p_\perp\sin\theta/p_\Vert\eta\sim
2p_\perp/p_\Vert\theta\ll 1$. One can see that integration of the kinetic equations (17) over $p_\perp$ by parts results in a change of the sign of the right-hand sides. Thus, in the course of the scattering the photons are mainly transferred from high harmonics, $\nu\sim s\omega_H/2\pi\gamma\eta$, to the zeroth one, $\nu_1\ll\omega_H/2\pi\gamma\eta_1$.
The harmonic number $s$ enters ${\rm d}I_\nu/{\rm d}r$ via $s^2J_s^2(s\xi)$ or $s^2J_S^{\prime^2}(s\xi)$, which peak at high harmonics, $s^{\rm max }\sim\gamma_0^3$. Note, however, that inside the light cylinder $s^{\rm max}$ corresponds to the frequencies lying in the optical or soft X-ray range [see, e.g., @p07c], and the intensity of pulsar radiation in these bands is much weaker than the radio intensity. Therefore the induced scattering from high harmonics, $s\sim s^{\rm max}$, is expected to be inefficient. In the present consideration, we leave aside this problem and concentrate on the induced scattering of the radio emission. As the magnetic field strength decreases with distance from the neutron star, a given radio frequency passes through the resonances of increasingly higher order. At the same time, the number density of the scattering particles and the radio intensity strongly decrease with distance, so that the scattering becomes much weaker.
Thus, we are interested in the radio beam scattering at several first harmonics of the gyrofrequency, which takes place within the light cylinder. Then $s\xi$ is still a small quantity and one can take approximately that $$J_s(s\xi)\approx\frac{1}{s!}\left (\frac{s\xi}{2}\right )^s,\quad
J_s^{\prime}(s\xi)\approx\frac{1}{2(s-1)!}\left
(\frac{s\xi}{2}\right )^{s-1}.$$ Performing integration in equation (17), taking into account that $\beta_0\approx 1$, $\theta\ll 1$, $\cos\theta^{(\rm
c)}=(\cos\theta -\beta_\Vert)/\eta\approx -1$, $\Delta\phi^{(\rm c
)}=\Delta\phi$ and noticing that $\theta_1^{\rm
max}\sim\gamma_0/\gamma$, one can come to the system $$\frac{{\rm d}I_\nu^i}{{\rm d}r}=-\alpha a_s^{ij}I_\nu^iI_{\nu_1}^j,$$ $$\frac{{\rm d}I_{\nu_1}^j}{{\rm d}r}=a_s^{ij}I_\nu^iI_{\nu_1}^j,$$ with $$a_s^{AA}=a_s^{BA}=\frac{r_e^2n_e}{8m\gamma^3\nu_1^2\theta^2}
\left (\frac{\omega_H/2\pi\gamma}{\nu_1\eta_1}\right
)\frac{s^{2s+2}}{(s!)^2}\left (\frac{\gamma_0}{\gamma\theta}\right
)^{2s-6},$$ $$a_s^{AB}=a_s^{BB}$$ $$=\frac{r_e^2n_e}{64m\gamma^3\nu_1^2\theta^2}
\left (\frac{\nu_1\eta_1}{\omega_H/2\pi\gamma}\right
)\frac{(4+2s)s^{2s+1}}{(s!)^2}\left
(\frac{\gamma_0}{\gamma\theta}\right )^{2s-6}.$$ One can see that $a_s^{AA}/a_s^{AB}=a_s^{BA}/a_s^{BB}\sim
(\omega_H/2\pi\gamma\nu_1\eta_1)^2\gg 1$, i.e. the scattering into the state with A-polarization is much more intense. Hence, the scattered component should be dominated by A-polarization, similarly to the scattering between the frequencies below the resonance (see Sect. 3.1). However, in contrast to that case, the initial intensities in the two polarizations are affected identically.
The general forms of the systems (13) and (19) differ by the presence of the factor $\alpha$ in the latter one. Taking notice that the substitution $\widetilde{I}_{\nu_1}=\alpha I_{\nu_1}$ makes equation (19) similar to equation (13), we immediately write the first integral, $$I\equiv I_\nu +\alpha I_{\nu_1}={\rm const},$$ and the solution $$I_\nu=\frac{I/y}{1+1/y},\quad I_{\nu_1}=\frac{I/\alpha}{1+1/y},$$ where $y\equiv[\alpha I_{\nu_1}^{(0)}/I_\nu^{(0)}]\exp (\Gamma_s)$ and $\Gamma_s\equiv Ia_sr$. One can see that the intensity transfer becomes significant at $y\ga 1$. Although the qualitative character of the solution (22) is the same as that in case of the scattering below the resonance, the quantitative difference is substantial. The maximum possible intensity of the scattered component appears much less, $I_{\nu_1}\sim I_\nu^{(0)}/\alpha\ll
I_\nu^{(0)}$, the rest of the energy being deposited to the scattering particles. Note that the scattered intensity may still be comparable with the original radio beam intensity at the same frequency $\nu_1$. Indeed, the conditions $\nu_1\ll\nu\eta/\eta_1\sim\nu\theta^2\gamma^2/\gamma_0^2$ and $\theta^2\gamma^2/\gamma_0^2\gg 1$ allow that $\nu_1\gg \nu$. Then, with the decreasing spectrum of the radio beam, even a small part of the beam intensity $I_\nu(\nu)$, which is transferred to the background, may be strong enough as compared to $I_\nu(\nu_1)$.
Application to the Crab pulsar
==============================
Now we turn to numerical estimates of the scattering efficiency and start from the scattering between the frequencies below the resonance. The growth of the scattered component at the frequency $\nu_1$ is characterized by the quantity $\Gamma_0=I_\nu^{(0)}a_0r$, where it is taken that $I\equiv
I_\nu^{(0)}+I_{\nu_1}^{(0)}\approx I_\nu^{(0)}$ and $a_0$ is given by equation (14) with $\nu=2\nu_1/\theta^2\gamma_\Vert^2$. The spectral intensity of the pulsar radio beam can be presented as $$I_\nu^{(0)}=I_{\nu_0}\left (\frac{\nu}{\nu_0}\right )^{-\alpha}.$$ Here $\nu_0\sim 10^8$ Hz and $$I_{\nu_0}=\frac{L}{\nu_0S},$$ where $L$ is the total radio luminosity of the pulsar, $S=\pi
r^2w^2/4$ is the cross-section of the pulsar beam at a distance $r$ and $w$ is the pulse width in the angular measure. The number density of the scattering particles can be written in terms of the multiplicity factor of the plasma, $\kappa$, $$n_e=\frac{\kappa B}{Pce},$$ where $P$ is the pulsar period. With the dipolar geometry of the magnetic field, $B\propto r^{-3}$, the scattering efficiency can be estimated as $$\Gamma_0=25\frac{0.1\,{\rm s}}{P}\frac{L}
{10^{30}\,{\rm erg\,s}^{-1}}\frac{B_\star}{10^{12}\,{\rm G}}
\frac{\kappa}{10^2}\left (\frac{10^9\,{\rm Hz}}{\nu_1}\right )^2
\left (\frac{\nu}{10^8\,{\rm Hz}}\right )^{-\alpha}$$ $$\times\left (\frac{w}{0.4}\right )^{-2}\left (\frac{r}{10^8\,{\rm
cm }}\right )^{-4}\left (\frac{\gamma}{10^2}\right )^{-5}\left
(\frac{\gamma_\Vert}{10}\right )^4\gamma_0^2,$$ where $B_\star$ is the magnetic field strength at the surface of the neutron star and the radius of the star is taken to be $10^6$ cm. The scattered component grows significantly provided that $x\equiv [I_{\nu_1}^{(0)}/I_\nu^{(0)}]\exp (\Gamma_0)\sim 1$. As has been shown in @p07a, this condition is satisfied for $\Gamma_0\sim$ 20–30. From equation (26) above one can conclude that such values of $\Gamma_0$ can indeed be characteristic of the Crab pulsar.
It should be noted that the pulsar radio beam is broadband and its angle of incidence increases with distance from the neutron star, $\theta\propto r$. Therefore at different altitudes $r$ the background component of a given frequency $\nu_1=\nu\theta^2(r)\gamma_\Vert^2/2$ is fed by the radiation of different frequencies $\nu\propto r^{-2}$. The number density of the scattering particles and the incident intensity entering $\Gamma_0$ are known to decrease with distance, $n_e\propto
r^{-3}$ and $I_\nu^{(0)}\propto r^{-2}$. However, at larger altitudes the feeding frequency $\nu$ is lower, $I_\nu^{(0)}$ is much larger, and for steep enough spectrum of the radio beam the scattering is more efficient. In case of the Crab pulsar $\alpha\approx 3$, so that $\Gamma_0\propto r^2$ and in the course of the scattering the intensity is transferred to the background from the lowest frequencies, the process taking place at distances of order of the cyclotron resonance radius.
The efficiency of the scattering from several first harmonics of the gyrofrequency, $\Gamma_s=I_\nu^{(0)}a_sr$, can be estimated analogously. Substituting equations (23)-(25) into the first line of equation (20), we obtain $$\Gamma_s=0.5\frac{s^{2s+2}}{(s!)^2}\frac{\omega_H/2\pi}
{\nu_1\eta_1\gamma}\left (\frac{\gamma_0}{\gamma\theta}\right
)^{2s-4}\frac{0.1\,{\rm s}}{P}\frac{L} {10^{30}\,{\rm
erg\,s}^{-1}}$$ $$\times\frac{B_\star}{10^{12}\,{\rm G}}
\frac{\kappa}{10^2}\left (\frac{10^9\,{\rm Hz}}{\nu_1}\right )^2
\left (\frac{\nu}{10^8\,{\rm Hz}}\right )^{-\alpha}$$ $$\times\left (\frac{w}{0.4}\right )^{-2}\left (\frac{r}{10^8\,{\rm
cm }}\right )^{-4}\left (\frac{\gamma}{10^2}\right )^{-5}\left
(\frac{\gamma_\Vert}{10}\right )^4\gamma_0^2.$$ The scattering of the lowest frequencies of the pulsar radio beam is most efficient and it occurs just beyond the corresponding resonance, at the distance satisfying the condition $\nu\theta^2\gamma/2=s\omega_H/2\pi+\nu_1\gamma/\gamma_\Vert^2$ ($\nu_1\gamma/\gamma_\Vert^2\ll\nu\theta^2\gamma/2,\,s\omega_H/2\pi$).
Comparison of equations (26) and (27) shows that $$\frac{\Gamma_s}{\Gamma_0}\sim\frac{\nu_s\eta}
{\nu_{1_s}\eta_1}\left (\frac{\gamma_0}{\gamma\theta}\right
)^{2s-4}\left (\frac{\nu_0}{\nu_s}\right )^\alpha\left
(\frac{\nu_{1_0}}{\nu_{1_s}}\right )^2,$$ where $\nu_s$ and $\nu_{1_s}$ are the frequencies involved in the scattering at the harmonics of the gyrofrequency, whereas $\nu_0$ and $\nu_{1_0}$ participate in the scattering below the resonance. Taking into account the relations $\nu_0\eta=\nu_{1_0}\eta_1$ and $\nu_s\eta\gg\nu_{1_s}\eta_1$, one can see that for the scattering of the same frequency, $\nu_s=\nu_0$, $\Gamma_s/\Gamma_0\sim(\nu_s\eta/\nu_{1_s}\eta_1)^3
(\gamma_0/\gamma\theta)^{2s-4}$. As $\gamma_0/\gamma\theta\ll 1$, we conclude that the scattering from several first harmonics of the gyrofrequency (at least for $s=1,2$) may dominate the scattering below the resonance. It should be kept in mind, however, that these scatterings take place at different altitudes in the magnetosphere, below and above the resonance, and because of the spatial dependence of $n_e$ and $I_\nu^{(0)}$ the scattering at the harmonics of the gyrofrequency is somewhat less efficient.
The original radio beam is known to be directed along the magnetic field in the emission region. Because of the magnetosphere rotation, at large enough altitudes $r$ it makes the angle $\sim
r/2r_L$ with the local magnetic field direction (here $r_L=5\times
10^9P$ cm is the light cylinder radius). In the scattering regimes considered in the present paper, the radio beam photons are predominantly scattered along the ambient magnetic field in the frame corotating with the neutron star. The rotational aberration in the scattering region shifts the wavevector of the scattered radiation by $\sim r/r_L$ in the direction of the magnetosphere rotation, so that in the laboratory frame it makes the angle $\sim
r/2r_L$ with the original radio beam. Correspondingly, in the pulse profile the scattered component precedes the MP in pulse longitude by $\Delta\lambda\sim r/2r_L$, where $r$ is the characteristic altitude of the scattering region [for more details see @p07a]. As the radio beam scattering at the harmonics of the gyrofrequency occurs at larger altitudes in the magnetosphere as compared to the scattering below the resonance, it should result in the components with larger separations from the MP.
To have a notion about the component location in the pulse profile let us estimate the radius of cyclotron resonance, $r_c$. Proceeding from the resonance condition $\nu\gamma\theta^2/2=\omega_H/2\pi$ and taking into account that $\omega_H\propto B\propto r^{-3}$ and $\theta\approx r/2r_L$, we find $$\frac{r_c}{r_L}=\left (\frac{0.1\,{\rm s}}{P}\right )^{3/5} \left
(\frac{B_\star}{10^{12}\,{\rm G}}\frac{10^9\,{\rm Hz}}{\nu}
\frac{10^2}{\gamma}\right )^{1/5}.$$ One can see that in the Crab pulsar the region of cyclotron resonance of the radio frequencies lies close to the light cylinder radius. Although for low enough frequencies the resonance is slightly beyond $r_L$, we assume that the estimate $\Delta\lambda\sim r/2r_L$ is still roughly applicable. The location of the LFC $\sim 30^\circ$ ahead of the MP implies that the scattering region is close to the light cylinder. As the PR precedes the MP by $\sim 15^\circ$, it should originate in the outer magnetosphere. Thus, we conclude that the PR component results from the scattering below the resonance, whereas the LFC arises in the course of the first-harmonic scattering.
Discussion
==========
We have examined the process of magnetized induced scattering off spiraling particles. Our consideration is restricted to the scattering of radio frequencies in the magnetic fields well below the critical value. In contrast to the classical problem on the scattering by rectilinearly moving particles, in our case the scattering at the harmonics of the particle gyrofrequency is not negligible. In the course of spontaneous scattering off the spiraling particles, the waves of the frequencies well below the first resonance are mainly scattered to high harmonics, into the range of the spectral maximum of the particle synchrotron emission [@p07c]. The induced scattering, however, appears most efficient for the pairs of states corresponding to close harmonics. We have concentrated at the low-frequency scattering, in which case one of the frequencies is well below the resonance and another one corresponds to one of the several first harmonics. It has been shown that the scattering in these regimes may dominate the scattering between the pair of frequencies below the resonance. The latter process is analogous to the scattering off the rectilinearly moving particles, but it appears $\sim
\gamma_0^2$ times more efficient provided that the scattering particles have the same total Lorentz-factor $\gamma$.
In application to pulsars, we are interested in the induced scattering of a narrow radio beam into the background. Given that the scattering is efficient, the photons are mainly scattered along the ambient magnetic field, in the direction corresponding to the maximum scattering probability, and the scattered component may grow roughly as large as the original radio beam. In the course of the scattering, the intensity is transferred from the harmonics of the gyrofrequency into the state below the resonance. The numerical estimates show that the scattering from several first harmonics and between the frequencies below the resonance can be substantial, especially for low enough frequencies of the radio beam, in which case the incident intensity is the largest because of the decreasing spectrum of the beam.
The scattered radiation makes the angle $\sim r/2r_L$ with the incident radio beam (where $r$ stands for the altitude of the scattering region) and precedes the MP in pulse longitude. The scattering from different harmonics takes place at different altitudes in the magnetosphere and therefore results in different components in the pulse profile. The scattering between the frequencies below the resonance has the lowest characteristic altitude and gives rise to the component closest to the MP. The components formed as a result of the scattering from increasingly high harmonics should have increasingly large separations from the MP. In application to the Crab pulsar, one can identify the component resulting from the scattering below the resonance with the PR and the component resulting from the first-harmonic scattering with LFC. The MP-LFC separation, $\Delta\lambda\sim
30^\circ$, implies that the first-harmonic scattering occurs close to the light cylinder. This is consistent with the estimate of the location of the resonance region in this pulsar.
The formation of the LFC close to the light cylinder is strongly supported by the polarization data: The position angle of linear polarization in the LFC is shifted by $\sim 30^\circ$ from that of the MP. Note that the position angle of the MP radiation is determined by the magnetic field direction in the emission region, whereas the position angle of the LFC radiation should reflect the magnetic field orientation in the scattering region. Because of the magnetosphere rotation, the ray emitted along the magnetic field makes the angle $\sim r/2r_L$ with the local magnetic field direction [see @p07a and Sect. 4 above]. Thus, the magnetic field orientations in the scattering and emission regions differ by $\sim r/2r_L$, the difference between the position angles of the original and scattered radiation being approximately the same. For the scattering taking place close to the light cylinder this difference is $\sim 30^\circ$. It is worthy to point out that in our consideration the position angle shift of the scattered component roughly equals its longitudinal separation from the MP. This is indeed the case for the LFC of the Crab pulsar [@mh98] and is believed to be a distinctive feature of the scattered components in other pulsars.
High percentage of linear polarization is another characteristic feature of the scattered components. In case of the scattering in a strong magnetic field, the scattered radiation is dominated by the waves of ordinary polarization, whose electric vector lies in the plane of the wavevector and the external magnetic field. In the original radio beam, only the ordinary waves are subject to the scattering below the resonance, whereas both types of waves, the ordinary and extraordinary ones, undergo equally efficient scattering at the harmonics of the gyrofrequency. In the Crab, as well as in other pulsars, the PR component is known to have almost complete linear polarization. Although the percentage of linear polarization of the LFC is lower, $\sim 40\%$, it still exceeds that of the MP.
One can expect that the LFC radiation suffers depolarization. This can be understood as follows. A noticeable sweep of the position angle across the LFC [@mh98] implies that this component is formed by the radiation coming from somewhat different altitudes in the magnetosphere, which results from the scattering of somewhat different frequencies. If one take into account the finite width of the MP, at a fixed pulse longitude within the LFC there should be radiation from different altitudes and therefore with different position angles. The superposition of the waves with different position angles may actually lead to a substantial depolarization of the resultant radiation.
The LFC and PR of the Crab pulsar are known to exhibit pronounced frequency evolution [@mh96]. The PR component is significant at the lowest frequencies, the LFC becomes strong at frequencies $\sim 1$ GHz, and at higher frequencies both components vanish. To analyze the spectral behaviour of the scattered components in our model let us turn to equation (28) and consider the ratio of the scattering efficiencies given that the frequencies of the scattered radiation are equal, $\nu_{1_0}=\nu_{1_s}$. Then we have $\Gamma_s/\Gamma_0\sim(\nu_s\eta/\nu_{1_s}\eta_1)^{1-\alpha}
(\gamma_0/\theta\gamma)^{2s-4}$, i.e. the role of the scattering at the harmonics of the gyrofrequency increases with frequency, $\Gamma_s/\Gamma_0\propto\nu_{1_s}^{\alpha-1}$. Thus, the LFC is expected to dominate at somewhat higher frequencies, which is in accordance with the observed trend.
On the way in the magnetosphere, both scattered components, the PR and LFC, may be further subject to scattering. Because of magnetosphere rotation their inclination to the ambient magnetic field rapidly increases with distance, while the magnetic field strength rapidly decreases. Similarly to the MP, the components may be involved in the induced transverse scattering in a moderately strong magnetic field, in which case the radiation is scattered backwards [for the general theory of this process see @p07b]. The consequences of the backward scattering of the components will be studied in detail in a separate paper. It will be shown that this process may give rise to the high-frequency components in the profile of the Crab pulsar: the backscattering of the PR can account for the IP’, whereas the scattering of the LFC can explain the HFC1 and HFC2. It will also be demonstrated that the efficiency of the induced transverse scattering increases with frequency and, correspondingly, at high enough frequencies the PR and LFC vanish, their intensities being almost completely transferred to the backward components.
The scattering efficiencies given by equations (26) and (27) depend on the intensity of the incident radio beam and on the number density and the characteristic Lorentz-factor of the scattering plasma particles. All these quantities may show marked pulse-to-pulse fluctuations, so that the scattering efficiencies may vary as well. If the scattering is so strong that the component growth is at the stage of saturation, $x\gg 1$ or $y\gg
1$, the fluctuations of $\Gamma$ do not affect the intensity of the scattered component significantly. On condition that $x\sim 1$ or $y\sim 1$, however, even small fluctuations of the scattering efficiency may lead to drastic variations of the scattered component. According to equations (26)-(27), this condition may be satisfied in a number of pulsars, which have large enough radio luminosities, strong magnetic fields and short periods. Such pulsars are believed to exhibit occasional activity at the pulse longitudes preceding the MP. Namely, these pulsars are expected to show the transient components with the spectral and polarization properties similar to those known for the PR and LFC of the Crab pulsar. Furthermore, the transient components resulting from the higher-harmonic scattering can also be present in pulsar profiles, in particular, in the Crab pulsar.
Conclusions
===========
The components of pulsar profiles outside of the MP are known to exhibit a number of peculiar properties, and at the same time the pulse-to-pulse fluctuations generally testify to a physical relation of these components to the MP. We believe that the components outside of the MP originate as a result of induced scattering of the pulsar radio beam into the background, with different types of the components corresponding to different scattering regimes. In the present paper, we have considered the magnetized induced scattering off the spiraling particles, which may be present in the outer magnetosphere of a pulsar. In this case the scattering at the harmonics of the particle gyrofrequency may be efficient. Our investigation is aimed at explaining, at least partially, the extremely complex radio emission pattern of the Crab pulsar. It has been demonstrated that the scattering from the first harmonic of the gyrofrequency into the state below the resonance can account for the formation of the LFC, whereas the scattering between the states below the resonance can explain the origin of the PR component.
As the scattered radiation concentrates along the local magnetic field direction in the scattering region, the scattering in different regimes, which hold at different altitudes, does form different components in the pulse profile because of the magnetosphere rotation. The observed LFC separation from the MP, $\Delta\lambda\sim 30^\circ$, implies that the first-harmonic scattering takes place close to the light cylinder. It is important to note that the formation of the LFC at large enough altitude in the magnetosphere and its orientation along the ambient magnetic field are strongly supported by the observed shift of the position angle of linear polarization with respect to that of the MP. The position angle shift appears approximately equal to the LFC-MP separation in pulse longitude. This is also expected from an analysis of the ray-magnetic field geometry in the rotating magnetosphere.
In the two regimes considered in the present paper, the scattering mainly results in the waves of the ordinary polarization, whereas both the ordinary and extraordinary waves are believed to be present in the original radio beam. Hence, the PR and LFC should be strongly polarized, which is consistent with the observational data.
As long as the scattering efficiency is large enough for the growth of the scattered component to reach the stage of saturation, the frequency dependence of $\Gamma$ and the variations of $\Gamma$ due to the pulse-to-pulse fluctuations of the incident radio beam intensity and of the parameters of the scattering plasma do not affect the scattered component substantially. This is believed to be the case for the PR and LFC of the Crab pulsar. However, the variations of the scattered component can be dramatic provided that $\Gamma$ is somewhat smaller and the scattering is about to reach the saturation stage. Therefore we conclude that a number of pulsars may have the transient components, which precede the MP and are analogous to the PR and LFC in the Crab pulsar. The presence of the transient components resulting from the higher-harmonic scattering is not excluded as well.
[55]{}
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[^1]: E-mail: [email protected]
| ArXiv |
---
abstract: 'We introduce an intuitive measure of genuine multipartite entanglement which is based on the well-known concurrence. We show how lower bounds on this measure can be derived that also meet important characteristics of an entanglement measure. These lower bounds are experimentally implementable in a feasible way enabling quantification of multipartite entanglement in a broad variety of cases.'
author:
- 'Zhi-Hao Ma$^{1}$, Zhi-Hua Chen$^{2}$, Jing-Ling Chen$^{3}$'
- 'Christoph Spengler, Andreas Gabriel, Marcus Huber'
title: Measure of genuine multipartite entanglement with computable lower bounds
---
Introduction
------------
Entanglement is an essential component in quantum information and at the same time a central feature of quantum mechanics [@Horodecki09; @Guhne09]. Its potential applications in quantum information processing vary from quantum cryptography [@Ekert91] and quantum teleportation [@Bennett93] to measurement-based quantum computing [@BRaussendorf01]. The use of entanglement as a resource not only bears the question of how it can be detected, but also how it can be quantified. For this purpose, several entanglement measures have been introduced, one of the most prominent of which is the concurrence [@Wootters98; @Horodecki09; @Guhne09]. However, beyond bipartite qubit systems [@Wootters98] and highly symmetric bipartite qudit states such as isotropic states and Werner states [@Terhal00; @Werner01] there exists no analytic method to compute the concurrence of arbitrary high-dimensional mixed states. For a bipartite pure state $|\psi\rangle$ in a finite-dimensional Hilbert space $\mathcal{H}
_1\otimes \mathcal{H}_2=\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$ the concurrence is defined as [@Mintert05] $C(|\psi\rangle)=\sqrt{2\left(1-\texttt{Tr}\rho_1^2\right)}$ where $\rho_1=\texttt{Tr}_2\rho$ is the reduced density matrix of $\rho={\ensuremath{| \psi \rangle}}{\ensuremath{\langle \psi |}}$. For mixed states $\rho$ the concurrence is generalized via the convex roof construction $C(\rho)=\inf_{\{p_i,|\psi_i\rangle\}} \sum_i p_i C({\ensuremath{| \psi_i \rangle}})$ where the infimum is taken over all possible decompositions of $\rho$, i.e. $\rho=\sum_i p_i |\psi_i\rangle\langle\psi_i|$. This generalization is well-defined, however, as it involves a nontrivial optimization procedure it is not computable in general. The concurrence is a useful measure with respect to a broad variety of tasks in quantum information which exploit entanglement between two parties. However, considering multipartite systems, a generalization of the concurrence is needed that strictly quantifies the amount of genuine multipartite entanglement - the type of entanglement that not only is the key resource of measurement-based quantum computing [@Briegel09] and high-precision metrology [@Giovannetti04] but also plays a central role in biological systems [@Sarovar; @Caruso], quantum phase transitions [@Oliv; @Afshin] and quantum spin chains [@spinchains]. Although many criteria detecting genuine multipartite entanglement have been introduced (see e.g. Refs. [@Huber10; @Huberqic; @HuberDicke; @Krammer; @HuberClass; @Deng09; @Deng10; @Chen10; @Bancal; @Horodeckicrit; @Yucrit; @Hassancrit; @Seevinckcrit; @Uffink; @Collins; @Guehnecrit]), there is still no computable measure quantifying the amount of genuine multipartite entanglement present in a system. There are only few quantities available for pure states (a set of possible measures is given in Ref. [@HHK1]) which, however, are in general incomputable for mixed states and corresponding computable lower bounds have not been found so far. In this paper, we define a generalized concurrence (analogously to a measure proposed for pure states in Ref. [@Milburn]) for systems of arbitrarily many parties as an entanglement measure which distinguishes genuine multipartite entanglement from partial entanglement. As a main result we show that strong lower bounds on this measure can be derived by exploiting close analytic relations between this concurrence and recently introduced detection criteria for genuine multipartite entanglement.
Genuine multipartite entanglement
---------------------------------
An $n$-partite pure state $|\psi\rangle\in \mathcal{H}_1\otimes
\mathcal{H}_2\otimes\cdots\otimes\mathcal{H}_n$ is called biseparable if it can be written as $|\psi\rangle=|\psi_A\rangle \otimes |\psi_B\rangle$, where $|\psi_A\rangle \in \mathcal{H}_{A} = \mathcal{H}_{j_1}\otimes \ldots \otimes \mathcal{H}_{j_k}$ and $|\psi_B\rangle \in \mathcal{H}_{B} = \mathcal{H}_{j_{k+1}}\otimes \ldots \otimes \mathcal{H}_{j_n}$ under any bipartition of the Hilbert space, i.e. a particular order $\{j_1,j_2,\ldots j_{k}|j_{k+1},\cdots
j_n \}$ of $\{1,2,\cdots, n\}$ (for example, for a 4-partite state, $\{1,3|2,4\}$ is a partition of $\{1,2,3,4\}$). An $n$-partite mixed state $\rho$ is biseparable if it can be written as a convex combination of biseparable pure states $\rho=\sum\limits_{i}p_i|\psi_i\rangle \langle\psi_i|$, wherein the contained $\{|\psi_i\rangle\}$ can be biseparable with respect to different bipartitions (thus, a mixed biseparable state does not need to be separable w.r.t. any particular bipartition of the Hilbert space). If an $n$-partite state is not biseparable then it is called genuinely $n$-partite entangled.\
If we denote the set of all biseparable states by $\mathcal{S}_2$ and the set of all states by $\mathcal{S}_1$ we can illustrate the convex nested structure of multipartite entanglement (see Fig. \[fig\_convex\]).\
![Illustration of the convex nested structure of multipartite entanglement. The set of biseparable states $\mathcal{S}_2$ is convexly embedded within the set $\mathcal{S}_1$ of all states ($\mathcal{S}_2 \subset \mathcal{S}_1$).[]{data-label="fig_convex"}](zwiebel.eps)
A measure of genuine multipartite entanglement (g.m.e.) $E(\rho)$ should at least satisfy:
- $E(\rho)=0 \,\forall\,\rho\in \mathcal{S}_2$ (zero for all biseparable states)
- $E(\rho)>0 \,\forall\,\rho\in \mathcal{S}_1$ (detecting all g.m.e. states)
- $E(\sum_ip_i\rho_i)\leq \sum_ip_iE(\rho_i)$ (convex)
- $E(\Lambda_{LOCC}[\rho])\leq E(\rho)$ (non-increasing under local operations and classical communication)[^1]
- $E(U_{local}\rho U^\dagger_{local})= E(\rho)$ (invariant under local unitary transformations)
There are of course further possible conditions which are sometimes required (such as e.g. additivity), but this set of conditions constitutes the minimal requirement for any entanglement measure. For a more detailed analysis of such requirements consult e.g. Refs. [@Mintertrep05; @HHK1].
Concurrence for genuine $n$-partite entanglement
------------------------------------------------
Let us now introduce a measure of multipartite entanglement satisfying all necessary conditions (M1-M5) for being a multipartite entanglement measure.\
[**Definition 1.**]{} For $n$-partite pure states ${\ensuremath{| \Psi \rangle}} \in \mathcal{H}_{1}\otimes \mathcal{H}_{2}\otimes\cdots\otimes \mathcal{H}_{n}$, where $dim(\mathcal{H}_{i})=d_{i},i=1,2, \cdots ,n$ we define the gme-concurrence as $$\begin{aligned}
\label{gmeconcurrence}
C_{gme}({\ensuremath{| \Psi \rangle}}):=\min\limits_{\gamma_i \in \gamma} \sqrt{2(1-{\mbox{Tr}}(\rho^{2}_{A_{\gamma_i}}))}\ ,\end{aligned}$$ where $\gamma=\{\gamma_i\}$ represents the set of all possible bipartitions $\{A_i|B_i\}$ of $\{1,2,\ldots,n\}$. The gme-concurrence can be generalized for mixed states $\rho$ via a convex roof construction, i.e. $$\begin{aligned}
C_{gme}(\rho)= \inf_{\{p_i,|\psi_i\rangle\}} \sum_{i}p_{i}C_{gme}({\ensuremath{| \psi_i \rangle}} ) \ ,
\label{4}\end{aligned}$$ where the infimum is taken over all possible decompositions $\rho=\sum_i p_i {\ensuremath{| \psi_i \rangle}} {\ensuremath{\langle \psi_i |}}$. For example, for a tripartite pure state ${\ensuremath{| \psi \rangle}}\in \mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \mathcal{H}_3$ there are three possible bipartitions $\gamma=\{ \{1|2, 3\}, \{2|1, 3\}, \{3|1, 2\} \}$. Consequently, the gme-concurrence is $C_{gme}(\psi)=\min\{\sqrt{2(1- Tr(\rho_{1}^2))},\sqrt{2(1- Tr(\rho_{2}^2))},\sqrt{2(1- Tr(\rho_{3}^2))}\}$.\
The definition of $C_{gme}(\rho)$ directly implies $C_{gme}(\rho)=0$ for all biseparable states (M1) and $C_{gme}(\rho)>0$ for all genuinely $n$-partite entangled states (M2). Convexity (M3) also follows directly from the fact that any mixture $\lambda\rho_1+(1-\lambda)\rho_2$ of two density matrices $\rho_1$ and $\rho_2$ is at least decomposable into states that attain the individual infima. As the concurrence of any subsystem was proven to be non-increasing under LOCC (see e.g. Ref. [@Mintert05]), the minimum of all possible concurrences will of course still remain non-increasing, thus proving (M4) also holds. Furthermore $\text{Tr}(\rho^2)$ is invariant under local unitary transformations for every reduced density matrix irrespective of the decomposition, which proves that also condition (M5) holds. For pure states the gme-concurrence is closely related to the entanglement of minimum bipartite entropy introduced for pure states in Ref. [@Milburn]. In contrast to the original definition using von Neumann entropies of reduced density matrices, we use linear entropies. In this way we can derive lower bounds even on the convex roof extension which had not been considered before.
Lower bounds on the gme-concurrence
-----------------------------------
As the computation of any proper entanglement measure is in general an NP-hard problem (see Ref. [@gurvits]), it is crucial for the quantification of entanglement that reliable lower bounds can be derived. These lower bounds should be computationally simple and also experimentally (locally) implementable to be of any use in practical applications. Let us now derive lower bounds on $C_{gme}$ which meet these requirements. Consider inequality II from Ref. [@Huber10], which is satisfied by all biseparable states (such that its violation implies genuine multipartite entanglement) $$\label{ineqII}
\underbrace{\sqrt{{\ensuremath{\langle \Phi |}}\rho^{\otimes 2}\Pi_{\{1,2,\cdots,n\}}{\ensuremath{| \Phi \rangle}}}-\sum\limits_{\gamma}\sqrt{{\ensuremath{\langle \Phi |}}\Pi_{\gamma}\rho^{\otimes 2}\Pi_{\gamma}{\ensuremath{| \Phi \rangle}}}}_{=: I[\rho,|\Phi\rangle]}\leq 0\, ,$$ where ${\ensuremath{| \Phi \rangle}}$ is any state separable with respect to the two copy Hilbert spaces and $\Pi_{\{\alpha\}}$ is the cyclic permutation operator acting on the twofold copy Hilbert space in the subsystems defined by $\{\alpha\}$, i.e. exchanging the vectors of the subsystems $\{\alpha\}$ of the first copy with the vectors of the subsystems $\{\alpha\}$ of the second copy. A simple example would be $\Pi_{\{1\}}|\phi_1\phi_2\rangle\otimes|\psi_1\psi_2\rangle=|\psi_1\phi_2\rangle\otimes|\phi_1\psi_2\rangle$.\
For sake of comprehensibility let us show how to derive lower bounds for three qubits and then generalize the result (in the appendix). If we consider a most general 3-qubit pure state in the computational basis $$\begin{aligned}
\label{explicit}
|\psi\rangle=&&a|000\rangle+b|001\rangle+c|010\rangle+d|011\rangle\nonumber\\&&+e|100\rangle+f|101\rangle+g|110\rangle+h|111\rangle\, ,\end{aligned}$$ the squared concurrences $C^2(\rho_\gamma)=2(1-\text{Tr}(\rho^2_\gamma))$ with respect to the three bipartitions read $$\begin{aligned}
C^2(\rho_1)=4|ah-de|^2+F_1\,,\\
C^2(\rho_2)=4|ah-cf|^2+F_2\,,\\
C^2(\rho_3)=4|ah-bg|^2+F_3\,,\end{aligned}$$ where $F_i$ are non-negative functions. The following relations thus hold $$\begin{aligned}
C(\rho_1)\geq 2|ah-de|\geq2|ah|-2|de|\,,\\
C(\rho_2)\geq 2|ah-cf|\geq2|ah|-2|cf|\,,\\
C(\rho_3)\geq 2|ah-bg|\geq2|ah|-2|bg|\,,\end{aligned}$$ and finally $$\begin{aligned}
&\min\{C(\rho_1),C(\rho_2),C(\rho_3)\} \nonumber \\ \geq &2|ah|-2\max\{|de|,|cf|,|bg|\}\nonumber\\\geq&2|ah|-2(|de|+|cf|+|bg|)\ =: B \,.\end{aligned}$$ Now for any given mixed state the convex roof construction is bounded by $$\begin{aligned}
C_{gme}(\rho)\geq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_iB_i\,.\end{aligned}$$ For the choice $|\Phi\rangle=|000111\rangle$ and the abbreviation $\rho_{uvwxyz}:=\langle uvw|\rho|xyz\rangle$, inequality (\[ineqII\]) reads $$\begin{aligned}
I[\rho,|000111\rangle]=|\rho_{000111}|-\sqrt{\rho_{001001}\rho_{110110}}\nonumber\\-\sqrt{\rho_{010010}\rho_{101101}}-\sqrt{\rho_{100100}\rho_{011011}}\leq 0\,.\end{aligned}$$ Now $$\begin{aligned}
2|\rho_{000111}|\leq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_i2|a_ih_i|\,,\end{aligned}$$ due to the triangle inequality and $$\begin{aligned}
2\sqrt{\rho_{001001}\rho_{110110}}\geq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_i2|b_ig_i|\,,\end{aligned}$$ holds due to the Cauchy-Schwarz inequality (of course for all the parts of the other bipartitions).\
This leads to a lower bound on the convex roof construction $$\begin{aligned}
C_{gme}(\rho)\geq 2I[\rho,|000111\rangle]\,.\end{aligned}$$ As $C_{gme}(\rho)$ is invariant under local unitary transformations, we can infer that indeed every $2I[\rho,|\Phi\rangle]$ constitutes a proper lower bound. By taking into account the set of all vectors $\{|\Phi\rangle\}$ we can thus define a computable lower bound which itself has many favorable properties (satisfying M1, M3, M4 and M5): $$\begin{aligned}
C_{gme}(\rho)\geq \max_{|\Phi\rangle}2I[\rho,|\Phi\rangle]\,.\label{lowerbound}\end{aligned}$$ As the lower bound is straightforwardly generalized (the structure of the proof essentially remains the same, see the appendix for details), eq.(\[lowerbound\]) is indeed a proper lower bound on (\[gmeconcurrence\]) for any $n$-partite qudit state.\
![Contour plot of the lower bound $\max_{|\Phi\rangle}2I[\rho,|\Phi\rangle]$ on the gme-concurrence for the set of three-qubit-states $\rho=c_1 {\ensuremath{| GHZ \rangle}}{\ensuremath{\langle GHZ |}} + c_2 {\ensuremath{| W \rangle}}{\ensuremath{\langle W |}}+\frac{1-c_1-c_2}{8}\mathbbm{1}$ given by convex mixtures of a GHZ state, W state and the maximally mixed state. The greyscale is related to the bound $\max_{|\Phi\rangle}2I[\rho,|\Phi\rangle]$ varying from $0$ to $1$ (where $0$ is white), while the blue region denotes states which are positive under partial transposition with respect to all bipartitions. The optimization over all $\{|\Phi\rangle\}$ was realized using the composite parametrization of the unitary group (see Ref. [@SHH2]).[]{data-label="fig_contourpl"}](contourpl.eps)
(0, 0)(0,0) (-0.8,0.2)
(0,0)\[\]
$1$
(-6.6,6.3)
(0,0)\[\]
$1$
(-0.8,1.6)
(0,0)\[\]
$c_1$
(-5.5,6.3)
(0,0)\[\]
$c_2$
\
**Discussion**\
The detection quality of our obtained bounds on the gme-concurrence is illustrated in Fig. \[fig\_contourpl\] for the family $\rho=c_1 {\ensuremath{| GHZ \rangle}}{\ensuremath{\langle GHZ |}} + c_2 {\ensuremath{| W \rangle}}{\ensuremath{\langle W |}}+\frac{1-c_1-c_2}{8}\mathbbm{1}$ of three-qubit-states, where $$\begin{aligned}
{\ensuremath{| GHZ \rangle}} = \frac{1}{\sqrt{2}}({\ensuremath{| 000 \rangle}}+{\ensuremath{| 111 \rangle}}) \quad \textrm{and} \nonumber \\
{\ensuremath{| W \rangle}} = \frac{1}{\sqrt{3}}( {\ensuremath{| 001 \rangle}} + {\ensuremath{| 010 \rangle}} + {\ensuremath{| 100 \rangle}}) \end{aligned}$$ are the well-known genuinely multipartite entangled $GHZ$- and $W$-state, respectively. It can be seen that the bounds are non-zero for a considerable amount of multipartite entangled states, especially in the vicinity of the GHZ-state.\
In fact, our bounds are exact for GHZ-like states, i.e. states of the form ${\ensuremath{| gGHZ \rangle}}=\alpha {\ensuremath{| 0' \rangle}}^{\otimes n}+\beta {\ensuremath{| 1' \rangle}}^{\otimes n}$ wherein ${\ensuremath{| 0' \rangle}}\in\mathcal{H}_i$ and ${\ensuremath{| 1' \rangle}}\in\mathcal{H}_i$ are arbitrary mutually orthogonal vectors. By expanding ${\ensuremath{| gGHZ \rangle}}$ in terms of ${\ensuremath{| 0' \rangle}}$ and ${\ensuremath{| 1' \rangle}}$ analogously to (\[explicit\]) one finds $C(\rho_{A_{\gamma_i}})=2|\alpha \beta|\,\forall\,\gamma_i$, hence $C_{gme}(\rho)=2|\alpha \beta|$. In order to prove the exactness of the bound we choose ${\ensuremath{| \phi \rangle}}={\ensuremath{| 0' \rangle}}^{\otimes n}{\ensuremath{| 1' \rangle}}^{\otimes n}$ for inequality II which then yields $2I\left[{\ensuremath{| gGHZ \rangle}}{\ensuremath{\langle gGHZ |}},{\ensuremath{| \phi \rangle}}\right]=2|\alpha \beta|$. In fact we already know from the results of Ref. [@Huber10], that the inequality will detect a huge amount of genuinely multipartite entangled mixed states in arbitrary high dimensional and multipartite systems. In all of these situations we thus also have a lower bound on the gme-concurrence.\
\
**Experimental Implementation**\
In order to be useful in practice, measures for multipartite entanglement need to be experimentally implementable by means of local observables (since all particles of composite quantum systems may not be available for combined measurements) without resorting to a full quantum state tomography (since the latter requires a vast number of measurements, which is unfeasible in practice). The bound (\[lowerbound\]) satisfies these demands, as for fixed ${\ensuremath{| \Phi \rangle}}$ its computation only requires at most the square root of the number of measurements needed for a full state tomography. Furthermore it can be implemented locally as explicitly shown in [@Huberqic]. In an experimental situation where one aims at producing a certain state ${\ensuremath{| \psi \rangle}}$, it is now possible to choose the corresponding ${\ensuremath{| \phi \rangle}}$ and not only detect the state as being genuinely multipartite entangled, but also have a reliable statement about the amount of multipartite entanglement the state exhibits. Even if the produced state deviates from the desired states, the criteria are astonishingly noise robust (as e.g. analyzed in Ref. [@Huber10]), as for example a GHZ state mixed with white noise is shown to be genuinely multipartite entangled with a white noise resistance of $\approx57\%$.\
\
**Conclusion**\
We introduced a measure of genuine multipartite entanglement, which can be lower bounded by means of one of the currently most powerful detection criteria. These bounds are experimentally implementable and computationally very efficient, allowing to not only detect, but also to quantify genuine multipartite entanglement in an experimental scenario. This has grave implications on applications where genuine multipartite entanglement is a crucial resource (as e.g. in quantum computing [@BRaussendorf01] or cryptopgraphy [@SHH3]) and might allow to give a good estimate of the relevance of genuine multipartite entanglement in other physical systems (as e.g. in biological systems [@Caruso] or quantum spin chains [@spinchains]).\
\
[**Acknowledgement**]{}: A. Gabriel, M. Huber and Ch. Spengler gratefully acknowledge the support of the Austrian FWF (Project P21947N16). J.L. Chen is supported by NSF of China (Grant No.10975075). Z.H. Ma is supported by NSF of China(10901103), partially supported by a grant of science and technology commission of Shanghai Municipality (STCSM, No.09XD1402500).\
\
**Appendix**\
Let us finish by proving the lower bound for the general n-qudit case. For the most general pure n-qudit state $|\psi\rangle=\sum_{i_1,i_2,(\cdots),i_n}c_{i_1,i_2,(\cdots),i_n}|i_1i_2(\cdots)i_n\rangle$ the squared concurrences $C^2(\rho_\gamma)=2(1-\text{Tr}(\rho^2_\gamma))$ with respect arbitrary bipartitions ($\gamma$) always take the form $$\begin{aligned}
C^2(\rho_\gamma)=4|c_{00(\cdots)0}c_{11(\cdots)1}-c_{\alpha(\gamma)}c_{\beta(\gamma)}|^2+F_\gamma\,,\\\end{aligned}$$ where $F_\gamma$ are non-negative functions (see e.g. Refs. [@HH2; @HHK1] for details on how to calculate the linear entropies of arbitrary subsystems). For every bipartition $\gamma$ there exists one pair $\alpha(\gamma)$ and $\beta(\gamma)$ that can be retrieved from $$\begin{aligned}
\{\alpha(\gamma),\beta(\gamma)\}=\pi_\gamma\{00(\cdots)0,11(\cdots)1\}\,\end{aligned}$$ where $\pi_\gamma$ permutes every number from the subset defined by $\gamma$ from the first half of the joint set with the second. Thus $$\begin{aligned}
C(\rho_\gamma)\geq 2|c_{00(\cdots)0}c_{11(\cdots)1}-c_{\alpha(\gamma)}c_{\beta(\gamma)}|\,\end{aligned}$$ will hold also for every $\gamma$. Now for the GME-concurrence we can infer $$\begin{aligned}
&\min_\gamma\{C(\rho_\gamma)\} \geq 2|c_{00(\cdots)0}c_{11(\cdots)1}|-(\sum_\gamma|c_{\alpha(\gamma)}c_{\beta(\gamma)}|)\, =: B \,.\end{aligned}$$ Now for any given mixed state the convex roof construction is bounded by $$\begin{aligned}
C_{gme}(\rho)\geq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_iB_i\,.\end{aligned}$$ For the choice $|\Phi\rangle=|0\rangle^{\otimes n}\otimes|1\rangle^{\otimes n}$, inequality (\[ineqII\]) reads $$\begin{aligned}
I[\rho,|\Phi\rangle]=|\rho_{00(\cdots)011(\cdots)1}|-\sum_\gamma\sqrt{\rho_{\alpha(\gamma)\alpha(\gamma)}\rho_{\beta(\gamma)\beta(\gamma)}}\leq 0\,.\end{aligned}$$ Now $$\begin{aligned}
2|\rho_{00(\cdots)011(\cdots)1}|\leq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_i2|c^i_{00(\cdots)0}c^i_{11(\cdots)1}|\,,\end{aligned}$$ due to the triangle inequality and $$\begin{aligned}
\sqrt{\rho_{\alpha(\gamma)\alpha(\gamma)}\rho_{\beta(\gamma)\beta(\gamma)}}\geq\inf_{\{p_i,|\psi_i\rangle\}}\sum_i p_i2|c_{\alpha(\gamma)}c_{\beta(\gamma)}|\,,\\\end{aligned}$$ due to the Cauchy-Schwarz inequality.\
This leads to a lower bound on the convex roof construction $$\begin{aligned}
C_{gme}(\rho)\geq 2I[\rho,|\Phi\rangle]\,.\end{aligned}$$ And again due to the local unitary invariance of $C_{gme}(\rho)$ this proves our lower bound for all $|\Phi\rangle$.
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[^1]: On a single copy. The property of being non-increasing under LOCC in general cannot be required of any measure of genuine multipartite entanglement, as it was shown (e.g. in Ref. [@Guhne09]) that genuine multipartite entanglement can locally be distilled out of a biseparable state if more copies are available.
| ArXiv |
---
abstract: 'We study the ground states of the single- and two-qubit asymmetric Rabi models, in which the qubit-oscillator coupling strengths for the counterrotating-wave and corotating-wave interactions are unequal. We take the transformation method to obtain the approximately analytical ground states for both models and numerically verify its validity for a wide range of parameters under the near-resonance condition. We find that the ground-state energy in either the single- or two-qubit asymmetric Rabi model has an approximately quadratic dependence on the coupling strengths stemming from different contributions of the counterrotating-wave and corotating-wave interactions. For both models, we show that the ground-state energy is mainly contributed by the counterrotating-wave interaction. Interestingly, for the two-qubit asymmetric Rabi model, we find that, with the increase of the coupling strength in the counterrotating-wave or corotating-wave interaction, the two-qubit entanglement first reaches its maximum then drops to zero. Furthermore, the maximum of the two-qubit entanglement in the two-qubit asymmetric Rabi model can be much larger than that in the two-qubit symmetric Rabi model.'
author:
- 'Li-Tuo Shen'
- 'Zhen-Biao Yang'
- Mei Lu
- 'Rong-Xin Chen'
- 'Huai-Zhi Wu'
title: Ground state of the asymmetric Rabi model in the ultrastrong coupling regime
---
Introduction
============
The Rabi model [@PR-49-324-1936], describing the interaction between a two-level system and a quantized harmonic oscillator, is a fundamental model in quantum optics. For the cavity quantum electrodynamics (QED) experiments, the qubit-oscillator coupling strength of the Rabi model is far smaller than the oscillator’s frequency and the corotating-wave approximation (RWA) works well, bringing in the ubiquitous Jaynes-Cummins model [@IEEE-51-89-1963; @JMO-40-1195-1993; @PRL-87-037902-2001; @PRA-71-013817-2005]. With recent experiment progresses in Rabi models [@PT-58-42-2005; @Science-326-108-2009; @PR-492-1-2010; @RPP-74-104401-2011; @Nature-474-589-2011; @RMP-84-1-2012; @RMP-85-623-2013; @arxiv1308-6253-2014] in the ultrastrong coupling regime [@PRB-78-180502-2008; @PRB-79-201303-2009; @Nature-458-178-2009; @Nature-6-772-2010; @PRL-105-237001-2010; @PRL-105-196402-2010; @PRL-106-196405-2011; @Science-335-1323-2012; @PRL-108-163601-2012; @PRB-86-045408-2012; @NatureCommun-4-1420-2013], in which the qubit-oscillator coupling strength becomes a considerable fraction of the oscillator’s or qubit’s frequency, the RWA breaks down but relatively complex quantum dynamics arises, bringing about many fascinating quantum phenomena [@NJP-13-073002-2011; @PRL-109-193602-2012; @PRA-81-042311-2010; @PRA-87-013826-2013; @PRA-59-4589-1999; @PRA-62-033807-2000; @PRB-72-195410-2005; @PRA-74-033811-2006; @PRA-77-053808-2008; @PRA-82-022119-2010; @PRL-107-190402-2011; @PRL-108-180401-2012; @PLA-376-349-2012].
Explicitly analytic solution to the Rabi model beyond the RWA is hard to obtain due to the non-integrability in its infinite-dimensional Hilbert space. Since it is difficult to capture the physics through numerical solution [@JPA-29-4035-1996; @EPL-96-14003-2011], various approximately analytical methods for obtaining the ground states of the symmetric Rabi models (SRM) have been tried [@RPB-40-11326-1989; @PRB-42-6704-1990; @PRL-99-173601-2007; @EPL-86-54003-2009; @PRA-80-033846-2009; @PRL-105-263603-2010; @PRA-82-025802-2010; @EPJD-66-1-2012; @PRA-86-015803-2012; @PRA-85-043815-2012; @PRA-86-023822-2012; @EPJB-38-559-2004; @PRB-75-054302-2007; @EPJD-59-473-2010; @arXiv-1303-3367v2-2013; @arXiv-1305-1226-2013; @PRA-87-022124-2013; @PRA-86-014303-2012; @arXiv-1305-6782-2013]. Especially, Braak [@PRL-107-100401-2011] used the method based on the $Z_{2}$ symmetry to analytically determine the spectrum of the single-qubit Rabi model, which was dependent on the composite transcendental function defined through its power series but failed to derive the concrete form of the system’s ground state. In Ref. [@PRA-81-042311-2010], Ashhab *et al.* applied the method of adiabatic approximation to treat two extreme situations to obtain the eigenstates and eigenenergies in the single-qubit SRM, i.e., the situation with a high-frequency oscillator or a high-frequency qubit. Ashhab [@PRA-87-013826-2013] used different order parameters to identify the phase regions of the single-qubit SRM and found that the phase-transition-like behavior appeared when the oscillator’s frequency was much lower than the qubit’s frequency. Lee and Law [@arXiv-1303-3367v2-2013] used the transformation method to seek the approximately analytical ground state of the two-qubit SRM in the near-resonance regime, and found that the two-qubit entanglement drops as the coupling strength further increased after it reached its maximum.
Previous studies consider the ground state of the SRM, i.e., the qubit-oscillator coupling strengths of the counterrotating-wave and corotating-wave interactions are equal. In this paper, we study the asymmetic Rabi models (ASRM), i.e., the coupling strengths for the counterrotating-wave and corotating-wave interactions are unequal, which helps to gain deep insight into the fundamentally physical property of such models. Different from Refs. [@PRA-81-042311-2010; @PRA-87-013826-2013], we here use the transformation method to obtain the ground state of the single-qubit ASRM under the near-resonance situation, where the oscillator’s frequency approximates the qubit’s frequency. Differ further from Ref. [@arXiv-1303-3367v2-2013], our investigation for the two-qubit ASRM intuitively identifies the collective contribution to its ground-state entanglement caused by the corotating-wave and counterrotating-wave interactions.
We investigate the single- and two-qubit ASRMs and show that their approximately analytical ground states agree well with the exactly numerical solutions for a wide range of parameters under the near-resonance situation, and the ground-state energy has an approximately quadratic dependence on the coupling strengths stemming from contributions of the counterrotating-wave and corotating-wave interactions. Besides, we show that the ground-state energy is mainly contributed by the counterrotating-wave interaction in both models. For the two-qubit ASRM, we obtain the approximately analytical negativity. Interestingly, for the two-qubit ASRM, we find that, with the increase of the coupling strength in the counterrotating-wave or corotating-wave interaction, the two-qubit entanglement first reaches its maximum then drops to zero.
The advantages of our result are the collective contributions to the ground state of the ASRM caused by the corotating-wave interaction and counterrotating-wave interaction can be determined approximately, and the contribution of the counterrotating-wave interaction on the ground state energy is larger than that of the corotating-wave interaction. We find that the maximal two-qubit entanglement of the ASRM is larger than that in the case of SRM. However, the transformation method here is applicable to the ASRM only under the near-resonant regime, where the oscillator’s frequency approximates the qubit’s frequency. When the corotating-wave and counterrotating-wave coupling constants are large enough in the ASRM, the result obtained by the transformation method has a big error compared with that obtained by the exactly numerical method. Such an investigation can also be generalized to the complex cases of three- and more-qubit ASRM. Note that the ASRM can be realized by using two unbalanced Raman channels between two atomic ground states induced by a cavity mode and two classical fields in theory [@PRA-75-013804-2007].
The single-qubit ASRM
=====================
Transformed ground state
------------------------
The Hamiltonian of the single-qubit ASRM is [@PRA-8-1440-1973]: (assume $\hbar=1$ for simplicity hereafter) $$\begin{aligned}
\label{1}
H_{1}&=&\frac{1}{2}w_{a}\sigma_{z}+w_{b}b^{\dagger}b\cr&&
+\frac{\lambda_{1}}{2}(b^{\dagger}\sigma_{-}+b\sigma_{+})
+\frac{\lambda_{2}}{2}(b^{\dagger}\sigma_{+}+b\sigma_{-}),\end{aligned}$$ where $w_{a}$ is the qubit’s frequency. $\sigma_{z}$ and $\sigma_{\pm}$ are the Pauli matrices, describing the qubit’s energy operator and the spin-flip operators, respectively. We assume that $|\downarrow\rangle_{A}$ and $|\uparrow\rangle_{A}$ are the eigenstates of $\sigma_{z}$, i.e., $\sigma_{z}$ $|\downarrow\rangle_{A}$ $=$ $-|\downarrow\rangle_{A}$ and $\sigma_{z}$ $|\uparrow\rangle_{A}$ $=$ $|\uparrow\rangle_{A}$. $b^{\dagger}$ ($b$) is the creation (annihilation) operator of the harmonic oscillator with the frequency $w_{b}$. The qubit-oscillator coupling strengths of the corotating-wave interaction $(b^{\dagger}\sigma_{-}+b\sigma_{+})$ and the counterrotating-wave interaction $(b^{\dagger}\sigma_{+}+b\sigma_{-})$ are denoted by $\lambda_{1}$ and $\lambda_{2}$, respectively. However, when $\lambda_{1}$ $\neq$ $\lambda_{2}$ (here $\lambda_{1}$, $\lambda_{2}$, $w_{a}$ $\neq0$), to our knowledge, there is still no analytical solution to the ground state of the single-qubit ASRM.
Our task in this paper is to determine the ground-state energy $E_g$ and the ground-state vector $|\phi_g\rangle$ for the single- (Section II) or two-qubit (Section III) ASRM, where $H_{1}|\phi_g\rangle$ $=$ $E_{g}|\phi_g\rangle$. In this paper, the subscripts $A$ and $F$ denote the vectors of the atomic state and field state, respectively.
To deal with the counterrotating-wave terms in Eq. (\[1\]), we apply a unitary transformation to the Hamiltonian $H_{1}$ [@EPJD-59-473-2010; @PRB-75-054302-2007; @PRA-82-022119-2010]: $$\begin{aligned}
\label{3}
H_{1}^{'}&=&e^{S_{1}}H_{1}e^{-S_{1}},\end{aligned}$$ with $$\begin{aligned}
\label{4}
S_{1}=\xi_{1}(b^{\dagger}-b)\sigma_{x},\end{aligned}$$ where $\xi_{1}$ is a variable to be determined later. Then the transformed Hamiltonian $H_{1}^{'}$ can be decomposed into three parts: $$\begin{aligned}
\label{5}
H_{1}^{'}&=&H_{1}^{a}+H_{1}^{b}+H_{1}^{c},\end{aligned}$$ with $$\begin{aligned}
\label{6-7-8}
H_{1}^{a}&=&\frac{1}{2}\big[w_{a}\eta_{1}-(\lambda_{1}-\lambda_{2})\xi_{1}\eta_{1}\big]\sigma_{z}\cr\cr&&+
\big[w_{b}-(\lambda_{1}-\lambda_{2})\xi_{1}\eta_{1}\sigma_{z}\big]b^{\dagger}b\cr\cr&&
+w_{b}\xi_{1}^{2}-\frac{1}{2}(\lambda_{1}+\lambda_{2})\xi_{1},\\
H_{1}^{b}&=&\big[\frac{1}{4}(\lambda_{1}+\lambda_{2})-w_{b}\xi_{1}\big](b^{\dagger}+b)\sigma_{x}\cr\cr&&
-i\big[\frac{1}{4}(\lambda_{1}-\lambda_{2})\eta_{1}+w_{a}\xi_{1}\eta_{1}\big](b^{\dagger}-b)\sigma_{y},\\
H_{1}^{c}&=&\frac{1}{2}w_{a}\sigma_{z}\bigg\{\cosh\big[2\xi_{1}(b^{\dagger}-b)\big]-\eta_{1}
\bigg\}\cr&&
-\frac{i}{2}w_{a}\sigma_{y}\bigg\{\sinh\big[2\xi_{1}(b^{\dagger}-b)\big]-2\xi_{1}\eta_{1}(b^{\dagger}-b)
\bigg\}\cr&&
-\frac{i}{4}(\lambda_{1}-\lambda_{2})(b^{\dagger}-b)\sigma_{y}\bigg\{\cosh\big[2\xi_{1}(b^{\dagger}-b)\big]\cr&&-\eta_{1}
\bigg\}+\frac{1}{4}(\lambda_{1}-\lambda_{2})(b^{\dagger}-b)\sigma_{z}\bigg\{\sinh\big[2\xi_{1}(b^{\dagger}
\cr&&-b)\big]-2\xi_{1}\eta_{1}(b^{\dagger}-b)
\bigg\}+O(b^{\dagger2},b^{2}),\end{aligned}$$ where $\eta_{1}$$=_{F}$$\langle
0|\cosh[2\xi_{1}(b^{\dagger}-b)]|0\rangle_{F}$ $=$ $e^{-2\xi_{1}^2}$ and $O(b^{\dagger2},b^{2})=\frac{1}{2}(\lambda_{1}-\lambda_{2})\xi_{1}\eta_{1}(b^{\dagger2}+b^2)\sigma_{z}$. The terms $\cosh[2\xi_{1}(b^{\dagger}-b)]$ and $\sinh[2\xi_{1}(b^{\dagger}-b)]$ in $H_{1}^{c}$ have the dominating expansions [@EPJD-59-473-2010]: $$\begin{aligned}
\label{9-10}
\cosh[2\xi_{1}(b^{\dagger}-b)]&\simeq&\eta_{1}+O(\xi_{1}^2),\\
\sinh[2\xi_{1}(b^{\dagger}-b)]&\simeq&2\xi_{1}\eta_{1}(b^{\dagger}-b)+O(\xi_{1}^3),\end{aligned}$$ where $O(b^{\dagger2},b^{2})$, $O(\xi_{1}^2)$ and $O(\xi_{1}^3)$ are higher-order terms of $b^{\dagger}$ and $b$, which represent the double- and three-photon transition processes and can be neglected as an approximation when $\xi_1$ and $|\lambda_1\pm\lambda_2|$ are much smaller than the frequency sum $w_a+w_b$ where $w_a\approx
w_b$. Thus, $H_{1}^{'}\simeq H_{1}^{a}+H_{1}^{b}$.
When the parameter $\xi_{1}$ is chosen such that it satisfies the condition: $$\begin{aligned}
\label{11}
e^{2\xi_{1}^2}\big[(\lambda_{1}+\lambda_{2})-4w_{b}\xi_{1}\big]=(\lambda_{1}-\lambda_{2})+4w_{a}\xi_{1},\end{aligned}$$ the qubit and the oscillator are coupled in the following form: $$\begin{aligned}
\label{12}
H_{1}^{b}&=&\frac{1}{2}\big[(\lambda_{1}+\lambda_{2})-4w_{b}\xi_{1}\big]\times\big(b^{\dagger}
\sigma_{-}+b\sigma_{+}\big).\end{aligned}$$ Note that $H_{1}^{b}$ in Eq. (\[12\]) contains no counterrotating-wave interactions in which the qubit excitation (deexcitation) is accompanied by the emission (absorption) of a photon. Therefore, the transformed Hamiltonian $H_{1}^{'}$ is exactly solvable when we eliminate the counterrotating-wave terms by choosing $\xi_{1}$ to satisfy Eq. (\[11\]) and by neglecting higher-order transition processes which are presented by terms $O(b^{\dagger2},b^{2})$, $O(\xi_{1}^2)$ and $O(\xi_{1}^3)$.
It is easy to show that the eigenvector $|\downarrow\rangle_A|0\rangle_F$ is the ground-state vector of the transformed Hamiltonian $H_{1}^{'}$, with $|0\rangle_{F}$ being the vacuum state of the harmonic oscillator, and the corresponding eigenenergy $E_{g1}$ is: $$\begin{aligned}
\label{13}
E_{g1}=\xi_1^2w_{b}-\frac{1}{2}(\lambda_1+\lambda_2)\xi_1-\frac{1}{2}\eta_1[w_a-\xi_1(\lambda_1-\lambda_2)].\cr&&\end{aligned}$$ We see that when $\lambda_{1}=\lambda_{2}$, $E_{g1}$ reduces to the transformed ground-state energy derived in Ref. [@EPJD-59-473-2010]. Therefore, the ground state of the original Hamiltonian (\[1\]) can be approximately constructed: $$\begin{aligned}
\label{14}
|\phi_{g1}\rangle&=&e^{-S_{1}}|\downarrow\rangle_{A}|0\rangle_{F}\cr
&=&\frac{1}{\sqrt{2}}(|\psi_{A}^{+}\rangle|-\xi_1\rangle_F-|\psi_{A}^{-}\rangle|\xi_1\rangle_F),\end{aligned}$$ with $|\xi_{1}\rangle_{F}$ and $|-\xi_{1}\rangle_{F}$ being the coherent states of the oscillator with the amplitudes $\xi_{1}$ and $-\xi_{1}$. $|\psi_{A}^{+}\rangle=(|\uparrow\rangle_A+|\downarrow\rangle_A)/\sqrt{2}$ and $|\psi_{A}^{-}\rangle=(|\uparrow\rangle_A-|\downarrow\rangle_A)/\sqrt{2}$ are the eigenstates of $\sigma_x$.
![(Color online) The ground-state energy for the single-qubit ASRM obtained by the transformation method $E_0=E_{g1}$ (red grid) and by the numerical solution $E_0=E_{g}$ (blue grid) versus the coupling strengths $\lambda_{1}$ and $\lambda_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. The energy deviation $\Delta E_{g1}=E_{g1}-E_g$ versus $\lambda_{1}$ and $\lambda_{2}$: (d) $w_{b}=0.8w_{a}$; (e) $w_{b}=w_{a}$; (f) $w_{b}=1.2w_{a}$. []{data-label="Fig.1."}](fig1.eps){width="0.9\columnwidth"}
The value of $\xi_{1}$ is obtained by numerically solving the nonlinear equation (\[11\]). $\xi_{1}$ has an approximately linear dependence on the counterrotating-wave coupling strength by neglecting high-order terms of the field mode as: $$\begin{aligned}
\label{29}
\xi_{1}&\simeq&\frac{\lambda_2}{2(w_a+w_b)}.\end{aligned}$$ In Fig. 1, we compare the ground-state energy obtained by the transformation method and that by the numerical solution. Especially, we find that the ground-state energy obtained by the transformation method coincides very well with the exactly numerical solution when $|\lambda_1-\lambda_2|\leq0.15w_a$. Therefore, when $\lambda_{1},\lambda_{2}\leq w_{a}$, the transformed ground-state energy $E_{g1}$ approximates: $$\begin{aligned}
\label{15}
E_{g1}\simeq-\frac{1}{2}w_a-\frac{\lambda_2^2}{4(w_a+w_b)}+\frac{\lambda_2^3(\lambda_1-\lambda_2)}{8(w_a+w_b)^3},\end{aligned}$$ which shows that the ground-state energy has an approximately quadratic dependence on the coupling strength by neglecting high-order terms of the field mode for the small factor $|\lambda_1-\lambda_2|$ and is mainly contributed by the counterrotating-wave interaction. This result differs further from that of the SRM [@EPJD-59-473-2010].
Considering the fidelity $F_{1}$ for the ground state $|\phi_{g1}\rangle$, where $F_{1}=\langle\phi_{g1}|\phi_{g}\rangle$ and $|\phi_g\rangle$ is the ground state obtained through numerical solutions [@arXiv-1303-3367v2-2013], we plot $F_1$ as a function of the coupling strengths $\lambda_{1}$ and $\lambda_{2}$ under different detunings in Fig. 2. The result shows that the fidelity is higher than $99.9\%$ when $\lambda_1\leq0.5w_{a}$ and $\lambda_2\leq0.5w_{a}$. Furthermore, the fidelity under the positive-detuning case ($w_{b}-w_{a}>0$) decreases slowest among all the cases in Fig. 2 (a) - (c) when $\lambda_1$ and $\lambda_2$ increase.
![(Color online) The fidelity $F_{1}$ of the ground state for the single-qubit ASRM obtained by the transformation method versus the coupling strengths $\lambda_{1}$ and $\lambda_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. []{data-label="Fig.2."}](fig2.eps){width="1\columnwidth"}
Ground-state entanglement
-------------------------
In this section, we focus on the entanglement between the qubit and the oscillator in the ground state of the single-qubit ASRM. Since the ground state is a pure state, we take the von Neumann entropy as an entanglement measure. If a pure state of a composite system $XY$ is given by the density matrix $\rho_{XY}$, the entropy of the subsystem $X$ is defined as: $$\begin{aligned}
\label{SS}
S_{\rho_{X}} = -Tr(\rho_{X}log_2\rho_{X}),\end{aligned}$$ where $\rho_{X}=Tr_{Y}(\rho_{XY})$ is the reduced density matrix for the subsystem $X$ by tracing out the freedom degree of the subsystem $Y$. Note that $S_{\rho_{X}}$ measures the entanglement between the subsystems $X$ and $Y$ of the system, which has a maximum value of $log_{2}K$ in a $K$-dimensional Hilbert space.
In the standard basis $\{|\uparrow\rangle_A,|\downarrow\rangle_A
\}$, the reduced density matrix of the qubit is $\rho_{A}=Tr_{F}(|\Phi_{G}\rangle\langle\Phi_{G}|)$, where $|\Phi_{G}\rangle$ is the exactly numerical ground state of the single-qubit ASRM. The entropy of the qubit $S_{\rho_{A}}$ = $-Tr(\rho_{A}log_2\rho_{A})$ is numerically plotted in Fig. 3, which shows that the entanglement between the qubit and the oscillator increases from as $\lambda_1$ and $\lambda_2$ increase from zero to values close to $w_a$ and $w_b$.
![(Color online) The degree of entanglement $S_{\rho_A}$ for the qubit in the ground state of the single-qubit ASRM obtained by the numerical simulation versus the coupling strengths $\lambda_{1}$ and $\lambda_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. []{data-label="Fig.3."}](fig3.eps){width="1\columnwidth"}
The two-qubit ASRM
==================
Transformed ground state
------------------------
The Hamiltonian of the two-qubit ASRM is [@PRA-8-1440-1973]: $$\begin{aligned}
\label{17}
H_{}&=&w_{a}J_{z}+w_{b}b^{\dagger}b+g_{1}(b^{\dagger}J_{-}+bJ_{+})\cr\cr&&
+g_{2}(b^{\dagger}J_{+}+bJ_{-}),\end{aligned}$$ where $w_{a}$ is the frequency of each qubit. $J_{l}\{
l=x,y,z,\pm\}$ describes the collective qubit operator of a spin-$1$ system. $b^{\dagger}$ ($b$) is the creation (annihilation) operator of the harmonic oscillator with the frequency $w_{b}$. The qubit-oscillator coupling strengths of the corotating-wave and counterrotating-wave interactions are $g_{1}$ and $g_{2}$, respectively. We denote the eigenstates of $J_{z}$ by $|-1\rangle_{A}$, $|0\rangle_{A}$, and $|1\rangle_{A}$, i.e., $J_{z}|m\rangle_{A}=m|m\rangle_{A}$ ($m=0,\pm1$). $|0\rangle_{F}$ is the vacuum state of the harmonic oscillator, and $|\alpha\rangle_{F}$ denotes the coherent-state field with the amplitude $\alpha$. When a rotation around the $y$ axis is performed, the Hamiltonian of the two-qubit ASRM can be written as : $$\begin{aligned}
\label{18}
H_{2}&=&w_{a}J_{x}+w_{b}b^{\dagger}b+(g_{1}+g_{2})(b^{\dagger}+b)J_{z}\cr\cr&&
+i(g_{1}-g_{2})(b^{\dagger}-b)J_{y}.\end{aligned}$$
To transform the Hamiltonian $H_{2}$ into a mathematical form without counterrotating-wave terms, we apply a unitary transformation to $H_{2}$: $$\begin{aligned}
\label{19}
H_{2}^{'}&=&e^{S_{2}}H_{2}e^{-S_{2}},\end{aligned}$$ with $$\begin{aligned}
\label{20}
S_{2}&=&\xi_{2}(b^{\dagger}-b)J_{z},\end{aligned}$$ where $\xi_{2}$ is a variable to be determined. Therefore, the transformed Hamiltonian $H_{2}^{'}$ is decomposed into three parts: $$\begin{aligned}
\label{21}
H_{2}^{'}&=&H_{2}^{a}+H_{2}^{b}+H_{2}^{c},\end{aligned}$$ with $$\begin{aligned}
\label{22-23-24}
H_{2}^{a}&=&w_{b}b^{\dagger}b+\bigg[
w_{a}\eta_{2}-(g_{1}-g_{2})\eta_{2}\xi_{2}\bigg]J_{x}\cr&&
+\bigg[w_{b}\xi_{2}^{2}-2\xi_{2}(g_{1}+g_{2}) \bigg]J_{z}^{2},\\
H_{2}^{b}&=&\bigg[
(g_{1}+g_{2})-w_{b}\xi_{2}\bigg](b^{\dagger}+b)J_{z}\cr&&+ i\bigg[
w_{b}\eta_{2}\xi_{2}+(g_{1}-g_{2})\eta_{2}\bigg](b^{\dagger}-b)J_{y},\\
H_{2}^{c}&=&w_{a}J_{x}\bigg\{
\cosh[\xi_{2}(b^{\dagger}-b)]-\eta_{2}\bigg\} \cr&&+
iw_{a}J_{y}\bigg\{
\sinh[\xi_{2}(b^{\dagger}-b)]-\eta_{2}\xi_{2}(b^{\dagger}-b)
\bigg\}\cr&&+ (g_{1}-g_{2})(b^{\dagger}-b)J_{x}\bigg\{
\sinh[\xi_{2}(b^{\dagger}-b)]\cr&&-\eta_{2}\xi_{2}(b^{\dagger}-b)
\bigg\}+i(g_{1}-g_{2})(b^{\dagger}-b)J_{y}\cr&&\times \bigg\{
\cosh[\xi_{2}(b^{\dagger}-b)]-\eta_{2}\bigg\}+
O(b^{\dagger2},b^{2}),\end{aligned}$$ where $\eta_{2}=$ $_{F}\langle
0|\cosh[\xi_{2}(b^{\dagger}-b)]|0\rangle_{F}$ $=$ $e^{-\xi_{2}^{2}/2}$ and $O(b^{\dagger2},b^{2})=(g_{1}-g_{2})\eta_{2}\xi_{2}J_{x}
(b^{\dagger2}-2b^{\dagger}b-b^{2})$. As shown in the single-qubit ASRM, when $\xi_2$ and $|g_1\pm g_2|$ are much smaller than the frequency sum $w_a+w_b$ where $w_a\approx w_b$, $H_{2}^{c}$ can be neglected, thus $H_{2}^{'}\simeq H_{2}^{a}+H_{2}^{b}$. Compared with $H_{1}^{a}$ in the single-qubit ASRM of Sec. II, the main difference is the presence of the $J_{z}^{2}$ operator term in $H_{2}^{a}$, but in the single-qubit ASRM the corresponding term $\sigma_z^2=1$ is just a constant. Therefore, $H_{2}^{a}$ here represents a renormalized three-level system in which we need to diagonalize $H_{2}^{a}$ to remove counterrotating-wave terms.
The eigenvalues $\nu_{k}$ ($k=1,2,3$) and eigenstates $|\varphi_{k}\rangle_{A}$ of the Hamiltonian $H_{2}^{''}=H_{2}^{a}-w_{b}b^{\dagger}b$ are: $$\begin{aligned}
\label{25}
\nu_{1}&=&\frac{A}{2}-\frac{1}{2}\sqrt{A^{2}+8B^2}, \cr
|\varphi_{1}\rangle_{A}&=&\frac{1}{N_{1}}\bigg\{|-1\rangle_{A}-\frac{(A+\sqrt{A^2+8B^2})}{2B}|0\rangle_{A}+|1\rangle_{A}\bigg\},\cr\cr
\nu_{2}&=&A, \cr\cr
|\varphi_{2}\rangle_{A}&=&\frac{1}{N_{2}}\bigg\{-|-1\rangle_{A}+|1\rangle_{A}\bigg\},\cr\cr
\nu_{3}&=&\frac{A}{2}+\frac{1}{2}\sqrt{A^{2}+8B^2}, \cr\cr
|\varphi_{3}\rangle_{A}&=&\frac{1}{N_{3}}\bigg\{|-1\rangle_{A}-\frac{(A-\sqrt{A^2+8B^2})}{2B}|0\rangle_{A}+|1\rangle_{A}\bigg\},\cr&&\end{aligned}$$ with $$\begin{aligned}
\label{26}
A&=&w_{b}\xi_{2}^2-2\xi_{2}(g_{1}+g_{2}), \cr
B&=&\frac{1}{\sqrt{2}}[w_{a}\eta_{2}-\eta_{2}\xi_{2}(g_{1}-g_{2})],\end{aligned}$$ where $N_{k}$ is the normalization factor of the eigenvector $|\varphi_{k}\rangle_{A}$. Here the eigenvalues are arranged in the decreasing order: $\nu_{1}<\nu_{2}<\nu_{3}$. Then $H_{2}^{'}$ can be expanded in terms of the renormalized eigenvectors: $$\begin{aligned}
\label{27}
H_{2}^{'}&\simeq&\sum_{k=1}^{3}\nu_{k}|\varphi_{k}\rangle_{A}\langle
\varphi_{k}|+\bigg[
(D_{1}b+D_{2}b^{\dagger})|\varphi_{1}\rangle_{A}\langle
\varphi_{2}|\cr&&+(D_{3}b+D_{4}b^{\dagger})|\varphi_{2}\rangle_{A}\langle
\varphi_{3}|+H.c.\bigg]+w_{b}b^{\dagger}b,\end{aligned}$$ where $D_{x}\ (x=1,2,3,4)$ is the coefficient depending on the variable $\xi_{2}$.
After transforming the Hamiltonian $H_{2}$ into $H_{2}^{'}$, we can eliminate counterrotating-wave terms describing the coupling between the lowest two eigenstates by setting: $$\begin{aligned}
\label{28}
D_{1}&=&\eta_{2}\bigg[
w_{a}\xi_{2}+(g_{1}-g_{2})\bigg]\bigg(A+\sqrt{A^2+8B^2}\bigg)\cr&&-2\sqrt{2}B\bigg[
(g_{1}+g_{2})-w_{b}\xi_{2}\bigg]=0.\end{aligned}$$ The value of $\xi_{2}$ is obtained by numerically solving the nonlinear equation (\[28\]). We find that when $g_{1}\leq0.5w_{a}$ and $g_{2}\leq0.5w_{a}$, $\xi_{2}$ has an approximately linear dependence on the coupling strengths: $$\begin{aligned}
\label{29}
\xi_{2}&\simeq&\frac{(w_{b}-w_{a})g_{1}+(w_{b}+w_{a})g_{2}}{w_{b}^2+w_{a}^2}.\end{aligned}$$
![(Color online) The ground-state energy for the two-qubit ASRM obtained by the transformation method $E_0=E_{g2}$ (red grid) and the numerical solution $E_0=E_{g}$ (blue grid) versus the coupling strengths $g_{1}$ and $g_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$, where $E_{g2}$ is plotted by using $\nu_{1}$ in Eq. (\[25\]). The energy deviation $\Delta E_{g2}=E_{g2}-E_g$ versus $g_{1}$ and $g_{2}$: (d) $w_{b}=0.8w_{a}$; (e) $w_{b}=w_{a}$; (f) $w_{b}=1.2w_{a}$. []{data-label="Fig.4."}](fig4.eps){width="1\columnwidth"}
In Fig. 4, we compare the ground-state energy obtained by the transformation method and that obtained by the numerical solution. We find that when $g_{1}\leq0.25w_{a}$ and $g_{2}\leq0.25w_{a}$, the ground-state energy obtained by the transformation method coincides very well with the exact value even for $|g_{1}-g_{2}|=0.24w_{a}$. Therefore, when $g_{1}\leq0.5w_{a}$ and $g_{2}\leq0.5w_{a}$, $|\varphi_{1}\rangle_{A}|0\rangle_{F}$ is expected to be the approximately analytical ground state of the transformed Hamiltonian, and the ground state $|\phi_{g}\rangle$ of the two-qubit ASRM can be expressed by the transformed ground state $|\phi_{g2}\rangle$: $$\begin{aligned}
\label{30}
|\phi_{g2}\rangle&=&e^{-S_{2}}|\varphi_{1}\rangle_{A}|0\rangle_{F}
\cr&&=\frac{1}{N_{1}}\big(
|-1\rangle_{A}|\xi_{2}\rangle_{F}-\frac{\nu_{3}}{B}|0\rangle_{A}|0\rangle_{F}+|1\rangle_{A}|-\xi_{2}\rangle_{F}\big),\cr&&\end{aligned}$$ and the ground-state energy $E_{g2}$ is: $$\begin{aligned}
\label{31}
E_{g2}&\simeq&\nu_{1}\simeq
-w_{a}-\frac{(g_{1}+g_{2})g_{2}}{w_{a}w_{b}},\end{aligned}$$ which directly shows that $E_{g2}$ has an approximately quadratic dependence on the qubit-oscillator coupling strengths by neglecting high-order terms of the field mode. This result differs further from that in the two-qubit SRM [@arXiv-1303-3367v2-2013].
![(Color online) The fidelity $F_{2}$ of the ground state for the two-qubit ASRM obtained by the transformation method versus the coupling strengths $g_{1}$ and $g_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. []{data-label="Fig.5."}](fig5.eps){width="1\columnwidth"}
The fidelity $F_{2}$ of the ground state as a function of the qubit-oscillator coupling strengths $g_{1}$ and $g_{2}$ under different detunings is plotted in Fig. 5. The result shows that $F_{2}$ keeps higher than $99.9\%$ when $g_{1}\leq0.25w_{a}$ and $g_{2}\leq0.25w_{a}$, which coincides with the behavior of the transformed ground-state energy shown in Fig. 5.
Ground-state entanglement
-------------------------
We also examine the ground-state entanglement of the two-qubit ASRM by taking into account both the transformation method and the exactly numerical treatment. Negativity is taken to quantify the entanglement for two qubits, which is defined as [@PRA-65-032314-2002]: $$\begin{aligned}
\label{32}
M_{\rho_A}&=&\frac{\|\rho_A^{T}\|-1}{2},\end{aligned}$$ where $\rho_A^{T}$ is the partially transposed matrix of the two-qubit reduced density matrix $\rho_A$, with $\rho_A=Tr_{F}(\rho_{AF})$ and $\rho_{AF}=|\phi_{g}\rangle\langle
\phi_{g}|$, and $\|\rho_A^{T}\|$ is the trace norm of $\rho_A^{T}$. Thus, $M_{\rho_A}$ alternatively equals the absolute value for the sum of the negative eigenvalues of $\rho_A^{T}$. For the transformed ground state $|\phi_{g2}\rangle$ in Eq. (\[30\]), the partially transposed matrix of the reduced density operator for the two qubits in the qubit basis $\Gamma_{q}$ $=$ $\{$ $|\uparrow_{1}\rangle|\uparrow_{2}\rangle,
|\uparrow_{1}\rangle|\downarrow_{2}\rangle,
|\downarrow_{1}\rangle|\uparrow_{2}\rangle,
|\downarrow_{1}\rangle|\downarrow_{2}\rangle$ $\}$, where $|\uparrow_{l}\rangle$ and $|\downarrow_{l}\rangle$ ($l=1,2$) correspond to the excited and ground states of the $l$th qubit respectively, is obtained as follows:
![(Color online) The negativity $M_{\rho_A}$ of two qubits in the ground state of the two-qubit ASRM versus the coupling strengths $g_{1}$ and $g_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. The results obtained by the transformation method $M_{\rho_A}=M_{\rho_A}^{t}$ and the numerical simulation $M_{\rho_A}=M_{\rho_A}^{n}$ are represented by the red grid and the blue grid, respectively. The deviation in the two-qubit negativity $\Delta M=M_{\rho_A}^{n}-M_{\rho_A}^{t}$ obtained by transformation method: (d) $w_{b}=0.8w_{a}$; (e) $w_{b}=w_{a}$; (f) $w_{b}=1.2w_{a}$. []{data-label="Fig.6."}](fig6.eps){width="0.95\columnwidth"}
$$\begin{aligned}
\label{33}
\rho_A^{T}&=&\frac{1}{(2+\beta^2)} \left(\begin{array}{cccc}
1 & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & \frac{\beta^2}{2} \\
\frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & \frac{\beta^2}{2} & e^{-2\alpha^2} & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}}\\
\frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & e^{-2\alpha^2} & \frac{\beta^2}{2} & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} \\
\frac{\beta^2}{2} & \frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} &
\frac{\beta}{\sqrt{2}}e^{-\frac{\alpha^2}{2}} & 1
\end{array}
\right),\cr&&\end{aligned}$$
where $\alpha=\xi_{2}$ and $\beta=-\frac{\nu_{3}}{B}$. With Eq. (\[33\]), we can calculate the negative $M_{\rho_A}$: $$\begin{aligned}
\label{34}
M_{\rho_A}&=&\max \bigg\{
\frac{2e^{-2\xi_{2}^2}-(\frac{\nu_{3}}{B})^2}{2[2+(\frac{\nu_{3}}{B})^2]},
0 \bigg\}.\end{aligned}$$ When $g_1\leq0.25w_{a}$ and $g_2\leq0.25w_{a}$, $M_{\rho_A}$ approximates: $$\begin{aligned}
\label{35}
M_{\rho_A}&\simeq&
\frac{w_{b}\big[(1-\frac{1}{\sqrt{2}})^2g_{2}^{2}+g_1g_2\big]}{4w_{a}(w_{a}+w_{b})^2}.\end{aligned}$$ From Eq. (\[35\]), we see that the two-qubit entanglement increases with $g_2^2$ and $g_1g_2$. The two-qubit negativity as a function of the qubit-oscillator coupling strengths $g_{1}$ and $g_{2}$ under different detunings has been plotted in Fig. 6 (a) - (c), and the corresponding deviation from the numerical simulation is plotted in Fig. 6 (d) - (f). For $0<g_1\leq0.25w_a$ and $0<g_2\leq0.25w_a$, the two-qubit negativity has a linear dependence on $g_1$ for the fixed $g_2$ and a quadratic dependence on $g_2$ for the fixed $g_1$; For $0<g_1\leq0.25w_a$ and $0.25w_a<g_2<0.5w_a$, the negativity keeps close to zero; However, for $0.25w_a<g_1<0.5w_a$ and $0<g_2<0.5w_a$, the negative has a similar dependence on $g_1$ and $g_2$ with the case of $0<g_1\leq0.25w_a$ and $0<g_2\leq0.25w_a$. We find that when $g_1\leq0.25w_{a}$ and $g_2\leq0.25w_{a}$ the deviation in the negativity is close to zero, meaning the ground state obtained by the transformation method agrees well with the exact one. This directly shows that the two-qubit entanglement is caused by the counterrotating-wave interaction in the Hamiltonian. Interestingly, after the negativity has reached its maximum, it will monotonically decrease when $g_{1}$ or $g_{2}$ further increases. Furthermore, the maximum of the two-qubit entanglement in the two-qubit ASRM is far larger than that in the two-qubit SRM, and the two-qubit entanglement mainly appears when the coupling strength of the corotating-wave interaction is bigger than that of the counterrotating-wave interaction, which is because the contribution to the two-qubit entanglement from the counterrotating-wave interaction is larger than that from the corotating-wave interaction in Eq. (\[35\]). As seen from Fig. 7, when $g_{1}>1.11w_{a}$ or $g_{2}>0.88w_{a}$ at $w_{b}=w_{a}$, $M_{\rho_A}$ decreases to zero and never increases again, and the maximum negativity is about $0.10$ which is only $3.5\times10^{-2}$ in the two-qubit SRM [@arXiv-1303-3367v2-2013].
![(Color online) The negativity $M_{\rho_A}$ of two qubits in the ground state of the two-qubit ASRM obtained by the numerical simulation versus the coupling strengths $g_{1}$ and $g_{2}$ when $w_{b}=w_{a}$. []{data-label="Fig.7."}](fig7.eps){width="0.5\columnwidth"}
In Fig. 8, we numerically plot the entropy $S_{\rho_A}$ of two qubits versus the coupling strengths $g_1$ and $g_2$ in the ground state of the two-qubit ASRM, where $S_{\rho_A}$ $=$ $-Tr(\rho_{A}log_2\rho_{A})$. The result shows that the entanglement between the qubit and the oscillator increases from as $g_1$ and $g_2$ increase from zero to values close to $w_a$ and $w_b$.
![(Color online) The degree of entanglement $S_{\rho_A}$ for the qubits in the ground state of the two-qubit ASRM obtained by numerical simulations versus the coupling strengths $g_{1}$ and $g_{2}$: (a) $w_{b}=0.8w_{a}$; (b) $w_{b}=w_{a}$; (c) $w_{b}=1.2w_{a}$. []{data-label="Fig.8."}](fig8.eps){width="1\columnwidth"}
Conclusion
==========
In conclusion, we have used the transformation method to obtain the approximately analytical ground states of the single- and two-qubit ASRMs, and shown that the transformed results coincided well with those obtained by numerical simulations for a wide range of parameters under the near-resonance condition. We find that the ground-state energy in either the single- or two-qubit ASRM has an approximately quadratic dependence on the qubit-oscillator coupling strengths, and the contribution of the counterrotating-wave interaction on the ground state energy is larger than that of the corotating-wave interaction. Interestingly, we also find that the two-qubit entanglement of the two-qubit ASRM decreases to zero and never increases again as long as the qubit-oscillator coupling strengths are large enough. Furthermore, the maximum of the two-qubit entanglement in the two-qubit ASRM is far larger than that in the two-qubit SRM, and the two-qubit entanglement mainly appears when the coupling strength of the corotating-wave interaction is bigger than that of the counterrotating-wave interaction.
Acknowledgement
===============
This work is supported by the Major State Basic Research Development Program of China under Grant No. 2012CB921601, the National Natural Science Foundation of China under Grant No. 11374054, No. 11305037, No. 11347114, and No. 11247283, the Natural Science Foundation of Fujian Province under Grant No. 2013J01012, and the funds from Fuzhou University under Grant No. 022513, Grant No. 022408, and Grant No. 600891.
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| ArXiv |
---
abstract: 'This paper presents a novel efficient receiver design for wireless communication systems that incorporate orthogonal frequency division multiplexing (OFDM) transmission. The proposed receiver does not require channel estimation or equalization to perform coherent data detection. Instead, channel estimation, equalization, and data detection are combined into a single operation, and hence, the detector is denoted as a direct data detector ($D^{3}$). The performance of the proposed system is thoroughly analyzed theoretically in terms of bit error rate (BER), and validated by Monte Carlo simulations. The obtained theoretical and simulation results demonstrate that the BER of the proposed $D^{3}$ is only $3$ dB away from coherent detectors with perfect knowledge of the channel state information (CSI) in flat fading channels, and similarly in frequency-selective channels for a wide range of signal-to-noise ratios (SNRs). If CSI is not known perfectly, then the $D^{3}$ outperforms the coherent detector substantially, particularly at high SNRs with linear interpolation. The computational complexity of the $D^{3}$ depends on the length of the sequence to be detected, nevertheless, a significant complexity reduction can be achieved using the Viterbi algorithm.'
author:
- 'A. Saci, , A. Al-Dweik, , A. Shami, [^1][^2][^3]'
title: Direct Data Detection of OFDM Signals Over Wireless Channels
---
OFDM, fading channels, data detection, Viterbi, sequence detection, channel estimation, equalization.
Introduction
============
Consequently, a low-complexity single-tap equalizer can be utilized to eliminate the impact of the multipath fading channel. Under such circumstances, the OFDM demodulation process can be performed once the fading parameters at each subcarrier, commonly denoted as channel state information (CSI), are estimated.
In general, channel estimation can be classified into blind [@One-Shot-CFO-2014]-[@blind-massive-mimo-acd], and pilot-aided techniques [@Robust-CE-OFDM-2015]-[@pilot-ce-pilot-freq-domain]. Blind channel estimation techniques are spectrally efficient because they do not require any overhead to estimate the CSI, nevertheless, such techniques have not yet been adopted in practical OFDM systems. Conversely, pilot-based CSI estimation is preferred for practical systems, because typically it is more robust and less complex. In pilot-based CSI estimation, the pilot symbols are embedded within the subcarriers of the transmitted OFDM signal in time and frequency domain; hence, the pilots form a two dimensional (2-D) grid [@LTE-A]. The channel response at the pilot symbols can be obtained using the least-squares (LS) frequency domain estimation, and the channel parameters at other subcarriers can be obtained using various interpolation techniques [@Rayleigh-Ricean-Interpolation-TCOM2008]. Optimal interpolation requires a 2-D Wiener filter that exploits the time and frequency correlation of the channel, however, it is substantially complex to implement [@Interpolation-TCOM-2010], [@Wiener]. The complexity can be reduced by decomposing the 2-D interpolation process into two cascaded 1-D processes, and then, using less computationally-involved interpolation schemes [@Adaptive-Equalization-IEEE-Broadcasting-2008], [@Comp-Pilot-VTC2007]. Low complexity interpolation, however, is usually accompanied by error rate performance degradation [@Comp-Pilot-VTC2007]. It is also worth noting that most practical OFDM-based systems utilize a fixed grid pattern structure [@LTE-A].
Once the channel parameters are obtained for all subcarriers, the received samples at the output of the fast Fourier transform (FFT) are equalized to compensate for the channel fading. Fortunately, the equalization for OFDM is performed in the frequency domain using single-tap equalizers. The equalizer output samples, which are denoted as the decision variables, will be applied to a maximum likelihood detector (MLD) to regenerate the information symbols.
In addition to the direct approach, several techniques have been proposed in the literature to estimate the CSI or detect the data symbols indirectly, by exploiting the correlation among the channel coefficients. For example, the per-survivor processing (PSP) approach has been widely used to approximate the maximum likelihood sequence estimator (MLSE) for coded and uncoded sequences [@PSP-Raheli], [@PSP-Zhu], [@Rev-1]. The PSP utilizes the Viterbi algorithm (VA) to recursively estimate the CSI without interpolation using the least mean squares (LMS) algorithm. Although the PSP provides superior performance when the channel is flat over the entire sequence, its performance degrades severely if this condition is not satisfied, even when the LMS step size is adaptive [@PSP-Zhu]. Multiple symbol differential detection (MSDD) can be also used for sequence estimation without explicit channel estimation. In such systems, the information is embedded in the phase difference between adjacent symbols, and hence, differential encoding is needed. Although differential detection is only $3$ dB worse than coherent detection in flat fading channels, its performance may deteriorate significantly in frequency-selective channels [@Divsalar], [@Diff-Xhang]. Consequently, Wu and Kam [@Wu; @2010] proposed a generalized likelihood ratio test (GLRT) receiver whose performance without CSI is comparable to the coherent detector. Although the GLRT receiver is more robust than differential detectors in frequency-selective channels, its performance is significantly worse than coherent detectors.
The signal at channel output is estimated with a minimum mean square error (MMSE) estimator from the knowledge of the received signal and the second order statistics of the channel and noay provide BER that is about $1$ dB from the ML coherent detector in flat fading channels but at the expense of a large number of pilots. Decision-directed techniques can also be used to avoid conventional channel estimation. For example, the authors in [@Saci-Tcom] proposed a hybrid frame structure that enables blind decision-directed channel estimation. Although the proposed system manages to offer reliable channel estimates and BER in various channel conditions, the system structure follows the typical coherent detector design where equalization and symbol detection are required.
Motivation and Key Contributions
--------------------------------
Unlike conventional OFDM detectors, this work presents a new detector to regenerate the information symbols directly from the received samples at the FFT output, which is denoted as the direct data detector ($D^{3})$. By using the $D^{3}$, there is no need to perform channel estimation, interpolation, equalization, or symbol decision operations. The $D^{3}$ exploits the fact that channel coefficients over adjacent subcarriers are highly correlated and approximately equal. Consequently, the $D^{3}$ is derived by minimizing the difference between channel coefficients of adjacent subcarriers. The main limitation of the $D^{3}$ is that it suffers from a phase ambiguity problem, which can be solved using pilot symbols, which are part of a transmission frame in most practical standards [@WiMax], [@LTE-A]. To the best of the authors’ knowledge, there is no work reported in the published literature that uses the proposed principle.
The $D^{3}$ performance is evaluated in terms of complexity, computational power, and bit error rate (BER), where analytic expressions are derived for several channel models and system configurations. The $D^{3}$ BER is compared to other widely used detectors such as the maximum likelihood (ML) coherent detector [@Proakis-Book-2001] with perfect and imperfect CSI, multiple symbol differential detector (MSDD) [@Divsalar], the ML sequence detector (MLSD) with no CSI [@Wu; @2010], and the per-survivor processing detector [@PSP-Raheli]. The obtained results show that the $D^{3}$ is more robust than all the other considered detectors in various cases of interest, particularly in frequency-selective channels at moderate and high SNRs. Moreover, the computational power comparison shows that the $D^{3}$ requires less than $35\%$ of the computational power required by the ML coherent detector.
Paper Organization and Notations
--------------------------------
The rest of this paper is organized as follows. The OFDM system and channel models are described in Section \[sec:Signal-and-Channel\]. The proposed $D^{3}$ is presented in Section \[sec:Proposed-System-Model\], and the efficient implementation of the $D^{3}$ is explored in Section \[sec:Efficient-Implementation-of-D3\]. The system error probability performance analysis is presented in Section \[sec:System-Performance-Analysis\]. Complexity analysis of the conventional pilot based OFDM and the $D^{3}$ are given in Section \[sec:Complexity-Analysis\]. Numerical results are discussed in Section \[sec:Numerical-Results\], and finally, the conclusion is drawn in Section \[sec:Conclusion\].
In what follows, unless otherwise specified, uppercase boldface and blackboard letters such as $\mathbf{H}$ and $\mathbb{H}$, will denote $N\times N$ matrices, whereas lowercase boldface letters such as $\mathbf{x}$ will denote row or column vectors with $N$ elements. Uppercase, lowercase, or bold letters with a tilde such as $\tilde{d}$ will denote trial values, and symbols with a hat, such as $\hat{\mathbf{x}}$, will denote the estimate of $\mathbf{x}$. Letters with apostrophe such as $\acute{v}$ are used to denote the next value, i.e., $\acute{v}\triangleq v+1$. Furthermore, $\mathrm{E}\left[\cdot\right]$ denotes the expectation operation.
Signal and Channel Models \[sec:Signal-and-Channel\]
====================================================
Consider an OFDM system with $N$ subcarriers modulated by a sequence of $N$ complex data symbols $\mathbf{d}=[d_{0}$, $d_{1}$, $....$, $d_{N-1}]^{T}$. The data symbols are selected uniformly from a general constellation such as $M$-ary phase shift keying (MPSK) or quadrature amplitude modulation (QAM). In conventional pilot-aided OFDM systems [@IEEE-AC], $N_{P}$ of the subcarriers are allocated for pilot symbols, which can be used for channel estimation and synchronization purposes. The modulation process in OFDM can be implemented efficiently using an $N$-point inverse FFT (IFFT) algorithm, where its output during the $\ell$th OFDM block can be written as $\mathbf{x(\ell)=F}^{H}\mathbf{d(\ell)}$ where $\mathbf{F}$ is the normalized $N\times N$ FFT matrix, and hence, $\mathbf{F}^{H}$ is the IFFT matrix. To simplify the notation, the block index $\ell$ is dropped for the remaining parts of the paper unless it is necessary to include it. Then, a CP of length $N_{\mathrm{CP}}$ samples, no less than the channel maximum delay spread ($\mathcal{D}_{\mathrm{h}}$), is appended to compose the OFDM symbol with a total length $N_{\mathrm{t}}=N+N_{\mathrm{CP}}$ samples and duration of $T_{\mathrm{t}}$s.
At the receiver front-end, the received signal is down-converted to baseband and sampled at a rate $T_{\mathrm{s}}=T_{\mathrm{t}}/N_{\mathrm{t}}$. In this work, the channel is assumed to be composed of $\mathcal{D}_{\mathrm{h}}+1$ independent multipath components each of which has a gain $h_{m}\sim\mathcal{CN}\left(0,2\sigma_{h_{m}}^{2}\right)$ and delay $m\times T_{\mathrm{s}}$, where $m\in\{0$, $1$,$...$, $\mathcal{D}_{\mathrm{h}}\}$. A quasi-static channel is assumed throughout this work, and thus, the channel taps are considered constant over one OFDM symbol, but they may change over two consecutive symbols. Therefore, the received sequence after dropping the CP samples and applying the FFT can be expressed as, $$\mathbf{r}=\mathbf{Hd+w}\label{eq:rx_sig_FD}$$ where $\left\{ \mathbf{r,w}\right\} \in\mathbb{C}^{N\times1}$, $w_{v}\sim\mathcal{CN}\left(0\text{, }2\sigma_{w}^{2}\right)$ is the additive white Gaussian noise (AWGN) vector and $\mathbf{\mathbf{H}}$ denotes the channel frequency response (CFR) $$\mathbf{\mathbf{H}}=\text{diag}\left\{ \left[H_{0}\text{, }H_{1}\text{,}\ldots\text{, }H_{N-1}\right]\right\} .$$ By noting that $\mathbf{r|}_{\mathbf{H,d}}\sim\mathcal{CN}\left(\mathbf{Hd}\text{, }2\sigma_{w}^{2}\mathbf{I}_{N}\right)$ where $\mathbf{I}_{N}$ is an $N\times N$ identity matrix, then it is straightforward to show that the MLD can be expressed as $$\mathbf{\hat{d}}=\arg\text{ }\min_{\tilde{\mathbf{d}}}\text{ }\left\Vert \mathbf{r-H}\tilde{\mathbf{d}}\right\Vert ^{2}\label{E-MLD-01}$$ where $\left\Vert \mathbf{\cdot}\right\Vert $ denotes the Euclidean norm, and $\tilde{\mathbf{d}}=\left[\tilde{d}_{0}\text{, }\tilde{d}_{1}\text{,}\ldots\text{, }\tilde{d}_{N1}\right]^{T}$ denotes the trial values of $\mathbf{d}$. As can be noted from (\[E-MLD-01\]), the MLD requires the knowledge of $\mathbf{\mathbf{H}}$. Moreover, because (\[E-MLD-01\]) describes the detection of more than one symbol, it is typically denoted as maximum likelihood sequence detector (MLSD). If the elements of $\mathbf{d}$ are independent, the MLSD can be replaced by a symbol-by-symbol MLD $$\hat{d}_{v}=\arg\text{ }\min_{\tilde{d}_{v}}\text{ }\left\vert r_{v}\mathbf{-}H_{v}\tilde{d}_{v}\right\vert ^{2}\text{.}\label{E-MLD-02}$$ Since perfect knowledge of $\mathbf{H}$ is infeasible, an estimated version of $\mathbf{H}$, denoted as $\hat{\mathbf{H}}$, can be used in (\[E-MLD-01\]) and (\[E-MLD-02\]) instead of $\mathbf{H}$**.** Another possible approach to implement the detector is to equalize $\mathbf{r}$, and then use a symbol-by-symbol MLD. Therefore, the equalized received sequence can be expressed as, $$\check{\mathbf{r}}=\left[\hat{\mathbf{H}}^{H}\hat{\mathbf{H}}\right]^{-1}\hat{\mathbf{H}}^{H}\mathbf{r}$$ and $$\hat{d}_{v}=\arg\min_{\tilde{d}_{v}}\left\vert \check{r}_{v}-\tilde{d}_{v}\right\vert ^{2}\text{, }\forall v\text{.}$$
It is interesting to note that solving (\[E-MLD-01\]) does not necessarily require the explicit knowledge of $\mathbf{H}$ under some special circumstances. For example, Wu and Kam [@Wu; @2010] noticed that in flat fading channels, i.e., $H_{v}=H$ $\forall v$, it is possible to detect the data symbols using the following MLSD, $$\mathbf{\hat{d}}=\arg\text{ }\max_{\tilde{\mathbf{d}}}\text{ }\frac{\left\vert \tilde{\mathbf{d}}^{H}\mathbf{r}\right\vert ^{2}}{\parallel\tilde{\mathbf{d}}\Vert}.\label{E-Wu}$$ Although the detector described in (\[E-Wu\]) is efficient in the sense that it does not require the knowledge of $\mathbf{H}$, its BER is very sensitive to the channel variations.
Proposed $D^{3}$ System Model\[sec:Proposed-System-Model\]
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One of the distinctive features of OFDM is that its channel coefficients over adjacent subcarriers in the frequency domain are highly correlated and approximately equal. The correlation coefficient between two adjacent subcarriers can be defined as $$\begin{aligned}
\varrho_{f} & \triangleq & \mathrm{E}\left[H_{v}H_{\acute{v}}^{\ast}\right]\nonumber \\
& = & \mathrm{E}\left[\sum_{n=0}^{\mathcal{D}_{\mathrm{h}}}h_{n}e^{-j2\pi\frac{nv}{N}}\sum_{m=0}^{\mathcal{D}_{\mathrm{h}}}h_{m}^{\ast}e^{j2\pi\frac{m\acute{v}}{N}}\right]=\sum_{m=0}^{\mathcal{D}_{\mathrm{h}}}\sigma_{h_{m}}^{2}e^{j2\pi\frac{m}{N}}\label{eq:rho-f}\end{aligned}$$ where $\sigma_{h_{m}}^{2}=\mathrm{E}\left[\left\vert h_{m}\right\vert ^{2}\right]$. The difference between two adjacent channel coefficients is $$\Delta_{f}=\mathrm{E}\left[H_{v}-H_{\acute{v}}\right]=\mathrm{E}\left[\sum_{m=0}^{\mathcal{D}_{\mathrm{h}}}h_{n}e^{-j2\pi\frac{mv}{N}}\left(1-e^{-j2\pi\frac{m}{N}}\right)\right]$$ For large values of $N$, it is straightforward to show that $\varrho_{f}\rightarrow1$ and $\Delta_{f}\rightarrow0$. Similar to the frequency domain, the time domain correlation defined according to the Jakes’ model can be computed as [@Jakes-Model], $$\varrho_{t}=\mathrm{E}\left[H_{v}^{\ell}\left(H_{v}^{\acute{\ell}}\right)^{\ast}\right]=J_{0}\left(2\pi f_{d}T_{\mathrm{s}}\right)\label{eq:rho-t}$$ where $J_{0}\left(\cdot\right)$ is the Bessel function of the first kind and $0$ order, $f_{d}$ is the maximum Doppler frequency. For large values of $N$, $2\pi f_{d}T_{\mathrm{s}}\ll1$, and hence $J_{0}\left(2\pi f_{d}T_{\mathrm{s}}\right)\approx1$, and thus $\varrho_{t}\approx1$. Using the same argument, the difference in the time domain $\Delta_{t}\triangleq\mathrm{E}\left[H_{v}^{\ell}-H_{v}^{\acute{\ell}}\right]\approx0$. Although the proposed system can be applied in the time domain, frequency domain, or both, the focus of this work is the frequency domain.
Based on the aforementioned properties of OFDM, a simple approach to extract the information symbols from the received sequence $\mathbf{r}$ can be designed by minimizing the difference of the channel coefficients between adjacent subcarriers, which can be expressed as $$\mathbf{\hat{d}}=\arg\min_{\tilde{\mathbf{d}}}\sum_{v=0}^{N-2}\left\vert \frac{r_{v}}{\tilde{d}_{v}}-\frac{r_{\acute{v}}}{\tilde{d}_{\acute{v}}}\right\vert ^{2}.\label{E-DDD-00}$$ As can be noted from (\[E-DDD-00\]), the estimated data sequence $\mathbf{\hat{d}}$ can be obtained without the knowledge of $\mathbf{H}$. Moreover, there is no requirement for the channel coefficients over the considered sequence to be equal, and hence, the $D^{3}$ should perform fairly well even in frequency-selective fading channels. Nevertheless, it can be noted that (\[E-DDD-00\]) does not have a unique solution because $\mathbf{d}$ and $-\mathbf{d}$ can minimize (\[E-DDD-00\]). To resolve the phase ambiguity problem, one or more pilot symbols can be used as a part of the sequence $\mathbf{d}$**.** In such scenarios, the performance of the $D^{3}$ will be affected indirectly by the frequency selectivity of the channel because the capability of the pilot to resolve the phase ambiguity depends on its fading coefficient. Another advantage of using pilot symbols is that it will not be necessary to detect the $N$ symbols simultaneously. Instead, it will be sufficient to detect $\mathcal{K}$ symbols at a time, which can be exploited to simplify the system design and analysis.
Using the same approach of the frequency domain, the $D^{3}$ can be designed to work in the time domain as well by minimizing the channel coefficients over two consecutive subcarriers, i.e., two subcarriers with the same index over two consecutive OFDM symbols, which is also applicable to single carrier systems. It can be also designed to work in both time and frequency domains, where the detector can be described as $$\mathbf{\hat{D}}_{\mathcal{L}\text{,}\mathcal{K}}\mathbf{=}\arg\min_{\mathbf{\tilde{\mathbf{D}}}_{\mathcal{L}\text{,}\mathcal{K}}}\text{ }J\left(\tilde{\mathbf{D}}_{\mathcal{L}\text{,}\mathcal{K}}\right)\label{eq:opt-D}$$ where $\mathbf{D}_{\mathcal{L}\text{,}\mathcal{K}}$ is an $\mathcal{L}\times\mathcal{K}$ data matrix, $\mathcal{L}$ and $\mathcal{K}$ are the time and frequency detection window size, and the objective function $J\left(\tilde{\mathbf{D}}\right)$ is given by $$J\left(\tilde{\mathbf{D}}_{\mathcal{L}\text{,}\mathcal{K}}\right)=\sum_{\ell=0}^{\mathcal{L}-1}\sum_{v=0}^{\mathcal{K}-2}\left\vert \frac{r_{v}^{\ell}}{\tilde{d}_{v}^{\ell}}-\frac{r_{\acute{v}}^{\ell}}{\tilde{d}_{\acute{v}}^{\ell}}\right\vert ^{2}+\left\vert \frac{r_{v}^{\ell}}{\tilde{d}_{v}^{\ell}}-\frac{r_{v}^{\acute{\ell}}}{\tilde{d}_{v}^{\acute{\ell}}}\right\vert ^{2}\text{.}\label{eq:objective-function}$$ For example, if the detection window size is chosen to be the LTE resource block, then, $\mathcal{L}=14$ and $\mathcal{K=}12$. Moreover, the system presented in can be extended to the multi-branch receiver scenarios, single-input multiple-output (SIMO) as, $$\begin{aligned}
\hat{\mathbf{D}} & =\arg\min_{\tilde{\mathbf{d}}}\sum_{n=1}^{\mathcal{N}}\sum_{\ell=0}^{\mathcal{L}-1}\sum_{v=0}^{\mathcal{K}-2}\left\vert \frac{r_{v}^{\ell,n}}{\tilde{d}_{v}}-\frac{r_{\acute{v}}^{\ell,n}}{\tilde{d}_{\acute{v}}^{\ell}}\right\vert ^{2}+\left\vert \frac{r_{v}^{\ell,n}}{\tilde{d}_{v}^{\ell}}-\frac{r_{v}^{\acute{\ell},n}}{\tilde{d}_{v}^{\acute{\ell}}}\right\vert ^{2}\end{aligned}$$ where $\mathcal{N}$ is the number of receiving antennas.
Efficient Implementation of $D^{3}$\[sec:Efficient-Implementation-of-D3\]
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It can be noted from and that solving for $\hat{\mathbf{D}}$, given that $N_{P}$ pilot symbols are used, requires an $M^{\mathcal{K}\mathcal{L-}N_{P}}$ trials if brute force search is adopted, which is prohibitively complex, and thus, reducing the computational complexity is crucial.
![Example of a 1-D segmentation over the frequency domain for an LTE-A resource block.\[fig:2D-to-1D\] ](graphics/fig_01_digram_1D_sigmentation_modified)
\[subsec:The-Viterbi-Algorithm\]The Viterbi Algorithm (VA)
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By noting that the expression in (\[E-DDD-00\]) corresponds to the sum of correlated terms, which can be modeled as a first-order Markov process, then MLSD techniques such as the VA can be used to implement the $D^{3}$ efficiently. For example, the trellis diagram of the VA with binary phase shift keying (BPSK) is shown in Fig. \[fig:Viterbi-D3\], and can be implemented as follows:
1. Initialize the path metrics $\left\{ \Gamma_{0}^{U},\acute{\Gamma}_{0}^{U},\Gamma_{0}^{L},\acute{\Gamma}_{0}^{L}\right\} =0$, where $U$ and $L$ denote the upper and lower branches, respectively. Since BPSK is used, the number of states is $2$.
2. Initialize the counter, $c=0$.
3. Compute the branch metric $J_{m,n}^{c}=\left\vert \frac{rc}{m}-\frac{r_{\acute{c}}}{n}\right\vert ^{2}$, where $m$ is current symbol index, $m=0\rightarrow\tilde{d}=-1$, and $m=1\rightarrow\tilde{d}=1$, and $n$ is the next symbol index using the same mapping as $m$.
4. Compute the path metrics using the following rules, $$\begin{array}{ccc}
\Gamma_{\acute{c}}^{U}=\min\left[\Gamma_{c}^{U}\text{, }\acute{\Gamma}_{c}^{U}\right]+J_{00}^{c} & & \Gamma_{\acute{c}}^{L}=\min\left[\Gamma_{c}^{L}\text{, }\acute{\Gamma}_{c}^{L}\right]+J_{01}^{c}\\
\acute{\Gamma}_{\acute{c}}^{U}=\min\left[\Gamma_{c}^{U}\text{, }\acute{\Gamma}_{c}^{U}\right]+J_{10}^{c} & & \acute{\Gamma}_{\acute{c}}^{L}=\min\left[\Gamma_{c}^{L}\text{, }\acute{\Gamma}_{c}^{L}\right]+J_{11}^{c}
\end{array}$$
5. Track the surviving paths, $2$ paths in the case of BPSK.
6. Increase the counter, $c=c+1$.
7. if $c=\mathcal{K}$, the algorithm ends. Otherwise, go to step 3.
![Trellis diagram of the $D^{3}$ detector for BPSK.\[fig:Viterbi-D3\]](graphics/fig_02_digram_trellis)
It is worth mentioning that placing a pilot symbol at the edge of a segment terminates the trellis. To simplify the discussion, assume that the pilot value is $-1$, and thus we compute only $J_{0,0}$ and $J_{1,0}$. Consequently, long data sequences can be divided into smaller segments bounded by pilots, which can reduce the delay by performing the detection over the sub-segments in parallel without sacrificing the error rate performance.
1. 2.
System Design with an Error Control Coding
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Forward error correction (FEC) coding can be integrated with the $D^{3}$ in two ways, based on the decoding process, i.e., hard or soft decision decoding. For the hard decision decoding, the integration of FEC coding is straightforward where the output of the $D^{3}$ is applied directly to the hard decision decoder (HDD).
For the soft decision decoding, we can exploit the coded data to enhance the performance of the $D^{3}$, and then use the $D^{3}$ output to estimate the channel coefficients in a decision-directed manner. The $D^{3}$ with coded data can be expressed as $$\mathbf{\hat{d}}=\arg\min_{\tilde{\mathbf{u}}\in\mathbb{U}}\sum_{v=0}^{N-2}\left\vert \frac{r_{v}}{\tilde{u}_{v}}-\frac{r_{\acute{v}}}{\tilde{u}_{\acute{v}}}\right\vert ^{2}\label{E-D3-Joint}$$ where $\mathbb{U}$ is the set of all codewords modulated using the same modulation used at the transmitter. Therefore, the trial sequences $\tilde{\mathbf{u}}$ are restricted to particular sequences. For the case of convolutional codes, the detection and decoding processes can be integrated smoothly since both of them are using the VA. Such an approach can be adopted with linear block codes as well because trellis-based decoding can be also applied to block codes [@Trellis-Block].
Error Rate Analysis of the $D^{3}$\[sec:System-Performance-Analysis\]
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The system BER analysis is presented for several cases according to the pilot and data arrangements. For simplicity, each case is discussed in separate subsections. To make the analysis tractable, we consider BPSK modulation in the analysis while the BER of higher-order modulations is obtained via Monte Carlo simulations.
![Double-sided pilot segment. \[fig:Double-sided-pilot\]](graphics/fig_03_single_sided_diagram){width="0.48\columnwidth"}
![Double-sided pilot segment. \[fig:Double-sided-pilot\]](graphics/fig_04_double_sided_diagram){width="0.48\columnwidth"}
Single-Sided Pilot \[subsec:Single-Sided-Pilot\]
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To detect a data segment that contains $\mathcal{K}$ symbols, at least one pilot symbol should be part of the segment in order to resolve the phase ambiguity problem. Consequently, the analysis in this subsection considers the case where there is only one pilot within the $\mathcal{K}$ symbols, as shown in Fig. \[fig:Single-sided-pilot\]. Given that the FFT output vector $\mathbf{r}=\left[r_{0}\text{, }r_{1}\text{,}\ldots,r_{N-1}\right]$ is divided into $L$ segments each of which consists of $\mathcal{K}$ symbols, including the pilot symbol, then the frequency domain $D^{3}$ detector can be written as, $$\hat{\mathbf{d}}_{l}=\arg\min_{\tilde{\mathbf{d}}}\sum_{v=l}^{\mathcal{K-}2+l}\left\vert \frac{r_{v}}{\tilde{d_{v}}}-\frac{r_{\acute{v}}}{\tilde{d}_{\acute{v}}}\right\vert ^{2}\,\,\,\,\mathcal{K}\in\left\{ 2,3,\dots,N-1\right\} \label{eq:d_hat}$$ where $l$ denotes the index of the first subcarrier in the segment, and without loss of generality, we consider that $l=0$. Therefore, by expanding we obtain, $$\begin{gathered}
\hat{\mathbf{d}}_{0}=\arg\min_{\tilde{\mathbf{d}}}\left(\frac{r_{0}}{\tilde{d_{0}}}-\frac{r_{1}}{\tilde{d}_{1}}\right)\left(\frac{r_{0}}{\tilde{d_{0}}}-\frac{r_{1}}{\tilde{d}_{1}}\right)^{\ast}+\cdots+\left(\frac{r_{\mathcal{K}-2}}{\tilde{d}_{\mathcal{K}-2}}-\frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}}\right)\left(\frac{r_{\mathcal{K}-2}}{\tilde{d}_{\mathcal{K}-2}}-\frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}}\right)^{\ast}\label{eq:analysis-expansion-01}\end{gathered}$$ which can be simplified to, $$\begin{gathered}
\hat{\mathbf{d}}_{0}=\arg\min_{\tilde{\mathbf{d}}}\left\vert \frac{r_{0}}{\tilde{d_{0}}}\right\vert ^{2}+\left\vert \frac{r_{1}}{\tilde{d_{1}}}\right\vert ^{2}+\dots+\left\vert \frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}}\right\vert ^{2}-\frac{r_{0}}{\tilde{d_{0}}}\frac{r_{1}}{\tilde{d}_{1}^{\ast}}-\frac{r_{0}}{\tilde{d_{0}^{\ast}}}\frac{r_{1}}{\tilde{d}_{1}}-\cdots\\
-\frac{r_{\mathcal{K}-2}}{\tilde{d}_{\mathcal{K}-2}}\frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}^{\ast}}-\frac{r_{\mathcal{K}-2}}{\tilde{d}_{\mathcal{K}-2}^{\ast}}\frac{r_{\mathcal{K}-1}}{\tilde{d}_{\mathcal{K}-1}}.\label{eq:analysis-expansion-02}\end{gathered}$$ For BPSK, $\left\vert r_{v}/\tilde{d_{v}}\right\vert ^{2}=\left\vert r_{v}\right\vert ^{2}$, which is a constant term with respect to the maximization process in , and thus, they can be dropped. Therefore, the detector is reduced to $$\hat{\mathbf{d}}_{0}=\arg\max_{\tilde{\mathbf{d}_{0}}}\sum_{v=0}^{\mathcal{K-}2}\Re\left\{ \frac{r_{v}r_{\acute{v}}}{\tilde{d_{v}}\tilde{d}_{\acute{v}}}\right\} .$$ Given that the pilot symbol is placed in the first subcarrier and noting that $d_{v}\in\left\{ -1,1\right\} $, then $\tilde{d_{0}}=1$ and $\hat{\mathbf{d}}_{0}$ can be written as $$\hat{\mathbf{d}}_{0}=\arg\max_{\tilde{d}_{0}\notin\tilde{\mathbf{d}}_{0}}\frac{1}{\tilde{d_{1}}}\Re\left\{ r_{0}r_{1}\right\} +\sum_{v=1}^{\mathcal{K-}2}\frac{1}{\tilde{d_{v}}\tilde{d}_{\acute{v}}}\Re\left\{ r_{v}r_{\acute{v}}\right\} .\label{eq:d_hat_single_sided}$$ The sequence error probability ($P_{S}$), conditioned on the channel frequency response over the $\mathcal{K}$ symbols ($\mathbf{H}_{0})$ and the transmitted data sequence $\mathbf{d}_{0}$ can be defined as, $$P_{S}|_{\mathbf{H}_{0},\mathbf{d}_{0}}\triangleq\left.\Pr\left(\hat{\mathbf{d}_{0}}\neq\mathbf{d}_{0}\right)\right\vert _{\mathbf{H}_{0},\mathbf{d}_{0}}\label{eq:SEP-definition}$$ which can be also written in terms of the conditional probability of correct detection $P_{C}$ as, $$P_{C}|_{\mathbf{H}_{0},\mathbf{d}_{0}}=1-\Pr\left(\hat{\mathbf{d}_{0}}=\mathbf{d}_{0}\right)\mid_{\mathbf{H}_{0},\mathbf{d}_{0}}.\label{eq:SEP-Analysis-01}$$ Without loss of generality, we assume that $\mathbf{d}_{0}\mathbf{=}[1$, $1$,…$,1]\triangleq\mathbf{1}$ . Therefore, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\Pr\left(\sum_{v=0}^{\mathcal{K-}2}\Re\left\{ r_{v}r_{\acute{v}}\right\} =\max_{\tilde{\mathbf{d}_{0}}}\left\{ \sum_{v=0}^{\mathcal{K-}2}\frac{\Re\left\{ r_{v}r_{\acute{v}}\right\} }{\tilde{d_{v}}\tilde{d}_{\acute{v}}}\right\} \right).\label{eq:probability-correct-sequence}$$ Since $\mathbf{d}_{0}$ has $\mathcal{K-}1$ data symbols, then there are $2^{\mathcal{K-}1}$ trial sequences, $\tilde{\mathbf{d}}_{0}^{(0)}$, $\tilde{\mathbf{d}}_{0}^{(1)}$,$\ldots$, $\tilde{\mathbf{d}}_{0}^{(\psi)}$, where $\psi=2^{\mathcal{K-}1}-1$, and $\tilde{\mathbf{d}}_{0}^{(\psi)}\mathbf{=}[1$, $1$,…$,1]$ . The first symbol in every sequence is set to $1$, which is the pilot symbol. By defining $\sum_{v=0}^{\mathcal{K-}2}\frac{\Re\left\{ r_{v}r_{\acute{v}}\right\} }{\tilde{d_{v}}\tilde{d}_{\acute{v}}}\triangleq A_{n}$, where $\tilde{d_{v}}\tilde{d}_{\acute{v}}\in\tilde{\mathbf{d}}_{0}^{(n)}$, then (\[eq:probability-correct-sequence\]) can be written as, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\Pr\left(A_{\psi}>A_{\psi-1},A_{\psi-2},\ldots,A_{0}\right)\label{E-PC-00}$$ which, as depicted in Appendix I, can be simplified to $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\prod\limits _{v=0}^{\mathcal{K-}2}\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} >0\right).\label{eq:pc-expansion-2}$$
To evaluate $P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}$ given in , it is necessary to compute $\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} >0\right)$, which can be written as $$\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} >0\right)=\Pr\left(\underbrace{r_{v}^{I}r_{\acute{v}}^{I}-r_{v}^{Q}r_{\acute{v}}^{Q}}_{r_{v,\acute{v}}^{\mathrm{SP}}}>0\right).\label{E-rSP}$$ Given that $\mathbf{d}_{0}\mathbf{=}[1$, $1$,…$,1]$ , then $r_{v}^{I}=\Re\left\{ r_{v}\right\} =H_{v}^{I}+w_{v}^{I}$ and $r_{v}^{Q}=\Im\left\{ r_{v}\right\} =H_{v}^{Q}+w_{v}^{Q}$. Therefore, $r_{v}^{I},$ $r_{v}^{Q}$, $r_{\acute{v}}^{I}$ and $r_{\acute{v}}^{Q}$ are independent conditionally Gaussian random variables with averages $H_{v}^{I}$, $H_{v}^{Q}$, $H_{\acute{v}}^{I}$ and $H_{\acute{v}}^{Q}$, respectively, and the variance for all elements is $\sigma_{w}^{2}$. To derive the PDF of $r_{v,\acute{v}}^{\mathrm{SP}}$, the PDFs of $r_{v}^{I}r_{\acute{v}}^{I}$ and $r_{v}^{Q}r_{\acute{v}}^{Q}$ should be evaluated, where each of which corresponds to the product of two Gaussian random variables. Although the product of two Gaussian variables is not usually Gaussian, the limit of the moment-generating function of the product has Gaussian distribution. Therefore, the product of two variables $X\sim\mathcal{N}(\mu_{x},\sigma_{x}^{2})$ and $Y\sim\mathcal{N}(\mu_{y},\sigma_{y}^{2})$ tends to be $\mathcal{N}(\mu_{x}\mu_{y},\mu_{x}^{2}\sigma_{y}^{2}+\mu_{y}^{2}\sigma_{x}^{2})$ as the ratios $\mu_{x}/\sigma_{x}$ and $\mu_{y}/\sigma_{y}$ increase [@Product; @of; @2RV]. By noting that in in (\[E-rSP\]) $\mathrm{E}\left[r_{y}^{x}\right]=H_{y}^{x}$, $x\in\left\{ I,Q\right\} $ and $y\in\left\{ v,\acute{v}\right\} $ and $\sigma_{r_{y}^{x}}=\sigma_{w}$, thus $\mathrm{E}\left[r_{y}^{x}\right]/\sigma_{r_{y}^{x}}\gg1$ $\forall\left\{ x,y\right\} $. Moreover, because the PDF of the sum or difference of two Gaussian random variables is also Gaussian, then, $r_{v,\acute{v}}^{\mathrm{SP}}\sim\mathcal{N}\left(\bar{\mu}_{\mathrm{SP}},\bar{\sigma}_{\mathrm{SP}}^{2}\right)$ where $\bar{\mu}_{\mathrm{SP}}=H_{v}^{I}H_{\acute{v}}^{I}+H_{v}^{Q}H_{\acute{v}}^{Q}$ and $\bar{\sigma}_{\mathrm{SP}}^{2}=\sigma_{w}^{2}\left(\left\vert H_{v}\right\vert ^{2}+\left\vert H_{\acute{v}}\right\vert ^{2}+\sigma_{w}^{2}\right)$. Consequently, $$P_{C}|_{\mathbf{H}_{0},\mathbf{1}}=\prod_{v=0}^{\mathcal{K}-2}\Pr\left(r_{v,\acute{v}}^{\mathrm{SP}}>0\right)=\prod_{v=0}^{\mathcal{K}-2}\left[1-Q\left(\sqrt{\frac{2\bar{\mu}_{\mathrm{SP}}}{\bar{\sigma}_{\mathrm{SP}}^{2}}}\right)\right]$$ and $$P_{S}|_{\mathbf{H}_{0},\mathbf{1}}=1-\prod_{v=0}^{\mathcal{K}-2}\left[1-Q\left(\sqrt{\frac{2\bar{\mu}_{\mathrm{SP}}}{\bar{\sigma}_{\mathrm{SP}}^{2}}}\right)\right]\label{eq:SEP}$$ where $Q\left(x\right)\triangleq\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\exp\left(-\frac{t^{2}}{2}\right)dt$. Since $H_{v}^{I}$ and $H_{v}^{Q}$ are independent, then, the condition on $\mathbf{H}_{0}$ in can be removed by averaging $P_{S}$ over the PDF of $\mathbf{H}_{0}^{I}$ and $\mathbf{H}_{0}^{Q}$ as, $$\begin{gathered}
\mathrm{SEP}\mid_{\mathbf{d}=1}=\underbrace{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}}_{2\mathcal{K}\text{ fold}}\mathrm{SEP}\mid_{\mathbf{H}_{0},\mathbf{d}=1}f_{\mathbf{H}_{0}^{I}}\left(H_{0}^{I},H_{1}^{I},\dots,H_{\mathcal{K}-1}^{I}\right)\times\\
f_{\mathbf{H}_{0}^{Q}}\left(H_{0}^{Q},H_{1}^{Q},\dots,H_{\mathcal{K}-1}^{Q}\right)dH_{0}^{I}dH_{1}^{I}\dots dH_{\mathcal{K}-1}^{I}dH_{0}^{Q}dH_{1}^{Q}\dots dH_{\mathcal{K}-1}^{Q}\text{.}\label{eq:unconditional-SER}\end{gathered}$$ Because the random variables $H_{i}^{I}$ and $H_{i}^{Q}$ $\forall i$ in are real and Gaussian, their PDFs are multivariate Gaussian distributions [@Proakis-Book-2001], $$f_{\mathbf{X}}\left(X_{0},X_{1},\dots,X_{\mathcal{K}-1}\right)=\frac{\exp\left(-\frac{1}{2}(\mathbf{X}-\boldsymbol{\mu})^{\mathrm{T}}\boldsymbol{\Sigma}^{-1}(\mathbf{X}-\boldsymbol{\mu})\right)}{\sqrt{(2\pi)^{\mathcal{K}}|\boldsymbol{\Sigma}|}}\label{eq:multi-variate-gaussian}$$ where $\boldsymbol{\mu}$ is the mean vector, which is defined as, $$\boldsymbol{\mu}=\mathrm{E}\left[\mathbf{X}\right]=\left[\mathrm{E}\left[X_{1}\right],\mathrm{E}\left[X_{2}\right],\dots,\mathrm{E}\left[X_{\mathcal{K}-1}\right]\right]^{T}$$ and $\boldsymbol{\Sigma}$ is the covariance matrix, $\boldsymbol{\Sigma}=\mathrm{E}\left[\left(\mathbf{X}-\mu\right)\left(\mathbf{X}-\mu\right)^{T}\right].$
Due to the difficulty of evaluating $2\mathcal{K}$ integrals, we consider the special case of flat fading, which implies that $H_{v}=H_{\acute{v}}\triangleq H$ and $\left(H^{I}\right)^{2}+\left(H^{Q}\right)^{2}\triangleq\alpha^{2}$, where $\alpha$ is the channel fading envelope, $\alpha=\left\vert H\right\vert $. Therefore, the SEP expression in becomes, $$P_{S}|_{\alpha,\mathbf{1}}=1-\left[1-Q\left(\sqrt{\frac{\alpha^{2}}{\sigma_{w}^{2}\left(\alpha^{2}+\sigma_{w}^{2}\right)}}\right)\right]^{\mathcal{K}-1}.\label{eq:SEP-conditional-general}$$ Recalling the Binomial Theorem, we get $$\left(a+b\right)^{n}=\sum_{v=0}^{n}\binom{n}{v}a^{n-v}b^{v}\text{, }\binom{n}{v}\triangleq\frac{n!}{\left(n-v\right)!v!}.\label{eq:binomial-theorem}$$ Then the SEP formula in using the Binomial Theorem in can be written as, $$P_{S}|_{\alpha,\mathbf{1}}=1-\sum_{v=0}^{\mathcal{K}-1}\binom{\mathcal{K}-1}{v}\left(-1\right)^{v}\left[Q\left(\sqrt{\frac{\alpha^{2}}{\sigma_{w}^{2}\left(\alpha^{2}+\sigma_{w}^{2}\right)}}\right)\right]^{v}.\label{eq:sep_cond_higher_k}$$ The conditioning on $\alpha$ can be removed by averaging over the PDF of $\alpha$, which is Rayleigh. Therefore, $$f\left(\alpha\right)=\frac{\alpha}{\sigma_{H}^{2}}e^{-\frac{\alpha^{2}}{2\sigma_{H}^{2}}}.\label{eq:rayleigh-pdf}$$ And hence, $$P_{S}|_{\mathbf{1}}=\int_{0}^{\infty}P_{S}|_{\alpha,\mathbf{1}}\text{ }f\left(\alpha\right)d\alpha.\label{E-Averaging}$$ Because the expression in contains high order of $Q$-function $Q^{n}\left(x\right)$, evaluating the integral analytically becomes intractable for $\mathcal{K}>2$. For the special case of $\mathcal{K}=2$, $P_{S}$ can be evaluated by substituting (\[eq:sep\_cond\_higher\_k\]) and (\[eq:rayleigh-pdf\]) into (\[E-Averaging\]) and evaluating the integral yields the following simple expression, $$P_{S}|_{\mathbf{1}}=\frac{1}{2\left(\bar{\gamma}_{s}+1\right)}\text{, \ \ }\bar{\gamma}_{s}\triangleq\frac{\mathrm{E}\left[\left\vert d_{v}\right\vert ^{2}\right]\mathrm{E}\left[\left\vert H\right\vert ^{2}\right]}{2\sigma_{w}^{2}}\label{E-Pe_K2}$$ where $\bar{\gamma}_{s}$ is the average signal-to-noise ratio (SNR). Moreover, because all data sequences have an equal probability of error, then $P_{S}|_{\mathbf{1}}=P_{S}$, which also equivalent to the bit error rate (BER). It is interesting to note that (\[E-Pe\_K2\]) is similar to the BER of the differential binary phase shift keying (DBPSK) [@Proakis-Book-2001]. However, the two techniques are essentially different as $D^{3}$ does not require differential encoding, has no constraints on the shape of the signal constellation, and performs well even in frequency-selective fading channels.
To evaluate $P_{S}$ for $\mathcal{K}>2$, we use an approximation for $Q\left(x\right)$ in [@Q-Func-Approx-02], which is given by $$Q\left(x\right)\approx\frac{1}{\sqrt{2\pi\left(x^{2}+1\right)}}e^{-\frac{1}{2}x^{2}},\text{ }x\in\lbrack0,\infty).\label{eq:Q-func-Approx}$$ Therefore, by substituting into the conditional SEP and averaging over the Rayleigh PDF , the evaluation of the SEP becomes straightforward. For example, evaluating the integral for $\mathcal{K}=3$ gives, $$P_{S}|_{\mathbf{1}}=\frac{\zeta_{1}}{\pi}\mathit{\mathrm{Ei}}\left(1,\zeta_{1}+1\right)e^{\zeta_{1}+1}\text{, \ \ }\zeta_{1}\triangleq\frac{1}{2\bar{\gamma}_{s}}\left(\frac{1}{\bar{\gamma}_{s}}+1\right)$$ where $\mathrm{Ei}\left(x\right)$ is the exponential integral (EI), $\mathrm{Ei}\left(x\right)\triangleq-\int_{-x}^{\infty}\frac{e^{-t}}{t}dt$. Similarly, $P_{S}$ for $\mathcal{K}=7$ can be evaluated to, $$P_{S}|_{\mathbf{1}}=\frac{\zeta_{2}}{64\pi^{3}}\left[e^{\zeta+3}\left(2\zeta_{2}+6\right)^{2}\text{ }\mathit{\mathrm{Ei}}\left(1,\zeta_{2}+3\right)-4\left(\zeta_{2}+1\right)\right]\text{, \ }\zeta_{2}\triangleq\frac{1}{2\bar{\gamma}_{s}}\left(\frac{1}{4\bar{\gamma}_{s}}+1\right).$$
Although the SEP is a very useful indicator for the system error probability performance, the BER is actually more informative. For a sequence that contains $\mathcal{K}_{D}$ information bits, the BER can be expressed as $P_{B}=\frac{1}{\Lambda}P_{S}$, where $\Lambda$ denotes the average number of bit errors given a sequence error, which can be defined as $$\Lambda=\sum_{m=1}^{\mathcal{K}_{D}}m\Pr\left(m\right).$$ Because the SEP is independent of the transmitted data sequence, then, without loss of generality, we assume that the transmitted data sequence is $\mathbf{d}_{0}^{(0)}$. Therefore, $$\Lambda=\sum_{m=1}^{\mathcal{K}_{D}}m\Pr\left(\left\Vert \mathbf{\hat{d}}_{0}\right\Vert ^{2}=m\right)$$ where $\left\Vert \mathbf{\hat{d}}_{0}\right\Vert ^{2}$, in this case, corresponds to the Hamming weight of the detected sequence $\mathbf{\hat{d}}_{0}$, which can be expressed as $$\Pr\left(\left\Vert \mathbf{\hat{d}}_{0}\right\Vert ^{2}=m\right)=\Pr\left(\mathbf{d}_{0}^{(0)}\rightarrow\bigcup\limits _{i}\mathbf{d}_{0}^{(i)}\right)\text{, }\left\Vert \mathbf{d}_{0}^{(i)}\right\Vert ^{2}=m$$ where $\mathbf{d}_{0}^{(0)}\rightarrow\mathbf{d}_{0}^{(i)}$ denotes the pairwise error probability (PEP). By noting that $\Pr\left(\mathbf{d}_{0}^{(0)}\rightarrow\mathbf{d}_{0}^{(i)}\right)\neq\Pr\left(\mathbf{d}_{0}^{(0)}\rightarrow\mathbf{d}_{0}^{(j)}\right)$ $\forall i\neq j$, then deriving the PEP for all cases of interest is intractable. As an alternative, a simple approximation is derived.
For a sequence that consists of $\mathcal{K}_{D}$ information bits, the BER is bounded by $$\frac{1}{\mathcal{K}_{D}}P_{S}\leq P_{B}\leq P_{S}\text{.}\label{E-Bounds}$$ In practical systems, the number of bits in the detected sequence is generally not large, which implies that the upper and lower bounds in (\[E-Bounds\]) are relatively tight, and hence, the BER can be approximated as the middle point between the two bounds as, $$P_{B}\approx\frac{P_{S}}{0.5\left(1+\mathcal{K}_{D}\right)}.\label{E_PB}$$
The analysis of the general $1\times\mathcal{N}$ SIMO system is a straightforward extension of the single-input single-output (SISO) case. To simplify the analysis, we consider the flat channel case where the conditional SEP can be written as, $$P_{S}|_{\mathbf{\alpha}}=1-\left[1-Q\left(\sqrt{\frac{\sum_{i=1}^{\mathcal{N}}\alpha_{i}^{2}}{\sigma_{w}^{2}\left(\mathcal{N}\sigma_{w}^{2}+\sum_{i=1}^{\mathcal{N}}\alpha_{i}^{2}\right)}}\right)\right]^{\mathcal{K}-1}.$$ Given that all the receiving branches are independent, the fading envelopes will have Rayleigh distribution $\alpha_{i}\sim\mathcal{R}\left(2\sigma_{H}^{2}\right)$ $\forall i$, and thus, $\sum_{i=1}^{\mathcal{N}}\alpha_{i}^{2}\triangleq a$ will have Gamma distribution, $a\sim\mathcal{G}\left(\mathcal{N},2\sigma_{H}^{2}\right)$, $$f\left(a\right)=\left(2\sigma_{H}^{2}\right)^{\mathcal{N}}e^{-2\sigma_{H}^{2}a}\frac{_{a^{\mathcal{N}-1}}}{\Gamma\left(\mathcal{N}\right)}.$$ Therefore, the unconditional SEP can be evaluated as, $$P_{S}=\int_{0}^{\infty}P_{S}|_{\mathbf{\alpha}}\text{ }f_{A}\left(a\right)da.$$ For the special case of $\mathcal{N=}2$, $\mathcal{K}=2$, $P_{S}$ can be evaluated as, $$P_{S}=\frac{1}{2}+Q\left(\frac{\varkappa}{\sqrt{\bar{\gamma}_{s}}}\right)\left[2\bar{\gamma}_{s}\left(\frac{\bar{\gamma}_{s}}{\sqrt{2}}+2\right)-e^{\varkappa^{2}}\right]-\bar{\gamma}_{s}\frac{\varkappa}{\sqrt{2\pi}}$$ where $\varkappa\triangleq\sqrt{2+\bar{\gamma}_{s}}.$ Computing the closed-form formulas for other values of $\mathcal{N}$ and $\mathcal{K}$ can be evaluated following the same approach used in the SISO case.
Double-Sided Pilot \[subsec:Double-Sided-Pilot\]
------------------------------------------------
Embedding more pilots in the detection segment can improve the detector’s performance. Consequently, it worth investigating the effect of embedding more pilots in the SEP analysis. More specifically, we consider double-sided segment, $\tilde{d}_{0}=1$, $\tilde{d}_{\mathcal{K}-1}=1$, as illustrated in Fig. \[fig:Double-sided-pilot\]. In this case, the detector can be expressed as, $$\hat{\mathbf{d}_{0}}=\arg\max_{\tilde{\mathbf{d}}_{0}}\frac{\Re\left\{ r_{0}r_{1}\right\} }{\tilde{d}_{1}}+\frac{\Re\left\{ r_{\mathcal{K}-2}r_{\mathcal{K}-1}\right\} }{\tilde{d}_{\mathcal{K}-2}}+\sum_{v=1}^{\mathcal{K-}3}\frac{\Re\left\{ r_{v}r_{\acute{v}}\right\} }{\tilde{d_{v}}\tilde{d}_{\acute{v}}},\text{\thinspace\thinspace\thinspace\thinspace\ensuremath{\mathcal{K}\in\left\{ 3,4,\dots,N-1\right\} .}}\label{eq:d_hat_double_sided}$$ From the definition in , the probability of receiving the correct sequence can be derived based on the reduced number of trials as compared to . Therefore, $$\begin{gathered}
P_{C}|_{\mathbf{H}_{0},\mathbf{1}}=\Pr\Big(\left(\Re\left\{ r_{0}r_{1}\right\} +\Re\left\{ r_{\mathcal{K}-2}r_{\mathcal{K}-1}\right\} \right)\cap\\
\Re\left\{ r_{1}r_{2}\right\} \cap\Re\left\{ r_{2}r_{3}\right\} \cap\dots\cap\Re\left\{ r_{\mathcal{K}-4}r_{\mathcal{K}-3}\right\} >0\Big)\label{eq:pc-double-sided}\end{gathered}$$ which, similar to the single-sided case, can be written as, $$P_{C}|_{\mathbf{H}_{0},\mathbf{1}}=\Pr\left(\left[\prod_{v=0}^{\mathcal{K}-3}\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} \right)+\prod_{v=1}^{\mathcal{K}-2}\Pr\left(\Re\left\{ r_{v}r_{\acute{v}}\right\} \right)\right]>0\right).$$ Therefore, $$P_{S}|_{\mathbf{H}_{0},\mathbf{1}}=1-\left[1-Q\left(\sqrt{\frac{2\sqrt{2}\bar{\mu}_{\mathrm{SP}}}{\bar{\sigma}_{\mathrm{SP}}^{2}}}\right)\right]\times\prod_{v=1}^{\mathcal{K}-3}\left[1-Q\left(\sqrt{\frac{2\bar{\mu}_{\mathrm{SP}}}{\bar{\sigma}_{\mathrm{SP}}^{2}}}\right)\right].\label{eq:SEP-1}$$ For flat fading channels, the SEP expression in can be simplified by following the same procedure in Subsection \[subsec:Single-Sided-Pilot\], for the special case of $\mathcal{K}=3$, the SEP becomes, $$P_{S}=\left(\frac{\Upsilon}{2}-\sqrt{2}\right)\frac{1}{\Upsilon}\text{, \ }\Upsilon\triangleq\sqrt{8\bar{\gamma}_{s}+\sqrt{2}\left(4+\frac{1}{\bar{\gamma}_{s}}\right)}.$$ For $\mathcal{K}>3$, the approximation of $Q^{n}\left(x\right)$, as illustrated in Subsection \[subsec:Single-Sided-Pilot\], can be used in to average over the PDF in . For example, the case $\mathcal{K}=4$ can be evaluated as, $$P_{S}=\frac{1}{8\pi\bar{\gamma}_{s}}\left(\Omega_{1}-1\right)e^{\Omega_{1}}\mathit{\mathrm{Ei}}\left(1,\Omega_{1}\right)\text{, \ }\Omega_{1}\triangleq1+\frac{\sqrt{2}}{4\bar{\gamma}_{s}}\left(1+\frac{1}{4\bar{\gamma}_{s}}\right).$$ For $\mathcal{K}=6$, $$P_{S}=\frac{\Omega_{1}-1}{4\pi^{2}}\left[1-\left[\left(\Omega_{1}-1\right)e^{\Omega_{2}}+2\right]\mathit{\mathrm{Ei}}\left(1,\Omega_{2}\right)\right]\text{, \ }\Omega_{2}\triangleq2+\frac{\sqrt{2}}{\bar{\gamma}_{s}}\left(8+\frac{1}{32\bar{\gamma}_{s}}\right)$$ For the double-sided pilot, $P_{B}=P_{S}$ for the case of $\mathcal{K}=3$, while it can be computed using (\[E\_PB\]) for $\mathcal{K}>3$.
Complexity Analysis\[sec:Complexity-Analysis\]
==============================================
The computational complexity is evaluated as the total number of primitive operations needed to perform the detection. The operations that will be used are the number of real additions ($R_{A}$), real multiplications ($R_{M}$), and real divisions ($R_{D}$) required to produce the set of detected symbols $\hat{\mathbf{d}}$ for each technique. It worth noting that one complex multiplication ($C_{M}$) is equivalent to four $R_{M}$ and three $R_{A}$ operations, while one complex addition ($C_{A}$) requires two $R_{A}$. To simplify the analysis, we first assume that constant modulus (CM) constellations such as MPSK is used, then, we evaluate the complexity for higher-order modulation such as quadrature amplitude modulation (QAM) modulation.
Complexity of Conventional OFDM Detectors\[subsec:Complexity-of-Conventional\]
------------------------------------------------------------------------------
The complexity of the conventional OFDM receiver that consists of the following main steps with the corresponding computational complexities:
1. Channel estimation of the pilot symbols, which computes $\hat{H}_{k}$ at all pilot subcarriers. Assuming that the pilot symbol $d_{k}$ is selected from a CM constellation, then $\hat{H}_{k}=r_{k}d_{k}^{*}$ and hence, $N_{P}$ complex multiplications are required. Therefore, $R_{A}^{\left(1\right)}=4N_{P}$ and $R_{M}^{\left(1\right)}=4N_{P}$.
2. Interpolation, which is used to estimate the channel at the non-pilot subcarriers. The complexity of the interpolation process depends on the interpolation algorithm used. For comparison purposes, we assume that linear interpolation is used, which is the least complex interpolation algorithm. The linear interpolation requires one complex multiplication and two complex additions per interpolated sample. Therefore, the number of complex multiplications required is $N-N_{P}$ and the number of complex additions is $2\left(N-N_{P}\right)$. And hence, $R_{A}^{\left(2\right)}=7\left(N-N_{P}\right)$ and $R_{M}^{\left(2\right)}=4\left(N-N_{P}\right)$.
3. Equalization, a single-tap equalizer requires $N-N_{P}$ complex division to compute the decision variables $\check{r}_{k}=\frac{r_{k}}{\hat{H}_{k}}=r_{k}\frac{\hat{H}_{k}^{*}}{\left|\hat{H}_{k}^{*}\right|^{2}}$. Therefore, one complex division requires two complex multiplications and one real division. Therefore, $R_{A}^{\left(3\right)}=6\left(N-N_{P}\right)$, $R_{M}^{\left(3\right)}=8\left(N-N_{P}\right)$ and $R_{D}^{\left(3\right)}=\left(N-N_{P}\right)$.
4. Detection, assuming symbol-by-symbol minimum distance detection, the detector can be expressed as $\hat{d}_{k}=\arg\min_{\tilde{d}_{i}}J\left(\tilde{d}_{i}\right),\,\,\forall i\in\left\{ 0,1,\dots,M-1\right\} $ where $J\left(\tilde{d}_{i}\right)=\left|\check{r}_{k}-\tilde{d}_{i}\right|^{2}$ . Assuming CM modulation is used, expanding the cost function and dropping the constant terms we can write $J\left(\tilde{d}_{k}\right)=-\check{r}_{k}\tilde{d}_{k}^{*}-\check{r}_{k}^{*}\tilde{d}_{k}$. We can also drop the minus sign from the cost function, and thus, the objective becomes maximizing the cost function $\hat{d}_{k}=\arg\min_{\tilde{d}_{i}}J\left(\tilde{d}_{i}\right)$. Since the two terms are complex conjugate pair, then $-\check{r}_{k}\tilde{d}_{k}^{*}-\check{r}_{k}^{*}\tilde{d}_{k}=2\Re\left\{ \check{r}_{k}\tilde{d}_{k}^{*}\right\} $, and thus we can write the detected symbols as, $$\hat{d}_{k}=\arg\max_{\tilde{d}_{k}}\left(\Re\left\{ \check{r}_{k}\right\} \Re\left\{ \tilde{d}_{k}^{*}\right\} -\Im\left\{ \check{r}_{k}\right\} \Im\left\{ \tilde{d}_{k}^{*}\right\} \right)$$ Therefore, the number of real multiplications required for each information symbol is $2M$, and the number of additions is $M$. Therefore, $R_{A}^{\left(4\right)}=\left(N-N_{P}\right)M$ and $R_{M}^{\left(4\right)}=2\left(N-N_{P}\right)M$.
Finally, the total computational complexity per OFDM symbol can be obtained by adding the complexities of the individual steps $1\rightarrow4$, as:
$$\begin{aligned}
R_{A}^{CM} & ={\displaystyle \sum_{i=1}^{4}R_{A}^{\left(i\right)}=\left(13+M\right)N-\left(10+M\right)N_{P}}\\
R_{M}^{CM} & =\sum_{i=1}^{4}R_{M}^{\left(i\right)}=2N\left(6+M\right)-2N_{P}\left(4+M\right)\\
R_{D}^{CM} & =\sum_{i=1}^{4}R_{D}^{\left(i\right)}=N-N_{P}.\end{aligned}$$
Complexity of the $D^{3}$
-------------------------
The complexity of the $D^{3}$ based on the VA is mostly determined by the branch and path metrics calculation. The branch metrics can be computed as $$J_{m,n}^{c}=\frac{\left\vert r_{c}\right\vert ^{2}}{\left\vert \tilde{d}_{m}\right\vert ^{2}}-\frac{r_{c}r_{\acute{c}}^{\ast}}{\tilde{d}_{m}\tilde{d}_{n}^{\ast}}-\frac{r_{c}^{\ast}r_{\acute{c}}}{\tilde{d}_{m}^{\ast}\tilde{d}_{n}}+\frac{\left\vert r_{c}\right\vert ^{2}}{\left\vert \tilde{d}_{n}\right\vert ^{2}}.$$ For CM constellation, the first and last terms are constants, and hence, can be dropped. Therefore, $$J_{m,n}^{c}=-\frac{r_{c}r_{\acute{c}}^{\ast}}{\tilde{d}_{m}\tilde{d}_{n}^{\ast}}+\frac{r_{c}^{\ast}r_{\acute{c}}}{\tilde{d}_{m}^{\ast}\tilde{d}_{n}}.\label{eq:branch-metric-viterbi}$$ By noting that the two terms in are the complex conjugate pair, then $$J_{m,n}^{c}=-2\Re\left\{ \frac{r_{c}r_{\acute{c}}^{\ast}}{\tilde{d}_{m}\tilde{d}_{n}^{\ast}}\right\} .\label{eq:branch-metric-viterbi-02}$$ From the expression in , the constant “$-2$ can be dropped from the cost function, however, the problem with be flipped to a maximization problem. Therefore, by expanding , we get, $$J_{m,n}^{c}=\Re\left\{ \frac{\Re\left\{ r_{c}\right\} \Re\left\{ r_{\acute{c}}^{\ast}\right\} -\Im\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} +j\left[-\Re\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} +\Im\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} \right]}{\Re\left\{ \tilde{d}_{m}\tilde{d}_{n}^{\ast}\right\} +j\Im\left\{ \tilde{d}_{m}\tilde{d}_{n}^{\ast}\right\} }\right\} .\label{eq:branch-metric-03}$$ By defining $\tilde{d}_{m}\tilde{d}_{n}^{\ast}\triangleq\tilde{u}_{m,n},$ and using complex numbers identities, we get , $$J_{m,n}^{c}=\frac{\left[\Re\left\{ r_{c}\right\} \Re\left\{ r_{\acute{c}}^{\ast}\right\} +\Im\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} \right]\Re\left\{ \tilde{u}_{m,n}\right\} -\left[-\Re\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} +\Im\left\{ r_{c}\right\} \Im\left\{ r_{\acute{c}}^{\ast}\right\} \right]\Im\left\{ \tilde{u}_{m,n}\right\} }{\Re\left\{ \tilde{u}_{m,n}\right\} ^{2}+\Im\left\{ \tilde{u}_{m,n}\right\} ^{2}}.\label{eq:branch-metric-04}$$ For CM, $\Re\left\{ \tilde{u}_{m,n}\right\} ^{2}+\Im\left\{ \tilde{u}_{m,n}\right\} ^{2}$ is constant, and hence, it can be dropped from the cost function, which implies that no division operations are required.
To compute $J_{m,n}^{c}$, it is worth noting that the two terms in brackets are independent of $\left\{ m,n\right\} $, and hence, they are computed only once for each value of $c$. Therefore, the complexity at each step in the trellis can be computed as $R_{A}=3\times2^{M}$, $R_{M}=4+2\times2^{M}$ and $R_{D}=0$, where $2^{M}$ is the number of branches at each step in the trellis. However, if the trellis starts or ends by a pilot, then only $M$ computations are required. By noting that the number of full steps is $N-2N_{P}-1$, and the number of steps that require $M$ computations is $2\left(N_{P}-1\right)$, then the total computations of the branch metrics (BM) are: $$\begin{aligned}
R_{A}^{BM} & =\left(3\times2^{M}\right)\left(N-2N_{P}-1\right)+2\left(3\times M\right)\left(N_{P}-1\right)\\
R_{M}^{BM} & =\left(4+2^{M+1}\right)\left(N-2N_{P}-1\right)+2\left(N_{P}-1\right)\left(4+2M\right)\\
R_{D}^{BM} & =0\end{aligned}$$ The path metrics (PM) require $R_{A}^{PM}=\left(N-2N_{P}-1\right)+M\left(N_{P}-1\right)$ real addition. Therefore, the total complexity is: $$\begin{aligned}
R_{A}^{CM} & =\left(N-2N_{P}-1\right)\left(5\times2^{M}\right)+7M\left(N_{P}-1\right)\\
R_{M}^{CM} & =\left(N-2N_{P}-1\right)\left(4+2^{M+1}\right)+2\left(N_{P}-1\right)\left(4+2M\right)\\
R_{D}^{CM} & =0\end{aligned}$$
[$N$]{} [$128$]{} [$256$]{} [$512$]{} [$1024$]{} [$2048$]{}
-------------------- ------------ ------------ ------------ ------------ ------------
[$\eta_{R_{A}}$]{} [$0.58$]{} [$1.07$]{} [$1.21$]{} [$1.27$]{} [$1.31$]{}
[$\eta_{R_{M}}$]{} [$0.77$]{} [$0.72$]{} [$0.68$]{} [$0.64$]{} [$0.61$]{}
[$R_{D}$]{} [$96$]{} [$192$]{} [$384$]{} [$768$]{} [$1536$]{}
[$\eta_{P}$]{} [$0.20$]{} [$0.21$]{} [$0.22$]{} [$0.26$]{} [$0.31$]{}
: Computational complexity comparison using different values of $N$, $N_{P}=N/4$, for BPSK.\[tab:Computational-power-analysis\]
To compare the complexity of the $D^{3}$, we use the conventional detector using LS channel estimation, linear interpolation, zero-forcing (ZF) equalization, and MLD, denoted as coherent-L, as a benchmark due to its low complexity. The relative complexity is denoted by $\eta$, which corresponds to the ratio of the $D^{3}$ complexity to the conventional detector, i.e., $\eta_{R_{A}}$ denotes the ratio of real additions and $\eta_{R_{M}}$ corresponds to the ratio of real multiplications. As depicted in Table \[tab:Computational-power-analysis\], $R_{A}$ for $D^{3}$ less than coherent-L only using BPSK for $N=128$, and then it becomes larger for all the other considered values of $N$. For $R_{M}$, $D^{3}$ is always less than the coherent-L, particularly for high values of $N$, where it becomes 0.61 for $N=2048$. It is worth noting that $R_{D}$ in the table corresponds to the number of divisions in the conventional OFDM since the $D^{3}$ does not require any division operations. For a more informative comparison between the two systems, we use the computational power analysis presented in [@computational_power], where the total power for each detector is estimated based on the total number of operations. Table \[tab:Computational-power-analysis\] shows the relative computational power $\eta_{P}$, which shows that the $D^{3}$ detector requires only $0.2$ of the power required by the coherent-L detector for $N=128$ and $0.31\%$ for $N=2048$.
-- -- -- -- --
-- -- -- -- --
Besides, it is worth noting that linear interpolation has lower complexity as compared to more accurate interpolation schemes such as the spline interpolation [@spline-interpolation], [@Spline], which comes at the expense of the error rate performance. Therefore, the results presented in Table \[tab:Computational-power-analysis\] can be generally considered as upper bounds on the relative complexity of the $D^{3}$, when more accurate interpolation schemes are used, the relative complexity will drop even further as compared to the results in Table \[tab:Computational-power-analysis\].
Complexity with Error Correction Coding
---------------------------------------
To evaluate the impact of the complexity reduction of the $D^{3}$ in the presence of FEC coding, convolutional codes are considered with soft and hard decision decoding using the VA. BPSK is the modulation considered for the complexity evaluation and the code rate is assumed to be $1/2$. For decoding of convolutional codes, the soft VA requires $n\times2^{K}$ addition or subtractions and multiplications per decoded bit, where $1/n$ is the code rate and $K$ is the constraint length [@P-Wu]. Therefore, for $1/2$ code rate, $R_{A}=R_{M}=2^{K+1}$. Given that each OFDM symbol has $N$ coded bits and $N/2$ information bits, the complexity per OFDM symbol becomes $R_{A}=R_{M}=N\times2^{K}$. For the hard VA, $N\times2^{K}$ XOR operations are required for the branch metric computation, while $N\times2^{K-1}$ additions are required for the path metric computations. Because the XOR operation is a bit operation, it’s complexity is much less than the addition. Assuming that addition is using an 8-bit representation, then the complexity of an addition operation is about eight times the XOR. Therefore, $R_{A}$, in this case, can be approximated as $N\left(2^{K}+2^{K-2}\right)$.
[$K$ ]{} [$3$ ]{} [$4$ ]{} [$5$]{} [$6$]{} [$7$]{}
---------- ------------- ------------- ------------- ------------- -------------
[$0.96$ ]{} [$0.97$ ]{} [$0.97$ ]{} [$0.98$ ]{} [$0.99$ ]{}
[$0.24$ ]{} [$0.26$ ]{} [$0.28$ ]{} [$0.33$ ]{} [$0.41$ ]{}
: Computational complexity comparison using hard and soft VA for different values of $K$, $N=2048$. \[T-coded\]
As can be noted from Table \[T-coded\], the complexity reduction when soft VA is used less significant as compared to the hard VA. Such a result is obtained because the soft VA requires the CSI to compute the reliability factors, which requires $N-N_{P}$ division operations when the $D^{3}$ is used. For hard decoding, the advantage of the $D^{3}$ is significant even for high constraint length values.
Numerical Results\[sec:Numerical-Results\]
==========================================
This section presents the performance of the $D^{3}$ detector in terms of BER for several operating scenarios. The system model follows the LTE-A physical layer (PHY) specifications [@LTE-A], where the adopted OFDM symbol has $N=512$, $N_{\mathrm{CP}}=64$, the sampling frequency $f_{s}=7.68$ MHz, the subcarrier spacing $\Delta f=15$ kHz, and the pilot grid follows that of Fig. \[fig:2D-to-1D\]. The total OFDM symbol period is $75$ $\mu\sec$, and the CP period is $4.69$ $\mu\sec$. The channel models used are the flat Rayleigh fading channel, the typical urban (TUx) multipath fading model [@Typical; @Urban] that consists of $6$ taps with normalized delays of $\left[0,2,3,9,13,29\right]$ and average taps gains are $\left[0.2,0.398,0.2,0.1,0.063,0.039\right]$, which corresponds to a severe frequency-selective channel. The TUx model is also used to model a moderate frequency-selective channel where the number of taps in the channel is $9$ with normalized delays of $[0$, $1$, $\ldots$, $8]$ samples, and the average taps gains are $[0.269$, $0.174$, $0.289$, $0.117$, $0.023$, $0.058$, $0.036$, $0.026$, $0.008]$. The channel taps gains are assumed to be independent and Rayleigh distributed. The Monte Carlo simulation results included in this work are obtained by generating $10^{6}$ OFDM symbols per simulation run. Throughout this section, the ML coherent detector with perfect CSI will be denoted as coherent, while the coherent with linear and spline interpolation will be denoted as coherent-L and coherent-S, respectively. Moreover, the results are presented for the SISO system, $\mathit{\mathcal{N}\mathrm{=1}}$, unless it is mentioned otherwise.
Fig. \[fig:BER-Single-Double-Sided-Flat\] shows the BER of the single-sided (SS) and double-sided (DS) $D^{3}$ over flat fading channels for $\mathcal{K}=2,6$ and $3,7$, respectively, and using BPSK. The number of data symbols $\mathcal{K}_{D}=\mathcal{K}-1$ for the SS and $\mathcal{K}_{D}=\mathcal{K}-2$ for the DS because there are two pilot symbols at both ends of the data segment for the DS case. The results in the figure for the SS show that $\mathcal{K}$ has a noticeable impact on the BER where the difference between the $\mathcal{K}=2$ and $6$ cases is about $1.6$ dB at BER of $10^{-3}$. For the DS segment, the BER has the same trends of the SS except that it becomes closer to the coherent case because having more pilots reduces the probability of sequence inversion due to the phase ambiguity problem. The figure shows that the approximated and simulation results match very well for all cases, which confirms the accuracy of the derived approximations.
The effect of the frequency selectivity is illustrated in Fig. \[fig:BER-SISO-D3-SS-6-taps\] for the SS and DS configurations using$\mathcal{K}_{D}=1$. As can be noted from the figure, frequency-selective channels introduce error floors at high SNRs, which is due to the difference between adjacent channel values caused by the channel frequency selectivity. Furthermore, the figure shows a close match between the simulation and the derived approximations. The approximation results are presented only for $\mathcal{K}=2$ because evaluating the BER for $\mathcal{K}>2$ becomes computationally prohibitive. For example, evaluating the integral for the $\mathcal{K}=3$ requires solving a $6$-fold integral. The results for the frequency-selective channels are quite different from the flat fading cases. In particular, the BER performance drastically changes when the DS pilot segment is used. Moreover, the impact of the frequency selectivity is significant, particularly for the SS pilot case.
![BER in frequency-selective channels using BPSK, $\mathcal{K}_{D}=1$ and $\mathcal{N}=1$. \[fig:BER-SISO-D3-SS-6-taps\] ](graphics/fig_05_ber_ss_ds_flat){width="0.4\paperwidth"}
![BER in frequency-selective channels using BPSK, $\mathcal{K}_{D}=1$ and $\mathcal{N}=1$. \[fig:BER-SISO-D3-SS-6-taps\] ](graphics/fig_07_ber_ss_ds_6_taps_9_taps_awgn){width="0.4\paperwidth"}
Fig. \[fig:BER-SIMO-D3-flat\] shows the BER of the $1\times2$ SIMO $D^{3}$ over flat fading channels for SS and DS pilot segments. It can be noted from the figure that the maximum ratio combiner (MRC) BER with perfect CSI outperforms the DS and SS systems by about $2$ and $3$ dB, respectively. Moreover, the figure shows that the MLSD [@Wu; @2010] and the $D^{3}$ have equivalent BER for the SISO and SIMO scenarios.
![BER of the SISO $D^{3}$ and MLSD [@Wu; @2010] over the 6-taps frequency-selective channel using QPSK, $\mathcal{K}_{D}=1$, $\mathcal{N}=1$, $2.$\[fig:BER-SISO-D3-QPSK\] ](graphics/fig_08_ber_ss_ds_flat_simo){width="0.4\paperwidth"}
![BER of the SISO $D^{3}$ and MLSD [@Wu; @2010] over the 6-taps frequency-selective channel using QPSK, $\mathcal{K}_{D}=1$, $\mathcal{N}=1$, $2.$\[fig:BER-SISO-D3-QPSK\] ](graphics/fig_10_ber_ds_6_taps_qpsk_siso_simo){width="0.4\paperwidth"}
Figs. \[fig:BER-SISO-D3-QPSK\] shows the BER of the SISO and $1\times2$ SIMO MLSD, coherent, coherent-S and coherent-L systems over frequency-selective channels. For both SISO and SIMO, the BER of all the considered techniques converges at low SNRs because the AWGN dominates the BER in the low SNR range. For moderate and high SNRs, the $D^{3}$ outperforms all the other considered techniques except for the coherent, where the difference is about $3.5$ and $2.75$ dB at BER of $10^{-3}$ for the SISO and SIMO systems, respectively.
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Conclusion and Future Work\[sec:Conclusion\]
============================================
This work proposed a new receiver design for OFDM-based broadband communication systems. The new receiver performs the detection process directly from the FFT output symbols without the need of experiencing the conventional steps of channel estimation, interpolation, and equalization, which led to a considerable complexity reduction. Moreover, the $D^{3}$ system can be deployed efficiently using the VA. The proposed system was analyzed theoretically where simple closed-form expressions were derived for the BER in several cases of interest. The analytical and simulation results show that the $D^{3}$ BER outperforms the coherent pilot-based receiver in various channel conditions, particularly in frequency-selective channels where the $D^{3}$ demonstrated high robustness.
Although the $D^{3}$ may perform well even in severe fading conditions, it is crucial to evaluate its sensitivity to various practical imperfections. Thus, we will consider in our future work the performance of the $D^{3}$ in the presence of various system imperfections such as phase noise, synchronization errors and IQ imbalance. Moreover, we will evaluate the $D^{3}$ performance in mobile fading channels, where the channel variation may introduce intercarrier interference.
Appendix I {#appendix-i .unnumbered}
==========
By defining the events $A_{\psi}>A_{n}\triangleq E_{\psi,n}$, $n\in\left\{ 0\text{, }1\text{, }\ldots,\psi-1\right\} $, then, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=P\left(\bigcap\limits _{n=0}^{\psi-1}E_{\psi,n}\right).\label{E-PC-01}$$ Using the chain rule, $P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}$ can be written as, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\Pr\left(\left.E_{\psi,\psi-1}\right\vert \bigcap\limits _{n=0}^{\psi-2}E_{\psi,n}\right)\Pr\left(\bigcap\limits _{n=0}^{\psi-2}E_{\psi,n}\right).$$ For $\mathcal{K}=2$, $\psi=1$, $\tilde{\mathbf{d}}_{0}^{(0)}=[1$, $-1]$, $\tilde{\mathbf{d}}_{0}^{(1)}=[1$,$1]$, and thus, $$\begin{aligned}
P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}} & = & \Pr\left(E_{1,0}\right)\nonumber \\
& = & \Pr\left(\Re\left\{ r_{1}r_{2}\right\} >\Re\left\{ -r_{1}r_{2}\right\} \right)=\Pr\left(\Re\left\{ r_{0}r_{1}\right\} >0\right).\end{aligned}$$ For $\mathcal{K}=3$, $\psi=4$, $\tilde{\mathbf{d}}_{0}^{(0)}=[1$, $1$, $-1]$, $\tilde{\mathbf{d}}_{0}^{(1)}=[1$, $-1$, $-1]$, $\tilde{\mathbf{d}}_{0}^{(2)}=[1$, $-1$, $1]$ and $\tilde{\mathbf{d}}_{0}^{(3)}=[1$, $1$,…$,1]$ . Using the chain rule $$\begin{aligned}
P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}} & = & \Pr\left(E_{3,2}|E_{3,1}\text{, }E_{3,0}\right)\Pr\left(E_{3,1},E_{3,0}\right)\nonumber \\
& = & \Pr\left(E_{3,2}|E_{3,1}\text{, }E_{3,0}\right)\Pr\left(E_{3,1}|E_{3,0}\right)\Pr\left(E_{3,0}\right)\label{E-PrA3A0}\end{aligned}$$ However, $\Pr\left(E_{3,0}\right)=\Pr\left(A_{3}>A_{0}\right)$, and thus $$\begin{aligned}
\Pr\left(E_{3,0}\right) & = & \Pr\left(\Re\left\{ r_{0}r_{1}+r_{1}r_{2}\right\} >\Re\left\{ r_{0}r_{1}-r_{1}r_{2}\right\} \right)\nonumber \\
& = & \Pr\left(\Re\left\{ r_{1}r_{2}\right\} >\Re\left\{ -r_{1}r_{2}\right\} \right)=\Pr\left(\Re\left\{ r_{1}r_{2}\right\} >0\right).\end{aligned}$$ The second term in (\[E-PrA3A0\]) can be evaluated by noting that the events $E_{3,1}$ and $E_{3,0}$ are independent. Therefore $\Pr\left(E_{3,1}|E_{3,0}\right)=\Pr\left(E_{3,1}\right)$, which can be computed as $$\begin{aligned}
\Pr\left(E_{3,1}\right) & = & \Pr\left(\Re\left\{ r_{0}r_{1}+r_{1}r_{2}\right\} >\Re\left\{ -r_{0}r_{1}+r_{1}r_{2}\right\} \right)\nonumber \\
& = & \Pr\left(\Re\left\{ r_{0}r_{1}\right\} >\Re\left\{ -r_{0}r_{1}\right\} \right)=\Pr\left(\Re\left\{ r_{0}r_{1}\right\} >0\right).\end{aligned}$$ The first term in (\[E-PrA3A0\]) $\Pr\left(E_{3,2}|E_{3,1}\text{, }E_{3,0}\right)=1$ because if $A_{3}>\left\{ A_{1},A_{0}\right\} $, then $A_{3}>A_{2}$ as well. Consequently, $$P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}=\Pr\left(\Re\left\{ r_{0}r_{1}\right\} >0\right)\Pr\left(\Re\left\{ r_{1}r_{2}\right\} >0\right).$$ By induction, it is straightforward to show that $P_{C}|_{\mathbf{H}_{0},\mathbf{\mathbf{1}}}$ can be written as, $$P_{C}\mid_{\mathbf{H},\mathbf{\mathbf{d}=\mathbf{1}}}=\prod\limits _{n=0}^{\mathcal{K-}2}\Pr\left(\Re\left\{ r_{n}r_{\acute{n}}\right\} >0\right).$$
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[^1]: A. Saci, A. Al-Dweik and A. Shami are with the Department of Electrical and Computer Engineering, Western University, London, ON, Canada, (e-mail: {asaci, aaldweik, abdallah.shami}@uwo.ca).
[^2]: A. Al-Dweik is also with the Department of Electrical and Computer Engineering, Khalifa University, Abu Dhabi, UAE, (e-mail: [email protected]).
[^3]: Part of this work is protected by the US patent: A. Al-Dweik “Signal detection in a communication system.” U.S. Patent No. 9,596,119. 14 Mar. 2017.
| ArXiv |
---
abstract: 'An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in this context. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, and the join construction, and we present an explicit algorithm for computing Noetherian operators.'
author:
- 'Yairon Cid-Ruiz, Roser Homs and Bernd Sturmfels'
title: Primary Ideals and their Differential Equations
---
Introduction {#sec1}
============
In his 1938 article [@GROBNER_MATH_ANN] on the foundations of algebraic geometry, Gröbner introduced differential operators to characterize membership in a polynomial ideal. He derived such characterizations for ideals that are prime or primary to a rational maximal ideal [@GROBNER_BOOK_AG_2 pages 174-178]. In a 1952 lecture [@GROBNER_LIEGE §1] he suggested that the same program can be carried out for any primary ideal. Gröbner was particularly interested in algorithmic solutions to this problem.
Substantial contributions in this subject area were made by analysts. In the 1960s, Ehrenpreis [@EHRENPREIS] stated his *Fundamental Principle* on solutions to linear partial differential equations (PDE) with complex constant coefficients. A main step was the characterization of primary ideals by differential operators. But, he incorrectly claimed that operators with constant coefficients suffice. Using Example (\[exam:Palamodov\]) below, Palamodov [@PALAMODOV] pointed out the error, and he gave a correct proof by introducing the representation by [*Noetherian operators*]{}. Details on the Ehrenpreis-Palamodov Fundamental Principle can also be found in [@BJORK; @HORMANDER].
The ball returned to algebra in 1978 when Brumfiel published the little-known paper [@BRUMFIEL_DIFF_PRIM]. In 1999, Oberst [@OBERST_NOETH_OPS] extended Palamodov’s Noetherian operators to polynomial rings over arbitrary fields. In 2007, Damiano, Sabadini and Struppa [@DAMIANO] gave a computational approach. A general theory for Noetherian commutative rings was developed recently in [@NOETH_OPS]. Building on this, the present article develops a theory of primary ideals as envisioned by Gröbner.
We now introduce a running example that serves to illustrate our title and results. The following prime ideal of codimension $c=2$ in $n=4$ variables is familiar to many algebraists: $$\label{eq:twistedcubic1}
P \quad = \quad \langle\,
x_1^2-x_2 x_3,\, x_1 x_2 - x_3 x_4, x_2^2 - x_1 x_4 \,\rangle \quad \subset
\quad {\mathbb{C} }[x_1,x_2,x_3,x_4].$$ This ideal defines the (affine cone over the) [*twisted cubic curve*]{} $\,V(P) = \bigl\{ \,(s^2t, s t^2,s^3, t^3) \,:\, s,t \in {\mathbb{C} }\,\bigr\}$; see [@CUBIC_LITTLE]. We identify the polynomials in (\[eq:twistedcubic1\]) with PDE with constant coefficients by setting $x_i = \partial_{z_i}$. Solving these PDE means describing all functions $\psi(z_1,z_2,z_3,z_4)$ with $$\label{eq:twistedcubic2}
\frac{\partial^2 \psi}{\partial z_1^2} = \frac{\partial^2 \psi}{\partial z_2 \partial z_3} \qquad {\rm and} \qquad
\frac{\partial^2 \psi}{\partial z_1 \partial z_2} = \frac{\partial^2 \psi}{\partial z_3 \partial z_4} \qquad {\rm and} \qquad
\frac{\partial^2 \psi}{\partial z_2^2} = \frac{\partial^2 \psi}{\partial z_1 \partial z_4}.$$ Results in analysis ensure that every solution comes from a measure $\mu$ on the $(s,t)$-plane: $$\label{eq:twistedcubic4}
\psi(z_1,z_2,z_3,z_4) \,\,\,\, = \,\,\,
\int {\rm exp} \bigl( z_1 s^2 t \,+\, z_2 s t^2 \,+\,z_3 s^3 \,+\, z_4 t^3
\bigr) \,\mu(s,t) \,{\rm d}s \,{\rm d}t .$$ For instance, if $\mu$ is the Dirac measure at the point $(2,3)$ then $\psi = {\rm exp}( 12 z_1 + 18 z_2 + 8 z_3 + 27 z_4)$. Thus, the functions $\psi$ are simply an analytic encoding of the affine surface $V(P) \subset {\mathbb{C} }^4$.
The situation becomes interesting when we consider a non-reduced scheme structure on our surface. Algebraically, this means replacing the prime $P$ by a $P$-primary ideal. We use differential operators to give compact representations of $P$-primary ideals $Q$. For instance, $$\label{eq:twistedcubic6}
\begin{matrix}
Q \,\,=\, \, \bigl\{ \,f \in {\mathbb{C} }[x_1,x_2,x_3,x_4]\,:\,
A_i \bullet f \in P \,\,\,{\rm for} \,\,\, i=1,2,3\, \bigr\}, \smallskip \\ {\rm where} \quad
\,A_1 \,=\, 1\,,\;\, A_2\,=\, \partial_{x_1}
\,\,\,{\rm and} \,\,\,A_3 \,=\, \partial_{x_1}^2 \,-\, 2 \,x_2\,\partial_{x_2} . \quad
\end{matrix}$$ Here $\bullet$ means applying a differential operator to a function. Note that a prime ideal is always represented by just one Noetherian operator $A_1=1$. We can encode (\[eq:twistedcubic6\]) by the ideal $$\label{eq:magic}
\!\! \bigl\langle
u_1^2 - u_2 u_3, u_1 u_2 -u_3 u_4,
u_2^2-u_1 u_4,
\, x_1-u_1-y_1, x_2-u_2-y_2, x_3-u_3, x_4-u_4, \,
\underline{ y_1^3, \, y_2 + u_2 \,y_1^2}
\bigr\rangle.$$ The minimal generators of $Q$ are obtained from (\[eq:magic\]) by eliminating $\{u_1,u_2,u_3,u_4,y_1,y_2\}$: $$\begin{small} \begin{matrix} Q \,\, = & \!\!\! \bigl\langle\,
3 x_1^2 x_2^2-x_2^3 x_3-x_1^3 x_4-3 x_1 x_2 x_3 x_4+2 x_3^2 x_4^2\,,\,\,
3 x_1^3 x_2 x_4-3 x_1 x_2^2 x_3 x_4-3 x_1^2 x_3 x_4^2+3 x_2 x_3^2 x_4^2\\ & +2 x_2^3 -2 x_3 x_4^2\,,\,\,
3 x_2^4 x_3-6 x_1 x_2^2 x_3 x_4+3 x_1^2 x_3 x_4^2+x_2^3-x_3 x_4^2\,,\,\,
4 x_1 x_2^3 x_3+x_1^4 x_4-6 x_1^2 x_2 x_3 x_4\\ & -3 x_2^2 x_3^2 x_4+4 x_1 x_3^2 x_4^2\,,\,\,
x_2^5-x_1 x_2^3 x_4-x_2^2 x_3 x_4^2+x_1 x_3 x_4^3\,,\,\,
x_1 x_2^4-x_2^3 x_3 x_4-x_1 x_2 x_3 x_4^2+x_3^2 x_4^3\,,\\ &
x_1^4 x_2-x_2^3 x_3^2-2 x_1^3 x_3 x_4+2 x_1 x_2 x_3^2 x_4\,,\,\,
x_1^5-4 x_1^3 x_2 x_3+3 x_1 x_2^2 x_3^2+2 x_1^2 x_3^2 x_4-2 x_2 x_3^3 x_4\,,\\ &
3 x_1^4 x_4^2-6 x_1^2 x_2 x_3 x_4^2+3 x_2^2 x_3^2 x_4^2+4 x_2^4-4 x_2 x_3 x_4^2\,,\,\,
x_2^3 x_3^2 x_4+x_1^3 x_3 x_4^2-3 x_1 x_2 x_3^2 x_4^2+x_3^3 x_4^3 \\ & +x_1 x_2^3-x_1 x_3 x_4^2\,,
3 x_1^4 x_3 x_4-6 x_1^2 x_2 x_3^2 x_4+3 x_2^2 x_3^3 x_4+2 x_1^3 x_2
+6 x_1 x_2^2 x_3-6 x_1^2 x_3 x_4-2 x_2 x_3^2 x_4\,,\\ &
4 x_2^3 x_3^3+4 x_1^3 x_3^2 x_4-12 x_1 x_2 x_3^3 x_4+4 x_3^4 x_4^2
-x_1^4+6 x_1^2 x_2 x_3+3 x_2^2 x_3^2-8 x_1 x_3^2 x_4\,
\bigr\rangle.
\end{matrix} \end{small}$$ As in (\[eq:twistedcubic1\]) and (\[eq:twistedcubic2\]), we can view $Q$ as a system of PDE by setting $x_i = \partial_{z_i}$. Its solutions are $$\label{eq:twistedcubic5}
\begin{matrix}
\!\! \psi(z_1,z_2,z_3,z_4) \, = \,\,
\sum_{i=1}^3 \int \! B_i(z_1,z_2,s,t) \cdot
{\rm exp} \bigl( z_1 s^2 t + z_2 s t^2 +z_3 s^3 + z_4 t^3 \bigr) \,\mu_i(s,t) \,{\rm d}s \,{\rm d}t ,
\smallskip \\ \qquad {\rm where} \quad
B_1\,=\, 1\,,\;\, B_2 = z_1 \,\,\, {\rm and} \,\,\, B_3 \,=\, z_1^2 - 2 st^2 z_2 ,
\end{matrix}$$ for suitable measures $\mu_1,\mu_2,\mu_3$ on the $(s,t)$-plane ${\mathbb{C} }^2$. Note that $Q$ has multiplicity $3$ over $P$.
The title of this paper refers to two ways of associating differential equations to a primary ideal in a polynomial ring. First, we use PDE with polynomial coefficients, namely Noetherian operators $A_i$ as in (\[eq:twistedcubic6\]), to give a compact encoding of $Q$. Second, we can interpret $Q$ itself as a system of PDE with constant coefficients, with solutions represented by [*Noetherian multipliers*]{} $B_i$ as in (\[eq:twistedcubic5\]). The dual roles played by the $A_i$ and $B_i$ is one of our main themes.
This paper is organized as follows. In Section \[sec2\] we present characterizations of primary ideals in terms of punctual Hilbert schemes and Weyl-Noether modules. The former offers a parametrization of all $P$-primary ideals of a given multiplicity, and the latter establishes the links to differential equations. In Section \[sec3\] we turn to the Ehrenpreis-Palamodov Fundamental Principle. We present a self-contained proof of the algebraic part, and we introduce algorithms for computing Noetherian operators. In Sections \[sec4\] and \[sec6\] we prove the results stated in Section \[sec2\]. Section \[sec5\] reviews differential operators in commutative algebra and supplies tools for our proofs. In Section \[sec7\] we study the join construction for primary ideals, which offers a new perspective on ideals that are similar to symbolic powers. Finally, in Section \[sec8\] we establish a connection to numerical algebraic geometry. We propose a definition of [*numerical primary decomposition*]{} that puts a focus on the representation of primary ideals.
Characterizing Primary Ideals {#sec2}
=============================
Irreducible varieties and their prime ideals are the basic building blocks in algebraic geometry. Solving systems of polynomial equations means extracting the associated primes from the system, and to subsequently study their irreducible varieties. However, if the given ideal is not radical then we seek the primary decomposition and not just the associated primes. We wish to gain a precise understanding of the primary ideals that make up the given scheme.
We furnish a representation theorem for primary ideals in a polynomial ring, extending the familiar case of zero-dimensional ideals (Macaulay’s inverse system [@Grobner37]). This combines a characterization via differential operators with a parametrization from a Hilbert scheme. Fix a field ${\mathbb{K}}$ of characteristic zero and a prime ideal $P$ of codimension $c$ in the polynomial ring $R = {\mathbb{K}}[x_1,\ldots,x_n]$. We write ${\mathbb{F}}$ for the field of fractions of the integral domain $R/P$.
\[thm:main\] The following four sets of objects are in a natural bijective correspondence:
(a) $P$-primary ideals $Q$ in $R$ of multiplicity $m$ over $P$,
(b) points in the punctual Hilbert scheme $\,{\rm Hilb}^m({\mathbb{F}}[[y_1,\ldots,y_c]])$,
(c) $m$-dimensional ${\mathbb{F}}$-subspaces of $\,{\mathbb{F}}[z_1,\ldots,z_c]$ that are closed under differentiation,
(d) $m$-dimensional ${\mathbb{F}}$-subspaces of the Weyl-Noether module $ {\mathbb{F}}\,\otimes_R \,D_{n,c}$ that are $R$-bi-modules.
Moreover, any basis of the ${\mathbb{F}}$-subspace in part (d) can be lifted to Noetherian operators $A_1,\ldots,A_m$ in the relative Weyl algebra $D_{n,c}$ that represent the ideal $Q$ in part (a) as in (\[eq:fromAtoQ\]).
The purpose of this section is to define and explain all the concepts in Theorem \[thm:main\]. Our aim is to state the promised bijections as explicitly as possible. The proof of Theorem \[thm:main\] will be divided into smaller pieces and given in Sections \[sec4\] and \[sec6\]. The encoding of $Q$ by Noetherian operators $A_i$ will be explained in Section \[sec3\]. We already saw an example in (\[eq:twistedcubic6\]). The Weyl-Noether module in part (d) is our stage for the PDE that portray primary ideals.
We begin by returning to Gröbner, whose 1937 article [@Grobner37] interpreted Macaulay’s inverse system as solutions to linear PDE. He considered the special case when $P = \langle x_1,\ldots,x_n \rangle $ is the maximal irrelevant ideal, so we have $c=n$ and ${\mathbb{F}}= {\mathbb{K}}$. The geometric intuition invoked in [@GROBNER_LIEGE §1] is captured by the punctual Hilbert scheme $\,{\rm Hilb}^m({\mathbb{K}}[[y_1,\ldots,y_n]])$, whose points are precisely the $P$-primary ideals of colength $m$. This zero-dimensional case is familiar to most commutative algebraists, especially the readers of [@MOURRAIN_DUALITY]. Here, parts (c) and (d) of Theorem \[thm:main\] refer to the $m$-dimensional ${\mathbb{K}}$-vector space of polynomial solutions to the PDE.
The general case of higher-dimensional primary ideals $Q$ was of great interest to Gröbner. In his 1952 Liège lecture [@GROBNER_LIEGE], he points to Severi [@Severi], and he writes: [*En ce sense la variété algébraique correspondante à un idéal primaire $Q$ pour l’idéal premier $P$ consiste en les points ordinaires de la variété $\,V(P)$ et en certain nombre $m$ des points infinitesiment voisins, c’est-à-dire dans $m$ conditions différentielles ajoutées à chaque point de la variété $\,V(P)$. Le nombre $m$ de ces conditions différentielles est égal à la longueur de l’idéal primaire $Q$.*]{} But Gröbner was never able to complete the program himself, in spite of the optimism he still expressed in his 1970 textbook [@GROBNER_BOOK_AG_2]. After the detailed treatment of Macaulay’s inverse systems for zero-dimensional ideals, he proclaims: [*Es dürfte auch nicht schwer sein den oben angegebenen Formalismus auf mehrdimensionale Primärideale auszudehnen*]{} [@GROBNER_BOOK_AG_2 page 178].
The issue was finally resolved by the theory of Ehrenpreis-Palamodov [@EHRENPREIS; @PALAMODOV], presented in Section \[sec3\], and the subsequent developments [@BRUMFIEL_DIFF_PRIM; @NOETH_OPS; @DAMIANO; @OBERST_NOETH_OPS] we discussed in the Introduction.
Theorem \[thm:main\] is our main contribution. We regard this as a definitive result on primary ideals in $R$. It captures the geometric spirit of Gröbner and Severi, as it relates their “infinitely near points” directly to current advances in numerical algebraic geometry (Section \[sec8\]).
Two essential ingredients in Theorem \[thm:main\] are the function field ${\mathbb{F}}$ and the Weyl-Noether module $\, {\mathbb{F}}\otimes_R D_{n,c}$. We start our technical discussion with some insights into these objects. By [*Noether normalization*]{}, after a linear change of coordinates, the quotient ring $R/P$ is a finitely generated module over the polynomial subring ${\mathbb{K}}[x_{c+1},\ldots,x_n]$. This implies that ${\mathbb{F}}$ is algebraic over the field ${\mathbb{K}}(x_{c+1},\ldots,x_n)$, a purely transcendental extension of ${\mathbb{K}}$.
Clear notation is very important for this article. This is why multiple letters $x,y,z,u$ are used to denote variables and differential operators. Elements in ${\mathbb{F}}$ are represented as fractions of polynomials in ${\mathbb{K}}[u_1,\dots,u_n]$, where $u_i$ denotes the residue class of $x_i$ modulo $P$. Whenever the number $n$ of variables is clear from the context, we use the multi-index notation $\mathbf{u}^\alpha=u_1^{\alpha_1}\cdots u_n^{\alpha_n}$. Elements $a(\mathbf{u})/b(\mathbf{u})$ of the field ${\mathbb{F}}$ can be uniquely represented by taking $a(\mathbf{u})$ and $b(\mathbf{u})$ coprime and in normal form with respect to a Gröbner basis of $P$. Arithmetic in ${\mathbb{F}}$ is performed via this Gröbner basis. The $R$-module structure of ${\mathbb{F}}$ is given by $\,\mathbf{x}^\alpha\cdot a(\mathbf{u})/b(\mathbf{u})=\mathbf{u}^\alpha a(\mathbf{u})/b(\mathbf{u})$. Alternatively, from the perspective of numerical algebraic geometry, a better approach to arithmetic in ${\mathbb{F}}$ is to work with generic points, obtained by realizing $R/P$ as a subring of a suitable field of functions on $V(P)$. In our running example, that suitable field could be $\,{\mathbb{K}}(s/t,t^3)$. It contains $R/P$ as the subring ${\mathbb{K}}[s^2 t, st^2, s^3, t^3]$.
The [*relative Weyl algebra*]{} $D_{n,c} = {\mathbb{K}}\langle x_1,\ldots,x_n, \partial_{x_1},\dots,\partial_{x_c}\rangle$ is the ${\mathbb{K}}$-algebra on $n{+}c$ generators $x_1,\ldots,x_n,\partial_{x_1},\ldots,\partial_{x_c}$ that commute except for $\partial_{x_i}x_i=x_i\partial_{x_i}+1$. This is a subalgebra of the usual Weyl algebra, so $D_{n,c}$ is non-commutative. Its elements are linear differential operators with polynomial coefficients, where derivatives occur with respect to the first $c$ variables. The set $\left\lbrace x_1^{\alpha_1}\cdots x_n^{\alpha_n}{\partial_{x_1}}^{\!\!\!\!\beta_1}\cdots
{\partial_{x_c}}^{\!\!\!\!\beta_c}:(\alpha,\beta)\in\mathbb{N}^n\times\mathbb{N}^c\right\rbrace$ is a ${\mathbb{K}}$-basis of $D_{n,c}$.
We define the [*Weyl-Noether module*]{} of the affine variety $V(P)$ to be the tensor product $$\label{eq:relativeweyl}
{\mathbb{F}}\,\otimes_R\,D_{n,c} \,\, = \,\, {\mathbb{F}}\,\otimes_R \,
R \langle \partial_{x_1},\ldots,\partial_{x_c}\rangle.$$ Since ${\mathbb{F}}$ is the field of fractions of the integral domain $R/P$, it is clearly an $R$-module. Note that the relative Weyl algebra $D_{n,c} = R\langle \partial_{x_1},\dots,\partial_{x_c}\rangle$ is non-commutative, and it has two distinct $R$-module structures: it is a left $R$-module and it is a right $R$-module. In the tensor product (\[eq:relativeweyl\]), for convenience of notation, we mean the left $R$-module structure on $D_{n,c}$. Later, in Remark \[rem\_isom\_restrict\_Weyl\_mod\], we shall give an intrinsic description of ${\mathbb{F}}\otimes_R D_{n,c}$ with differential operators.
By construction, the Weyl-Noether module (\[eq:relativeweyl\]) has both right and left $R$-module structures. The action by $R$ on the left is easy to write using the standard ${\mathbb{K}}$-basis above: $$\label{eq:leftaction}
\mathbf{x}^\alpha \cdot \biggl( \frac{a(\mathbf{u})}{b(\mathbf{u})} \,\otimes_R\, \partial_{\mathbf{x}}^\beta\ \biggr) \,\,\, = \,\,\,
\frac{\mathbf{u}^{\alpha} a(\mathbf{u})}{b(\mathbf{u})} \,\otimes_R\, \partial_{\mathbf{x}}^\beta.$$ For the action on the right we need the commutation identities in the Weyl algebra: $$\partial_{\mathbf{x}}^\beta \mathbf{x}^\alpha\,\,=\,\,\sum_{\gamma, \delta} \lambda_{\gamma,\delta}
\, \mathbf{x}^\gamma \partial_\mathbf{x}^\delta.$$ Here $\lambda_{\gamma,\delta}$ are the positive integers derived in [@SatStu Problem 4]. With this, the right action is $$\label{eq:rightaction}
\biggl( \frac{a(\mathbf{u})}{b(\mathbf{u})} \,\otimes_R \, \partial_{\mathbf{x}}^\beta \biggr)
\cdot \mathbf{x}^\alpha \, \,\, = \,\,\, \frac{a(\mathbf{u})}{b(\mathbf{u})}
\, \otimes_R \, \partial_{\mathbf{x}}^\beta \mathbf{x}^\alpha \,\,\,=\,\,\,
\sum_{\gamma,\delta} \lambda_{\gamma,\delta} \, \frac{\mathbf{u}^{\gamma}a(\mathbf{u})}{b(\mathbf{u})} \, \otimes_R \,
\partial_\mathbf{x}^\delta.$$ This means that the requirement to be an $R$-bi-module in Theorem \[thm:main\] (d) is very strong.
From the action (\[eq:leftaction\]) we deduce that $\,{\mathbb{F}}\,\otimes_R\,D_{n,c}\,$ is a left ${\mathbb{F}}$-vector space with basis $\,\left\lbrace 1 \otimes_R \partial_{\mathbf{x}}^\beta: \beta\in\mathbb{N}^c\right\rbrace$, so we could also write ${\mathbb{F}}\langle \partial_{x_1},\dots,\partial_{x_c}\rangle$ for (\[eq:relativeweyl\]). However, we prefer the previous notation because it highlights that there are two distinct structures. The Weyl-Noether module is a left ${\mathbb{F}}$-vector space via (\[eq:leftaction\]) and it is a right $R$-module via (\[eq:rightaction\]). It is not a right ${\mathbb{F}}$-vector space because the right $R$-action is not compatible with passing to $R/P$:
Fix the maximal ideal $P=\langle x_1, \ldots,x_n\rangle$ so that ${\mathbb{F}}={\mathbb{K}}$ and $c=n$. Since $\overline{x_j} = 0 \in R/P$, we have $\,x_j \cdot \left(1 \otimes_R \partial_{x_j}\right) = 0 \,$ and hence $\,\left(1 \otimes_R \partial_{x_j}\right) \cdot x_j = 1 \otimes_R 1$ holds in $\,{\mathbb{F}}\,\otimes_R\,D_{n,c}$. This shows that there is no right ${\mathbb{F}}$-action on the Weyl-Noether module $\,{\mathbb{F}}\,\otimes_R\,D_{n,c}$.
We now come to our parameter space in part (b), namely the punctual Hilbert scheme $$\label{eq:hilbertscheme}
{\rm Hilb}^m \bigl( \,{\mathbb{F}}[[y_1,\ldots,y_c]] \,\bigr).$$ This is a quasiprojective scheme over the function field ${\mathbb{F}}$. Its classical points are ideals of colength $m$ in the local ring ${\mathbb{F}}[[y_1,\ldots,y_c]]$. By Cohen’s Structure Theorem, this ring is the completion of $R_P$, the localization of $R$ at the prime $P$. To connect parts (a) and (b), we recall that the multiplicity $m$ of a primary ideal $Q$ over its prime $P = \sqrt{Q}$ is the length of the artinian local ring $\,R_P/Q R_P$. In symbols, using the command [degree]{} in [Macaulay2]{} [@MAC2], $$m \,\,=\,\, {\rm length}\bigl( R_P/QR_P \bigr) \,\,=\,\, \frac{{\tt degree}(Q)}{{\tt degree}(P)}.$$
The punctual Hilbert scheme (\[eq:hilbertscheme\]) is familiar to algebraic geometers, but its structure is very complicated when $c \geq 3$. We refer to Iarrobino’s article [@IARROBINO_HILB] as a point of entry. While the punctual Hilbert scheme is trivial for $c=1$, Briançon [@Bri77] undertook a detailed study for $c=2$. He showed that $\,{\rm Hilb}^m \bigl( {\mathbb{F}}[[y_1,y_2]] \bigr)\,$ is smooth and irreducible of dimension $m-1$. A dense subset is given by the $(m-1)$-dimensional family of $\langle y_1,y_2 \rangle$-primary ideals of the form $$\label{eq:HSfamily}
\qquad
\bigl\langle \,\, y_1^m\,,\, \,y_2 \,+\, a_1 y_1 + a_2 y_1^2 + \cdots + a_{m-1} y_1^{m-1} \,
\bigr\rangle, \qquad
{\rm where}\,\,a_1,a_2,\ldots, a_{m-1} \in {\mathbb{F}}.$$ For instance, for $m=3$, the Hilbert scheme (\[eq:hilbertscheme\]) is a surface over ${\mathbb{F}}$. Each of its points encodes a scheme structure of multiplicity $3$ on the variety $V(P)$. This is the generic point on $V(P)$ together with two “infinitely near points”, in the language of Gröbner and Severi.
To see that the family (\[eq:HSfamily\]) is a proper subset of $\,{\rm Hilb}^m \bigl( {\mathbb{F}}[[y_1,y_2]] \bigr)$, we consider the points $$\qquad \langle\, y_1^3 \,, \,y_2 + \epsilon^{-1} y_1^2 \,\rangle\,\, =\,\,
\langle \,y_1^2 + \epsilon y_2\, , \,y_1y_2\, ,\,y_2^2 \,\rangle \quad \in \,\,\,
{\rm Hilb}^3 \bigl( {\mathbb{F}}[[y_1,y_2]] \bigr).$$ For $\epsilon \in {\mathbb{F}}\backslash \{0\}$, this $\langle y_1,y_2 \rangle $-primary ideal is in the family (\[eq:HSfamily\]), but for $\epsilon = 0$ it is not.
In the zero-dimensional case, when $P = \langle x_1,\ldots,x_n \rangle$, the correspondences in Theorem \[thm:main\] are well-known since the 1930’s. Wolfgang Gröbner tells us: [*Die noch verbleibende Aufgabe, die Integrale eines Primärideals aus denjenigen für das zugehörige Primideal abzuleiten, wollen wir hier wenigstens für null-dimensionale Primärideale allgemein lösen*]{} [@Grobner39 page 272]. In our current understanding, the $P$-primary ideals are points in ${\rm Hilb}^m({\mathbb{K}}[[y_1,\dots,y_n]])$, subspaces closed by differentiation are Macaulay’s inverse systems, and these account for polynomial solutions to linear PDE with constant coefficients [@MOURRAIN_DUALITY; @STURMFELS_SOLVING].
The idea behind Theorem \[thm:main\] is to reduce the study of arbitrary primary ideals in $R = {\mathbb{K}}[x_1,\ldots,x_n]$ to a zero-dimensional setting over the function field ${\mathbb{F}}$. Recall that coordinates were chosen so that $R/P$ is finite over ${\mathbb{K}}[x_{c+1},\ldots,x_n]$. We define the inclusion map $$\label{eq_map_gamma}
\gamma:R \hookrightarrow {\mathbb{F}}[y_1,\dots,y_c]\, , \qquad
\begin{matrix}
x_i &\mapsto & y_i+u_i, & \!\!\!\!\! \mbox{ for }1\leq i\leq c,\\
x_j & \mapsto & u_j,& \quad \mbox{ for }c+1\leq j\leq n,
\end{matrix}$$ where $u_i$ denotes the class of $x_i$ in ${\mathbb{F}}$, for $1\leq i\leq n$. With this, we can give an explicit description of the correspondence between the objects in parts (a) and (b) of Theorem \[thm:main\]: $$\label{eq:corr12}
\begin{array}{ccc}
\left\lbrace\begin{array}{c}
\mbox{$P$-primary ideals of $R$}\\
\mbox{with multiplicity $m$ over $P$}
\end{array}\right\rbrace
& \longleftrightarrow &
\left\lbrace\begin{array}{c}
\mbox{points in }{\rm Hilb}^m({\mathbb{F}}[[y_1,\ldots,y_c]])\\
\end{array}\right\rbrace\\
Q & \longrightarrow & I=\langle y_1,\dots,y_c\rangle^m+\gamma(
Q){\mathbb{F}}[y_1,\dots,y_c]\\
Q\,=\,\gamma^{-1}(I) & \longleftarrow & I.
\end{array}$$
\[Bij:1,2\] Fix $P $ and $Q$ as in the Introduction, with $n=4$, $m=3$, $c=2$, where $R/P$ is finite over ${\mathbb{C} }[x_3,x_4]$. The primary ideal $Q$ corresponds to a point in ${\rm Hilb}^3({\mathbb{F}}[[y_1,y_2]])$. See [@Bri77 Section IV.2] for a detailed description of points in the Hilbert scheme of degree 3 in two variables. The bijection in (\[eq:corr12\]) gives us the following point in the punctual Hilbert scheme: $$\label{eq:punctual2}
I \,\,=\,\,\langle y_2^2,y_1y_2,y_1^2+u_2^{\,-1}y_2\rangle
\,\,\,\subset \,\,{\mathbb{F}}[[y_1,y_2]].$$ Note that this ideal is also generated by $y_1^3$ and $y_2+u_2y_1^2$, as in (\[eq:magic\]).
The bijection between (b) and (c) is Macaulay’s duality between zero-dimensional ideals in a power series ring and finite-dimensional subspaces in a polynomial ring that are closed under differentiation. To interpret polynomials in $I$ as PDE, we replace $y_i$ by $\partial_{z_i}$. So, by slight abuse of notation, we shall write ${\mathbb{F}}[[y_1,\ldots,y_c]]$ and ${\mathbb{F}}[[\partial_{z_1}, \ldots, \partial_{z_c}]]$ interchangeably. With this, the [*inverse system*]{} of a zero-dimensional ideal $I$ in the local ring ${\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]$, denoted by $I^\perp$, is the ${\mathbb{F}}$-vector space of solutions $\,\left\lbrace F\in {\mathbb{F}}[z_1,\ldots,z_c]: f\bullet F=0 \mbox{ for all }f\in I\right\rbrace$.
Inverse systems furnish an explicit bijection between items (b) and (c) in Theorem \[thm:main\]: $$\label{eq:HilbBijection}
\begin{array}{ccc}
\left\lbrace\begin{array}{c}
\mbox{points in }{\rm Hilb}^m\left({\mathbb{F}}[[\partial_{z_1},\ldots,\partial_{z_c}]]\right)\\
\end{array}\right\rbrace
& \longleftrightarrow &
\left\lbrace\begin{array}{c}
\mbox{$m$-dimensional ${\mathbb{F}}$-subspaces} \\
\mbox{of $ {\mathbb{F}}[z_1,\dots,z_c]$} \\ \mbox{closed under differentiation}
\end{array}\right\rbrace\\
I & \longrightarrow & V=I^\perp\\
I={\rm Ann}_{{\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]}(V) & \longleftarrow & V.\\
\end{array}$$
Setting $y_i=\partial_{z_i}$, the ideal in Example \[Bij:1,2\] is $I=\langle\partial_{z_2}^2,\partial_{z_1}\partial_{z_2},\partial_{z_1}^2+u_2^{\, -1}\partial_{z_2}\rangle\subset {\mathbb{F}}[[\partial_{z_1},\partial_{z_2}]]$. Note that $\,z_1^2-2u_2z_2\,$ belongs to the inverse system $I^\perp$ because this polynomial is annihilated by all operators in $I$. Applying the differential operators $\partial_{z_1}$ and $\partial_{z_1}^2$ to $B_3=z_1^2-2u_2z_2$ we obtain an ${\mathbb{F}}$-basis of the inverse system: $B_1=1$, $B_2=z_1$ and $B_3$. Moreover, $I^\perp$ is generated by $B_3$ as an ${\mathbb{F}}[[\partial_{z_1},\partial_{z_2}]]$-module. Hence $I$ is a Gorenstein ideal.
The correspondence between items (c) and (d) in Theorem \[thm:main\] links generators of the inverse system of $I$ with Noetherian operators for $Q$. These will be discussed in depth in Section \[sec3\]. Suppose we are given an ${\mathbb{F}}$-basis $\{B_1,\ldots,B_m\}$ of the inverse system $I^\perp$ in (c). After clearing denominators, we can write $B_i(\mathbf{u},\mathbf{z})=\sum_{\vert\alpha\vert\leq m}\lambda_\alpha(\mathbf{u})\mathbf{z}^\alpha$ where $\lambda_\alpha(\mathbf{u})$ is a polynomial in $R$ that represents a residue class modulo $P$. We now replace the unknown $z_i$ in these polynomials with the differential operator $\partial_{x_i}$. This gives the Noetherian operators $$\label{eq:resultingDO} \qquad
A_i(\mathbf{x},\partial_{x_1},\dots,\partial_{x_c})\,\,\,=\,\,\,\sum_{\vert\alpha\vert\leq m} \lambda_\alpha(\mathbf{x}) \partial_{x_1}^{\alpha_1}\cdots \partial_{x_c}^{\alpha_c}
\qquad {\rm for} \,\,\, i =1,\ldots,m.$$ The transition from the $B_i$’s to the $A_i$’s is invertible, giving the bijection between (c) and (d).
\[ex:noeth\] Consider the ideal $Q$ in (\[eq:twistedcubic6\]) and $I$ in (\[eq:punctual2\]). From the generators $B_1(u,z)=1$, $B_2(u,z)=z_1$ and $B_3(u,z)=z_1^2-2u_2z_2$ of the inverse system $I^\perp$ in ${\mathbb{F}}[z_1,z_2]$, we obtain the three Noetherian operators $A_1=1$, $A_2=\partial_{x_1}$ and $A_3=\partial_{x_1}^2-2x_2\partial_{x_2}$ that encode $Q$. Note that $A_3$ alone does not determine $Q$, although $B_3$ is enough to generate the inverse system.
An Algebraic View on Ehrenpreis-Palamodov {#sec3}
=========================================
In this section we derive the Noetherian differential operators that are central to the Fundamental Principle of Ehrenpreis [@EHRENPREIS] and Palamodov [@PALAMODOV]. In particular, we present a practical algorithm that computes these operators for arbitrary primary ideals in a polynomial ring over a field ${\mathbb{K}}$ of characteristic zero. Our approach extends the algebraic theory in [@BRUMFIEL_DIFF_PRIM; @NOETH_OPS; @OBERST_NOETH_OPS] and the first algorithmic steps taken in [@DAMIANO; @STURMFELS_SOLVING]. For analytic aspects of the Ehrenpreis-Palamodov Theorem we refer to [@EHRENPREIS; @PALAMODOV] and to the books by Björk [@BJORK] and Hörmander [@HORMANDER].
Our point of departure is a prime ideal $P$ in the polynomial ring $R = {\mathbb{K}}[x_1,\ldots,x_n]$. We are interested in $P$-primary ideals. Later on we shall interpret these ideals as systems of linear PDE, by replacing each variable $x_i$ by a differential operator $ \partial_{z_i} = \partial / \partial z_i$. First, however, we take a different path, aimed to turn part (d) in Theorem \[thm:main\] into an algorithm.
After applying Noether normalization, $R/P$ is a finitely generated ${\mathbb{K}}[x_{c+1},\ldots,x_n$\]-module, where $c={\rm codim}(P)$. The relative Weyl algebra $D_{n,c} = {\mathbb{K}}\langle x_1,\ldots,x_n,\partial_{x_1},\ldots,\partial_{x_c} \rangle$ consists of linear differential operators with polynomial coefficients, where only derivatives for the first $c$ variables appear. Every operator $A = A(\mathbf{x}, \partial_\mathbf{x})$ in $D_{n,c}$ is a unique ${\mathbb{K}}$-linear combination of [*normal monomials*]{} $\,\mathbf{x}^\alpha \partial_\mathbf{x}^\beta = x_1^{\alpha_1} \cdots x_n^{\alpha_n} \partial_{x_1}^{\beta_1} \cdots \partial_{x_c}^{\beta_c}$, where $\alpha \in \mathbb{N}^n$, $\beta \in {\mathbb{N}}^c$. We write $A \bullet f$ for the natural action of $D_{n,c}$ on polynomials $f \in R$., which is defined by $$x_i \bullet f = x_i \cdot f \quad {\rm and} \quad \partial_{x_i} \bullet f = \partial f / \partial x_i .$$
Suppose we are given $A_1,\ldots,A_m$ in the relative Weyl algebra $D_{n,c}$. This specifies $$\label{eq:fromAtoQ}
Q \,\,=\,\, \big\{\, f \in R \,: A_l \bullet f \in P \;\text{ for } \,
l = 1,2,\ldots,m \,\big\}.$$ The set $Q$ is a ${\mathbb{K}}$-vector space. However, in general, the subspace $Q$ is not an ideal in $R$.
\[ex:leftright\] Fix $n=m=2$, $P = \langle x_1,x_2 \rangle$ and $A_1 = \partial_{x_1}$. If $A_2 = \partial_{x_2}$ then $Q $ is the space of polynomials $f$ in ${\mathbb{K}}[x_1,x_2]$ such that $x_1$ and $x_2$ do not appear in the expansion of $f$. That space is not an ideal. However, if $A_2 = 1$ then the formula (\[eq:fromAtoQ\]) gives the ideal $\,Q = \langle x_1^2, x_2 \rangle$.
\[rem:contains\_power\] The space $Q$ always contains a power of $P$. Namely, if $k$ is the maximal order among the operators $A_i$ then $P^{k+1} \subseteq Q$. This follows from the product rule of calculus.
We next present a necessary and sufficient condition for $m$ operators in $D_{n,c}$ to specify a primary ideal via (\[eq:fromAtoQ\]). We abbreviate $S= {\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$. The point in (\[eq:leftequalsright\]) below is that the relative Weyl algebra $D_{n,c}$ is both a left $R$-module and a right $R$-module.
\[thm:leftequalsright\] The space $Q$ is a $P$-primary ideal in the polynomial ring $R$ if and only if $$\label{eq:leftequalsright}
\qquad A_i \cdot x_j\,\,\in \,\,
S\cdot \{A_1,\ldots,A_m\} \,+\, PS \cdot D_{n,c}
\qquad \text{for}\,\, \, i =1,\ldots,m \,\,\text{and} \,\,j =1,\ldots,n.$$
In Example \[ex:leftright\] with $\{A_1,A_2\} = \{\partial_{x_1},\partial_{x_2} \}$ we have $ R = S$. Here $Q$ is not an ideal, and (\[eq:leftequalsright\]) fails indeed for $i=j=1$. To see this, one checks that $ \partial_{x_1} x_1 \not\in R\cdot \{\partial_{x_1},\partial_{x_2}\} + \langle x_1,x_2 \rangle D_{2,2}$. It would be desirable to turn the criterion in Theorem \[thm:leftequalsright\] into a general practical algorithm.
Suppose (\[eq:leftequalsright\]) holds and let $f \in Q$. By hypothesis, there exist $\,h_1,\ldots,h_m \in S$ such that $\,A_i x_j \,=\, \sum_{k=1}^m h_k \,A_k \,$ modulo $ PS\cdot D_{n,c}$. Since $A_k \bullet f \in P$, we see that $A_i \bullet (x_j f) = (A_i \,x_j) \bullet f$ lies in $P$ for all $i,j$, and hence $x_j f \in Q$. Thus, $Q$ is an ideal.
Next we show that $Q$ is $P$-primary, by the following direct argument. Let $f,g \in R$ such that $f \cdot g \in Q$ and $g \not\in Q$. We claim that $f \in P$. We select an operator $A$ of minimal order among those inside $S\cdot \{A_1,\ldots,A_m\} + PS \cdot D_{n,c}$ that satisfy $A \bullet g \not\in PS$. The element $\,A \bullet (fg) \,=\, f \cdot (A \bullet g) \,+\, (A f - f A) \bullet g \,$ lies in $PS$. The commutator $A f - f A$ is a differential operator of order smaller than that of $A$. By (\[eq:leftequalsright\]), it is inside $S\cdot \{A_1,\ldots,A_m\} + PS \cdot D_{n,c}$. This ensures that $(A f - f A) \bullet g$ is in $PS$. We conclude that $f \cdot (A \bullet g) \in PS$. But, we know that $A \bullet g$ is not in $PS$, and hence $f$ is in the prime ideal $P$. Remark \[rem:contains\_power\] ensures that $\sqrt{Q}$ contains $ P$. Our argument shows that $Q$ is primary with $\sqrt{Q} = P$. The if-direction follows.
For the only-if-direction we utilize the isomorphism in Remark \[rem\_isom\_restrict\_Weyl\_mod\] and Lemma \[lem\_prim\_ideal\_implies\_bimod\]. The condition (\[eq:leftequalsright\]) is equivalent to the bi-module condition in Lemma \[lem\_prim\_ideal\_implies\_bimod\].
The following result is the key algebraic ingredient in the Ehrenpreis-Palamodov theory.
\[thm\_Noeth\_ops\] For every $P$-primary ideal $Q$ of multiplicity $m$ over $P$, there exist operators $A_1,\ldots,A_m$ in the relative Weyl algebra $D_{n,c}$ such that (\[eq:fromAtoQ\]) holds.
Theorem \[thm\_Noeth\_ops\] follows from Theorem \[thm:main\], to be proved in the next three sections. Indeed, if we are given a $P$-primary ideal $Q$ of multiplicity $m$ over $P$, then $Q$ specifies an $m$-dimensional $R$-bi-module inside the ${\mathbb{F}}$-vector space ${\mathbb{F}}\otimes_R D_{n,c}$. We choose elements $A_1,\ldots,A_m$ in $D_{n,c}$ whose images form an ${\mathbb{F}}$-basis for that $R$-bi-module. These operators satisfy (\[eq:fromAtoQ\]).
Following Palamodov [@PALAMODOV], we call $A_1,\ldots,A_m$ the [*Noetherian operators*]{} that encode the primary ideal $Q$. It is an essential feature that these are linear differential operators with polynomial coefficients. Operators with constant coefficients do not suffice. In other words, the Weyl algebra is essential in describing primary ideals. This key point is due to Palamodov. It had been overlooked initially by Gröbner and Ehrenpreis. For instance, consider the ideal $Q$ for $n=4, m=3$ in the Introduction. Three Noetherian operators $A_1,A_2,A_3$ are given in (\[eq:twistedcubic6\]), and it is instructive to verify condition (\[eq:leftequalsright\]). Algorithms for passing back and forth between Noetherian operators and ideal generators of $Q$ will be presented later in this section.
Our problem is to solve a homogeneous system of linear PDE with constant coefficients. This is given by the generators of a primary ideal $Q$ in ${\mathbb{K}}[x_1,\ldots,x_n]$, where $x_j$ stands for the differential operator $\partial_{z_j} = \partial / \partial z_j$ with respect to a new unknown $z_j$. Our aim is to characterize all sufficiently differentiable functions $\psi(z_1,\ldots,z_n)$ that are solutions to these PDE. This characterization is the content of the Ehrenpreis-Palamodov Theorem, to be stated below. Note that, if we are given an arbitrary system $J \subset R$ of such PDE then we can reduce to the case discussed here by computing a primary decomposition of the ideal $J$.
For the analytic aspects that follow, we work over the field ${\mathbb{K}}= {\mathbb{C} }$ of complex numbers. Suppose $Q = \langle p_1,p_2,\ldots,p_r \rangle$, where $p_k = p_k(\mathbf{x})$. The PDE we need to solve take the form: $$\label{eq:mustsolvethis}
\qquad p_k(\partial_\mathbf{z})\bullet \psi(\mathbf{z})\,\, =\,\, 0 \qquad \text{ for } k=1,2,\ldots,r.$$ Let $\mathcal{K} \subset {\mathbb{R}}^n$ be a compact convex set. We seek all functions $\psi(\mathbf{z})$ in $\, C^\infty(\mathcal{K})\,$ that satisfy (\[eq:mustsolvethis\]). Here we also use vector notation, namely $\mathbf{z} = (z_1,\ldots,z_n)$ and $\partial_\mathbf{z} = (\partial_{z_1},\ldots,\partial_{z_n})$. According to Theorem \[thm\_Noeth\_ops\], there exist Noetherian operators $A_1(\mathbf{x},\partial_\mathbf{x}),\ldots,A_m(\mathbf{x},\partial_\mathbf{x})$ which encode the primary ideal $Q$ in the sense of (\[eq:fromAtoQ\]). In symbols, $\, A_l( \mathbf{x}, \partial_\mathbf{x}) \bullet f \in P\,$ for all $l$.
Each $A_l$ is an element in the relative Weyl algebra $D_{n,c}$, given as a unique ${\mathbb{C} }$-linear combination of normal monomials $\,\mathbf{x}^\alpha \partial_\mathbf{x}^\beta $. This is important since $D_{n,c}$ is non-commutative. We now replace $\partial_\mathbf{x}$ by $\mathbf{z}$ in the normal monomials. This results in commutative polynomials $$\label{eq:thisresults}
B_l (\mathbf{x},\mathbf{z}) \,\, := \,\,
A_l(\mathbf{x},\partial_\mathbf{x})|_{\partial_{x_1} \mapsto z_1,\ldots,
\partial_{x_c} \mapsto z_c}
\qquad {\rm for} \quad l=1,2,\ldots,m .$$ We call $B_1,\ldots,B_m$ the [*Noetherian multipliers*]{} of the primary ideal $Q$. These are polynomial in $n+c$ variables, obtained by reinterpreting the Noetherian (differential) operators. Note that $B_1,\ldots,B_m$ span the inverse system in Theorem \[thm:main\] (c) when viewed inside ${\mathbb{F}}[z_1,\ldots,z_c]$.
The Noetherian operators and Noetherian multipliers in the Introduction are $$\label{eq:NoetMult}
\begin{matrix}
A_1 \,=\, 1\,,\;\, A_2\,=\, \partial_{x_1}
\,\,\,{\rm and} \,\,\,A_3 \,=\, \partial_{x_1}^2 \,-\, 2 \,x_2\,\partial_{x_2}, \\
B_1 \,=\, 1\,,\;\,\, B_2\,=\, z_1\,
\,\,\,{\rm and}\, \,\,\,B_3 \,=\, z_1^2 \,-\, 2 \,x_2\,z_2. \qquad
\end{matrix}$$ We note that this is consistent with (\[eq:twistedcubic5\]) because $\,x_2 = st^2\,$ holds on the variety $\,V(P)$.
Here is now the celebrated result on solutions to linear PDE with constant coefficients:
\[thm:Palamodov\_Ehrenpreis\] Fix the system (\[eq:mustsolvethis\]) of PDE given by the $P$-primary ideal $Q$. Any solution $\psi$ in $C^\infty(\mathcal{K})$ has an integral representation $$\label{eq:anysolution}
\psi(\mathbf{z}) \,\,\,= \,\,\, \sum_{l=1}^m\,\int_{V(P)} \!\! B_l\left(\mathbf{x},\mathbf{z}\right)
\exp\left( \mathbf{x}^t \,\mathbf{z} \right) d\mu_l(\mathbf{x})$$ for suitable measures $\mu_l$ supported in $V(P)$. And, conversely, all such functions are solutions.
We follow the conventions used in analysis (cf. [@BJORK Chapter 8]) and we write our system in terms of the differential operators $D_{z_j} = -i\partial_{z_j}$, where $i=\sqrt{-1}$. We can account for this in the Noetherian multipliers by replacing $\mathbf{x}$ with $-i \mathbf{x}$. It is shown in [@BJORK Theorem 1.3, page 339] that any solution in $C^\infty(\mathcal{K})$ to the system (\[eq:mustsolvethis\]) can be written as $$\psi(\mathbf{z}) \,\,=\,\, \sum_{l=1}^m\int_{V(P)} B_l\left(-i\mathbf{x},\mathbf{z}\right)
\exp\left(-i \mathbf{x}^t \, \mathbf{z} \right) d\mu_l(\mathbf{x}).$$ We can now change variables, by incorporating the multiplication with $-i$ into the measures, to get the formula (\[eq:anysolution\]). Conversely, to see that any such integral $\psi(\mathbf{z})$ is a solution to the PDE (\[eq:mustsolvethis\]) given by $Q$, we differentiate under the integral sign and use the Fourier transform.
Consider the system of PDE determined by the ideal $Q$ in the Introduction. The Noetherian multipliers in (\[eq:NoetMult\]) furnish integral representations for all of its solutions: $$\psi(\mathbf{z}) \,\,= \,\,
\int_{V(P)} \!\!\!\! \exp\left(\mathbf{x}^t \mathbf{z}\right) d\mu_1(\mathbf{x}) \,+\,
\int_{V(P)}\!\!\!\! z_1 \exp\left(\mathbf{x}^t \mathbf{z}\right) d\mu_2(\mathbf{x}) \, +\,
\int_{V(P)} \!\!\!\! (z_1^2-2x_2 z_2) \exp\left(\mathbf{x}^t \mathbf{z}\right) d\mu_3(\mathbf{x}).$$ Here $\mu_1,\mu_2,\mu_3$ are measures supported on the variety $\,V(P) = \bigl\{ \,(s^2t, s t^2,s^3, t^3) \,:\, s,t \in {\mathbb{C} }\,\bigr\}$. The assertion in (\[eq:twistedcubic5\]) is obtained by pulling the integrals back to the $(s,t)$-plane via the parametrization of $V(P)$. This replaces the measures $\mu_i$ by their pull-backs to that plane.
We next present two algorithms for Theorem \[thm\_Noeth\_ops\]. The first is for computing Noetherian operators from the generators of $Q$, and the second for going in the reverse direction. A key ingredient is the map $\gamma$ in (\[eq\_map\_gamma\]) which we encode in the ideal $$\label{eq:weencode}
\big\langle \,x_1-y_1-u_1,\,\ldots\,,\,x_c-y_c-u_c\,,\,\, x_{c+1}-u_{c+1}\,,
\,\ldots\,,\,x_n-u_n \,\bigr\rangle.$$ This technique was used for encoding the differential operators in our running example in (\[eq:magic\]).
\[alg:forward\]\
[Input:]{} Generators $p_1,p_2,\ldots,p_r$ of a $P$-primary ideal $Q$ in $R ={\mathbb{K}}[x_1,\ldots,x_n]$.\
[Output:]{} Elements $A_1,A_2,\ldots,A_m $ in the relative Weyl algebra $D_{n,c}$ that satisfy (\[eq:fromAtoQ\]).\
1. Compute polynomials in ${\mathbb{F}}[y_1,\ldots,y_c]$ that generate the zero-dimensional ideal $I$ in (\[eq:corr12\]).\
2. Using linear algebra over ${\mathbb{F}}$, compute a basis $\{B_1,\ldots,B_m\}$ for the inverse system $I^\perp$.\
3. Lift each $B_i(\mathbf{u},\mathbf{z})$ to obtain the Noetherian multipliers $B_i(\mathbf{x},\mathbf{z})$.\
4. Replace $\mathbf{z}$ by $\partial_\mathbf{x}$ to get the Noetherian operators $A_i(\mathbf{x},\partial_\mathbf{x}) $ in (\[eq:resultingDO\]).
\[alg:backward\]\
[Input:]{} Elements $A_1,A_2,\ldots,A_m $ in the relative Weyl algebra $D_{n,c}$ that satisfy (\[eq:leftequalsright\]).\
[Output:]{} Generators $p_1,p_2,\ldots,p_r$ of a $P$-primary ideal $Q$ that is defined as in (\[eq:fromAtoQ\]).\
1. In each $A_i(\mathbf{x},\partial_\mathbf{x})$ replace $\partial_\mathbf{x}$ by $\mathbf{z}$ to obtain the $m$ Noetherian multipliers $B_i(\mathbf{x},\mathbf{z})$ in (\[eq:thisresults\]).\
2. Replace $\mathbf{x}$ by $\mathbf{u}$ to obtain an ${\mathbb{F}}$-basis $\{B_1,\ldots,B_m\}$ for the inverse system $I^\perp$.\
3. Using $\,{\mathbb{F}}$-linear algebra in ${\mathbb{F}}[y_1,\ldots,y_c]$, find generators for the zero-dimensional ideal $I$.\
4. Add the ideal $I$ to (\[eq:weencode\]) and eliminate $\{y_1,\ldots,y_c,u_1,\ldots,u_n\}$ to obtain generators of $Q$.
We implemented both of these algorithms in [Macaulay2]{}. The code is made available at <https://software.mis.mpg.de>. We hope to develop this further into a [Macaulay2]{} package.
We close this section by presenting a new example that explains the algorithms.
To illustrate Algorithm \[alg:forward\], let $n = 4$ and fix the prime $P = \langle x_1,x_2,x_3 \rangle$ that defines a line in $4$-space ${\mathbb{K}}^4$. The following ideal is $P$-primary of multiplicity $m = 4$: $$Q \,\,=\,\, \bigl\langle\, x_1^2, \,x_1 x_2,\, x_1 x_3, \,x_1 x_4-x_3^2+x_1, \,
x_3^2 x_4-x_2^2, \,x_3^2 x_4-x_3^2-x_2 x_3+2 x_1 \,\bigr\rangle .$$ In Step 1 we replace $x_1,x_2,x_3$ by $y_1,y_2,y_3$ and $x_4$ by $u_4$ to get a zero-dimensional ideal $I$ in ${\mathbb{F}}[y_1,y_2,y_3]$, where ${\mathbb{F}}= {\mathbb{K}}(u_4)$. Note that $I$ contains $\langle y_1,y_2,y_3 \rangle^4$. To check that $I$ is a point in ${\rm Hilb}^4({\mathbb{F}}[[y_1,y_2,y_3]])$, we exhibit a flat deformation to the square of the maximal ideal: $$I \,\,=\,\,\bigl\langle\, y_1^2\,,\,y_1 y_2\,,\, y_1 y_3\, ,\,
y_2^2 -(u_4^2+u_4) \,y_1\,,\,
y_2 y_3 - (u_4^2 + 1) \,y_1\,,\,
y_3^2 - (u_4+1)\, y_1\,\bigr\rangle.$$ The inverse system $I^\perp$ lives in ${\mathbb{F}}[z_1,z_2,z_3]$. It is the $4$-dimensional ${\mathbb{F}}$-vector space with basis $$B_1 \,=\, (u_4^2+u_4) z_2^2 + 2 (u_4^2+1)z_2 z_3 + (u_4+1) z_3^2 +2z_1\,,\,
B_2 = z_2\,,\, B_3 = z_3\,, \,B_4 = 1.$$ Note that this space is closed under differentiation. The Noetherian operators in Step 4 are $$A_1 \,=\,
(x_4^2+x_4) \partial_{x_2}^2 + 2 (x_4^2+1)
\partial_{x_2} \partial_{x_3} + (x_4+1) \partial_{x_3}^2 +2 \partial_{x_1},\,\,
A_2 = \partial_{x_2\,}, \,\, A_3 = \partial_{x_3} \,, \,\, A_4 = 1 .$$ We can now check that these four operators in $D_{4,3}$ represent the given primary ideal: $$Q \,\,= \,\,\bigl\{\,f \in {\mathbb{K}}[x_1,x_2,x_3,x_4]:\, A_i \bullet f\in\langle x_1,x_2,x_3\rangle
\hbox{ for $i=1,2,3,4$} \,\bigr\} .$$
Reversing this entire computation is the point of Algorithm \[alg:backward\]. Starting from the operators $A_1,A_2,A_3,A_4$, we compute the polynomials $B_1,B_2,B_3,B_4$ in ${\mathbb{F}}[z_1,z_2,z_3]$, which span the inverse system $I^\perp$. In Step 3, we find generators of the ideal $I$ in ${\mathbb{F}}[y_1,y_2,y_3$\]. And, finally, from this one obtains generators of $Q$ by the elimination process described in Step 4.
Hilbert Schemes and Inverse Systems {#sec4}
===================================
In this section we provide a proof of the bijections between parts (a), (b) and (c) of Theorem \[thm:main\]. Here the key players are punctual Hilbert schemes and Macaulay’s inverse systems.
We retain the notation from Sections \[sec2\] and \[sec3\], and we write ${\mathfrak{p}}=PS$ for the extension of our prime ideal $P$ in $R = {\mathbb{K}}[x_1,\ldots,x_c,x_{c+1},\ldots,x_n]$ to $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$. By Noether Normalization, we assume that ${\mathbb{K}}[x_{c+1},\ldots,x_n] \hookrightarrow R/P$ is an integral extension, and this implies that ${\mathfrak{p}}$ is a maximal ideal in $S$. Our first goal is to parametrize $P$-primary ideals of fixed multiplicity $m$ over $P$ by the punctual Hilbert scheme ${\rm Hilb}^m\bigl({\mathbb{F}}[[y_1,\ldots,y_c]]\bigr)$. A special role is played by the inclusion map $\gamma:R \hookrightarrow {\mathbb{F}}[y_1,\dots,y_c]$ in (\[eq\_map\_gamma\]). This induces an inclusion $\,\gamma_S : S \hookrightarrow {\mathbb{F}}[y_1,\ldots,y_c]$, also given by $\,x_i \mapsto y_i+u_i $ for $i \leq c\,$ and $\,x_j \mapsto u_j$ for $j > c$.
\[rem\_local\_primary\_ideals\] Since ${\mathbb{K}}[x_{c+1},\ldots,x_n] \cap P=0$, the canonical map $R \hookrightarrow S$ gives a bijection between $P$-primary ideals and ${\mathfrak{p}}$-primary ideals (see, e.g., [@MATSUMURA Theorem 4.1]).
The maximal irrelevant ideal in ${\mathbb{F}}[y_1,\dots,y_c]$ is denoted by ${\mathcal{M}}=\langle y_1,\ldots,y_c\rangle$. For any $f(\mathbf{x})=f(x_1,\ldots,x_n) \in P$, we have $f(\mathbf{u})=f(u_1,\ldots,u_n)=0$ in $ {\mathbb{F}}$. A Taylor expansion yields $$f(\mathbf{u}+\mathbf{y})
\,\,=\,\,f(u_1+y_1,\ldots,u_c+y_c,u_{c+1},\ldots,u_n)
\,\,=\,\, \sum_{\substack{\lambda \in {\mathbb{N}}^c\\\lvert\lambda \rvert > 0}} \frac{\partial^{\lvert\lambda\rvert} f}{\partial_{x_1}^{\lambda_1}\cdots\partial_{x_c}^{\lambda_c}}(\mathbf{u})\,\mathbf{y}^\lambda.$$ This shows that $\gamma(P) \subseteq {\mathcal{M}}$, and therefore $\gamma_S( {\mathfrak{p}}) \subseteq {\mathcal{M}}$. The next proposition establishes a bijection between ${\mathfrak{p}}$-primary ideals containing ${\mathfrak{p}}^m$ and ${\mathcal{M}}$-primary ideals containing ${\mathcal{M}}^m$.
\[prop\_corespondence\_primary\_ideals\] For all $m \ge 1$, the inclusion $\gamma_S$ induces the isomorphism of local rings $$S/{\mathfrak{p}}^m \,\,\xrightarrow{\cong}\,\, {\mathbb{F}}[y_1,\ldots,y_c]/{\mathcal{M}}^m.$$
This result has also appeared in [@BRUMFIEL_DIFF_PRIM Proposition 4.1] and [@NOETH_OPS Proposition 3.9]. In these sources it was assumed that ${\mathbb{K}}$ is a perfect field. This holds here since ${\rm char}({\mathbb{K}}) = 0$.
\[rem\_ident\_Hilb\] (i) Any ideal of colength $m$ in ${\mathbb{F}}[[y_1,\ldots,y_c]]$ contains the ideal $\langle y_1,\ldots,y_c \rangle^m $. Therefore, ${\rm Hilb}^m\bigl({\mathbb{F}}[[y_1,\ldots,y_c]]\bigr)$ can be identified with $\,{\rm Hilb}^m\bigl({\mathbb{F}}[[y_1,\ldots,y_c]]/ \langle y_1,\ldots,y_c \rangle^m \bigr). $ (ii) Any $ \langle y_1,\ldots,y_c\rangle$-primary ideal of colength $m$ in the polynomial ring ${\mathbb{F}}[y_1,\ldots,y_c]$ contains the ideal $\langle y_1,\ldots,y_c \rangle^m \subset {\mathbb{F}}[y_1,\ldots,y_c]$. For all $m>0$, we have the natural isomorphism $$\frac{{\mathbb{F}}[[y_1,\ldots,y_c]]}{{ \langle y_1,\ldots,y_c \rangle}^m}
\,\,\cong \,\,\frac{{\mathbb{F}}[y_1,\ldots,y_c]}{{ \langle y_1,\ldots,y_c \rangle }^m}.$$ Therefore, the $\langle y_1,\ldots,y_c \rangle$-primary ideals of colength $m$ in ${\mathbb{F}}[y_1,\ldots,y_c]$ are parametrized by ${\rm Hilb}^m \bigl( {\mathbb{F}}[[y_1,\ldots,y_c]] \bigr)$. From now on, $\langle y_1,\ldots,y_c \rangle $-primary ideals in the polynomial ring ${\mathbb{F}}[y_1,\ldots,y_c]$ will automatically be identified with ideals in the power series ring ${\mathbb{F}}[[y_1,\ldots,y_c]]$.
Now we are ready to prove the correspondence between parts (a) and (b) in Theorem \[thm:main\].
\[thm:param\_primary\] As asserted in (\[eq:corr12\]), there is a bijective correspondence $$\begin{array}{ccc}
\left\lbrace\begin{array}{c}
\mbox{$P$-primary ideals of $R$}\\
\mbox{with multiplicity $m$ over $P$}
\end{array}\right\rbrace
& \longleftrightarrow &
\left\lbrace\begin{array}{c}
\mbox{points in }{\rm Hilb}^m({\mathbb{F}}[[y_1,\ldots,y_c]])\\
\end{array}\right\rbrace\\
Q & \longrightarrow & I=\langle y_1,\dots,y_c\rangle^m+\gamma(
Q){\mathbb{F}}[y_1,\dots,y_c]\\
Q=\gamma^{-1}(I) & \longleftarrow & I.
\end{array}$$
The canonical map $R \hookrightarrow S$ gives a bijection between $P$-primary ideals and ${\mathfrak{p}}$-primary ideals (Remark \[rem\_local\_primary\_ideals\]). Also, for any $P$-primary ideal $Q \subset R$ we have $R_P/QR_P \cong S_{{\mathfrak{p}}}/Q S_{{\mathfrak{p}}}$. So, nothing is changed if we take $S$ and ${\mathfrak{p}}$ instead of $R$ and $P$. We have the commutative diagram
\(m) \[matrix of math nodes,row sep=3em,column sep=9em,minimum width=1.7em, text height=1.5ex, text depth=0.25ex\] [ S & \[\]\
S/\^[m]{} & \[\]/\^[m]{}.\
]{}; (m-1-1) edge node \[above\] [$\gamma_S$]{} (m-1-2) (m-2-1) edge node \[above\] [$\cong$]{} (m-2-2) ; (m-1-1) – (m-2-1); (m-1-2) – (m-2-2);
The map in the bottom row is the isomorphism in Proposition \[prop\_corespondence\_primary\_ideals\]. This gives an inclusion-preserving bijection between ${\mathfrak{p}}$-primary ideals containing ${\mathfrak{p}}^m$ and ${\mathcal{M}}$-primary ideals containing ${\mathcal{M}}^{m}$, in particular, colength does not change under this correspondence. In explicit terms, the ${\mathcal{M}}$-primary ideal $I$ corresponding to a ${\mathfrak{p}}$-primary ideal $QS \supseteq {\mathfrak{p}}^m$ is $$I \,\,=\,\, {\mathcal{M}}^{m} \,+\, \gamma_S(QS)\big({\mathbb{F}}[\mathbf{y}]\big).$$ And, the ${\mathfrak{p}}$-primary ideal $QS$ corresponding to an ${\mathcal{M}}$-primary ideal $I \supseteq {\mathcal{M}}^m$ is given by $$QS \,\,=\,\, \gamma_S^{-1}(I).$$ Finally, the result now follows from Remark \[rem\_ident\_Hilb\].
We next show the correspondence between parts (b) and (c) in Theorem \[thm:main\]. This follows from the usual Macaulay duality. Although this argument is well-known, we will need a short discussion to later connect parts (c) and (d) of Theorem \[thm:main\]. Consider the injective hull $E=E_{{\mathbb{F}}[[y_1,\ldots,y_c]]}({\mathbb{F}})$ of the residue field ${\mathbb{F}}\cong {\mathbb{F}}[[y_1,\ldots,y_c]]/\langle y_1,\ldots,y_c \rangle$ of ${\mathbb{F}}[[y_1,\ldots,y_c]]$. Since ${\mathbb{F}}[[y_1,\ldots,y_c]]$ is a formal power series ring, this equals the module of inverse polynomials: $$\label{eq_isom_E_inv_sys}
E \,\,\cong \,\, {\mathbb{F}}[y_1^{-1},\ldots,y_c^{-1}].$$ For a derivation see e.g. [@Brodmann_Sharp_local_cohom Lemma 11.2.3, Example 13.5.3] or [@BRUNS_HERZOG Theorem 3.5.8].
Consider the polynomial ring ${\mathbb{F}}[z_1,\ldots,z_c]$ as an ${\mathbb{F}}[[y_1,\ldots,y_c]]$-module by setting that $y_i$ acts on ${\mathbb{F}}[z_1,\ldots,z_c]$ as $\partial_{z_i}$, that is, $y_i \cdot F = \partial_{z_i} \bullet F$ for any $F \in {\mathbb{F}}[z_1,\ldots,z_c]$. Since the field ${\mathbb{F}}$ has characteristic zero, we have the following isomorphism of ${\mathbb{F}}[[y_1,\ldots,y_c]]$-modules $$\label{eq_isom_inv_sys_char_zero}
{\mathbb{F}}[y_1^{-1},\ldots,y_c^{-1}] \;\xrightarrow{\cong}\; {\mathbb{F}}[z_1,\ldots,z_c], \quad \frac{1}{\mathbf{y}^{\alpha}} \;\mapsto\; \frac{\mathbf{z}^\alpha}{\alpha!}.$$
Now, Macaulay’s duality is simply performed via Matlis duality. We use ${\left(-\right)}^\vee$ to denote Matlis dual ${\left(-\right)}^\vee={{\normalfont\text{Hom}}}_{{\mathbb{F}}[[y_1,\ldots,y_c]]}\left(-,E\right)$. This is a contravariant exact functor which establishes an anti-equivalence between the full-subcategories of artinian ${\mathbb{F}}[[y_1,\ldots,y_c]]$-modules and finitely generated ${\mathbb{F}}[[y_1,\ldots,y_c]]$-modules (see, e.g., [@BRUNS_HERZOG Theorem 3.2.13]).
For any zero-dimensional ideal $I $ in the power series ring $
{\mathbb{F}}[[y_1,\ldots,y_c]]$, the isomorphisms (\[eq\_isom\_E\_inv\_sys\]) and (\[eq\_isom\_inv\_sys\_char\_zero\]) together with Matlis duality yield the following identifications: $$I^{\perp} \,\,=\,\, \left\lbrace F\in {\mathbb{F}}[z_1,\ldots,z_c]: f\bullet F=0 \mbox{ for all }
f\in I\right\rbrace \,\,\cong \,\, (0 :_{E} I) \,\,\cong \,\, \bigl({\mathbb{F}}[[y_1,\ldots,y_c]]/I \bigr)^\vee.$$ On the other hand, consider any ${\mathbb{F}}[[y_1,\ldots,y_c]]$-submodule $V$ of $ {\mathbb{F}}[z_1,\ldots,z_c] \cong E$. Then $V$ is an ${\mathbb{F}}$-subspace of ${\mathbb{F}}[z_1,\ldots,z_c]$ that is closed by differentiation, as $y_i$ is identified with the operator $\partial_{z_i}$. Again, the isomorphisms (\[eq\_isom\_E\_inv\_sys\]) and (\[eq\_isom\_inv\_sys\_char\_zero\]) with Matlis duality give identifications $${\rm Ann}_{{\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]}(V) \,\, \cong \,\,
{\rm Ann}_{{\mathbb{F}}[[y_1,\ldots,y_c]]}(V) \, \, \cong \,\,
{\bigl( E /V \bigr)}^{\vee} \, \,\subset \,\, {\mathbb{F}}[[y_1,\ldots,y_c]].$$ Hence, from the above discussions, we get the connection between (b) and (c) in Theorem \[thm:main\].
\[thm:Macaulay\_dual\] As asserted in (\[eq:HilbBijection\]), there is a bijective correspondence $$\begin{array}{ccc}
\left\lbrace\begin{array}{c}
\mbox{points in }{\rm Hilb}^m\left({\mathbb{F}}[[\partial_{z_1},\ldots,\partial_{z_c}]]\right)\\
\end{array}\right\rbrace
& \longleftrightarrow &
\left\lbrace\begin{array}{c}
\mbox{$m$-dimensional ${\mathbb{F}}$-subspaces of}\\
\mbox{${\mathbb{F}}[z_1,\dots,z_c]$ closed by differentiation}
\end{array}\right\rbrace\\
I & \longrightarrow & V=I^\perp\\
I={\rm Ann}_{{\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]}(V) & \longleftarrow & V.\\
\end{array}$$
Differential Operators Revisited {#sec5}
================================
In this section we review basic material on differential operators in commutative algebra. This is used in Section \[sec6\] to complete the proof of Theorem \[thm:main\]. Even though the Noetherian operators $A_i$ live in the Weyl algebra, we need the abstract perspective to link them to the Weyl-Noether module (\[eq:relativeweyl\]). As before, ${\mathbb{K}}$ is a field of characteristic zero and $R={\mathbb{K}}[x_1,\ldots,x_n]$.
For two $R$-modules $M$ and $N$, we regard ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$ as an $(R\otimes_{\mathbb{K}}R)$-module, by setting $$\left((r \otimes_{\mathbb{K}}s) \delta\right)(w)\,\, =\,\, r \delta(sw) \quad \text{ for all }\,\, \delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N), \; w \in M,\; r,s \in R.$$ This is equivalent to saying that ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M,N)$ is an $R$-bi-module, where the action on the left is given by post-composing $(r \cdot \delta)(w)=r\delta(w)$ and the action on the right is given by pre-composing $(\delta \cdot s)(w)=\delta(sw)$, for all $\delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N), \; w \in M,\; r,s \in R$. We use the bracket notation $[\delta,r](w) = \delta(rw)-r\delta(w)$ for all $\delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$, $r \in R$ and $w \in M$.
\[nota\_T\_mod\_struct\] We write $T=R \otimes_{\mathbb{K}}R = {\mathbb{K}}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ as a polynomial ring in $2n$ variables, where $x_i$ represents $x_i \otimes_{\mathbb{K}}1$ and $y_i$ represents $1 \otimes_{\mathbb{K}}x_i - x_i \otimes_{\mathbb{K}}1$. The action of $T$ on ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M,N)$ is thus given as follows. For all $\delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$ and $w \in M$, we have $$(x_i \cdot \delta) (w) = x_i \delta(w) \quad \text{ and } \quad (y_i \cdot \delta) (w) = \delta(x_i w) - x_i\delta(w) = \left[\delta,x_i\right](w) \qquad {\rm for} \,\,i=1,\ldots,n.$$
Any $T$-module is regarded as an $R$-module via the canonical map $R \hookrightarrow T, x_i \mapsto x_i $. Thus, any $T$-module is given an $R$-module structure by using the left factor $R\otimes_{\mathbb{K}}1 \subset T = R \otimes_{\mathbb{K}}R$. The ${\mathbb{K}}$-linear differential operators form a $T$-submodule of ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$, defined as follows.
\[def\_diff\_ops\] Let $M, N$ be $R$-modules. The *$m$-th order ${\mathbb{K}}$-linear differential operators ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(M, N) \subseteq {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$ from $M$ to $N$* form a $T$-module that is defined inductively by:
(i) ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{0}(M,N) \,:=\, {{\normalfont\text{Hom}}}_R(M,N)$.
(ii) ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}(M, N) \,:= \,
\big\lbrace \delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M,N) : \,[\delta, r] \in {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(M, N)
\,\text{ for all }\, r \in R \big\rbrace$.
The set of all *${\mathbb{K}}$-linear differential operators from $M$ to $N$* is the $T$-module $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(M, N) \,\,:=\,\, \bigcup_{m=0}^\infty {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(M,N).$$ Subsets $\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(M, N)$ are viewed as differential equations. Their solutions spaces are $$\label{eq:solE}
{\rm Sol}(\mathcal{E}) \,\,:= \,\,\big\lbrace w \in M : \delta(w) = 0 \text{ for all } \delta \in \mathcal{E} \big\rbrace
\,\,= \,\,\bigcap_{\delta \in \mathcal{E} } {\rm Ker}(\delta).$$
Following the approach in [@NOETH_OPS Section 2], we now introduce the module of principal parts. By construction, the ideal $ \Delta_{R/{\mathbb{K}}} = \langle y_1, \ldots,y_n \rangle$ in $T$ is the kernel of the multiplication map $$T = R \otimes_{\mathbb{K}}R \,\rightarrow \,R\,, \quad r \otimes_{\mathbb{K}}s \,\mapsto\, rs.$$
Let $M$ be an $R$-module. The module of *$m$-th principal parts of $M$* equals $$P_{R/{\mathbb{K}}}^m(M) \,\,:=\,\, \frac{R \otimes_{\mathbb{K}}M}{\Delta_{R/{\mathbb{K}}}^{m+1} \left(R \otimes_{\mathbb{K}}M\right)}.$$ This is a $T$-module. It comes with the natural map $\,d^m : \,M \rightarrow P_{R/{\mathbb{K}}}^m(M),\,
w \mapsto \overline{1 \otimes_{\mathbb{K}}w}$. In the special case $M=R$ we abbreviate $\,P_{R/{\mathbb{K}}}^m \,:=\, P_{R/{\mathbb{K}}}^m(R)=T/\Delta_{R/{\mathbb{K}}}^{m+1}$, and the map becomes $$\label{eq_univ_diff}
d^m : R \rightarrow P_{R/{\mathbb{K}}}^m, \;\;x_i \, \mapsto \,\overline{1 \otimes_{\mathbb{K}}x_i} \,=\, \overline{x_i+y_i} .$$
The following proposition offers a fundamental characterization of differential operators.
\[prop\_represen\_diff\_opp\] Let $m\ge 0$ and let $M, N$ be $R$-modules. Then, the following map is an isomorphism of $R$-modules: $$\begin{aligned}
{\left(d^m\right)}^*\, :\, {{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m(M), N\right)
&\,\,\xrightarrow{\cong} \,\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(M, N), \\
\varphi \quad &\,\,\mapsto \,\quad \varphi \circ d^m.
\end{aligned}$$
This is a very general result for commutative rings $R$. What we are interested in here is the polynomial ring $R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field ${\mathbb{K}}$ of characteristic zero. In this case, the $R$-module $P_{R/{\mathbb{K}}}^m=T/\Delta_{R/{\mathbb{K}}}^{m+1}$ is free, and a basis is given by $\bf y$-monomials of degree at most $m$: $$\label{eq_direct_sum_Prin}
P_{R/{\mathbb{K}}}^m \,\,\,= \,\,\bigoplus_{\lvert \alpha \rvert \le m} R\mathbf{y}^\alpha \quad =
\bigoplus_{\alpha_1 + \cdots + \alpha_r \le m} \!\!\!\! R y_1^{\alpha_1}\cdots y_n^{\alpha_n}.$$ Proposition \[prop\_represen\_diff\_opp\] implies that $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R)\, \cong\,
{{\normalfont\text{Hom}}}_R\bigl(P_{R/{\mathbb{K}}}^m, R\bigr)\,$ is a free $R$-module with basis $$\label{eq_basis_diff_ops}
\big\{ {(y_1^{\alpha_1}\cdots y_n^{\alpha_n})}^* \circ d^m : \alpha_1 + \cdots + \alpha_n \le m \big\}.$$ For any polynomial $f(\mathbf{x})$ in $ R$, the operator $d^m$ in (\[eq\_univ\_diff\]) computes the Taylor expansion $$d^m(f(\mathbf{x})) \,\,= \,\,f(1\otimes_{\mathbb{K}}\mathbf{x}) \,\,= \,\,f(\mathbf{x}+\mathbf{y}) \,\,=\,\,
\sum_{\lambda \in {\mathbb{N}}^n} \left(D_\mathbf{x}^\lambda f\right)\!(\mathbf{x})\,\mathbf{y}^\lambda,$$ where $\,D_\mathbf{x}^{\lambda}:R\rightarrow R\,$ is the differential operator we all know from calculus: $$D_\mathbf{x}^{\lambda}\,\, =\,\, \frac{1}{\lambda!}\partial_\mathbf{x}^\lambda \,\,= \,\,
\frac{1}{\lambda_1!\cdots \lambda_n!}\partial_{x^1}^{\lambda_1}\cdots \partial_{x^n}^{\lambda_n}.$$ For any $\alpha \in {\mathbb{N}}^n$ we thus have $\,
\left({(\mathbf{y}^\alpha)}^* \circ d^m\right)(f(\mathbf{x})) = \left(D_\mathbf{x}^\alpha f\right)(\mathbf{x})$. The equation (\[eq\_basis\_diff\_ops\]) implies $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R)\,\, =\,\,\bigoplus_{\lvert \alpha \rvert \le m} R D_\mathbf{x}^\alpha
\,\,=\,\, \bigoplus_{\lvert \alpha \rvert \le m} R \partial_\mathbf{x}^\alpha.$$ By letting $m$ go to infinity, we now recover the Weyl algebra in its well-known role:
\[lem\_Weyl\_as\_diff\_ops\] $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R, R)$ coincides with the Weyl algebra ${\mathbb{K}}\langle x_1,\ldots,x_n,\partial_{x_1},\ldots,\partial_{x_n} \rangle$.
Let $J $ be an ideal in $R= {\mathbb{K}}[x_1,\ldots,x_n]$. The canonical projection $\pi : R \rightarrow R/J$ induces a natural map of differential operators. This is the following homomorphism of $T$-modules: $$\label{eq:pimap}
{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi) \,: \,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R) \,\rightarrow \,
{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R/J), \quad \delta \,\mapsto\, \pi \circ \delta.$$
\[lem\_diff\_ops\_R/J\] We have the following explicit description of the objects in (\[eq:pimap\]):
(i) ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R/J)$ is a free $R/J$-module with direct summands decomposition $$\qquad
{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J) \,\,= \,\,\bigoplus_{\lvert \alpha \rvert \le m} (R/J) \overline{D_\mathbf{x}^\alpha}, \qquad {\rm where}\,\,\,
\overline{D_\mathbf{x}^\alpha} = \pi \circ D_\mathbf{x}^\alpha.$$
(ii) The map ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi)$ is surjective. Explicitly, any differential operator $$\epsilon \,\,=\, \sum_{\lvert \alpha \rvert \le m} \overline{r_\alpha}
\overline{D_\mathbf{x}^\alpha} \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J), \quad \text{ where } r_\alpha \in R,$$ can be lifted to an operator $\,\delta=\sum_{\lvert \alpha \rvert \le m} r_\alpha D_\mathbf{x}^\alpha \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R)\,$ with $\,\epsilon = {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi)(\delta)$.
$(i)$ From Proposition \[prop\_represen\_diff\_opp\] and the Hom-tensor adjunction we obtain the isomorphisms $$\begin{aligned}
\label{eq_isoms_Diff_R/J}
\begin{split}
{{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_R P_{R/{\mathbb{K}}}^m, R/J\right) &
\,\,\cong \,\, {{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m, {{\normalfont\text{Hom}}}_{R/J}\left(R/J, R/J\right)\right)\\
& \,\,\cong \,\,{{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m, R/J\right)\\
&\,\,\cong \,\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J).
\end{split}
\end{aligned}$$ The isomorphism from the first row to the second row in (\[eq\_isoms\_Diff\_R/J\]) is given by $$\psi \in {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_R P_{R/{\mathbb{K}}}^m, R/J\right) \;\;\mapsto \;\;\psi\circ h_m \in {{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m, R/J\right),$$ where $\,h_m\,$ is the canonical map $\, P_{R/{\mathbb{K}}}^m \rightarrow R/J \otimes_R P_{R/{\mathbb{K}}}^m$. Therefore, the isomorphism from the first to the third row in (\[eq\_isoms\_Diff\_R/J\]) is given explicitly as $ \,\psi\, \mapsto \, \psi \circ h_m \circ d^m $. By using equation (\[eq\_direct\_sum\_Prin\]) we get that $R/J \otimes_R P_{R/{\mathbb{K}}}^m$ is a free $R/J$-module with direct summands decomposition $$R/J \otimes_R P_{R/{\mathbb{K}}}^m \,\,=
\,\, \bigoplus_{\lvert \alpha \rvert \le m} (R/J)\mathbf{y}^\alpha.$$ Our explicit isomorphism for (\[eq\_isoms\_Diff\_R/J\]) shows that ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J)$ is a free $R/J$-module with basis $$\big\{ {(y_1^{\alpha_1}\cdots y_n^{\alpha_n})}^* \circ h_m \circ d^m : \alpha_1 + \cdots + \alpha_n \le m \big\}.$$ Now, for any polynomial $f(\mathbf{x})$ in $ R$, we obtain the equations $$\begin{aligned}
\label{eq_diff_opp_z_alpha}
\begin{split}
\left({(\mathbf{y}^\alpha)}^* \circ h_m \circ d^m\right)(f(\mathbf{x})) &\,\,
= \,\,\left({(\mathbf{y}^\alpha)}^* \circ h_m\right)\left(\sum_{\lambda \in {\mathbb{N}}^n} \left(D_\mathbf{x}^\lambda f\right)(\mathbf{x})\mathbf{y}^\lambda\right)\\
&\,\,=\,\, \left({(\mathbf{y}^\alpha)}^*\right)\left(\sum_{\lambda \in {\mathbb{N}}^n} \pi\left(\left(D_\mathbf{x}^\lambda f\right)(\mathbf{x})\right)\mathbf{y}^\lambda\right)
\,\,=\,\,\,\pi\big(\left(D_\mathbf{x}^\alpha f\right)(\mathbf{x})\big).
\end{split}
\end{aligned}$$ This implies that the operators $\overline{D_\mathbf{x}^\alpha}=\pi \circ D_\mathbf{x}^\alpha$ with $\lvert\alpha\rvert \le m$ give a basis of ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R/J)$. Part $(ii)$ follows directly from part $(i)$. This concludes the proof of Lemma \[lem\_diff\_ops\_R/J\].
Since $R$ is a polynomial ring, the process of lifting differential operators is easy and explicit. However, the surjectivity of ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi)$ is a subtle property, and it is not always satisfied over more general types of rings. For an illustration see [@NOETH_OPS Example 5.2].
Proof of the Representation Theorem {#sec6}
===================================
We here finish the proof of Theorem \[thm:main\] by connecting part (d) with parts (a), (b), and (c). The section is divided into two subsections. In the first one, we treat the zero-dimensional situation, where $c=n$. In the second one, we use Noether normalization and the results on differential operators in Section \[sec5\] to reduce the general case to the zero-dimensional case.
The zero-dimensional case
-------------------------
We here restrict ourselves to ideals in $R= {\mathbb{K}}[x_1,\ldots,x_n]$ that are primary to a maximal ideal $P$. Hence $c=n$ and ${\mathbb{F}}=R/P$. Since the base field ${\mathbb{K}}$ is assumed to have characteristic zero, an adaptation of Gröbner’s classical approach via Macaulay’s inverse system will be valid.
Using the notation $\,T=R\otimes_{\mathbb{K}}R={\mathbb{K}}[x_1,\ldots,x_n,y_1,\ldots,y_n]\,$ from Section \[sec5\], we now have $$\label{eq:FFRT}
{\mathbb{F}}\otimes_R T \,\,=\,\, {\mathbb{F}}\otimes_R \left(R\otimes_{\mathbb{K}}R\right) \,\,\cong\,\,
R/P \otimes_{\mathbb{K}}{\mathbb{K}}[y_1,\ldots,y_n] \,\,\cong\,\, {\mathbb{F}}[y_1, \ldots, y_n].$$ This endows ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}\left(R, {\mathbb{F}}\right)$ with the structure of an ${\mathbb{F}}[y_1,\ldots,y_n]$-module. Applying Lemma \[lem\_diff\_ops\_R/J\] with $J=P$, we see that ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}\left(R, {\mathbb{F}}\right)$ is a finite-dimensional ${\mathbb{F}}$-vector space. In the sequel, the irrelevant maximal ideal $
{\mathcal{M}}= \langle y_1,\ldots,y_n \rangle \subset {\mathbb{F}}[y_1,\ldots,y_n]
$ will play an important role. This ideal is also given as ${\mathcal{M}}=\Delta_{R/{\mathbb{K}}} \left( {\mathbb{F}}\otimes_R T \right)$. For any $m \ge 0$ we identify $$\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{{\mathcal{M}}^{m+1}} \,\,\,=\,\,
\bigoplus_{\vert\alpha\rvert \le m} {\mathbb{F}}\mathbf{y}^{\alpha}.$$ For any ${\mathbb{F}}[y_1,\ldots,y_n]$-module $M$, the ${\mathbb{F}}$-dual ${{\normalfont\text{Hom}}}_{{\mathbb{F}}}(M, {\mathbb{F}})$ is naturally an ${\mathbb{F}}[y_1,\ldots,y_n]$-module as follows: if $\psi \in {{\normalfont\text{Hom}}}_{{\mathbb{F}}}(M, {\mathbb{F}})$, then $y_i \cdot \psi$ is the ${\mathbb{F}}$-linear map $\,\psi(y_i \cdot -) : w \in M \mapsto \psi(y_iw) \in {\mathbb{F}}$.
The next result relates submodules of ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}\left(R, {\mathbb{F}}\right)$ to ${\mathcal{M}}$-primary ideals in ${\mathbb{F}}[y_1,\ldots,y_n]$.
\[lem\_descrip\_diff\_opp\] The following statements hold for all positive integers $m$:
(i) We have an isomorphism of ${\mathbb{F}}[y_1,\ldots,y_n]$-modules $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}\left(R, {\mathbb{F}}\right)\,\, \cong \,\,
{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\Bigg(\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{{{\mathcal{M}}}^{m}}, {\mathbb{F}}\Bigg).$$
(ii) The following map gives a bijective correspondence between ${\mathcal{M}}$-primary ideals $I $ in $ {\mathbb{F}}[y_1,\ldots,y_n]$ that contain ${\mathcal{M}}^{m}$ and ${\mathbb{F}}[y_1,\ldots,y_n]$-submodules of $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}\left(R,{\mathbb{F}}\right)$: $$\label{eq:ImapHom}
I \;\mapsto\; {{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{I}, {\mathbb{F}}\right).$$
(iii) Let $\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$ be the image under (\[eq:ImapHom\]) of an ${\mathcal{M}}$-primary ideal $I \supseteq {\mathcal{M}}^{m}$. Then $${\rm Sol}(\mathcal{E}) \,\,=\,\, \gamma^{-1}(I),$$ with notation as in (\[eq:solE\]), where $\gamma$ is the inclusion $ R \hookrightarrow {\mathbb{F}}[y_1,\ldots,y_n], x_i \mapsto y_i{+}u_i$ in (\[eq\_map\_gamma\]).
This is essentially [@NOETH_OPS Lemma 3.8]. We provide a proof for the sake of completeness.
$(i)$ Since ${\mathbb{F}}=R/P$, from equation (\[eq\_isoms\_Diff\_R/J\]) we obtain the isomorphism $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R, {\mathbb{F}})\,\, \cong \,\,{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\bigl({\mathbb{F}}\otimes_R P_{R/{\mathbb{K}}}^{m-1}, {\mathbb{F}}\bigr).$$ Thus, the result follows from the fact that $\,{\mathbb{F}}\otimes_R P_{R/{\mathbb{K}}}^{m-1} \,\cong\, {\mathbb{F}}[\mathbf{y}]/ {\mathcal{M}}^{m} $.
$(ii)$ Since ${\mathbb{F}}[\mathbf{y}]/{\mathcal{M}}^{m}$ is a finite-dimensional vector space over ${\mathbb{F}}$, the functor ${{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(-,{\mathbb{F}}\right)$ gives a bijection between quotients of ${\mathbb{F}}[\mathbf{y}]/{\mathcal{M}}^{m}$ and ${\mathbb{F}}[\mathbf{y}]$-submodules of ${{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^{m}},{\mathbb{F}}\right)}$. So, the claim follows from $(i)$.
$(iii)$ By assumption, $\,\mathcal{E} \,=\,{{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I},{\mathbb{F}}\right)} \,$ is in $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$. Consider the canonical map $ \,\Phi_I : \frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^{m}} \twoheadrightarrow \frac{{\mathbb{F}}[\mathbf{y}]}{I}\,$ given by the $\mathcal{M}$-primary ideal $I \supseteq {\mathcal{M}}^{m}$. From the isomorphism (\[eq\_isoms\_Diff\_R/J\]) we get $${\rm Sol}(\mathcal{E}) \,\,=\,\,
\bigl\{ f \in R : \left(\psi \circ \Phi_I \circ h_{m-1} \circ d^{m-1}\right)(f)=0
\text{ for all } \psi \in {{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left({\mathbb{F}}[\mathbf{y}]/I ,{\mathbb{F}}\right)} \bigr\}.$$ The composition $\Phi_I \circ h_{m-1} \circ d^{m-1}$ coincides with the map $\,R \mapsto {\mathbb{F}}[\mathbf{y}]/ I,\, x_i \mapsto \overline{y_i+u_i}$. Hence $$\begin{aligned}
{\rm Sol}(\mathcal{E}) &\,\,=\,\, \bigl\{\,f \in R :
\psi\bigl(\,\overline{f(
\mathbf{y}+\mathbf{u}})\, \bigr)=0\, \text{ for all } \,\psi \in {{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left({\mathbb{F}}[\mathbf{y}]/ I ,{\mathbb{F}}\right)}\bigr\} \\
&\,\,=\,\, \bigl\{\, f \in R : f\bigl(\mathbf{y}+\mathbf{u}\bigr) \in I \, \bigr\}
\,\,=\,\, \gamma^{-1}(I).
\end{aligned}$$ This completes the proof of Proposition \[lem\_descrip\_diff\_opp\].
Next, under the assumption of $P$ being maximal, we relate part (d) with the other parts in Theorem \[thm:main\]. By Definition \[def\_diff\_ops\] and Lemma \[lem\_Weyl\_as\_diff\_ops\], the Weyl-Noether module has the filtration $${\mathbb{F}}\,\otimes_R \, R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle \,\;= \;\, {\mathbb{F}}\, \otimes_R \, \biggl(\lim\limits_{\substack{\longrightarrow\\m}} {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R) \biggr) \,\;\cong \; \,\lim\limits_{\substack{\longrightarrow\\m}}\Big({\mathbb{F}}\otimes_R {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R)\Big).$$ Applying Lemma \[lem\_diff\_ops\_R/J\] with $J=P$ gives ${\mathbb{F}}\otimes_R {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R) \cong {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,{\mathbb{F}}) \cong \bigoplus_{\vert\alpha\rvert\le m}{\mathbb{F}}\overline{\partial_{\mathbf{x}}^\alpha}$. This gives rise to the following isomorphisms of ${\mathbb{F}}$-vector spaces: $$\label{eq_isom_relWeyl_diff}
{\mathbb{F}}\,\otimes_R \, R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle \;\cong \; {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}}) \;\cong \; \bigoplus_{ \alpha \in {\mathbb{N}}^n} {\mathbb{F}}\overline{\partial_\mathbf{x}^\alpha}.$$ When the Weyl-Noether module was introduced in (\[eq:relativeweyl\]), we gave a purely algebro-symbolic treatment and we noticed that an ${\mathbb{F}}$-basis is given by $\,\left\lbrace 1 \otimes_R \partial_{\bf x}^\alpha: \alpha \in{\mathbb{N}}^n\right\rbrace$. Now, with the isomorphism (\[eq\_isom\_relWeyl\_diff\]), the elements $1 \otimes_R \partial_{\mathbf{x}}^\alpha $ are seen as the differential operators $\overline{\partial_{\mathbf{x}}^\alpha} \in {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R, {\mathbb{F}})$.
The following map is an isomorphism of ${\mathbb{F}}$-vector spaces: $$\label{eq_map_omega}
\omega : {\mathbb{F}}[z_1,\ldots,z_n] \;\rightarrow\; {\mathbb{F}}\,\otimes_R \, R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle\;\cong\;{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}}), \quad \mathbf{z}^\alpha \mapsto \partial_{\mathbf{x}}^\alpha.$$ From (\[eq:FFRT\]) and Notation \[nota\_T\_mod\_struct\] we get the following actions. For any $\alpha \in {\mathbb{N}}^n$ and $1 \le i \le n$, $$\label{eq_deriv_z_bracket_partial}
\partial_{z_i} \bullet \mathbf{z}^\alpha \,=\, \alpha_iz_1^{\alpha_1}\cdots z_i^{\alpha_i-1}\cdots z_n^{\alpha_n} \quad\text{ and }\quad y_i\cdot\partial_{\mathbf{x}}^\alpha
\,=\,\left[\partial_{\mathbf{x}}^\alpha,x_i\right]\,=\,
\alpha_i\partial_{x_1}^{\alpha_1}\cdots \partial_{x_i}^{\alpha_i-1}\cdots \partial_{x_n}^{\alpha_n}.$$ Hence the map $\omega$ in (\[eq\_map\_omega\]) gives a bijection between ${\mathbb{F}}$-vector subspaces of ${\mathbb{F}}[z_1,\ldots,z_n]$ closed under differentiation and ${\mathbb{F}}[y_1,\ldots,y_n]$-submodules of ${\mathbb{F}}\otimes_R R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle$. The latter structure as a submodule is equivalent to being an $R$-bi-submodule of the Weyl-Noether module.
\[lem\_prim\_ideal\_implies\_bimod\] Let $\mathcal{E}$ be a finite dimensional ${\mathbb{F}}$-vector subspace of $ {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}})$. If $\,Q = {\rm Sol}(\mathcal{E})\,$ is a $P$-primary ideal in $R = {\mathbb{K}}[x_1,\ldots,x_n]$ then $\mathcal{E}$ is an $R$-bi-module.
Fix $m \in {\mathbb{N}}$ such that $Q \supseteq P^m$ and $\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$. The map $\gamma$ in (\[eq\_map\_gamma\]) defines the ideal $I={\mathcal{M}}^m+\gamma(
Q){\mathbb{F}}[y_1,\dots,y_n]$. Let $\mathcal{E}^\prime \subseteq {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m},{\mathbb{F}}\right)$ be the ${\mathbb{F}}$-vector subspace coming from $\mathcal{E}$ under the isomorphism of Proposition \[lem\_descrip\_diff\_opp\]$(i)$. The hypothesis $\,Q={\rm Sol}(\mathcal{E})\,$ implies $$\label{eq_functionals_sols}
I/{\mathcal{M}}^m\,\, = \,\,\bigl\{ \,w \in {\mathbb{F}}[\mathbf{y}]/ {\mathcal{M}}^m
\,:\, \delta(w)=0 \,\text{ for all } \,\delta \in \mathcal{E}^\prime\, \bigr\}.$$ Dualizing the inclusion $\mathcal{E}^\prime \hookrightarrow
{{\normalfont\text{Hom}}}_{\mathbb{F}}\left( {\mathbb{F}}[\mathbf{y}]/{\mathcal{M}}^m,{\mathbb{F}}\right)$ we get the short exact sequence $$\label{eq_dualize_short_E}
0 \,\,\rightarrow \,\,Z \,\,\rightarrow\,\, {\mathbb{F}}[\mathbf{y}] / {\mathcal{M}}^m
\,\, \rightarrow \,\,{{\normalfont\text{Hom}}}_{\mathbb{F}}(\mathcal{E}^\prime, {\mathbb{F}})\,\, \rightarrow \,\, 0,$$ where $Z=\Big\lbrace w \in \frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m}
\,:\, \delta(w)=0 \text{ for all } \delta \in \mathcal{E}^\prime\Big\rbrace$. Therefore, equations (\[eq\_functionals\_sols\]) and (\[eq\_dualize\_short\_E\]) yield the isomorphism $\,{{\normalfont\text{Hom}}}_{\mathbb{F}}(\mathcal{E}^\prime,{\mathbb{F}}) \cong {\mathbb{F}}[\mathbf{y}]/I$, and we conclude that $\mathcal{E} \cong \mathcal{E}^\prime$ is an $R$-bi-module.
Finally, to complete the proof of Theorem \[thm:main\], it will suffice to prove the following.
\[thm\_noeth\_ops\_zero\_dim\] Let $P$ be a maximal ideal in $R = {\mathbb{K}}[x_1,\ldots,x_n]$, and let $Q \subset R$ be a $P$-primary ideal of multiplicity $m$ over $P$. Then $\,Q = {\rm Sol}(\mathcal{E})$, where $\mathcal{E}$ is obtained by the following steps:
(i) As in Theorem \[thm:param\_primary\], set $\,I=\langle y_1,\dots,y_n\rangle^m+\gamma(
Q){\mathbb{F}}[y_1,\dots,y_n]$.
(ii) As in Theorem \[thm:Macaulay\_dual\], set $\,V = I^\perp \,\subset\, {\mathbb{F}}[z_1,\ldots,z_n]$.
(iii) Using the map $\omega$ in (\[eq\_map\_omega\]), set $\,\,\mathcal{E}=\omega(V) \;\subset \;{\mathbb{F}}\otimes_R R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle\;\cong\;{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}})$.
We claim that the correspondence in Proposition \[lem\_descrip\_diff\_opp\]$(ii)$ gives $$\mathcal{E} \,\,\cong\, \,{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\bigl( {\mathbb{F}}[\mathbf{y}] / I, {\mathbb{F}}\bigr)
\,\, \,\hookrightarrow\, \,\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}}).$$ The isomorphism (\[eq\_isom\_inv\_sys\_char\_zero\]) implies that $V\cong V^\prime = \left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} I\right)$. Since $I \supseteq {\mathcal{M}}^m$, it follows that $V^\prime \subseteq \left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right)$. For each $0 \le j < m$, there is a perfect pairing $$\label{eq_perf_pairing}
{\left[\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m}\right]}_j \;\otimes_{\mathbb{F}}\; {\left[\left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right)\right]}_{-j} \;\rightarrow \;{\mathbb{F}}, \quad \mathbf{y}^\alpha \otimes_{\mathbb{F}}\frac{1}{\mathbf{y}^\beta} \,\mapsto\, \mathbf{y}^\alpha\cdot\frac{1}{\mathbf{y}^\beta} = \begin{cases}
1 \;\;\text{ if } \alpha = \beta\\
0 \;\;\text{ otherwise},
\end{cases}$$ where $\vert\alpha\rvert=\vert\beta\rvert=j$, induced by the usual multiplication. Hence, we get the isomorphisms $$\label{eq_isom_inv_sys_diff_ops}
\left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right) \;\cong\; {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m},{\mathbb{F}}\right) \;\cong\; {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}}) .$$ The second isomorphism follows from Proposition \[lem\_descrip\_diff\_opp\]$(i)$. The Hom-tensor adjunction yields $$\label{eq_isom_V_prime_dual_quot_I}
\!\! \!\! V^\prime = \left(0 :_{\left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right)} I\right)
\cong\;{{\normalfont\text{Hom}}}_{{\mathbb{F}}[[\mathbf{y}]]}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I},
{{\normalfont\text{Hom}}}_{\mathbb{F}}\left( \frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m},{\mathbb{F}}\right) \right)
\cong\; {{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I}, {\mathbb{F}}\right).$$ The isomorphism $V^\prime \cong {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I},{\mathbb{F}}\right)$ also follows from the duality in [@EISEN_COMM Proposition 21.4].
By the isomorphism (\[eq\_isom\_inv\_sys\_char\_zero\]) and the map $\omega$ in (\[eq\_map\_omega\]), $\mathcal{E}$ can be obtained from $V^\prime$ via the map $$V^\prime \xrightarrow{\cong} \mathcal{E}, \quad
\frac{1}{\mathbf{y}^\alpha} \mapsto \frac{1}{\alpha!}\partial_{\mathbf{x}}^\alpha.$$ On the other hand, by (\[eq\_diff\_opp\_z\_alpha\]), (\[eq\_perf\_pairing\]) and (\[eq\_isom\_inv\_sys\_diff\_ops\]), the dual monomial ${(\mathbf{y}^\alpha)}^* \in {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{{\mathcal{M}}^m},{\mathbb{F}}\right)$ is identified with the inverted monomial $\frac{1}{\mathbf{y}^\alpha} \in {\mathbb{F}}[\mathbf{y^{-1}}]$ and with the differential operator $\overline{D_\mathbf{x}^\alpha} = \frac{1}{\alpha!}\overline{\partial_{\mathbf{x}}^\alpha}\in {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$. Therefore, the isomorphisms in (\[eq\_isom\_V\_prime\_dual\_quot\_I\]) imply that $\mathcal{E}$ is indeed determined by $I$ via the correspondence in Proposition \[lem\_descrip\_diff\_opp\]$(ii)$.
After this identification, Proposition \[lem\_descrip\_diff\_opp\]$(iii)$ and Theorem \[thm:param\_primary\] imply that $${\rm Sol}(\mathcal{E})\, =\, \gamma^{-1}(I) \,=\,Q.$$ This completes the proof of Theorem \[thm\_noeth\_ops\_zero\_dim\], and we obtain Theorem \[thm:main\] for $P$ maximal.
The general case
----------------
In this subsection, we complete the proof of Theorem \[thm:main\]. As before, $R={\mathbb{K}}[x_1,\ldots,x_n]$, ${\rm char}({\mathbb{K}}) = 0$, and $P \subset R$ is a prime ideal of height $c$. We use the notation from Section \[sec4\], where $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$ and ${\mathfrak{p}}= PS$. By Noether normalization, ${\mathbb{K}}[x_{c+1},\ldots,x_n] \hookrightarrow R/P$ is an integral extension. The ideal ${\mathfrak{p}}\subset S $ is maximal and ${\mathbb{F}}= S/{\mathfrak{p}}$. The following remarks will allow us to derive Theorem \[thm:main\] from Theorems \[thm:param\_primary\], \[thm:Macaulay\_dual\] and \[thm\_noeth\_ops\_zero\_dim\].
\[rem\_clear\_fractions\_diff\_ops\] By Lemma \[lem\_diff\_ops\_R/J\], any operator $A^\prime\in {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}^{m-1}(S,S/{\mathfrak{p}})$ can be written as $$A^\prime \,\,\,= \sum_{\substack{\beta \in {\mathbb{N}}^c\\ \lvert \beta \rvert \le m-1}}
\overline{h_\beta}\,\, \overline{\partial_{x_1}^{\beta_1}\cdots\partial_{x_c}^{\beta_c}}
\quad \text{ for some }\,\, h_\beta \in S.$$ We choose $h \in {\mathbb{K}}[x_{c+1},\ldots,x_n]$ such that $h\cdot h_\beta \in R$ for all $\beta$. Hence, we can consider $$A \,\,\, = \sum_{\substack{\beta \in {\mathbb{N}}^c\\ \lvert \beta \rvert \le m-1}} \overline{h\cdot h_\beta}\, \overline{\partial_{x_1}^{\beta_1}\cdots\partial_{x_c}^{\beta_c}} \;\in \; {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/P).$$ This differential operator satisfies $\,{\rm Sol}(A)={\rm Sol}(A^\prime) \cap R$.
\[rem\_lift\_diff\_ops\_sol\] Let $A^\prime = \sum_{\lvert \alpha \rvert \le m-1} \overline{r_\alpha} \overline{\partial_{\mathbf{x}}^\alpha} \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/P)$ be a differential operator. By Lemma \[lem\_diff\_ops\_R/J\], we can lift this to $A = \sum_{\lvert \alpha \rvert \le m-1}
r_\alpha \partial_\mathbf{x}^\alpha \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R)$. Then, it follows that $${\rm Sol}(A^\prime)=\lbrace f \in R : A \bullet f \in P \rbrace .$$
The next remark describes the Weyl-Noether module in terms of differential operators.
\[rem\_isom\_restrict\_Weyl\_mod\] We have the following isomorphisms $$\begin{aligned}
{\mathbb{F}}\otimes_R D_{n,c} \,\,=\,\,
{\mathbb{F}}\otimes_R R\langle\partial_{x_1},\ldots,\partial_{x_c} \rangle \,&
\,\cong\,\, {\mathbb{F}}\otimes_S \left( S \otimes_R R\langle \partial_{x_1},\ldots,\partial_{x_c}\rangle\right)\\
& \,\cong \,\, {\mathbb{F}}\otimes_S S\langle \partial_{x_1},\ldots,\partial_{x_c}\rangle\\
& \,\cong\,\, {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}\left(S,{\mathbb{F}}\right).
\end{aligned}$$ The last isomorphism follows from (\[eq\_isom\_relWeyl\_diff\]) by applying this to the polynomial ring $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$ and the maximal ideal ${\mathfrak{p}}= PS$ in $S$.
The correspondences between parts (a), (b) and (c) have already been established in Theorems \[thm:param\_primary\] and \[thm:Macaulay\_dual\]. Using Remark \[rem\_isom\_restrict\_Weyl\_mod\], we identify the Weyl-Noether module $\,{\mathbb{F}}\otimes_R D_{n,c} \,$ with $\, {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}\left(S,{\mathbb{F}}\right)$. As in (\[eq\_map\_omega\]), we consider the map $$\begin{aligned}
\label{eq_map_omega_S}
\begin{split}
\omega_S \,:\, {\mathbb{F}}[z_1,\ldots,z_c] \;&\rightarrow\; {\mathbb{F}}\,\otimes_R \, D_{n,c}\;\cong\;{{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}(S,{\mathbb{F}})\\
z_1^{\alpha_1}\cdots z_c^{\alpha_c} \;&\mapsto\; \partial_{x_1}^{\alpha_1}\cdots\partial_{x_c}^{\alpha_c},
\end{split}
\end{aligned}$$ but now applied to the polynomial ring $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$ and its maximal ideal ${\mathfrak{p}}\subset S$. This map $\omega_S$ yields the correspondence between parts (c) and (d), that is, between $m$-dimensional ${\mathbb{F}}$-vector subspaces of ${\mathbb{F}}[z_1,\ldots,z_c]$ that are closed under differentiation and $m$-dimensional ${\mathbb{F}}$-vector subspaces of ${\mathbb{F}}\otimes_R D_{n,c}$ that are $R$-bi-modules under the action (\[eq\_deriv\_z\_bracket\_partial\]).
It remains to show that a basis of an ${\mathbb{F}}$-vector subspace in part (d) can be lifted to a set of Noetherian operators for the $P$-primary ideal in part (a). For that, let $Q$ be a $P$-primary ideal with multiplicity $m$ over $P$, and set $I=\gamma(Q)$, $V = I^\perp$ and $\mathcal{E} = \omega_S(V)$, by using Theorem \[thm:param\_primary\], Theorem \[thm:Macaulay\_dual\] and (\[eq\_map\_omega\_S\]), respectively. Then, Theorem \[thm\_noeth\_ops\_zero\_dim\] implies that, for any basis $A_1^{\prime\prime},\ldots,A_m^{\prime\prime}$ of the ${\mathbb{F}}$-vector subspace $\mathcal{E} \subset {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}(S,{\mathbb{F}})$, we get the equality $
QS = {\rm Sol}(A_1^{\prime\prime},\ldots,A_m^{\prime\prime}).
$ From Remark \[rem\_clear\_fractions\_diff\_ops\], we can choose differential operators $$A_i^\prime \,\,=\,\, \sum_{\alpha \in {\mathbb{N}}^c} \overline{r_{i,\alpha}} \,\,
\overline{\partial_{x_1}^{\alpha_1}\cdots\partial_{x_c}^{\alpha_c}}
\,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,R/P), \quad \text{ where } 1\le i \le m \text{ and } r_{i,\alpha} \in R,$$ such that $Q={\rm Sol}(A_1^\prime,\ldots,A_m^\prime)$. Finally, by Remark \[rem\_lift\_diff\_ops\_sol\], the lifted differential operators $$A_i \,\,=\,\, \sum_{\alpha \in {\mathbb{N}}^c} r_{i,\alpha} \partial_{x_1}^{\alpha_1}\cdots\partial_{x_c}^{\alpha_c} \,\in\, D_{n,c}
$$ are Noetherian operators for $Q$, which means that (\[eq:fromAtoQ\]) holds. This completes the proof.
Symbolic Powers and other Joins {#sec7}
===============================
The symbolic power of an ideal is a fundamental construction in commutative algebra. We here work in the polynomial ring $R={\mathbb{K}}[x_1,\ldots,x_n]$ over a field ${\mathbb{K}}$ of characteristic zero, with irrelevant maximal ideal $\mathfrak{m} = \langle x_1,\ldots,x_n \rangle$. The $r$-th [*symbolic power*]{} of an ideal $J$ in $R$ equals $$J^{(r)} \,\,\,\,:= \,\,\bigcap_{{\mathfrak{p}}\in {\rm Ass}(J)} \!\! J^r R_{\mathfrak{p}}\cap R.$$ Hence, if $P$ is a prime ideal in $R$ then $P^{(r)}$ is the $P$-primary component of the usual power $P^r$. If $\,{\rm codim}(P) = c\,$ then the primary ideal $P^{(r)}$ has multiplicity $m = \binom{c+r-1}{c}$ over $P$, and in Theorem \[thm:param\_primary\] it is represented by the zero-dimensional ideal $\,I = \langle y_1,\ldots,y_c \rangle^r\, \subset \, {\mathbb{F}}[y_1,\ldots,y_c]$.
Our point of departure in this section is a formula due to Sullivant [@SULLIVANT_SYMB Proposition 2.8]: $$\label{eq:sulli1}
J^{(r)} \,\, = \,\, J \star \mathfrak{m}^r .$$ Here, $J$ is any radical ideal in $R$, and $\star$ denotes the join of ideals. This is a reformulation of the [*Zariski-Nagata Theorem*]{} which expresses the symbolic power via differential equations: $$\label{eq:sulli2}
J^{(r)} \,\, = \,\, \,
\biggl\{ \,f \,\in\, R \,\,\,\bigg\vert \,\,\,
\frac{\partial^{i_1+i_2+\cdots+i_n} f}{
\partial x_1^{i_1}
\partial x_2^{i_2} \cdots
\partial x_n^{i_n}
} \,\in\, J \quad \text{whenever}\, \,\,i_1+i_2 + \cdots + i_n < r \, \biggr\}.$$ The goal of this section is to generalize the equivalence between (\[eq:sulli1\]) and (\[eq:sulli2\]). We construct $P$-primary ideals by means of joins and we relate this to the results seen in earlier sections.
If $J$ and $K$ are ideals in $R$, then their *join* is the new ideal $$J \star K \,\,\,:=\,\,\, \Big( J(\mathbf{v}) \,+\, K(\mathbf{w}) \,+\,
\langle x_i - v_i - w_i : 1 \le i \le n \rangle \Big) \,\,\cap \,\,R,$$ where $J(\mathbf{v})$ is the ideal $J$ with new variables $v_i$ substituted for $x_i$ and $K(\mathbf{w})$ is the ideal $K$ with $w_i$ substituted for $x_i$. The parenthesized ideal lives in a polynomial ring in $3n$ variables.
\[rem\_kernel\_map\_join\] Following Simis and Ulrich [@SIMIS_ULRICH_JOIN], the join $J \star K$ equals the kernel of the map $$\begin{aligned}
R \;&\,\rightarrow \,\; \frac{{\mathbb{K}}[v_1,\ldots,v_n,w_1,\ldots,w_n]}{J(\mathbf{v})+K(\mathbf{w})} \,\;\;\xleftrightarrow{\cong}\; R/J \otimes_{\mathbb{K}}R/K\\
x_i \;&\, \mapsto \,\,\;\; \overline{v_i}+\overline{w_i} \qquad\qquad\qquad\qquad\leftrightarrow \;\;\overline{x_i} \otimes_{\mathbb{K}}1 + 1 \otimes_{\mathbb{K}}\overline{x_i}.
\end{aligned}$$ Hence, the quotient $\,R/\left(J\star K\right)\,$ can be identified with a subring of $\,R/J \otimes_{\mathbb{K}}R/K$.
The following result summarizes a few basic properties of the join construction.
\[prop\_properties\_join\] Let $J$ and $K$ be ideals in $R$. Then, the following statements hold:
(i) If $J = J_1 \cap J_2$, where $J_1,J_2 \subset R$ are ideals, then $J \star K = (J_1 \star K) \cap (J_2 \star K)$.
(ii) $\sqrt{J \star K} = \sqrt{J} \star \sqrt{K}$; in particular, $J \star K$ is radical when $J$ and $K$ are.
(iii) Suppose that ${\mathbb{K}}$ is algebraically closed. If $P_1$ and $P_2$ are prime ideals, then $P_1 \star P_2$ is a prime ideal. If $J$ and $K$ are primary ideals, then $J \star K$ is a primary ideal.
(iv) If $M$ is an $\mathfrak{m}$-primary ideal, then $P \star M$ is a $P$-primary ideal.
This is an adaptation of [@SIMIS_ULRICH_JOIN Proposition 1.2] for non-necessarily homogeneous ideals.
$(i)$ The join distributes over intersections by [@SULLIVANT_SYMB Lemma 2.6].
$(ii)$ The ring $R/\sqrt{J} \otimes_{\mathbb{K}}R/\sqrt{K}$ is reduced by [@GORTZ_WEDHORN Corollary 5.57]. As the kernel of the map $R/J\otimes_{\mathbb{K}}R/K \twoheadrightarrow R/\sqrt{J} \otimes_{\mathbb{K}}R/\sqrt{K}$ is nilpotent, the claim follows from Remark \[rem\_kernel\_map\_join\].
$(iii)$ Since ${\mathbb{K}}$ is algebraically closed, $R/P_1 \otimes_{\mathbb{K}}R/P_2$ is an integral domain [@GORTZ_WEDHORN Lemma 4.23]. By Remark \[rem\_kernel\_map\_join\], $R/(P_1 \star P_2)$ is a subring of this domain. Thus, $P_1 \star P_2$ is a prime ideal. Suppose ${\rm Ass}(R/J)=\{P_1\}$ and ${\rm Ass}(R/K)=\{P_2\}$. From [@MATSUMURA Theorem 23.2] we infer $$\label{eq_equality_ass_primes}
{\rm Ass}(R/J \otimes_{\mathbb{K}}R/K)\,\, = \,\, {\rm Ass}(R/P_1 \otimes_{\mathbb{K}}R/P_2).$$ We already saw that $R/P_1 \otimes_{\mathbb{K}}R/P_2$ is an integral domain. Therefore, $R/J \otimes_{\mathbb{K}}R/K$ has only one associated prime, and hence its subring $R/(J \star K)$ has only one associated prime.
$(iv)$ The equality in (\[eq\_equality\_ass\_primes\]) is valid for any field. From this we get ${\rm Ass}(R/P \otimes_{\mathbb{K}}R/M)
= {\rm Ass}(R/P \otimes_{\mathbb{K}}R/\mathfrak{m}) = \{ P \star \mathfrak{m} \} = \{P\}$. We hence conclude $\,{\rm Ass}(R / (P \star M))= \{P\}$.
In Proposition \[prop\_properties\_join\] (iii) we need the hypothesis that ${\mathbb{K}}$ is algebraically closed. If ${\mathbb{K}}= {\mathbb{R}}$ then $P_1 = \langle x_1^2+1, x_2 \rangle$ and $P_2 = \langle x_1 , x_2^2+1 \rangle $ are prime but their join is not primary: $$P_1 \star P_2 \,\, = \,\, \langle x_1^2+1, x_2^2+1 \rangle \,\, =\,\,
\langle x_1 - x_2 , x_2^2+1 \rangle \,\, \cap \,\, \langle x_1 + x_2 , x_2^2+1 \rangle .$$
In what follows we focus on the $P$-primary ideals $Q = P \star M$ in Proposition \[prop\_properties\_join\] (iv). These will be characterized by differential equations derived from the $\mathfrak{m}$-primary ideal $M$.
Let $M$ be an $\mathfrak{m}$-primary ideal. We shall encode $M$ by a system $\mathfrak{A}(M)$ of linear PDE with constant coefficients. This is computed by the performing following steps:
(i) Interpret $M$ as PDE by replacing the variables $x_i$ with $\partial_{z_i}$ for $i=1,\ldots,n$.
(ii) Compute the inverse system $M^\perp=\left\lbrace F\in {\mathbb{K}}[z_1,\ldots,z_n]: f\bullet F=0 \mbox{ for all }f\in M\right\rbrace$.
(iii) Let $\mathfrak{A}(M) \subset {\mathbb{K}}[\partial_{x_1},\ldots,\partial_{x_n}]$ be the image of $M^\perp$ under the map $\mathbf{z}^\alpha \mapsto \partial_{\mathbf{x}}^\alpha$.
We say that the ${\mathbb{K}}$-subspace $\mathfrak{A}(M)$ comprises the *differential operators associated to $M$*.
\[rem\_joins\_props\] (i) The space $\mathfrak{A}(M)$ is closed under taking brackets as in (\[eq\_deriv\_z\_bracket\_partial\]) and Theorem \[thm:Macaulay\_dual\].
\(ii) For any $r \ge 1$, we have $ \mathfrak{A}\left(\mathfrak{m}^r\right) = \bigoplus_{\lvert \alpha \rvert \le r-1} {\mathbb{K}}\, \partial_{\mathbf{x}}^\alpha $. Thus, $\mathfrak{A}(\mathfrak{m}^r)$ comprises the differential operators used in the Zariski-Nagata formula for symbolic powers; see (\[eq:sulli2\]) and [@EISEN_COMM §3.9].
The following result is a generalization of the classical Zariski-Nagata Theorem, to ideals obtained with the join construction. Of main interest is the situation when $J=P$ is prime.
\[thm:ourZN\] Let $J$ be any ideal in $R = {\mathbb{K}}[x_1,\ldots,x_n]$ and let $M$ be an $\mathfrak{m}$-primary ideal.
(i) The join of $J$ and $M$ equals $J \star M \;=\; \big\lbrace f \in R : A \bullet f \in J \;
\text{ for all }\; A \in \mathfrak{A}(M) \big\rbrace$.
(ii) If $J$ is radical and $r \in {\mathbb{N}}$ then $\,
J^{(r)} \,=\, J \star \mathfrak{m}^r \,=\,
\big\lbrace f \in R : \partial_{\mathbf{x}}^\alpha \bullet f \in J \;
\text{ for all }\; \lvert \alpha \rvert \le r-1 \big\rbrace $.
Let $n=4,c=2$, fix the prime ideal $P$ in (\[eq:twistedcubic1\]), and consider the $\mathfrak{m}$-primary ideal $ M = \langle x_1^2,x_2^2,x_3^2,x_4^2 \rangle$. The join $Q = P \star M$ is a $P$-primary ideal of multiplicity $m=11$. It is minimally generated by eight octics such as $\,x_1^8-4 x_1^6 x_2 x_3+6 x_1^4 x_2^2 x_3^2-4x_1^2 x_2^3 x_3^3+x_2^4 x_3^4$. The differential equations from $\mathfrak{A}(M)$ are simply the squarefree partial derivatives, so that $$\label{eq:repQ1}
Q \,\,\,= \,\,\,\biggl\{ \,f \,\in\, R \,\,\,\bigg\vert \,\,\,
\frac{\partial^{i_1+i_2+i_3+i_4} f}{
\partial x_1^{i_1}
\partial x_2^{i_2}
\partial x_3^{i_3}
\partial x_4^{i_4}
} \,\in\, P \quad \text{whenever}\, \,\,i_1,i_2,i_3,i_4 \in \{0,1\}\, \biggr\}.$$ This should be compared to the representation by Noetherian operators found in Algorithm \[alg:forward\]. In Step 1, we obtain the ideal $I = \langle y_1^4, u_2 y_1^3 y_2 - u_3 y_1 y_2^3, 3 u_1 y_1^2 y_2^2 - 5 u_3 y_1 y_2^3, y_2^4 \rangle$. The inverse system $I^\perp$ in Step 2 is the $11$-dimensional subspace of ${\mathbb{F}}[y_1,y_2]$ spanned by $$B({\bf u},{\bf z}) \,\,\,= \,\,\, 2 u_1 u_3 \,z_1^3 z_2\, +\, 5 u_2 u_3 \,z_1^2 z_2^2
\,+\, 2 u_1 u_2 \,z_1 z_2^3$$ together with all ten monomials $z_1^{j_1} z_2^{j_2} $ of degree $j_1+j_2\leq 3$. From Steps 3 and 4 we obtain $$A({\bf x},\partial_{\bf x}) \,\, = \,\,
2 x_1 x_3 \partial_{x_1}^3 \partial_{x_2 }
+ 5 x_2 x_3 \partial_{x_1}^2 \partial_{x_2}^2
+ 2 x_1 x_2 \partial_{x_1} \partial_{x_2}^3 ,$$ and this gives rise to the following alternative representation of $Q$ by differential equations: $$\label{eq:repQ2}
Q \,\,\,= \,\,\,\biggl\{ \,f \,\in\, R \,\,\,\bigg\vert \,\,\,
A \bullet f \in P \,\,\, \,{\rm and} \,\,\,
\frac{\partial^{j_1+j_2} f}{
\partial x_1^{j_1}
\partial x_2^{j_2}
} \,\in\, P \quad \text{whenever}\, \,j_1 + j_2 \leq 3
\, \biggr\}.$$ The two representations (\[eq:repQ1\]) and (\[eq:repQ2\]) differ in two fundamental ways. The operators in (\[eq:repQ1\]) have constant coefficients but differentiation involves all four variables. In (\[eq:repQ2\]) we are using an operator from $D_{4,2}$ with polynomial coefficients but we differentiate only two variables.
The next example shows that not every primary ideal arises from the join construction.
\[exam:Palamodov\] Let $n = 3$ and $c = 2$, and consider the primary ideal $Q = \langle x_1^2, x_2^2, x_1 - x_2x_3 \rangle$ with $P = \sqrt{Q}=\langle x_1, x_2 \rangle$. From [@BJORK Proposition 4.8 and Example 4.9, page 352] we know that $Q$ cannot be described by differential operators with constant coefficients only. Theorem \[thm:ourZN\] (i) implies that $Q$ does not arise from the join construction, i.e. we cannot find an $\mathfrak{m}$-primary ideal $M$ such that $Q = P \star M$. On the other hand, Algorithm \[alg:forward\] applied to $Q$ gives the two Noetherian operators $A_1 = 1, A_2 = x_3\partial_{x_1} + \partial_{x_2}$.
$(i)$ We use the notation and results from Section \[sec5\]. We begin by fixing an integer $m$ such that $\mathfrak{m}^m \subseteq M$. In (\[eq\_isoms\_Diff\_R/J\]) we obtained the explicit isomorphism $$\label{eq_join_isom_diff_ops}
{{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_R P_{R/{\mathbb{K}}}^{m-1}, R/J\right) \xrightarrow{\cong} {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J), \quad \psi \mapsto \psi \circ h_{m-1} \circ d^{m-1}$$ where $h_{m-1}$ is the canonical map $ P_{R/{\mathbb{K}}}^{m-1} \rightarrow R/J \otimes_R P_{R/{\mathbb{K}}}^{m-1}$ and $d^{m-1}$ is the map in (\[eq\_univ\_diff\]). Setting $\,T=R \otimes_{\mathbb{K}}R ={\mathbb{K}}[x_1, \ldots, x_n, y_1, \ldots, y_n]\,$ as in Section \[sec5\], we have the following isomorphisms: $$\label{eq_isom_tensor_prods_join}
R/J \otimes_R P_{R/{\mathbb{K}}}^{m-1} \;\cong\; \frac{T}{J(\mathbf{x}) \,+
\, \mathfrak{m}^m(\mathbf{y})} \;\cong\; R/J \otimes_{\mathbb{K}}R/\mathfrak{m}^m.$$ Recall that this ${\mathbb{K}}$-vector space is considered as an $R$-module via the left factor $R/J \otimes_{\mathbb{K}}1$.
Using (\[eq\_join\_isom\_diff\_ops\]) and (\[eq\_isom\_tensor\_prods\_join\]), the surjection $\,R/J \otimes_{\mathbb{K}}R/\mathfrak{m}^m \twoheadrightarrow R/J \otimes_{\mathbb{K}}R/M\,$ induces the inclusion $$\label{eq_inclusion_join_diffs}
{{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_{\mathbb{K}}R/M, R/J\right)
\,\,\hookrightarrow\,\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J).$$ Since $ R/J \otimes_{\mathbb{K}}R/M$ is a finitely generated free $R/J$-module, we have $$\big\lbrace w \in R/J \otimes_{\mathbb{K}}R/M : \psi(w)=0 \;\text{ for all }\; \psi \in {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_{\mathbb{K}}R/M, R/J\right) \big\rbrace \;=\; \{0\}.$$ Let $\,\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J)\,$ denote the image of (\[eq\_inclusion\_join\_diffs\]). So, the isomorphism (\[eq\_join\_isom\_diff\_ops\]) implies $${\rm Sol}(\mathcal{E}) = {\rm Ker}\left(\overline{d^{m-1}}\right), \;\text{ where }\; \overline{d^{m-1}} : R \rightarrow R/J \otimes_{\mathbb{K}}R/M, \;\; x_i \mapsto \overline{x_i} \otimes_{\mathbb{K}}1 + 1 \otimes_{\mathbb{K}}\overline{x_i}.$$ Therefore, Remark \[rem\_kernel\_map\_join\] yields that ${\rm Sol}(\mathcal{E})=J \star M$.
By [@MATSUMURA Theorem 7.11], the inclusion (\[eq\_inclusion\_join\_diffs\]) can be written equivalently as $$\begin{aligned}
{{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_{\mathbb{K}}R/M, R/J\right)
\,\, \cong \,\, R/J &\otimes_{\mathbb{K}}{{\normalfont\text{Hom}}}_{\mathbb{K}}(R/M,{\mathbb{K}})\\ &
\hookrightarrow\; R/J \otimes_{\mathbb{K}}{{\normalfont\text{Hom}}}_{\mathbb{K}}(R/\mathfrak{m}^m,{\mathbb{K}}) \,\, \cong \,\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J).\end{aligned}$$ The Hom-tensor adjunction and the perfect pairing in (\[eq\_perf\_pairing\]) give the following isomorphisms: $${{\normalfont\text{Hom}}}_{\mathbb{K}}\left(R/M,{\mathbb{K}}\right) \,\,\cong\,\, {{\normalfont\text{Hom}}}_R\left(R/M, {{\normalfont\text{Hom}}}_{{\mathbb{K}}}\left(R/\mathfrak{m}^m,{\mathbb{K}}\right)\right) \,\,\cong\,\, \left(0 :_{{\mathbb{K}}[\mathbf{x^{-1}}]} M\right).$$ Then, by arguments almost verbatim to those used in the proof of Theorem \[thm\_noeth\_ops\_zero\_dim\], we find that $\,\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R, R/J)\,$ is a finitely generated free $R/J$-module, and it is generated by $\,\bigl\{ \,\overline{A} \,:\, A \in \mathfrak{A}(M) \subset {\mathbb{K}}[\partial_{\mathbf{x}}] \cap {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R)\bigr\} \subset {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J)$. Summing up, we conclude $$J \star M \,\,=\,\, {\rm Sol}(\mathcal{E}) \,\,=\,\,
\big\lbrace f \in R : A \bullet f \in J \;\text{ for all }\; A \in \mathfrak{A}(M) \big\rbrace.$$
$(ii)$ Since $J$ is radical, $J=P_1 \cap \cdots \cap P_k$ for some prime ideals $P_j \subset R$, and so we have $J^{(r)}=P_1^{(r)} \cap \cdots \cap P_k^{(r)}$. Proposition \[prop\_properties\_join\]$(i)$ implies $J \star \mathfrak{m}^r = \left(P_1 \star \mathfrak{m}^r\right) \cap \cdots \cap \left(P_k \star \mathfrak{m}^r \right)$. Therefore, to finish the proof, it suffices to consider the case where $J=P$ is a prime ideal. The Zariski-Nagata Theorem implies $\,
P^{(r)} = \big\lbrace f \in R : \partial_{\mathbf{x}}^\alpha \bullet f \in J \; \text{ for all }\; \lvert \alpha \rvert \le r-1 \big\rbrace $. The conclusion now follows from part $(i)$ applied to $M=\mathfrak{m}^r$. This establishes Theorem \[thm:ourZN\].
Decomposition and Fusion in a Numerical Future {#sec8}
==============================================
This closing section takes the perspective of applied and computational mathematics. We consider a system of polynomial equations over the complex numbers ${\mathbb{C} }$, viewed as an ideal $I$ in the polynomial ring $R = {\mathbb{C} }[x_1,\ldots,x_n]$. This ideal has a minimal primary decomposition $$\label{eq:primdeco}
I \,\,\, = \,\,\, Q_1 \,\cap \, Q_2 \,\cap \, \cdots \,\cap \, Q_s .$$ Each associated prime $P_i = \sqrt{Q_i}$ defines an irreducible variety $X_i = V(P_i)$ in ${\mathbb{C} }^n$. Solving the equations means identifying the varieties $X_i$ corresponding to the associated primes $P_i$.
Computing the primary decomposition (\[eq:primdeco\]) from generators of $I$ thus refines the problem of solving polynomial systems. Algorithms for this task are a well-developed subject in computer algebra [@DECKER]. However, most studies focus on the irreducible components $X_i$ and the associated primes $P_i$, and they pay less attention to the primary ideals $Q_i$ themselves.
The past decade has seen significant advances in numerical algebraic geometry [@BATES], and this has led to the design of numerical techniques for primary decomposition [@KRONE; @LEYKIN]. A paramount ingredient is the identification of all minimal primes $P_i$ from the generators of $I$. Algorithms and implementations for this are now well-established; see e.g. [@BATES Chapter 8]. In the output, each irreducible variety $X_i$ is represented by a finite [*witness set*]{} of the form $X_i \cap L_i$, where $L_i$ is a general affine-linear subspace of dimension $c_i = {\rm codim}(X_i)$ in ${\mathbb{C} }^n$.
Numerical identification of embedded primes $P_i$ is more subtle. This topic was pioneered by Krone and Leykin [@KRONE; @LEYKIN] who proposed algorithms based on a technique known as [*inflation*]{}. However, the concluding paragraph in [@KRONE] indicates that more work is needed. Furthermore, their articles do not address the description of the primary ideals $Q_i$ in (\[eq:primdeco\]).
The following definitions pave the way for future numerical algorithms. By Theorem \[thm:main\], each primary ideal $Q_i$ is encoded by a pair $(X_i,\mathfrak{A}_i$) where $\mathfrak{A}_i$ is an $m_i$-dimensional ${\mathbb{F}}_i$-vector subspace of ${\mathbb{F}}_i \otimes_R D_{n,c_i}$, where $X_i = V(P_i) = V(Q_i)$ and ${\mathbb{F}}_i $ denotes the field of fractions of $R/P_i$. The numerical representation of the prime ideal $P_i$ or the associated function field ${\mathbb{F}}_i$ is the same as that of $X_i$, namely it is simply a witness set as in [@BATES Chapter 8]. The space $\mathfrak{A}_i$ provides a set of Noetherian operators $A_{ij}({\bf x},\partial_{\bf x})$ for $Q_i$, where $j=1,2,\ldots,m_i$.
We propose to use (\[eq:fromAtoQ\]) as the numerical encoding of primary ideals in future algorithms: $$\label{eq:fromAtoQnum}
Q_i \,\, = \,\, \{ \, f \in R \,: \, A_{ij} \bullet f \,\,\text{vanishes on} \,\, X_i \,
\,\text{for all} \,\, j \,\}.$$ Here $A_{ij}$ is an element in the relative Weyl algebra $D_{n,c_i}$ and its coefficients are given in floating point arithmetic. Likewise, the vanishing condition in (\[eq:fromAtoQnum\]) is meant to be inexact.
\[def:numprimdec\] Given an ideal $I$ in $R = {\mathbb{C} }[x_1,\ldots,x_n]$, we define a [*numerical primary decomposition*]{} of $I$ to be a list $(X_1,\mathfrak{A}_1), \ldots, (X_s,\mathfrak{A}_s)$ of representations of primary ideals, where the $X_i$ are precisely the irreducible varieties that are associated to $I$, and we have $$\label{eq:fusionI}
I \,\, = \,\, \{ \, f \in R \,: \, A \bullet f \,\,\,\text{vanishes on}\, \, X_i\,
\,\,\text{for all $A \in \mathfrak{A}_i $ and all $i=1,\ldots,s$} \,\}.$$ By an abuse of notation, here each $\mathfrak{A}_i$ is also identified with an appropriate finite subset of Noetherian operators for $Q_i$. If $X_i$ is a geometric component then this subset is simply obtained from a basis of the relevant $R$-bi-module in part (d) of Theorem \[thm:main\], and its cardinality is the multiplicity $m_i$ of the primary ideal $Q_i$. However, if $X_i$ is an embedded component, say $X_i \subset X_j$, then we may use a subset of cardinality strictly less than $m_i$.
Let $n=2$ and $I = \langle x_1^3, x_1^2 x_2^2\rangle\, = \, \langle x_1^2 \rangle \,\cap \,
\langle x_1^3, x_2^2 \rangle$. A numerical primary decomposition consists of $(X_1,\mathfrak{A}_1)$ and $(X_2,\mathfrak{A}_2)$, where $X_1 $ is the $x_2$-axis with $\mathfrak{A}_1 = \{ 1, \partial_1 \}$, and $X_2 = \{(0,0)\}$ with $\mathfrak{A}_2 = \{
\partial_1^2 \partial_2, \partial_1^2 , \partial_1 \partial_2, \partial_2 \}$. Note that $|\mathfrak{A}_2|=4 < 6 = m_2 = {\rm mult}(\langle x_1^3,x_2^2\rangle) $.
The computation of a numerical primary decomposition should be carried out by combining existing methods for numerical irreducible decomposition [@BATES; @KRONE; @LEYKIN] with an appropriate adaptation of Algorithm \[alg:forward\]. For each associated irreducible variety $X_i$ one must identify the inverse system in Step 2 using linear algebra over the function field ${\mathbb{F}}_i$ of the component $X_i$. Linear algebra over ${\mathbb{F}}_i$ is to be carried out not from equations but from the witness set alone.
One task that arises naturally in this setting is the converse to primary decomposition. This process, which we propose to call [*primary fusion*]{}, amounts to combining a finite collection of primary ideals by their intersection. Let $Q_1$ and $Q_2$ be primary ideals in $R$, encoded by pairs $(X_1,\mathfrak{A}_1)$ and $(X_2,\mathfrak{A}_2)$ as above. Here $\mathfrak{A}_i$ is an $R$-bi-submodule of ${\mathbb{F}}_i \otimes_R D_{n,c_i}$. The first case to consider is when the underlying varieties agree, so $X_1 = X_2$ with $c = c_1 = c_2$.
If $Q_1$ and $Q_2$ are $P$-primary ideals, then $Q_1 \cap Q_2$ is also $P$-primary. Its bi-module of Noetherian operators in ${\mathbb{F}}\otimes_R D_{n,c}$ is $\,\mathfrak{A}_1 + \mathfrak{A}_2$. This is the primary fusion.
Next consider the situation when $P_1 = \sqrt{Q_1}$ and $P_2 = \sqrt{Q_2}$ are distinct. Suppose first that there is no containment between the varieties $X_1 $ and $X_2$. We certify this from their witness sets. In this case, the primary fusion of $(X_1,\mathfrak{A}_1)$ and $(X_2,\mathfrak{A}_2)$ is the union of their representations by Noetherian operators, that is, the primary fusion is (\[eq:fusionI\]) with $s=2$.
The most interesting case arises when $X_1 \subset X_2$. The codimensions satisfy $c_1 > c_2$ and coordinates are chosen so that the Noether normalizations are compatible. Note that $\mathfrak{A}_1 \subset {\mathbb{F}}_1 \otimes_R D_{n,c_1} $ and $\mathfrak{A}_2 \subset {\mathbb{F}}_2 \otimes_R D_{n,c_2} $. We wish to replace the ${\mathbb{F}}_1$-vector space $\mathfrak{A}_1$ by a proper subspace in order to turn (\[eq:fusionI\]) into a minimal representation for $Q_1 \cap Q_2$. It would be desirable to develop an algorithm for doing this in practice, not just for two components but for an arbitrary number $s$ of numerically represented primary ideals $(X_i,\mathfrak{A}_i)$.
\[prob:fusion\] Develop a practical numerical method for primary fusion in ${\mathbb{C} }[x_1,\ldots,x_n]$.
The numerical solution of partial differential equations is a vast area whose importance for the sciences and engineering can hardly be overestimated. In this paper we explored one special aspect, namely systems of homogeneous linear PDE on ${\mathbb{C} }^n$ with constant coefficients. Such PDE are polynomials in the operators $ \partial_{z_1},
\ldots, \partial_{z_n}$, and their solutions are functions $\psi(z_1,\ldots,z_n)$. We seek numerical algorithms for computing and manipulating these $\psi({\bf z})$ via their integral representations (\[eq:anysolution\]), promised to us by Ehrenpreis [@EHRENPREIS] and Palamodov [@PALAMODOV]. These should go well beyond the zero-dimensional case, studied by Gröbner in the 1930’s.
The given PDE form an ideal $I$ in the polynomial ring $R$. We view this input and the desired output in the spirit of numerical algebraic geometry [@BATES]. Exploiting the primary decomposition (\[eq:primdeco\]), our task is to numerically compute the objects of Theorem \[thm:Palamodov\_Ehrenpreis\] for each primary ideal $Q_i$. The varieties $X_i$ are given by witness sets. These need to be enhanced by measures $\mu_{ij}$ for the integral representation (\[eq:anysolution\]). The key algebraic objects are the Noetherian multipliers $B_{ij}({\bf x},{\bf z})$. Their construction is described in Step 3 of Algorithm \[alg:forward\], but this must now be done in a numerical setting. Moreover, to combine solutions $\psi_1({\bf z})$ and $\psi_2({\bf z})$ whose supports are nested, say $X_1 \subset X_2$, we also need primary fusion (Problem \[prob:fusion\]).
The problem of solving linear PDE with constant coefficients was discussed in [@STURMFELS_SOLVING Chapter 10]. The author of [@STURMFELS_SOLVING] worked out several nice examples, like the one of page 144, but he was unable to go further, because he lacked the necessary tools from commutative algebra.
Overcoming that barrier is precisely the contribution of the present paper. We here develop the tools from commutative algebra that were needed to advance [@STURMFELS_SOLVING Chapter 10]. Theorem \[thm:main\] offers a new characterization of primary ideals and their differential equations. This leads to Algorithms \[alg:forward\] and \[alg:backward\], and these lay the foundation for future development of the Ehrenpreis-Palamodov Fundamental Principle within numerical algebraic geometry.
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| ArXiv |
---
abstract: 'We report a measurement of the branching fractions of $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ decays based on 417 fb$^{-1}$ of data collected at the $\Upsilon(4S)$ resonance with the detector at the 2 $e^+e^-$ storage rings. Events are selected by fully reconstructing one of the $B$ mesons in a hadronic decay mode. A fit to the invariant mass differences $m(D^{(*)}\pi)-m(D^{(*)})$ is performed to extract the signal yields of the different $D^{**}$ states. We observe the $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ decay modes corresponding to the four $D^{**}$ states predicted by Heavy Quark Symmetry with a significance greater than six standard deviations including systematic uncertainties.'
title: 'Measurement of the Branching Fractions of [$\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ ]{} Decays in Events Tagged by a Fully Reconstructed [$B$]{} Meson'
---
authors\_jun2008.tex
Semileptonic $B$ decays to orbitally-excited P-wave charm mesons ($D^{**}$) are of interest for several reasons. Improved knowledge of the branching fractions for these decays is important to reduce the systematic uncertainty in the measurements of the Cabibbo-Kobayashi-Maskawa [@CKM] matrix elements $|V_{cb}|$ and $|V_{ub}|$. For example, one of the leading sources of systematic uncertainty on $|V_{cb}|$ measurements from $\Bbar \to D^* \ell^- \bar{\nu}_{\ell}$ decays [@ell] is the limited knowledge of the background due to $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ [@BaBarHQET].
The $D^{**}$ mesons contain one charm quark and one light quark with relative angular momentum $L=1$. According to Heavy Quark Symmetry (HQS) [@IW], they form one doublet of states with angular momentum $j \equiv s_q + L= 3/2$ $\left[D_1(2420), D_2^*(2460)\right]$ and another doublet with $j=1/2$ $\left[D^*_0(2400), D_1'(2430)\right]$, where $s_q$ is the light quark spin. Parity and angular momentum conservation constrain the decays allowed for each state. The $D_1$ and $D_2^*$ states decay through a D-wave to $D^*\pi$ and $D^{(*)}\pi$, respectively, and have small decay widths, while the $D_0^*$ and $D_1'$ states decay through an S-wave to $D\pi$ and $D^*\pi$ and are very broad.
$\Bbar \rightarrow D^{**} \ell^- \bar{\nu}_{\ell}$ decays constitute a significant fraction of $B$ semileptonic decays [@pdg] and may help to explain the discrepancy between the inclusive $\Bbar \to X\ell^- \bar{\nu}_{\ell}$ rate and the sum of the measured exclusive decay rates [@pdg; @babar-2; @babar-3]. The measured decay properties for $\Bbar \rightarrow D^{**} \ell^- \bar{\nu}_{\ell}$ can be compared with the predictions of the Heavy Quark Effective Theory (HQET) [@LLSW]. QCD sum rules [@uraltsev] imply the strong dominance of $B$ decays to the narrow $D^{**}$ states over those to the wide ones, while some experimental data show the opposite trend [@belle; @delphi2005].
In this letter, we present the observation of $B$ semileptonic decays into the four excited $D$ mesons predicted by HQS and measure the ${\cal B}(\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell})$ branching fractions. The analysis is based on data collected with the detector [@detector] at the 2 asymmetric-energy $e^+e^-$ storage rings at SLAC. The data consist of a total of 417 fb$^{-1}$ recorded at the $\Upsilon(4S)$ resonance, corresponding to approximately 460 million pairs. An additional 40 fb$^{-1}$, taken at a center-of-mass (CM) energy 40 MeV below the $\Upsilon(4S)$ resonance, is used to study background from $e^+e^- \to f\bar{f}~(f=u,d,s,c,\tau)$ continuum events. A detailed GEANT4-based Monte Carlo (MC) simulation [@Geant] of and continuum events is used to study the detector response, its acceptance, and to validate the analysis techniques. The simulation describes $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ decays using the ISGW2 model [@ISGW], and non-resonant $\Bbar
\to D^{(*)} \pi \ell^- \bar{\nu}_{\ell}$ decays using the model of Goity and Roberts [@Goity].
We select semileptonic $\Bbar \to D^{**}\ell^-\bar{\nu}_{\ell}$ decays with $\ell=e, \mu$ in events containing a fully reconstructed $B$ meson ($B_\mathrm{tag}$), which allows us to constrain the kinematics, reduce the combinatorial background, and determine the charge and flavor of the signal $B$ meson. $D^{**}$ mesons are reconstructed in the $D^{(*)}\pi^{\pm}$ decay modes and the different $D^{**}$ states are identified by a fit to the invariant mass differences $m(D^{(*)}\pi) -
m(D^{(*)})$.
We first reconstruct the semileptonic $B$ decay, selecting a lepton with momentum $p^*_{\ell}$ in the CM frame larger than 0.6 GeV/$c$. We search for pairs of oppositely-charged tracks that form a vertex and remove those with an invariant mass consistent with a photon conversion or a $\pi^0$ Dalitz decay. Candidate $D^0$ mesons that have the correct charge correlation with the lepton are reconstructed in the $K^-\pi^+$, $K^- \pi^+ \pi^0$, $K^- \pi^+ \pi^+ \pi^-$, $K^0_S \pi^+ \pi^-$, $K^0_S \pi^+ \pi^- \pi^0$, $K^0_S \pi^0$, $K^+ K^-$, $\pi^+ \pi^-$, and $K^0_S K^0_S$ channels, and $D^+$ mesons in the $K^- \pi^+ \pi^+$, $K^- \pi^+ \pi^+ \pi^0$, $K^0_S \pi^+$, $K^0_S \pi^+ \pi^0$, $K^+ K^- \pi^+$, $K^0_S K^+$, and $K^0_S \pi^+ \pi^+ \pi^-$ channels. In events with multiple $D\ell^-$ combinations, the candidate with the best $D$-$\ell$ vertex fit is selected. Candidate $D^*$ mesons are reconstructed by combining a $D$ candidate with a pion or a photon in the $D^{*+} \rightarrow D^0 \pi^+ $, $D^{*+} \rightarrow D^+ \pi^0$, $D^{*0} \rightarrow D^0 \pi^0$, and $D^{*0} \rightarrow D^0 \gamma$ channels. In events with multiple $D^{*}\ell^-$ combinations, we choose the candidate with the smallest $\chi^2$ based on the deviations from the nominal values of the $D$ invariant mass and the invariant mass difference between the $D^*$ and the $D$, using the resolution measured in each mode.
We reconstruct $B_\mathrm{tag}$ decays [@BrecoVub] in charmed hadronic modes $\Bbar \rightarrow D Y$, where $Y$ represents a collection of hadrons, composed of $n_1\pi^{\pm}+n_2 K^{\pm}+n_3 K^0_S+n_4\pi^0$, where $n_1+n_2 =1,3,5$, $n_3
\leq 2$, and $n_4 \leq 2$. Using $D^0(D^+)$ and $D^{*0}(D^{*+})$ as seeds for $B^-(\Bzb)$ decays, we reconstruct about 1000 different decay chains.
The kinematic consistency of a $B_\mathrm{tag}$ candidate with a $B$ meson decay is evaluated using two variables: the beam-energy substituted mass $m_{ES} \equiv \sqrt{s/4-|p^*_B|^2}$, and the energy difference $\Delta E \equiv E^*_B -\sqrt{s}/2$. Here $\sqrt{s}$ is the total CM energy, and $p^*_B$ and $E^*_B$ denote the momentum and energy of the $B_\mathrm{tag}$ candidate in the CM frame. For correctly identified $B_\mathrm{tag}$ decays, the $m_{ES}$ distribution peaks at the $B$ meson mass, while $\Delta E$ is consistent with zero. We select $B_\mathrm{tag}$ candidates in the signal region defined as 5.27 GeV/$c^2$ $< m_{ES} <$ 5.29 GeV/$c^2$, excluding those with daughter particles in common with the charm meson or the lepton from the semileptonic $B$ decay. In the case of multiple $B_\mathrm{tag}$ candidates in an event, we select the one with the smallest $|\Delta E|$ value. The $B_\mathrm{tag}$ and the $D^{(*)}\ell$ candidates are required to have the correct charge-flavor correlation. We account for mixing effects in the $\Bzb$ sample as described in Ref. [@BBmixing]. Cross-feed effects, $i.e.$, $B^-_\mathrm{tag} (\Bzb_\mathrm{tag})$ candidates erroneously reconstructed as a neutral (charged) $B$, are subtracted using estimates from the simulation.
We reconstruct $B^- \to D^{(*)+}\pi^- \ell^- \bar{\nu}_{\ell}$ and $\Bzb \to D^{(*)0}\pi^+ \ell^- \bar{\nu}_{\ell}$ decays starting from the corresponding $B_\mathrm{tag}+D^{(*)} \ell^-$ combinations. We select events with only one additional reconstructed charged track, correctly matched to the $D^{(*)}$ flavor, that has not been used for the reconstruction of the $B_\mathrm{tag}$, the signal $D^{(*)}$, or the lepton. $D(D^{*})$ candidates are selected within 2$\sigma$ (1.5-2.5$\sigma$, depending on the $D^*$ decay mode) of the $D$ mass ($D^{*}-D$ mass difference), where the resolution $\sigma$ is typically around 8 (1-7) MeV$/c^{2}$. For the $\Bzb \rightarrow D^{(*)0}\pi^+ \ell^- \bar{\nu}_{\ell}$ decay, we additionally require the invariant mass difference $m(D^0\pi^+)-m(D^0)$ to be greater than 0.18 GeV/$c^2$ to veto $\Bzb \rightarrow D^{*+} \ell^- \bar{\nu}_{\ell}$ events.
Semileptonic $\Bbar \rightarrow D^{**}\ell^- \bar{\nu}_{\ell}$ decays are identified by the missing mass squared in the event, $m^2_\mathrm{miss} = \left[p(\Upsilon(4S)) -p(B_\mathrm{tag}) - p(D^{(*)}\pi) - p(\ell)\right]^2$, defined in terms of the particle four-momenta. For correctly reconstructed signal events, the only missing particle is the neutrino, and $m^2_\mathrm{miss}$ peaks at zero. Other $B$ semileptonic decays, where one particle is not reconstructed (feed-down) or is erroneously added to the charm candidate (feed-up), exhibit higher or lower values in $m^2_\mathrm{miss}$ [@babar-3]. In feed-down cases where both a $D$ and a $D^*$ candidate have been reconstructed, we keep only the latter candidate.
Mode Selection Criteria
------------------------------------------------ ------------------------------------------------
$B^- \to D^{*+}\pi^- \ell^- \bar{\nu}_{\ell}$ $-0.25 < m^2_\mathrm{miss} < 0.25$ GeV$^2/c^4$
$B^- \to D^{+}\pi^- \ell^- \bar{\nu}_{\ell}$ $-0.25 < m^2_\mathrm{miss} < 0.8$ GeV$^2/c^4$
$\Bzb \to D^{*0}\pi^+ \ell^- \bar{\nu}_{\ell}$ $-0.2 < m^2_\mathrm{miss} < 0.35$ GeV$^2/c^4$
$\Bzb \to D^{0}\pi^+ \ell^- \bar{\nu}_{\ell}$ $-0.15 < m^2_\mathrm{miss} < 0.85$ GeV$^2/c^4$
: $m^2_\mathrm{miss}$ selection criteria.
\[tab:MMcuts\]
The $m^2_\mathrm{miss}$ selection criteria are listed in Table \[tab:MMcuts\]. The $m^2_\mathrm{miss}$ region between 0.2 and 1 GeV$^{2}/c^{4}$ for $\Bbar \rightarrow D \pi\ell^- \bar{\nu}_{\ell}$ events is dominated by feed-down from $\Bbar \to D^{**} (\to D^* \pi) \ell^- \bar{\nu}_{\ell}$ semileptonic decays where the soft pion from the $D^*$ decay is not reconstructed. In order to retain these events we apply an asymmetric cut on $m^2_\mathrm{miss}$ for these modes.
The signal yields for the $\Bbar \to D^{**}\ell^- \bar{\nu}_{\ell}$ decays are extracted through a simultaneous unbinned maximum likelihood fit to the four $m(D^{(*)}\pi) - m(D^{(*)})$ distributions. With the current statistics, validation studies on MC samples show that our sensitivity to non-resonant $\Bbar \to D^{(*)}\pi \ell^- \bar{\nu}_{\ell}$ decays is limited. Including hypotheses for these components results in a fitted contribution that is consistent with zero. Thus we assume that these non-resonant contributions are negligible. The probability that $\Bbar \to D^{**} (\to D^* \pi) \ell^- \bar{\nu}_{\ell}$ decays are reconstructed as $\Bbar \to D^{**} (\to D \pi) \ell^- \bar{\nu}_{\ell}$ is determined with the MC simulation to be 26%(59%) for the $B^-$($\Bzb$) sample and held fixed in the fit.
{width="50.00000%"} \[fig:Fit\]
The Probability Density Functions (PDFs) for the $D^{**}$ signal components are determined using MC $\Bbar \to D^{**}\ell^- \bar{\nu}_{\ell}$ signal events. A convolution of a Breit-Wigner function with a Gaussian, whose resolution is determined from the simulation, is used to model the $D^{**}$ resonances. The $D^{**}$ masses and widths are fixed to measured values [@pdg]. We rely on the MC prediction for the shape of the combinatorial and continuum background. A non-parametric KEYS function [@keys] is used to model this component for the $D^* \pi \ell^- \bar{\nu}_{\ell}$ sample, while for the $D \pi \ell^- \bar{\nu}_{\ell}$ sample we use the convolution of an exponential with a Gaussian to model the tail from virtual $D^*$ mesons. The combinatorial and continuum background yields are estimated from data. We fit the hadronic $B_\mathrm{tag}$ $m_{ES}$ distributions for $\Bbar \to D^{**}\ell^- \bar{\nu}_{\ell}$ events as described in [@babar-3], and we obtain the number of background events from the integral of the background function in the $m_{ES}$ signal region.
Table \[tab:results\] summarizes the results from two fits: one in which we fit the charged and neutral $B$ samples separately, and one in which we impose the isospin constraints ${\cal B} (B^- \to D^{**}\ell^- \bar{\nu}_{\ell})/{\cal B} (\Bzb \to D^{**}\ell^- \bar{\nu}_{\ell}) = \tau_{B^-}/\tau_{\Bzb}$. The latter fit yields a significance greater than 6 standard deviations for all four $D^{**}$ states including systematic uncertainties. The results of this fit are shown in Fig. 1\[fig:Fit\].
The $D_2^*$ contributes to both the $D\pi$ and the $D^*\pi$ samples. In the nominal fit we fix the ratio ${\cal B} (D^*_2 \to D\pi)/{\cal B} (D^*_2 \to D^*\pi)$ to $2.2$ [@pdg]. When we allow this ratio to float we obtain $1.9 \pm 0.6$.
To reduce systematic uncertainties we measure the ratios of the ${\cal B} (\Bbar \rightarrow D^{**} \ell^- \bar{\nu}_{\ell})$ branching fractions to the inclusive $\Bzb$ and $B^-$ semileptonic branching fractions. A sample of $\Bbar \to X \ell^- \bar{\nu}_{\ell}$ events is selected by identifying a charged lepton with $p^*_{\ell}>0.6$ GeV/$c$ and the correct charge correlation with the $B_\mathrm{tag}$ candidate. In the case of multiple $B_\mathrm{tag}$ candidates in an event, we select the one reconstructed in the decay channel with the highest purity, defined as the fraction of signal events in the $m_{ES}$ signal region. Background components that peak in the $m_{ES}$ signal region include cascade $B$ meson decays ($i.e.$, the lepton does not come directly from the $B$) and hadronic decays, and are subtracted using the corresponding MC predictions.
Decay Mode Yield $\epsilon_\mathrm{sig} (\times 10^{-4})$ ${\cal B}$ ($\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ ) $\times$ ${\cal B} (D^{**} \to D^{(*)} \pi^{\pm})$ % $S_\mathrm{tot} (S_\mathrm{stat})$ ${\cal B}$ % $S_\mathrm{tot} (S_\mathrm{stat})$
--------------------------------------------- -------------- ------------------------------------------ --------------------------------------------------------------------------------------------------------------- ------------------------------------ -------------------------- ------------------------------------
$B^- \to D^{0}_1 \ell^- \bar{\nu}_{\ell}$ $165 \pm 18$ 1.24 $0.29 \pm 0.03 \pm 0.03$ 9.9 (12.7) $0.29 \pm 0.03 \pm 0.03$ 10.7 (15.2)
$B^- \to D^{*0}_2 \ell^- \bar{\nu}_{\ell}$ $97 \pm 16$ 1.44 $0.15 \pm 0.02 \pm 0.01$ 6.3 (7.3) $0.12 \pm 0.02 \pm 0.01$ 6.0 (7.4)
$B^- \to D^{'0}_1 \ell^- \bar{\nu}_{\ell}$ $142 \pm 21$ 1.13 $0.27 \pm 0.04 \pm 0.05$ 5.4 (8.0) $0.30 \pm 0.03 \pm 0.04$ 6.4 (10.0)
$B^- \to D^{*0}_0 \ell^- \bar{\nu}_{\ell}$ $137 \pm 26$ 1.15 $0.26 \pm 0.05 \pm 0.04$ 4.5 (5.8) $0.32 \pm 0.04 \pm 0.04$ 6.1 (8.3)
$\Bzb \to D^{+}_1 \ell^- \bar{\nu}_{\ell}$ $88 \pm 14$ 0.70 $0.27 \pm 0.04 \pm 0.03$ 7.0 (8.4)
$\Bzb \to D^{*+}_2 \ell^- \bar{\nu}_{\ell}$ $29 \pm 13$ 0.91 $0.07 \pm 0.03 \pm 0.01$ ($< 0.11$ @90% CL) 2.3 (2.5)
$\Bzb \to D^{'+}_1 \ell^- \bar{\nu}_{\ell}$ $86 \pm 18$ 0.60 $0.31 \pm 0.07 \pm 0.05$ 4.6 (5.8)
$\Bzb \to D^{*+}_0 \ell^- \bar{\nu}_{\ell}$ $142 \pm 26$ 0.70 $0.44 \pm 0.08 \pm 0.06$ 4.7 (6.0)
\[tab:results\]
The total yield for the inclusive $\Bbar \to X \ell^- \bar{\nu}_{\ell}$ decays is obtained from a maximum likelihood fit to the $m_{ES}$ distribution of the $B_\mathrm{tag}$ candidates, as described in [@babar-3]. The fit yields 198,897 $\pm$ 1,578 events for the $B^- \to X \ell^- \bar{\nu}_{\ell}$ sample and 120,168 $\pm$ 1,036 events for the $\Bzb \to X \ell^- \bar{\nu}_{\ell}$ sample.
The ratios ${\cal B}(\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell})/{\cal B}(\Bbar \to X \ell^- \bar{\nu}_{\ell})= (N_\mathrm{sig}/\epsilon_\mathrm{sig})\cdot (\epsilon_\mathrm{sl}/N_\mathrm{sl})$ are obtained by correcting the signal yields for the reconstruction efficiencies (estimated from MC events). Here, $N_\mathrm{sig}$ is the number of $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ signal events, reported in Table \[tab:results\] together with the corresponding reconstruction efficiencies $\epsilon_\mathrm{sig}$, $N_\mathrm{sl}$ is the $\Bbar \to X \ell^-
\bar{\nu}_{\ell}$ signal yield, and $\epsilon_\mathrm{sl}$ is the corresponding reconstruction efficiency including the $B_\mathrm{tag}$ reconstruction, equal to 0.39% and 0.25% for the $B^- \to X \ell^- \bar{\nu}_{\ell}$ and $\Bzb \to X \ell^- \bar{\nu}_{\ell}$ decays, respectively. The absolute branching fractions ${\cal B} (\Bbar \rightarrow D^{**} \ell^- \bar{\nu}_{\ell})$ are then determined using the semileptonic branching fraction ${\cal B}(\Bbar \to X \ell^- \bar{\nu}_{\ell})= ( 10.78 \pm 0.18)\%$ and the ratio of the $\Bzb$ and the $B^-$ lifetimes $\tau_{B^-}/\tau_{\Bzb} = 1.071 \pm 0.009$ [@pdg].
Numerous sources of systematic uncertainties have been investigated. The largest uncertainty is due to the determination of the $\Bbar \to D^{**}\ell^- \bar{\nu}_{\ell}$ signal yields (resulting in 5.5-17.0% relative systematic uncertainty depending on the $D^{**}$ state). This uncertainty is estimated using ensembles of fits to the data in which the input parameters are varied within the known uncertainties in the PDF parameterization (0.2-8.7%), the shape and yield of the combinatorial and continuum background (0.2-10.4%), the modeling of the broad $D^{**}$ states (4.5-13.8%), and the $D^{*}$ feed-down rate (0.5-4.0%). We check that the combinatorial and continuum background shape is well reproduced by the simulation by verifying that the MC samples of right-sign and wrong-sign $D^{(*)}\pi$ combinations have similar shapes, and that the wrong-sign distribution in the data agrees well with that in the simulation. We observe an excess of events in the low invariant mass difference region in the four samples that is not accounted for by the background PDF. We study $\Bbar \to D^{(*)}n\pi \ell^-\bar{\nu}_{\ell}$ ($n>1$) decays, not included in our standard MC simulation, as a possible source of this excess. We use different MC models for these decays, and find that they do not account for all the observed excess. We evaluate a corresponding systematic uncertainty (0.1-3.2%), included in the yield uncertainty above. The uncertainties due to the detector simulation are determined by varying, within bounds given by data control samples, the charged track reconstruction efficiency (1.3-2.0%), the photon reconstruction efficiency (0.2-4.8%), the lepton identification efficiency (1.2-1.6%), and the reconstruction efficiency for low momentum charged (1.2%) and neutral pions (1.3%). We use an HQET model [@LLSW] to test the model dependence of the $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ simulation (0.8-2.5%). We include the uncertainty on the branching fractions of the reconstructed $D$ and $D^{*}$ modes (3.0-4.5%), and on the absolute branching fraction ${\cal B} (\Bbar \to X \ell^- \bar{\nu}_{\ell})$ used for the normalization (1.9%). We also include a systematic uncertainty due to differences in the efficiency of the $B_\mathrm{tag}$ selection in the exclusive selection of $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ decays and the inclusive $\Bbar \to X \ell^- \bar{\nu}_{\ell}$ reconstruction (4.0-5.6%).
In conclusion, we report the simultaneous observation of $\Bbar \to D^{**}\ell^-\bar{\nu}_{\ell}$ decays into the four $D^{**}$ states predicted by HQS. The measured branching fractions are reported in Table \[tab:results\]. We find results consistent with Ref. [@babar-3] for the sum of the different $D^{**}$ branching fractions. The rate for the $D^{**}$ narrow states is in good agreement with recent measurements [@D0]; the one for the broad states is in agreement with DELPHI [@delphi2005] but does not agree with the $D'_1$ limit of Belle [@belle]. The rate for the broad states is found to be large. If these broad states are indeed due to $\Bbar \to D'_1 \ell^- \bar{\nu}_{\ell}$ and $\Bbar \to D^*_0 \ell^- \bar{\nu}_{\ell}$ decays, this is in conflict with the expectations from QCD sum rules.
acknow\_PRL.tex
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| ArXiv |
---
abstract: 'A system of nested dichotomies is a method of decomposing a multi-class problem into a collection of binary problems. Such a system recursively applies binary splits to divide the set of classes into two subsets, and trains a binary classifier for each split. Many methods have been proposed to perform this split, each with various advantages and disadvantages. In this paper, we present a simple, general method for improving the predictive performance of nested dichotomies produced by any subset selection techniques that employ randomness to construct the subsets. We provide a theoretical expectation for performance improvements, as well as empirical results showing that our method improves the root mean squared error of nested dichotomies, regardless of whether they are employed as an individual model or in an ensemble setting.'
author:
- |
Tim Leathart, Eibe Frank, Bernhard Pfahringer and Geoffrey Holmes\
Department of Computer Science, University of Waikato, New Zealand
bibliography:
- 'multi\_subset\_nd.bib'
title: Ensembles of Nested Dichotomies with Multiple Subset Evaluation
---
Introduction
============
Multi-class classification problems are commonplace in real world applications. Some models, like neural networks and random forests, are inherently able to operate on multi-class data directly, while other models, such as classic support vector machines, can only be used for binary (two-class) problems. The standard way to bypass this limitation is to convert the multi-class classification problem into a series of binary problems. There exist several methods of performing this decomposition, the most well-known including one-vs-rest [@rifkin2004defense], pairwise classification [@hastie1998classification] and error-correcting output codes [@dietterich1995solving]. Models that are directly capable of working with multi-class problems may also see improved accuracy from such a decomposition [@mayoraz1997decomposition; @furnkranz2002round; @pimenta2005study].
The use of ensembles of nested dichotomies is one such method for decomposing a multi-class problem into several binary problems. It has been shown to outperform one-vs-rest and perform competitively compared to the aforementioned methods [@frank2004ensembles]. In a nested dichotomy [@fox1997applied], the set of classes is recursively split into two subsets in a tree structure. Two examples of nested dichotomies for a four class problem are shown in Figure \[fig:nd\_example\]. At each split node of the tree, a binary classifier is trained to discriminate between the two subsets of classes. Each leaf node of the tree corresponds to a particular class. To obtain probability estimates for a particular class from a nested dichotomy, assuming the base learner can produce probability estimates, one can simply compute the product of the binary probability estimates along the path from the root node to the leaf node corresponding to the class.
For non-trivial multi-class problems, the space of potential nested dichotomies is very large. An ensemble classifier can be formed by choosing suitable decompositions from this space. In the original formulation of ensembles of nested dichotomies, decompositions are sampled with uniform probability [@frank2004ensembles], but several other more sophisticated methods for splitting the set of classes have been proposed [@dong2005ensembles; @duarte2012nested; @leathart2016building]. Superior performance is achieved when ensembles of nested dichotomies are trained using common ensemble learning methods like bagging or boosting [@rodriguez2010forests].
In this paper, we describe a simple method that can improve the predictive performance of nested dichotomies by considering several splits at each internal node. Our technique can be applied to nested dichotomies built with almost any subset selection method, only contributing a constant factor to the training time and no additional cost when obtaining predictions. It has a single hyperparameter $\lambda$ that gives a trade-off between predictive performance and training time, making it easy to tune for a given learning problem. It is also very easy to implement.
The paper is structured as follows. First, we describe existing methods for class subset selection in nested dichotomies. Following this, we describe our method and provide a theoretical expectation of performance improvements. We then discuss related work, before presenting and discussing empirical results for our experiments. Finally, we conclude and discuss future research directions.
Class Subset Selection Methods\[sec:subset\_selection\_methods\]
================================================================
At each internal node $i$ of a nested dichotomy, the set of classes present at the node $\mathcal{C}_i$ is split into two non-empty, non-overlapping subsets, $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$. In this section, we give an overview of existing class subset selection methods for nested dichotomies. Note that other methods than those listed here have been proposed for constructing nested dichotomies—these are not suitable for use with our method and are discussed later in Related Work.
Random Selection
----------------
The most basic form of class subset selection method, originally proposed in [@frank2004ensembles], is to split the set of classes into two subsets such that each member of the space of nested dichotomies has an equal probability of being sampled. This approach has several attractive qualities. It is simple to compute, and does not scale with the size of the dataset, making it suitable for datasets of any size. Furthermore, for an $n$-class problem, the number of possible nested dichotomies is very large, given by the recurrence relation $$\begin{aligned}
T(n) = (2n-3) \times T(n-1)\end{aligned}$$
where $T(1) = 1$. This ensures that, in an ensemble of nested dichotomies, there is a high level of diversity amongst ensemble members. We refer to this function that relates the number of classes to the size of the sample space of nested dichotomies for a given subset selection method as the *growth function*. Growth functions for each method discussed in this section are compared in Figure \[fig:growth\].
Balanced Selection
------------------
An issue with random selection is that it can produce very unbalanced tree structures. While the number of internal nodes (and therefore, binary models) is the same in any nested dichotomy for the same number of classes, an unbalanced tree often implies that the internal binary models are trained on large datasets near the leaves, which has a negative effect on the time taken to train the full model. Deeper subtrees also provide more opportunity for estimation errors to accumulate. Dong *et. al.* mitigate this effect by enforcing $\mathcal{C}_i$ to be split into two subsets $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ such that ${abs}(|\mathcal{C}_{i1}| - |\mathcal{C}_{i2}|) \leq 1$ [@dong2005ensembles]. This has been shown empirically to have little effect on the accuracy in most cases, while reducing the time taken to train nested dichotomies. Balanced selection has greater benefits for problems with many classes.
It is clear that the sample space of random nested dichotomies is larger than that of class balanced nested dichotomies, but it is still large enough to ensure sufficient ensemble diversity. The growth function for class balanced nested dichotomies is given by $$\begin{aligned}
T_{CB}(n) =
\begin{cases}
\frac{1}{2} \binom{n}{n/2} T_{CB}(\frac{n}{2}) T_{CB}(\frac{n}{2}), & \text{if } n \text{ is even} \\
\binom {n}{(n+1)/2} T_{CB}(\frac{n+1}{2}) T_{CB}(\frac{n-1}{2}), & \text{if } n \text{ is odd} \\
\end{cases}\end{aligned}$$
where $T_{CB}(2) = T_{CB}(1) = 1$ [@dong2005ensembles].
Dong *et. al.* also explored a form of balancing where the amount of data in each subset is roughly equal, which gave similar results for datasets with unbalanced classes [@dong2005ensembles].
Random-Pair Selection
---------------------
Random-pair selection provides a non-deterministic method of creating $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ that groups similar classes together [@leathart2016building]. In random-pair selection, the base classifier is used directly to identify similar classes in $\mathcal{C}_i$. First, a random pair of classes $c_1, c_2 \in \mathcal{C}_i$ is selected, and a binary classifier is trained on just these two classes. Then, the remaining classes are classified with this classifier, and its predictions are stored as a confusion matrix $M$. $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ are constructed by $$\begin{aligned}
\mathcal{C}_{i1} &= \{ c \in \mathcal{C}_i \setminus \{c_1, c_2\} : M_{c, c_1} \leq M_{c, c_2} \} \cup \{c_1\} \\
\mathcal{C}_{i2} &= \{ c \in \mathcal{C}_i \setminus \{c_1, c_2\} : M_{c, c_1} > M_{c, c_2} \} \cup \{c_2\}\end{aligned}$$ where $M_{i,j}$ is defined as the number of examples of class $j$ that were classified as class $i$ by the binary classifier. In other words, a class is assigned to $\mathcal{C}_{i1}$ if it is less frequently confused with $c_1$ than with $c_2$, and to $\mathcal{C}_{i2}$ otherwise. Finally, the binary classifier is re-trained on the new meta-classes $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$. This way, each binary split is more easily separable for the base learner than a completely random split, but also exhibits a degree of randomness, which leads to diverse and high-performing ensembles.
Due to the fact that the size of the sample space of nested dichotomies under random-pair selection is dependent on the dataset and base learner (different initial random pairs may lead to the same split), it is not possible to provide an exact expression for the growth function $T_{RP}(n)$; using logistic regression as the base learner [@leathart2016building], it has been empirically estimated to be $$\begin{aligned}
T_{RP}(n) = p(n)T_{RP}(\frac{n}{3})T_{RP}(\frac{2n}{3}) \label{eqn:random_pair_estimation}\end{aligned}$$ where $$\begin{aligned}
p(n) = 0.3812n^2 - 1.4979n + 2.9027\end{aligned}$$ and $T_{RP}(2) = T_{RP}(1) = 1$.
Multiple Subset Evaluation\[sec:multiple\_subset\_selection\]
=============================================================
In existing class subset selection methods, at each internal node $i$, a single class split $(\mathcal{C}_{i1}, \mathcal{C}_{i2})$ of $\mathcal{C}_i$ is considered, produced by some splitting function $S(\mathcal{C}_i) : \mathbb{N}^n \rightarrow \mathbb{N}^a \times \mathbb{N}^b$ where $a+b=n$. Our approach for improving the predictive power of nested dichotomies is a simple extension. We propose to, at each internal node $i$, consider $\lambda$ subsets $\{(\mathcal{C}_{i1}, \mathcal{C}_{i2})_1 \dots (\mathcal{C}_{i1}, \mathcal{C}_{i2})_\lambda\}$ and choose the split for which the corresponding model has the lowest training root mean squared error (RMSE). The RMSE is defined as the square root of the Brier score [@brier1950verification] divided by the number of classes: $$\textrm{RMSE} = \sqrt{\frac{1}{nm}\sum_{i=1}^n \sum_{j=1}^m (\hat{y}_{ij} - y_{ij})^2 }$$ where $n$ is the number of instances, $m$ is the number of classes, $\hat{y}_{ij}$ is the estimated probability that instance $i$ is of class $j$, and $y_{ij}$ is $1$ if instance $i$ actually belongs to class $j$, and $0$ otherwise. RMSE is chosen over other measures such as classification accuracy because it is smoother and a better indicator of generalisation performance. Previously proposed methods with single subset selection can be considered a special case of this method where $\lambda = 1$.
Although conceptually simple, this method has several attractive qualities, which are now discussed.
#### Predictive Performance.
It is clear that by choosing the best of a series of models at each internal node, the overall performance should improve, assuming the size of the sample space of nested dichotomies is not hindered to the point where ensemble diversity begins to suffer.
#### Generality.
Multiple subset evaluation is widely applicable. If a subset selection method $S$ has some level of randomness, then multiple subset evaluation can be used to improve the performance. One nice feature is that advantages pertaining to $S$ are retained. For example, if class-balanced selection is chosen due to a learning problem with a very high number of classes, we can boost the predictive performance of the ensemble while keeping each nested dichotomy in the ensemble balanced. If random-pair selection is chosen because the computational budget for training is high, then we can improve the predictive performance further than single subset selection in conjunction with random-pair selection.
#### Simplicity.
Implementing multiple subset evaluation is very simple. Furthermore, the computational cost for evaluating multiple subsets of classes scales linearly in the size of the tuneable hyperparameter $\lambda$, making the tradeoff between predictive performance and training time easy to navigate. Additionally, multiple subset evaluation has no effect on prediction times.
Higher values of $\lambda$ give diminishing returns on predictive performance, so a value that is suitable for the computational budget should be chosen. When training an ensemble of nested dichotomies, it may be desirable to adopt a *class threshold*, where single subset selection is used if fewer than a certain number of classes is present at an internal node. This reduces the probability that the same subtrees will appear in many ensemble members, and therefore reduce ensemble diversity. In the lower levels of the tree, the number of possible binary problems is relatively low (Fig. \[fig:growthrate\]).
\[sec:growth\_functions\_effect\]Effect on Growth Functions
-----------------------------------------------------------
Performance of an ensemble of nested dichotomies relies on the size of the sample space of nested dichotomies, given an $n$-class problem, to be relatively large. Multiple subset evaluation removes the $\lambda-1$ class splits that correspond to the worst-performing binary models at each internal node $i$ from being able to be used in the tree. The effect of multiple subset evaluation on the growth function is non-deterministic for random selection, as the sizes of $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ affect the values of the growth function for the subtrees that are children of $i$. The upper bound occurs when all worst-performing splits isolate a single class, and the lower bound is given when all worst-performing splits are class-balanced. Class-balanced selection, on the other hand, is affected deterministically as the size of $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ are the same for the same number of classes.
Growth functions for values of $\lambda \in \{1, 3, 5, 7\}$, for random, class balanced and random-pair selection methods, are plotted in Figure \[fig:growthrate\]. The growth curves for random and class balanced selection were generated using brute-force computational enumeration, while the effect on random-pair selection is estimated.
{width="95.00000%"}
Analysis of error\[sec:theoretical\]
------------------------------------
In this section, we provide a theoretical analysis showing that performance of each internal binary model is likely to be improved by adopting multiple subset evaluation. We also show empirically that the estimates of performance improvements are accurate, even when the assumptions are violated.
Let $E$ be a random variable for the training root mean squared error (RMSE) for some classifier for a given pair of class subsets $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$, and assume $E \sim N(\mu, \sigma^2)$ for a given dataset under some class subset selection scheme. For a given set of $\lambda$ selections of subsets $\mathcal{S} = \{(\mathcal{C}_{i1}, \mathcal{C}_{i2})_1, \dots, (\mathcal{C}_{i1}, \mathcal{C}_{i2})_\lambda\}$ and corresponding training s $\mathcal{E} = \{E_1, \dots, E_\lambda\}$, let $\hat{E}_\lambda = min(\mathcal{E})$. There is no closed form expression for the expected value of $\hat{E}_\lambda$, the minimum of a set of normally distributed random variables, but an approximation is given by $$\mathbb{E}[\hat{E}_\lambda] \approx \mu + \sigma \Phi^{-1} \Bigg( \frac{1-\alpha}{\lambda-2\alpha + 1}\Bigg) \label{eqn:expected_order_statistics}$$ where $\Phi^{-1}(x)$ is the inverse normal cumulative distribution function [@royston1982algorithm], and the *compromise value* $\alpha$ is the suggested value for $\lambda$ given by Harter ([-@harter1961expected]).[^1]
Figure \[fig:norm\_drawn\] illustrates how this expected value changes when increasing values of $\lambda$ from $1$ to $5$. The first two rows show the distribution of $E$ and estimated $\mathbb{E}[\hat{E}_\lambda]$ on the UCI dataset `mfeat-fourier`, for a logistic regression model trained on 1,000 random splits of the class set $\mathcal{C}$. These rows show the training and testing RMSE respectively, using 90% of the data for training and the rest for testing. Note that as $\lambda$ increases, the distribution of the train and test error shifts to lower values and the variance decreases.
This reduction in error affects each binary model in the tree structure, so the effects accumulate when constructing a nested dichotomy. The third row shows the distribution of RMSE of 1,000 nested dichotomies trained with multiple subset evaluation on `mfeat-fourier`, using logistic regression as the base learner, considering increasing values of $\lambda$. As expected, a reduction in error with diminishing returns is seen as $\lambda$ increases.
In order to show an example of how the estimate from (\[eqn:expected\_order\_statistics\]) behaves when the error is not normally distributed, the distribution of $E$ for logistic regression trained on the `segment` UCI data is plotted in the bottom row. This assumption is commonly violated in real datasets, as the distribution is often skewed towards zero error. As with the other examples, 1,000 different random choices for $\mathcal{C}_1$ and $\mathcal{C}_2$ were used to generate the histogram. Although the distribution in this case is not very well modelled by a Gaussian, the approximation of $\mathbb{E}[\hat{E}_\lambda]$ from (\[eqn:expected\_order\_statistics\]) still closely matches the empirical mean. This shows that even when the normality assumption is violated, performance gains of the same degree can be expected. This example is not cherry picked; the same behaviour was observed on the entire collection of datasets used in this study.
Related Work\[sec:related\_work\]
=================================
Splitting a multi-class problem into several binary problems in a tree structure is a general technique that has been referred to by different names in the literature. For example, in a multi-class classification context, nested dichotomies in the broadest sense of the term have been examined as filter trees, conditional probability trees, and label trees. @beygelzimer2009conditional proposed algorithms which build balanced trees and demonstrate the performance on datasets with very large numbers of classes. Filter trees, with deterministic splits [@beygelzimer2009error], as well as conditional probability trees, with probabilistic splits [@beygelzimer2009conditional], were explored. @bengio2010label ([-@bengio2010label]) define a tree structure and optimise all internal classifiers simultaneously to minimise the tree loss. They also propose to learn a low-dimensional embedding of the labels to improve performance, especially when a very large number of classes is present. @melnikov2018effectiveness ([-@melnikov2018effectiveness]) also showed that a method called bag-of-$k$ models—simply sampling $k$ random nested dichotomies and choosing the best one based on validation error—gives competitive predictive performance to the splitting heuristics discussed so far for individual nested dichotomies (i.e., not trained in an ensemble). However, it is very expensive at training time, as $k$ independent nested dichotomies must be constructed and tested on a validation set.
A commonality of these techniques is that they attempt to build a single nested dichotomy structure with the best performance. Nested dichotomies that we consider in this paper, while conceptually similar, differ from these methods because they are intended to be trained in an ensemble setting, and as such, each individual nested dichotomy is not built with optimal performance in mind. Instead, a group of nested dichotomies is built to maximise ensemble performance, so diversity amongst the ensemble members is key [@kuncheva2003measures].
Nested dichotomies based on clustering [@duarte2012nested], are deterministic and used in an ensemble by resampling or reweighting the input. They are built by finding the two classes in $\mathcal{C}_i$ for which the class centroids are furthest from each other by some distance metric. The remainder of the classes are grouped based on the distance of their centroids from the initial two centroids.
@wever2018ensembles ([-@wever2018ensembles]) utilise genetic algorithms to build nested dichotomies. In their method, a population of random nested dichotomies is sampled and runs through a genetic algorithm for several generations. The final nested dichotomy is chosen as the best performing model on a held-out validation set. An ensemble of $k$ nested dichotomies is produced by initialising $k$ individual populations, independently evolving each population, and taking the best-performing model from each population.
Experimental Results\[sec:experiments\]
=======================================
All experiments were conducted in WEKA 3.9 [@hall2009weka], and performed with 10 times 10-fold cross validation. We use class-balanced nested dichotomies and nested dichotomies built with random-pair selection and logistic regression as the base learner.
For both splitting methods, we compare values of $\lambda \in \{1,3,5,7\}$ in a single nested dichotomy structure, as well as in ensemble settings with bagging [@breiman1996bagging] and AdaBoost [@freund1996game]. The default settings in WEKA were used for the `Logistic` classifier as well as for the `Bagging` and `AdaBoostM1` meta-classifiers. We evaluate performance on a collection of datasets taken from the UCI repository [@lichman2013uci], as well as the MNIST digit recognition dataset [@lecun1998gradient]. Note that for MNIST, we report results of 10-fold cross-validation over the entire dataset rather than the usual train/test split. Datasets used in our experiments, and their number of classes, instances and features, are listed in Table \[tab:datasets\].
We provide critical difference plots [@demvsar2006statistical] to summarise the results of the experiments. These plots present average ranks of models trained with differing values of $\lambda$. Models producing results that are not significantly different from each other at the 0.05 significance level are connected with a horizontal black bar. Full results tables showing RMSE for each experimental run, including significance tests, are available in the supplementary materials.
Individual Nested Dichotomies
-----------------------------
**Dataset** **Classes** **Instances** **Features**
---------------- ------------- --------------- --------------
audiology 24 226 70
krkopt 18 28056 7
LED24 10 5000 25
letter 26 20000 17
mfeat-factors 10 2000 217
mfeat-fourier 10 2000 77
mfeat-karhunen 10 2000 65
mfeat-morph 10 2000 7
mfeat-pixel 10 2000 241
MNIST 10 70000 784
optdigits 10 5620 65
page-blocks 5 5473 11
pendigits 10 10992 17
segment 7 2310 20
usps 10 9298 257
vowel 11 990 14
yeast 10 1484 9
: \[tab:datasets\]The datasets used in our experiments.
Restricting the sample space of nested dichotomies through multiple subset evaluation is expected to have a greater performance impact on smaller ensembles than larger ones. This is because in a larger ensemble, a poorly performing ensemble member does not have a large impact on the overall performance. On the other hand, in a small ensemble, one poorly performing ensemble member can degrade the ensemble performance significantly. In the extreme case, where a single nested dichotomy is trained, there is no need for ensemble diversity, so a technique for improving the predictive performance of an individual nested dichotomy should be effective. Therefore, we first compare the performance of single nested dichotomies for different values of $\lambda$.
Figure \[fig:cd\_individual\] shows critical difference plots for both subset selection methods. Class balanced selection shows a clear trend that increasing $\lambda$ improves the RMSE, with the average rank for $\lambda=1$ being exactly 4. For random-pair selection, choosing $\lambda=3$ is shown to be statistically equivalent to $\lambda=1$, while higher values of $\lambda$ give superior results on average.
Ensembles of Nested Dichotomies
-------------------------------
Typically, nested dichotomies are utilised in an ensemble setting, so we investigate the predictive performance of ensembles of ten nested dichotomies with multiple subset evaluation, with bagging and AdaBoost employed as the ensemble methods.
### Class Threshold. {#class-threshold. .unnumbered}
[l]{}\
As previously discussed, the number of binary problems is reduced when multiple subset evaluation is applied. This could have negative a effect on ensemble diversity, and therefore potentially reduce predictive performance. To investigate this effect, we built ensembles of nested dichotomies with multiple subset evaluation by introducing a *class threshold*, the number of classes present at a node required to perform multiple subset evaluation, and varying its value from one to seven. We plot the test RMSE, relative to having a class threshold of one, averaged over the datasets from Table \[tab:datasets\], including standard errors, in Figure \[fig:threshold\]. Surprisingly, the RMSE increases monotonically, showing that the potentially reduced ensemble diversity does not have a negative effect on the RMSE for ensembles of this size. Therefore, we use a class threshold of one in our subsequent experiments. However, note that increasing the class threshold has a positive effect on training time, so it may be useful to apply it in practice.
### Number of Subsets. {#number-of-subsets. .unnumbered}
We now investigate the effect of $\lambda$ when using bagging and boosting. Figure \[fig:cd\_bagging\] shows critical difference plots for bagging. Both subset selection methods improve when utilising multiple subset selection. In the case when class-balanced selection is used, as was observed for single nested dichotomies, the average ranks across all datasets closely correspond to the integer values, showing that increasing the number of subsets evaluated consistently improves performance. For random-pair selection, a more constrained subset selection method, each value of $\lambda > 1$ is statistically equivalent and superior to the single subset case.
The critical difference plots in Figure \[fig:cd\_boosting\] (top) show boosted nested dichotomies are significantly improved by increasing the number of subsets sufficiently when class-balanced nested dichotomies are used. Results are less consistent for random-pair selection, with few significant results in either direction. This is reflected in the critical differences plot (Fig. \[fig:cd\_boosting\], bottom), which shows single subset evaluation statistically equivalent to multiple subset selection for all values of $\lambda$, with $\lambda = 7$ performing markedly worse on average. As RMSE is based on probability estimates, this may be in part due to poor probability calibration, which is known to affect boosted ensembles [@niculescu2005predicting] and nested dichotomies [@leathart2018calibration].
\
Conclusion\[sec:conclusion\]
============================
Multiple subset selection in nested dichotomies can improve predictive performance while retaining the particular advantages of the subset selection method employed. We present an analysis of the effect of multiple subset selection on expected RMSE and show empirically in our experiments that adopting our technique can improve predictive performance, at the cost of a constant factor in training time.
The results of our experiments suggest that for class-balanced selection, performance can be consistently improved significantly by utilising multiple subset evaluation. For random-pair selection, $\lambda=3$ yields the best trade-off between predictive performance and training time, but when AdaBoost is used, our experiments show that multiple subset evaluation is not generally beneficial.
Avenues of future research include comparing multiple subset evaluation with base learners other than logistic regression. It is unlikely that training RMSE of the internal models will be a reliable indicator when selecting splits based on more complex models such as decision trees or random forests, so other metrics may be needed. Also, it may be beneficial to choose subsets such that maximum ensemble diversity is achieved, possibly through information theoretic measures such as variation of information [@meilua2003comparing]. Existing meta-heuristic approaches to constructing individual nested dichotomies like genetic algorithms [@lee2003binary; @wever2018ensembles] could also be adapted to optimise ensembles in this way.
Acknowledgements
================
This research was supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.
[^1]: Appropriate values for $\alpha$ for a given $\lambda$ can be found in Table 3 of [@harter1961expected].
| ArXiv |
---
abstract: |
The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation $$\partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi)
=\mu\Delta\psi+f$$ is studied. The nonlinear transfer of this system conserves the two quadratic quantities $\Psi_1=\langle[(-\Delta)^{1/4}\psi]^2\rangle/2$ and $\Psi_2=\langle[(-\Delta)^{1/2}\psi]^2\rangle/2$ (kinetic energy), where $\langle\cdot\rangle$ denotes a spatial average. The energy density $\Psi_2$ is bounded and its spectrum $\Psi_2(k)$ is shallower than $k^{-1}$ in the inverse-transfer range. For bounded turbulence, $\Psi_2(k)$ in the low-wavenumber region can be bounded by $Ck$ where $C$ is a constant independent of $k$ but dependent on the domain size. Results from numerical simulations confirming the theoretical predictions are presented.
author:
- 'CHUONGV. TRAN[^1]'
- 'JOH NC.BOWMAN'
date: 12 May 2004 and in revised form 05 November 2004
title: 'Large-scale energy spectra in surface quasi-geostrophic turbulence'
---
Introduction
============
The dynamics of a three-dimensional stratified rapidly rotating fluid is characterized by the geostrophic balance between the Coriolis force and pressure gradient. The nonlinear dynamics governed by the first-order departure from this linear balance is known as quasi-geostrophic dynamics and is inherently three-dimensional. The theory of quasi-geostrophy is interesting and the research performed on this subject constitutes a rich literature (see, for example, Charney 1948, 1971; Rhines 1979; Pedlosky 1987). This theory renders a variety of two-dimensional models that are appealing for their relative simplicity and yet sufficiently sophisticated to capture the underlying dynamics of geophysical fluids. One such model, the so-called surface quasi-geostrophic (SQG) equation, is the subject of the present study.
Quasi-geostrophic flows can be described in terms of the geostrophic streamfunction $\psi(\x,t)$. The vertical dimension $z$ is usually taken to be semi-infinite and the horizontal extent may be either bounded or unbounded. Normally, decay conditions are imposed as $z\rightarrow\infty$. At the flat surface boundary $z=0$, the vertical gradient of $\psi(\x,t)$ matches the temperature field $T(\x,t)$, i.e. $T(\x,t)|_{z=0}=\partial_z\psi(\x,t)|_{z=0}$. For flows with zero potential vorticity, this surface temperature field can be identified with $(-\Delta)^{1/2}\psi$, where $\Delta$ is the (horizontal) two-dimensional Laplacian. Here, the operator $(-\Delta)^{1/2}$ is defined by $(-\Delta)^{1/2}\widehat\psi(\k)=k\widehat\psi(\k)$, where $k=|\k|$ is the wavenumber and $\widehat\psi(\k)$ is the Fourier transform of $\psi(\x)$. The conservation equation governing the advection of the temperature $(-\Delta)^{1/2}\psi$ by the surface flow is (Blumen 1978; Pedlosky 1987; Pierrehumbert, Held & Swanson 1994; Held 1995) $$\begin{aligned}
\label{Tadvection}
\partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi)&=&0,\end{aligned}$$ where $J(\varphi,\phi)=\partial_x\varphi\partial_y\phi
-\partial_y\varphi\partial_x\phi$. This equation is known as the SQG equation.
In this paper a forced-dissipative version of (\[Tadvection\]) is studied. A dissipative term of the form $\mu\Delta\psi$, where $\mu>0$, which results from Ekman pumping at the surface, is considered (Constantin 2002; Tran 2004). Since $(-\Delta)^{1/2}\psi$ is the advected quantity, this physical dissipation mechanism corresponds to the (hypoviscous) dissipation operator $\mu(-\Delta)^{1/2}$. The dissipation coefficient $\mu$ has the dimension of velocity and is not vanishingly small in the atmospheric context (Constantin 2002). The system is assumed to be driven by a forcing $f$, for which the spectral support is confined to wavenumbers $k\ge s>0$ (in bounded turbulence, wavenumber zero is replaced by the minimum wavenumber). Thus, the forced-dissipative SQG equation can be written as $$\begin{aligned}
\label{governing}
\partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi)
&=&\mu\Delta\psi+f.\end{aligned}$$ It is customary in the classical theory of turbulence to consider a doubly periodic domain of size $L$; the unbounded case is obtained [*via*]{} the limit $L\rightarrow\infty$.
The Jacobian operator $J(\cdot,\cdot)$ admits the identities $$\begin{aligned}
\label{id}
\langle\chi J(\varphi,\phi)\rangle=-\langle\varphi J(\chi,\phi)\rangle
=-\langle\phi J(\varphi,\chi)\rangle,\end{aligned}$$ where $\langle\cdot\rangle$ denotes the spatial average. As a consequence, the nonlinear term in (\[governing\]) obeys the conservation laws $$\begin{aligned}
\label{conservation}
\langle\psi J(\psi,(-\Delta)^{1/2}\psi)\rangle=
\langle(-\Delta)^{1/2}\psi J(\psi,(-\Delta)^{1/2}\psi)\rangle=0.\end{aligned}$$ It follows that the two quadratic quantities $\Psi_\theta=\langle
|(-\Delta)^{\theta/4}\psi|^2\rangle/2=\int\Psi_\theta(k)\,\dk$, where $\theta=1,2$, are conserved by nonlinear transfer. Here, $\Psi_\theta(k)$ is defined by $\Psi_\theta(k)=k^\theta\Psi(k)$, $\Psi(k)$ is the power density of $\psi$ associated with wavenumber $k$ and $\theta$ is a real number. Note that $\Psi_2(k)$ is the kinetic energy spectrum and $\Psi_2$ is the kinetic energy density.
The simultaneous conservation of two quadratic quantities by advective nonlinearities is a common feature in incompressible fluid systems in two dimensions. Some familiar systems in this category are the Charney–Hasegawa–Mima equation (Hasegawa & Mima 1978; Hasegawa, Maclennan & Kodama 1979) and the class of $\alpha$ turbulence equations (Pierrehumbert 1994), which includes both the Navier–Stokes and the SQG equations. These conservation laws, together with the scale-selectivity of the dissipation and unboundedness of the domain, are the building block of the classical dual-cascade theory (Fj[ø]{}rtoft 1953; Kraichnan 1967, 1971; Leith 1968; Batchelor 1969). This theory, when applied to the present case, implies that $\Psi_1$ cascades to low wavenumbers (inverse cascade) and $\Psi_2$ cascades to high wavenumbers (direct cascade). For some recent discussion on the possibility of a dual cascade in various two-dimensional systems, including the Navier–Stokes and SQG equations, see [@TS02], Tran & Bowman (2003b,2004) and [@T04]. The inverse cascade toward wavenumber $k=0$ would eventually evade viscous dissipation altogether because the spectral dissipation rate vanishes as $k\rightarrow0$. Hence, according to the classical picture, $\Psi_1$ necessarily grows unbounded, by a steady growth rate $\d\Psi_1/\dt>0$, as $t\rightarrow\infty$. Strictly speaking, one may have to address the possibility of a dissipated inverse cascade, i.e. one for which the dissipation of $\Psi_1$ occurs at scales much larger than the forcing scale and for which $\d\Psi_1/\dt$ has a zero time mean. Such a cascade is not a plausible scenario (and is not the traditional undissipated inverse cascade) in fluid systems, dissipated by a single viscous operator, where the viscous dissipation rate diminishes toward the large scales. A discussion of this issue can be found in [@T04].
In this study, upper bounds are derived for the time averages of the kinetic energy density $\Psi_2$ and of the large-scale spectrum $\Psi_2(k)$. These bounds are derived from the governing equation, involving simple but rigorous estimates. The bound on $\Psi_2$ is valid in both unbounded and bounded cases, and a straightforward consequence of this bound is a bound on the energy spectrum, which also applies to both unbounded and bounded turbulence. Another bound on the large-scale energy spectrum is derived by estimating the nonlinear triple-product term representing the inverse transfer of $\Psi_1$. This result applies to bounded turbulence since upper bounds for the triple-product term are inherently domain-size dependent. The difficulties of extending this result to the unbounded case are discussed. Some numerical results confirming the theoretical predictions are presented.
Bounded dynamical quantities
============================
A notable feature of unbounded incompressible fluid turbulence in two dimensions is the appearance of infinite quadratic quantities (per unit area): namely, the kinetic energy density $\Psi_2$ for Navier–Stokes turbulence and $\Psi_1$ for the SQG case. According to the classical theory (applied to the SQG case), a (steady) injection of $\Psi_1$, applied around some finite wavenumber $s$, cascades to ever-larger scales, leading to an unbounded growth of $\Psi_1$ (this is presumably the case for the general quadratic invariant $\Psi_\alpha$ in the so-called $\alpha$ turbulence; [*cf.*]{} Tran 2004). In other words, if the classical inverse cascade is realizable, unbounded incompressible fluid turbulence in two dimensions constitutes an ill-posed problem, in the sense that a key quadratic invariant becomes infinite. Of course, there still exist finite quadratic quantities, in particular the dissipation agent for each quadratic invariant. This section is concerned with these quantities.
On multiplying (\[governing\]) by $\psi$ and $(-\Delta)^{1/2}\psi$ and taking the spatial average of the resulting equations, noting from the conservation laws that the nonlinear terms identically vanish, one obtains evolution equations for $\Psi_1$ and $\Psi_2$, $$\begin{aligned}
\label{Psi1evolution}
\frac{\d}{\dt}\Psi_1&=&-2\mu\Psi_2+\langle f\psi\rangle,\\
\label{Psi2evolution}
\frac{\d}{\dt}\Psi_2&=&-2\mu\Psi_3
+\langle f(-\Delta)^{1/2}\psi\rangle.\end{aligned}$$ Using the Cauchy–Schwarz and geometric–arithmetic mean inequalities, one obtains upper bounds on the injection terms in (\[Psi1evolution\]) and (\[Psi2evolution\]): $$\begin{aligned}
\label{forcebounds}
\langle f\psi\rangle&\le&\langle|(-\Delta)^{1/2}\psi|^2\rangle^{1/2}
\langle|(-\Delta)^{-1/2}f|^2\rangle^{1/2}
\le\mu\Psi_2+\mu^{-1}F_{-2},\nonumber\\
\langle f(-\Delta)^{1/2}\psi\rangle &\le&
\langle|(-\Delta)^{3/4}\psi|^2\rangle^{1/2}
\langle|(-\Delta)^{-1/4}f|^2\rangle^{1/2}
\le\mu\Psi_3+\mu^{-1}F_{-1},\end{aligned}$$ where the ‘integration by parts’ rule $\langle(-\Delta)^\theta\phi
\chi\rangle=\langle(-\Delta)^{\theta'}\phi(-\Delta)^{\theta''}\chi\rangle$, for $\theta=\theta'+\theta''$, has been used and $F_\theta=\langle
|(-\Delta)^{\theta/4}f|^2\rangle/2$. Substituting (\[forcebounds\]) in (\[Psi1evolution\]) and (\[Psi2evolution\]) yields $$\begin{aligned}
\label{evolbound1}
\frac{\d}{\dt}\Psi_1&\le&-\mu\Psi_2+\mu^{-1}F_{-2},\\
\label{evolbound2}
\frac{\d}{\dt}\Psi_2&\le&-\mu\Psi_3+\mu^{-1}F_{-1}.\end{aligned}$$ To avoid unnecessary complications, zero initial conditions are assumed, so that for $T>0$ the time means $\langle\d\Psi_1/\dt\rangle_t=
\Psi_1(T)/T$ and $\langle\d\Psi_2/\dt\rangle_t$ are non-negative. One can then deduce upper bounds on the time means $\langle\Psi_2\rangle_t$ and $\langle\Psi_3\rangle_t$, which are valid regardless of whether or not $\Psi_1$ remains finite in the limit $t\rightarrow\infty$: $$\begin{aligned}
\label{averagebound1}
\langle\Psi_2\rangle_t &\le& \mu^{-2}\langle F_{-2}\rangle_t,\\
\label{averagebound2}
\langle\Psi_3\rangle_t &\le& \mu^{-2}\langle F_{-1}\rangle_t.\end{aligned}$$
For $\theta\in(2,3)$, $\langle\Psi_\theta\rangle_t$ is also bounded. Indeed, from the Hölder inequalities $\Psi_\theta\le\Psi_2^{3-\theta}\Psi_3^{\theta-2}$ ([*cf.*]{} Tran 2004) and $\langle\Psi_2^{3-\theta}\Psi_3^{\theta-2}\rangle_t\le
\langle\Psi_2\rangle_t^{3-\theta}\langle\Psi_3\rangle_t^{\theta-2}$, one can deduce from (\[averagebound1\]) and (\[averagebound2\]) that $$\begin{aligned}
\langle\Psi_\theta\rangle_t &\le&
\langle\Psi_2\rangle_t^{3-\theta}\langle\Psi_3\rangle_t^{\theta-2}
\le \mu^{-2}
\langle F_{-2}\rangle_t^{3-\theta}\langle F_{-1}\rangle_t^{\theta-2}.\end{aligned}$$ This result implies that for $\theta\in(2,3)$, $\langle\Psi_\theta\rangle_t$ is bounded, provided that both $\langle F_{-1}\rangle_t$ and $\langle F_{-2}\rangle_t$ are bounded. This condition is assured if $s>0$ and $F_0$ is bounded, a condition normally required of the forcing, because $F_{-2}\le F_{-1}/s\le F_0/s^2$. One may even consider a class of forcing for which $F_0=\infty$ and $F_{-2}\le F_{-1}/s<\infty$.
Upper bounds of the above type on dynamical quantities are rather trivial for bounded turbulence. However, they are important in the unbounded case, for two reasons. First, the scale-selective viscous dissipation allows for the possibility of unbounded growth of certain quadratic quantities toward the low wavenumbers. Hence, rigorous bounds on dynamical quantities are not as abundant as in the bounded case. Second, analytic studies of the nonlinear triple-product transfer function are difficult in unbounded domains. In the absence of pointwise estimates for the spectrum, these bounds are particularly useful for qualitative estimates of the large-scale distribution of energy. For example, [@T04] uses inequality (\[averagebound1\]) to argue that the energy spectrum $\Psi_2(k)$ should be shallower than $k^{-1}$, as $k\rightarrow0$.
Large-scale energy spectrum
===========================
In this section, it is shown that the physical laws of SQG dynamics admit only large-scale energy spectra shallower than $k^{-1}$. This result is due in part to the fact that the simultaneous conservation of $\Psi_1$ and $\Psi_2$ allows virtually no kinetic energy to get transferred toward the low wavenumbers, so that only large-scale kinetic energy spectra shallower than $k^{-1}$ are possible.
Shell-averaged energy spectrum
------------------------------
For a given wavenumber $r$, let us denote by $S=S(r)$ the wavenumber shell between $k=r/2$ and $k=3r/2$, i.e. $S(r)=\{\k : r/2 \le k \le 3r/2\}$. The shell-averaged energy spectrum $\overline\Psi_2(r)$ over $S(r)$ is defined by $$\begin{aligned}
\label{spectrum}
\overline\Psi_2(r)&=&\frac{1}{r}\int_{r/2}^{3r/2}\Psi_2(k)\,\dk.\end{aligned}$$ In the present case of a doubly periodic domain of size $L$, the Fourier representation of the stream function is $\psi(\x)=\sum_{\k}\exp\{i\k\cdot\x\}\widehat\psi(\k)$, where $\k=2\pi L^{-1}(k_x,k_y)$ with $k_x$ and $k_y$ being integers not simultaneously zero. Let $\psi(S)$ denote the component of $\psi$ spectrally supported by $S$, i.e. $\psi(S)=\sum_{\k\in S}\exp
\{i\k\cdot\x\}\widehat\psi(\k)$. One has $$\begin{aligned}
\label{ineq}
\sup_{\x}|\nabla\psi(S)| &\le& \sum_{\k\in S}k|\widehat\psi(\k)|
\le \left(\sum_{\k\in S}1\sum_{\k\in S}k^2|\widehat\psi(\k)|^2\right)^{1/2}
\le cLr\Psi_2^{1/2}(S),\end{aligned}$$ where the Cauchy–Schwarz inequality is used, the sum $\sum_{\k\in S}1=(cLr)^2$ is the number of wavevectors in $S$, $c$ is an absolute constant of order unity and $\Psi_2(S)$ is the contribution to the kinetic energy from $S$.
Upper bounds for the energy spectrum
------------------------------------
A simple upper bound for $\overline\Psi_2(k)$, which is applicable to both the unbounded and bounded cases, can be derived from (\[averagebound1\]). In fact, it follows from (\[averagebound1\]) and (\[spectrum\]) that $$\begin{aligned}
\label{spectbound1}
\langle\overline\Psi_2(k)\rangle_t&=&\frac{1}{k}\int_{k/2}^{3k/2}
\langle\Psi_2(\kappa)\rangle_t\,\d\kappa
\le \mu^{-2}\langle F_{-2}\rangle_tk^{-1}.\end{aligned}$$ This bound is supposed to apply to $k$ in the inverse-transfer region. For $k$ in the direct-transfer region, (\[averagebound2\]) yields $$\begin{aligned}
\label{spectbound2}
\langle\overline\Psi_2(k)\rangle_t&=&\frac{1}{k}\int_{k/2}^{3k/2}
\langle\Psi_2(\kappa)\rangle_t\,\d\kappa
\le \frac{2}{k^2}\int_{k/2}^{3k/2}
\langle\Psi_3(\kappa)\rangle_t\,\d\kappa
\le 2\mu^{-2}\langle F_{-1}\rangle_tk^{-2}.\end{aligned}$$
The upper bound (\[spectbound1\]) suggests that dimensional analysis arguments, which predict a large-scale $k^{-1}$ energy spectrum, are not well justified. If a persistent inverse cascade of $\Psi_1$ exists ($\d \Psi_1/\dt > 0$), then the energy $\Psi_2$ ought to acquire a value such that $\Psi_2<\mu^{-2}F_{-2}$. In the unbounded case, the large-scale energy spectrum then needs to be strictly shallower than $k^{-1}$, to ensure that the dissipation of $\Psi_1$ does not increase without bound as the inverse cascade proceeds toward $k=0$. On the other hand, if no inverse cascade of $\Psi_1$ exists, then a $k^{-1}$ energy spectrum with limited extent is possible. If viscous dissipation mechanisms with degrees higher than that of the natural dissipation are considered, then the upper bounds derived above are not valid. Nevertheless, diminishing energy transfer towards the lowest wavenumbers appears to be consistent only with spectra shallower than $k^{-1}$ (for low-wavenumber convergence of the energy integral). The numerical results reported in §4 are well suited to this expectation.
An upper bound for the large-scale energy spectrum, based on the nonlinear transfer term, can be derived for the bounded case. This analysis employs elementary but rigorous estimates of the triple-product term. For $3k/2<s$, the evolution of $\Psi_1(S(k))$ is governed by $$\begin{aligned}
\frac{\d}{\dt}\Psi_1(S)&=&-\langle\psi(S)
J(\psi,(-\Delta)^{1/2}\psi)\rangle-2\mu\Psi_2(S) \nonumber\\
&=&\langle(-\Delta)^{1/2}\psi J(\psi,\psi(S))\rangle-2\mu\Psi_2(S)\nonumber\\
&\le&\langle|(-\Delta)^{1/2}\psi||\nabla\psi||\nabla\psi(S)|\rangle
-2\mu\Psi_2(S)\nonumber\\
&\le&\sup_{\x}|\nabla\psi(S)|\langle|(-\Delta)^{1/2}\psi||\nabla\psi|\rangle
-2\mu\Psi_2(S)\nonumber\\
&\le&2cLk\Psi_2^{1/2}(S)\Psi_2-2\mu\Psi_2(S)\nonumber\\
&\le&c^2\mu^{-1}L^2k^2\Psi_2^2-\mu\Psi_2(S)\nonumber\\
&=&c^2\mu^{-1}L^2k^2\Psi_2^2-\mu k\overline\Psi_2(k),\end{aligned}$$ where the second equality is a consequence of (\[id\]) and the second last and last inequalities follow from (\[ineq\]) and the geometric–arithmetic mean inequality, respectively. It follows that $$\begin{aligned}
\label{spectbound3}
\langle\overline\Psi_2(k)\rangle_t&\le&c^2\mu^{-2}L^2k\langle\Psi_2^2\rangle_t.\end{aligned}$$ A notable feature of (\[spectbound3\]) is its dependence on the fluid domain size. The presence of $L$ in this upper bound is natural: the upper bound $\sup_{\x}|\nabla\psi(S)|$, which is associated with the fluid velocity at scales $\approx k^{-1}$, is inherently domain-size dependent. There are no known analytic estimates that allow one to derive an upper bound on the nonlinear transfer function $\langle\psi(S)J(\psi,(-\Delta)^{1/2}\psi)\rangle$ in terms of ‘intensive quantities’ only. This difficulty arises not only in the present estimate but also in other analytic estimates of the transfer function. In other words, the nonlinear triple-product term is intrinsically domain-size dependent. This problem considerably limits our ability to assess the nonlinear transfer in unbounded systems. Finally, it is worth mentioning that although the upper bound (\[spectbound3\]) has a linear dependence on $k$, it may be more excessive than the bound $\mu^{-2}\langle F_{-2}\rangle_t k^{-1}$ derived earlier (even for very low wavenumbers). The reason is that $L^2k\ge k^{-1}$ and the prefactor $c^2\langle\Psi_2^2\rangle_t$ may not be as optimal as $\langle F_{-2}\rangle_t$.
Numerical results
=================
This section reports results from numerical simulations that illustrate the realization of large-scale spectra shallower than $k^{-1}$. Equation (\[governing\]) is simulated in a doubly periodic square of side $2\pi$, where the forcing $\widehat{f}(\bm k)$ is nonzero only for those wavevectors $\bm k$ having magnitudes lying in the interval $K=[59,61]$: $$\begin{aligned}
\label{forcing}
\widehat{f}(\bm k)&=&\frac{\epsilon}{N}\frac{\widehat{\psi}(\bm{k})}
{2\Psi_1(k)}.\end{aligned}$$ Here $\epsilon=1$ is the constant energy injection rate and $N$ is the number of distinct wavenumbers in $K$. The (constant) injection rate of $\Psi_1$ is $\epsilon/s\approx1/60$, where $1/s\approx1/60$ is the mean of $k^{-1}$ over $K$. This type of forcing was used by [@Shepherd87], [@T04] and [@TB04] in numerical simulations of a large-scale zonal jet on the so-called beta-plane and of Navier–Stokes turbulence. The attractive aspect of (\[forcing\]), as noted in [@Shepherd87], is that it is steady. Dealiased $683^2$ and $1365^2$ pseudospectral simulations ($1024^2$ and $2048^2$ total modes) were performed. Three dissipative forms were considered: $2.5\times10^{-2}\Delta\psi$, $-4\times10^{-4}(-\Delta)^{3/2}\psi$, and $-6\times10^{-6}\Delta^2\psi+\mu\Delta\psi$ for several values of $\mu$. The first case represents the natural dissipation of the SQG dynamics due to Ekman pumping, as mentioned earlier. The second case represents thermal (molecular) diffusion since $(-\Delta)^{1/2}\psi$ is equivalent to the fluid temperature. The third case—the mixed hyperviscous/Ekman dissipation form—is considered in order to demonstrate that even slight amounts of Ekman damping will inhibit the formation of an inverse cascade. Unlike [@Smith02], the case of mechanical friction \[$\propto(-\Delta)^{1/2}\psi$\] was not considered. The higher resolution was used for the first (natural dissipation) case and the lower resolution was used for the second and third cases. All simulations were initialized with the spectrum $\Psi_2(k)=10^{-5}\pi k/(60^2+k^2)$.
Figure \[sqg2\] shows the time-averaged steady-state kinetic energy spectrum for the case of the natural dissipation term $2.5\times10^{-2}\Delta\psi$. The dissipation agents of $\Psi_1$ and $\Psi_2$ are, respectively, $\Psi_2$ (energy) and $\Psi_3$. The value of the energy, $0.3333$, implies that the dissipation of $\Psi_1$, averaged in the same period, is $0.01666$. This amounts to virtually all of the injection rate $1/60$. Hence, there exists no inverse cascade of $\Psi_1$ to the large scales and both $\Psi_1$ and $\Psi_2$ are steady. The small-scale energy spectrum scales as $k^{-3.5}$, so that the spectrum $\Psi_3(k)$ scales as $k^{-2.5}$. This scaling means that the energy dissipation occurs mainly around the forcing region and is consistent with the bound (\[spectbound2\]).
Unlike Navier–Stokes turbulence, for which the inverse energy cascade is robust and can be simulated at relatively low resolution, it was noticed that no choice for the value of $\mu$ at the present resolution could be used to simulate an inverse cascade of $\Psi_1$. It is not known whether an inverse cascade of $\Psi_1$ is realizable at higher resolutions, using a smaller value of $\mu$. Nevertheless, this observation suggests that $\Psi_1$ is ‘reluctant’ to cascade to the large scales, as compared with the more robust inverse energy cascade in Navier–Stokes turbulence.
![The time-averaged steady-state energy spectrum $\Psi_2(k)$ [*vs.*]{} $k$ for the dissipation term $2.5\times10^{-2}\Delta\psi$.[]{data-label="sqg2"}](sqg2)
Figure \[sqgviscous1\] shows the kinetic energy spectrum averaged between $t=37.3$ and $t=38.7$, for a lower viscous degree. The dissipation agents of $\Psi_1$ and $\Psi_2$ are, respectively, $\Psi_3$ and $\Psi_4$ (enstrophy). The value of $\Psi_3$ is $20$, implying that the dissipation of $\Psi_1$ is $1.6\times10^{-2}$. This amounts to about $96\%$ of the injection rate $1/60$. The inverse cascade then carries only a few percent of the injection of $\Psi_1$ to the large scales.
The small-scale energy spectrum scales as $k^{-4.5}$, so that the enstrophy spectrum $\Psi_4(k)$ scales as $k^{-2.5}$. Most of the energy dissipation occurs around the forcing region, consistent with a ‘weak’ inverse cascade (one that does not carry virtually all of the injection of $\Psi_1$ toward $k=0$; [*cf.*]{} Tran and Bowman 2004, Tran 2004). No direct cascade is possible for bounded turbulence in equilibrium or for unbounded turbulence in the presence of a weak inverse cascade.
![The quasisteady energy spectrum $\Psi_2(k)$ [*vs.*]{} $k$ averaged between $t=37.3$ and $t=38.7$ for the dissipation term $-4\times10^{-4}(-\Delta)^{3/2}\psi$.[]{data-label="sqgviscous1"}](sqgviscous1)
Similarly, Figure \[sqgviscous2mixed\] shows the kinetic energy spectrum averaged between $t=15.7$ and $t=16.5$ for the mixed dissipation $-6\times10^{-6}\Delta^2\psi+\mu\Delta\psi$, using three different values of $\mu$. When $\mu=0$, the dissipation agents of $\Psi_1$ and $\Psi_2$ are, respectively, $\Psi_4$ (enstrophy) and $\Psi_5$. The value of the enstrophy, $1208$, implies that the dissipation of $\Psi_1$ is $1.45\times10^{-2}$, amounting to about $87\%$ of the injection rate $1/60$. The small-scale energy spectrum scales as $k^{-5}$, so that $\Psi_5(k)$ scales as $k^{-2}$. Again, this scaling means that most of the energy dissipation occurs around the forcing region and that the inverse cascade is weak. We note that as $\mu$ is increased, the inverse cascade becomes increasingly weak. We emphasize this behaviour by plotting in Fig. \[invstrengthvnuL\] the inverse cascade [*strength*]{} $r=1-2s(\mu\Psi_2+6\times10^{-6}\Psi_4)/\epsilon$ for six different values of $\mu$.
![The quasisteady energy spectrum $\Psi_2(k)$ [*vs.*]{} $k$ averaged between $t=15.67$ and $t=16.52$ for the dissipation term $-6\times10^{-6}\Delta^2\psi+\mu\Delta\psi$, using three different values of $\mu$. []{data-label="sqgviscous2mixed"}](sqgviscous2mixed)
![The decay of the inverse cascade strength $r$ for the dissipation term $-6\times10^{-6}\Delta^2\psi+\mu\Delta\psi$ as $\mu$ is increased.[]{data-label="invstrengthvnuL"}](invstrengthvnuL)
Unlike Navier–Stokes turbulence, for which the enstrophy acquires its near-equilibrium value once a discernible inverse-transfer range has formed, the energy in SQG turbulence can remain significantly less than its equilibrium value until a very wide inverse-transfer range has developed. For example, for a one-decade Navier–Stokes inverse-transfer range (achievable in numerical simulations), the enstrophy acquires 95% of its projected equilibrium value (calculated with a $k^{-5/3}$ energy spectrum extrapolated to $k=0$). On the other hand, for a one-decade SQG inverse-transfer range, the energy acquires only 66% of its projected equilibrium value (calculated with a $k^{-0.7}$ energy spectrum extrapolated to $k=0$, as realized in the present simulations; [*cf.*]{} the $\mu=0$ case of Figure \[sqgviscous2mixed\]). This means that one needs a considerably wider inverse-transfer region for SQG turbulence than for Navier–Stokes turbulence, in order to approach a quasi-steady state. This problem is in addition to the resolution limitations at the small scales for both cases.
Due to the steep spectrum in the inverse-transfer region, the energy in the $\mu=0$ case of Figure \[sqgviscous2mixed\] has not acquired a value considerably close to its equilibrium value. This means that the system is still well within the transient phase, However, the dissipation of $\Psi_1$ (proportional to the enstrophy) cannot grow considerably (without significant change to the existing spectrum), because of the high degree of viscosity, which makes the dissipation of $\Psi_1$ relatively insensitive to growth of the large-scale energy.
Conclusion and discussion
=========================
In this paper, the kinetic energy density of SQG turbulence and its large-scale spectrum have been studied. For the unbounded case, upper bounds are derived for the time means of the kinetic energy density and of the large-scale energy spectrum, averaged over a narrow window of wavenumbers. Another result is an upper bound on the the time mean of the large-scale energy spectrum, which is derived for the bounded case. Numerical results confirming the predicted slopes of the large-scale energy spectrum are presented and discussed.
An important feature in SQG turbulence that gives rise to the rigorous upper bound on the time mean of the kinetic energy density in the unbounded case is that the kinetic energy is the dissipation agent of the inverse-cascading candidate $\Psi_1$. This fact is due to the hypoviscous nature of the dissipation operator $(-\Delta)^{1/2}$, a natural physical dissipation mechanism of SQG dynamics (Ohkitani 1997; Constantin 2002; Tran 2004). If $(-\Delta)^{1/2}$ is replaced by an operator of the form $(-\Delta)^{\eta}$, where $\eta>1/2$, then the simple analysis of Section 2 fails to show that the time mean of the energy density $\langle\Psi_2\rangle_t$ is bounded, although it may remain so for low degrees of viscosity $\eta$. The reason is that the amount of energy getting transferred to wavenumbers lower than a given wavenumber $k$ decreases at least as rapidly as $k$, so that the spectral dissipation rate $\propto k^{2\eta}$, a consequence of the dissipation operator $(-\Delta)^{\eta}$, may be sufficiently strong to balance the diminishing inverse energy transfer and keep the energy from growing unbounded.
Numerical simulations of SQG turbulence were performed, using the natural dissipation operator $(-\Delta)^{1/2}$ and two viscous operators $\Delta$ and $(-\Delta)^{3/2}$. The results show large-scale energy spectra shallower than $k^{-1}$, consistent with the theoretical prediction.
There have been attempts to explain, within the context of SQG turbulence (Constantin 2002; Tung & Orlando 2003), the kinetic energy spectra observed in the laboratory experiment of [@Baroud02] and in the atmosphere. In the former case, the turbulence in a rotating tank is driven at a sufficiently small scale to allow for a wide inverse-transferring range. A $k^{-2}$ spectrum extending over nearly two wavenumber decades lower than the forcing wavenumber is observed. In the latter case, a $k^{-5/3}$ spectrum is observed in the mesoscales (see Frisch 1995 and Tung & Orlando 2003 and references therein), which correspond to wavenumbers higher (lower) than the forcing wavenumber if the energy released from baroclinic instability (thunderstorms) is considered to be the driving force. The $-2$ power-law scaling observed in [@Baroud02] for the wavenumber range lower than the forcing wavenumber is excessively steeper than the permissible scalings derived in this work. The $-5/3$ slope in the atmosphere is either steeper (if considered to be on the wavenumber range lower than the forcing wavenumber) or shallower (if considered to be on the wavenumber range higher than the forcing wavenumber) than the permissible slopes. According to the present analysis, these data cannot be attributed to SQG turbulence.
We would like to thank two anonymous referees for their comments, which were helpful in improving this manuscript. This work was funded by a Pacific Institute for the Mathematical Sciences Postdoctoral Fellowship, an Alexander von Humboldt Research Fellowship, and the Natural Sciences and Engineering Research Council of Canada.
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[^1]: Present address: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
| ArXiv |
---
abstract: |
Let $n,k\in\mathbb{N}$ and let $p_{n}$ denote the $n$th prime number. We define $p_{n}^{(k)}$ recursively as $p_{n}^{(1)}:=p_{n}$ and $p_{n}^{(k)}=p_{p_{n}^{(k-1)}}$, that is, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime.
In this note we give answers to some questions and prove a conjecture posed by Miska and Tóth in their recent paper concerning subsequences of the sequence of prime numbers. In particular, we establish explicit upper and lower bounds for $p_{n}^{(k)}$. We also study the behaviour of the counting functions of the sequences $(p_{n}^{(k)})_{k=1}^{\infty}$ and $(p_{k}^{(k)})_{k=1}^{\infty}$.
author:
- 'B[ł]{}ażej Żmija'
title: A note on primes with prime indices
---
[^1]
Introduction
============
Let $(p_{n})_{n=1}^{\infty}$ be the sequence of consecutive prime numbers. In a recent paper [@MT] Miska and Tóth introduced the following subsequences of the sequence of prime numbers: $p_{n}^{(1)}:=p_{n}$ and for $k\geq 2$ $$\begin{aligned}
p_{n}^{(k)}:=p_{p_{n}^{(k-1)}}.\end{aligned}$$ In other words, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime. They also defined $$\begin{aligned}
{{\rm Diag}\mathbb{P}}:= & \{\ p_{k}^{(k)}\ |\ k\in\mathbb{N}\ \}, \\
\mathbb{P}_{n}^{T}:= & \{\ p_{n}^{(k)}\ |\ k\in\mathbb{N}\ \}\end{aligned}$$ for each positive integer $n$.
The main motivation in [@MT] was the known result that the set of prime numbers is ($R$)-dense, that is, the set $\{\ \frac{p}{q}\ |\ p,q\in\mathbb{P}\ \}$ is dense in $\mathbb{R}_{+}$ (with respect to the natural topology on $\mathbb{R}_{+}$). It was proved in [@MT] that for each $k\in\mathbb{N}$ the sequence $\mathbb{P}_{k}:=(p_{n}^{(k)})_{n=1}^{\infty}$ is ($R$)-dense. This result might be surprising, because the sequences $\mathbb{P}_{k}$ are very sparse. In fact, for each $k$ set $\mathbb{P}_{k+1}$ is a zero asymptotic density subset of $\mathbb{P}_{k}$. On the other hand, it was showed, that the sequences $(p_{n}^{(k)})_{k=1}^{\infty}$ for each fixed $n\in\mathbb{N}$, and $(p_{k}^{(k)})_{k=1}^{\infty}$ are not ($R$)-dense.
Results of another type that were proved in [@MT] concern the asymptotic behaviour of $p_{n}^{(k)}$ as $n\rightarrow\infty$, or as $k\rightarrow\infty$. In particular, as $n\rightarrow\infty$, we have for each $k\in\mathbb{N}$ $$\begin{aligned}
p_{n}^{(k)}\sim n(\log n)^{k},\ \ \ \ p_{n+1}^{(k)}\sim p_{n}^{(k)}, \ \ \ \ \log p_{n}^{(k)}\sim \log n\end{aligned}$$ by [@MT Theorem 1]. Some results from [@MT] concerning $p_{n}^{(k)}$ as $k\rightarrow\infty$ are mentioned later.
For a set $A\subseteq \mathbb{N}$ let $A(x)$ be its counting function, that is, $$\begin{aligned}
A(x):=\# \left(A\cap [1,x]\right).\end{aligned}$$ Miska and Tóth posed four questions concerning the numbers $p_{n}^{(k)}$:
1. Is it true that $p_{k+1}^{(k)}\sim p_{k}^{(k)}$ as $k\rightarrow\infty$?
2. Are there real constants $c>0$ and $\beta$ such that $$\begin{aligned}
\exp\mathbb{P}_{n}^{T}(x)\sim cx(\log x)^{\beta}
\end{aligned}$$ for each $n\in\mathbb{N}$?
3. Are there real constants $c>0$ and $\beta$ such that $$\begin{aligned}
\exp{{\rm Diag}\mathbb{P}}(x)\sim cx(\log x)^{\beta}?
\end{aligned}$$
4. Is it true that $$\begin{aligned}
{{\rm Diag}\mathbb{P}}(x)\sim\mathbb{P}_{n}^{T}(x)
\end{aligned}$$ for each $n\in\mathbb{N}$?
The aim of this paper it to give answers to question B, C and D.
The main ingredients of our proofs are the following inequalities: $$\begin{aligned}
\label{ineqPrimes}
n\log n<p_{n}<2n\log n.\end{aligned}$$ The first inequality holds for all $n\geq 2$, and the second one for all $n\geq 3$. For the proofs, see [@RS]. In Section \[Results\] we use (\[ineqPrimes\]) in order to show explicit bounds for $p_{n}^{(k)}$. In particular, for all $n>e^{4200}$ we have: $$\begin{aligned}
\log p_{n}^{(k)}= & k(\log k+\log\log k+O_{n}(1)), \\
\log p_{k}^{(k)}= & k(\log k+\log\log k+O(\log\log\log k)),\end{aligned}$$ as $k\rightarrow\infty$, where the implied constant in the first line may depend on $n$, see Theorem \[MAIN\] below. In consequence, we improve the (in)equalities $$\begin{aligned}
\lim_{k\rightarrow\infty}\frac{p_{n}^{(k)}}{k\log k}= & 1, \\
1\leq \liminf_{k\rightarrow\infty}\frac{p_{k}^{(k)}}{k\log k}\leq & \limsup_{k\rightarrow\infty}\frac{p_{k}^{(k)}}{k\log k}\leq 2.\end{aligned}$$ that appeared in [@MT]. Then we show in Section \[Coro\] that the answers to questions B and C are negative (Corollary \[CoroBC\]), while the one for question D is affirmative (Theorem \[AsympEqThm\]). In fact, we find the following relation: $$\begin{aligned}
\mathbb{P}_{n}^{T}(x)\sim{{\rm Diag}\mathbb{P}}(x)\sim\frac{\log x}{\log\log x}\end{aligned}$$ for all positive integers $n$.
In their paper, Miska and Tóth also posed a conjecture, that we state here as a proposition, since it is in fact a consequence of a result that had already appeared in [@MT].
Let $n\in\mathbb{N}$ be fixed. Then $$\begin{aligned}
\frac{p_{n}^{(k)}}{p_{k}^{(k)}}\longrightarrow 0\end{aligned}$$ as $k\longrightarrow\infty$.
Let $k> p_{n}$. Then $$\begin{aligned}
0\leq \frac{p_{n}^{(k)}}{p_{k}^{(k)}}<\frac{p_{n}^{(k)}}{p_{p_{n}}^{(k)}}=\frac{p_{n}^{(k)}}{p_{n}^{(k+1)}}.\end{aligned}$$ The expression on the right goes to zero as $k$ goes to infinity, as was proved in [@MT Corollary 3].
It is worth to note, that primes with prime indices have already appeared in the literature, for example in [@BB] and [@BKO]. However, according to our best knowledge, our paper is the second one (after [@MT]), where the number of iterations of indices, that is, the number $k$ in $p_{n}^{(k)}$, is not fixed.
Throughout the paper we use the following notation: $\log x$ denotes the natural logarithm of $x$, and for functions $f$ and $g$ we write $f\sim g$ if $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=1$, $f=O(g)$ if there exists a positive constants $c$ such that $f(x)<cg(x)$, and $f=o(g)$ if $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=0$.
Upper and lower bounds for $p_{n}^{(k)}$ {#Results}
========================================
In this Section, we find explicit upper and lower bounds for $p_{n}^{(k)}$. We start with the upper bound.
\[lemUP\] Let $n\geq 9$. Then for each $k\in\mathbb{N}$ we have: $$\begin{aligned}
p_{n}^{(k)}<2^{2k-1}\cdot n\cdot (k-1)!\cdot\big(\log(\max\{k,n\})\big)^{k}. \end{aligned}$$ In particular, $$\begin{aligned}
p_{n}^{(k)}< \big(4\cdot k\log k\big)^{k}\end{aligned}$$ for $k\geq n$.
We proceed by induction on $k$. For $k=1$ it is a simple consequence of (\[ineqPrimes\]). Then the second induction step goes as follows: let us denote $m:=\max\{n,k\}$. Observe that $(k-1)!<(k-1)^{k-1}<m^{k-1}$ and $4\log m<m$ for $m\geq 9$. Hence, $$\begin{aligned}
p_{n}^{(k+1)}\leq & 2p_{n}^{k}\log p_{n}^{(k)}<2\cdot 2^{2k-1}\cdot n\cdot (k-1)!\cdot(\log m)^{k}\cdot \log\left[2^{2k-1}\cdot n\cdot (k-1)!\cdot (\log m)^{k}\right] \\
< & 2^{2k}\cdot n\cdot (k-1)!\cdot(\log m)^{k} \log\left[4^{k}\cdot m\cdot m^{k-1}\cdot (\log m)^{k}\right]=2^{2k}\cdot n\cdot (k-1)!\cdot(\log m)^{k} \log\left[4\cdot m\cdot \log m\right]^{k} \\
\leq & 2^{2k}\cdot n\cdot k!\cdot (\log m)^{k}\cdot \log[m]^{2}\leq 2^{2k+1}\cdot n\cdot k!\cdot (\log m)^{k+1}.\end{aligned}$$
The second part of the statement is an easy consequence of the first part and the inequalities $(k-1)!<k^{k-1}$ and $n\leq k$.
In order to prove a lower bound for $p_{n}^{(k)}$ we will need the following fact.
\[lemL\] Let $$\begin{aligned}
L(x):=\left(\frac{x}{x+1}\right)^{x+1}\left(\frac{\log x}{\log (x+1)}\right)^{x+1}.\end{aligned}$$ Then we have $$\begin{aligned}
L(x)>0.32627\end{aligned}$$ for all $x\geq 4200$.
Observe, that the function $\left(\frac{x}{x+1}\right)^{x+1}$ is increasing. Indeed, if $$\begin{aligned}
f(x):=\log\left(\frac{x}{x+1}\right)^{x+1}=(x+1)\left(\log x-\log (x+1)\right),\end{aligned}$$ then $$\begin{aligned}
f'(x)=\log x-\log (x+1)+(x+1)\left(\frac{1}{x}-\frac{1}{x+1}\right)=\frac{1}{x}-\log \left(1+\frac{1}{x}\right)>0,\end{aligned}$$ where the last inequality follows from the well-known inequality $y>\log (1+y)$ used with $y=\frac{1}{x}$. Hence, we can bound $$\begin{aligned}
\label{L1}
\left(\frac{x}{x+1}\right)^{x+1}\geq \left(\frac{4200}{4201}\right)^{4201}\end{aligned}$$ for all $x\geq 4200$.
Now we need to find a lower bound for $\left(\frac{\log x}{\log (x+1)}\right)^{x+1}$. Let us write $$\begin{aligned}
\left(\frac{\log x}{\log (x+1)}\right)^{x+1}=\left[\left(1-\frac{1}{\frac{\log (x+1)}{\log (x+1)-\log x}}\right)^{\frac{\log (x+1)}{\log (x+1)-\log x}}\right]^{\log\left(1+\frac{1}{x}\right)^{x}\cdot\frac{1}{\log (x+1)}\cdot\left(1+\frac{1}{x}\right)}.\end{aligned}$$ At first, we prove that functions $g(t):=\left(1-\frac{1}{t}\right)^{t}$ and $h(x):=\frac{\log (x+1)}{\log (x+1)-\log x}$ are increasing. For the function $g(t)$ it is enough to observe, that $\log g(t)=f(t-1)$ and the function $f(x)$ is increasing. For the function $h(x)$ we have: $$\begin{aligned}
h'(x)= & \frac{1}{(\log (x+1)-\log x)^{2}}\left[\frac{\log (x+1)-\log x}{x+1}-\log (x+1)\left(\frac{1}{x+1}-\frac{1}{x}\right)\right] \\
= & \frac{1}{(\log (x+1)-\log x)^{2}}\left[\frac{\log (x+1)}{x}-\frac{\log x}{x+1}\right] >0.\end{aligned}$$
The fact that the functions $g(t)$ and $h(x)$ are increasing, together with the properties $g(h(4200))\in (0,1)$ and $\log\left(1+\frac{1}{x}\right)^{x}<1$, give us $$\label{L2}
\begin{aligned}
\left(\frac{\log x}{\log (x+1)}\right)^{x+1}\geq & \big[g(h(4200))\big]^{\log\left(1+\frac{1}{x}\right)^{x}\cdot\frac{1}{\log (x+1)}\cdot\left(1+\frac{1}{x}\right)}
> \big[g(h(4200))\big]^{\frac{1}{\log (x+1)}\cdot\left(1+\frac{1}{x}\right)} \\
\geq & \big[g(h(4200))\big]^{\frac{1}{\log (4200+1)}\cdot\left(1+\frac{1}{4200}\right)}=\left(\frac{\log 4200}{\log 4201}\right)^{\frac{4201}{4200}\cdot\frac{1}{\log 4201 -\log 4200}}
\end{aligned}$$ for all $x\geq 4200$.
Combining (\[L1\]) and (\[L2\]) we get the inequality $$\begin{aligned}
L(x)\geq \left(\frac{4200}{4201}\right)^{4201}\left(\frac{\log 4200}{\log 4201}\right)^{\frac{4201}{4200}\cdot\frac{1}{\log 4201 -\log 4200}}\approx 0.3262768>0.32627.\end{aligned}$$ The proof is finished.
In the next lemma we provide a lower bound for $p_{n}^{(k)}$.
\[lemDOWN\] If $n>e^{4200}$, then for all $k\geq\lfloor\log n\rfloor$ we have $$\begin{aligned}
p_{n}^{(k)}>\left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}.\end{aligned}$$
First, let us observe that a simple induction argument on $k$ implies the inequality $$\begin{aligned}
\label{ineqDOWN}
p_{n}^{(k)}>n(\log n)^{k}.\end{aligned}$$ Indeed, for $k=1$ this follows from left inequality in (\[ineqPrimes\]). Using the same inequality we get also $$\begin{aligned}
p_{n}^{(k+1)}>p_{n}^{(k)}\log p_{n}^{(k)}>n (\log n)^{k}\log(n (\log n)^{k})>n(\log n)^{k+1},\end{aligned}$$ and hence (\[ineqDOWN\]).
Now we show that the inequality from the statement is true for $k=\lfloor \log n\rfloor$. Because of (\[ineqDOWN\]) it is enough to show: $$\begin{aligned}
n(\log n)^{k}>\left(\frac{e\cdot k\log k}{\log\log n}\right)^{k},\end{aligned}$$ or equivalently, after taking logarithms we get $$\begin{aligned}
\log n+\lfloor\log n\rfloor\log\log n>\lfloor\log n\rfloor+\lfloor\log n\rfloor\log\lfloor\log n\rfloor+\lfloor\log n\rfloor\log\log\lfloor\log n\rfloor -\lfloor\log n\rfloor\log\log\log n.\end{aligned}$$ This is equivalent to the inequality $$\begin{aligned}
\big(\log n-\lfloor\log n\rfloor\big)+\lfloor\log n\rfloor\big(\log\log n-\log\lfloor\log n\rfloor\big)+\lfloor\log n\rfloor\big(\log\log\log n-\log\log\lfloor\log n\rfloor\big)>0,\end{aligned}$$ which is obviously true.
In order to finish the proof, we again use the induction argument. The inequality from the statement of our lemma is true for $k=\lfloor \log n\rfloor$. Assume it holds for some $k\geq \lfloor\log n\rfloor$. Then by (\[ineqPrimes\]) and the induction hypothesis we get $$\begin{aligned}
p_{n}^{(k+1)}> & p_{n}^{(k)}\log p_{n}^{(k)}>\left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}\log \left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}.\end{aligned}$$ It is enough to show that for all $n>e^{4200}$ and all $k\geq\lfloor\log n\rfloor$ we have $$\begin{aligned}
\left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}\log \left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}>\left(\frac{e\cdot (k+1)\log (k+1)}{\log\log n}\right)^{k+1}.\end{aligned}$$ This is equivalent to $$\begin{aligned}
k^{k+1}(\log k)^{k}\left[\log k+\log\left(\frac{e\log k}{\log\log n}\right)\right]>\frac{e}{\log\log n}(k+1)^{k+1}(\log(k+1))^{k+1}.\end{aligned}$$ Recall, that we assume that $k\geq \lfloor\log n\rfloor$. Thus $k^{e}>\log n$, that is, $e\log k>\log\log n$. Therefore, it is enough to show the following inequalities: $$\begin{aligned}
k^{k+1}(\log k)^{k+1}>\frac{e}{\log\log n}(k+1)^{k+1}(\log(k+1))^{k+1},\end{aligned}$$ or equivalently $$\begin{aligned}
\left(\frac{k}{k+1}\right)^{k+1}\left(\frac{\log k}{\log (k+1)}\right)^{k+1}>\frac{e}{\log\log n}.\end{aligned}$$ Notice that the left-hand side expression of the last inequality is equal to $L(k)$, where the function $L(x)$ is defined in the statement of Lemma \[lemL\]. If $n>e^{4200}$, then $k\geq \lfloor\log n\rfloor \geq 4200$, and Lemma \[lemL\] implies $L(k)>0.32627$. Therefore, if $N:=\max\left\{\lfloor e^{4200}\rfloor,\lceil e^{e^{e/0.32627}}\rceil\right\}=\lfloor e^{4200}\rfloor$, then for all $n>N$ and $k\geq \lfloor\log n\rfloor$ we have: $$\begin{aligned}
L(k)>0.32627\geq \frac{e}{\log\log N}>\frac{e}{\log\log n}.\end{aligned}$$ This finishes the proof.
Main results {#Coro}
============
We begin this section by a theorem that provides good information about asymptotic growth of $\log p_{n}^{(k)}$ for large fixed $n$, and for $\log p_{k}^{(k)}$ as $k\rightarrow\infty$.
\[MAIN\]
1. Let $n>e^{4200}$. Then $$\begin{aligned}
\log p_{n}^{(k)}=k(\log k+\log\log k+O_{n}(1))
\end{aligned}$$ as $k\rightarrow\infty$, where the implied constant may depend on $n$.
2. We have $$\begin{aligned}
\log p_{k}^{(k)}=k(\log k+\log\log k+O(\log\log\log k))
\end{aligned}$$ as $k\rightarrow\infty$.
If $n>e^{4200}$ and $k\geq n$, Lemmas \[lemUP\] and \[lemDOWN\] give us: $$\begin{aligned}
\left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}<p_{n}^{(k)}<\left(4\cdot k\log k\right)^{k}.\end{aligned}$$ After taking logarithms, we simply get the first part of our theorem. In order to get the second part, we need to put $n=k$ and repeat the reasoning.
Now we give the answer to Question D.
\[AsympEqThm\] For each $n\in\mathbb{N}$ we have $$\begin{aligned}
{{\rm Diag}\mathbb{P}}(x)\sim \mathbb{P}_{n}^{T} (x)\end{aligned}$$ as $x\rightarrow\infty$.
From [@MT Theorem 6] we know that $$\begin{aligned}
\mathbb{P}_{m}^{T}(x)\sim \mathbb{P}_{n}^{T}(x)\end{aligned}$$ for each $m,n\in\mathbb{N}$. Therefore, it is enough to prove ${{\rm Diag}\mathbb{P}}(x)\sim \mathbb{P}_{n}^{T} (x)$ for some sufficiently large $n$.
Let $n=\lfloor e^{4200}\rfloor +100$. We use the idea from the proof of [@MT Theorem 17]. Let $x$ be a large real number. Let $k$ be such that $p_{k}^{(k)}\leq x<p_{k+1}^{(k+1)}$. Then ${{\rm Diag}\mathbb{P}}(x)=k$. By [@MT Theorem 8] and Theorem \[MAIN\] above we have $$\begin{aligned}
\frac{\mathbb{P}_{n}^{T}(x)}{{{\rm Diag}\mathbb{P}}(x)}\leq & 1+\frac{\log p_{k+1}^{(k+1)}}{k\log\log p_{n}^{(k)}}-\frac{\log p_{n}^{(k)}}{k\log\log p_{n}^{(k)}} \\
= & 1+\frac{(1+o(1))(k+1)\log (k+1)}{k\log \big[(1+o(1))k\log k\big]}-\frac{(1+o(1))k\log k}{k\log\big[(1+o(1))k\log k\big]} \\
= & 1+(1+o(1))\left(1+\frac{1}{k}\right)\frac{\log (k+1)}{\log k+\log\left[(1+o(1))\log k\right]}-(1+o(1))\frac{\log k}{\log k+\log\left[(1+o(1))\log k\right]}.\end{aligned}$$ The whole last expression goes to $1$ as $k$ goes to infinity. On the other hand, ${{\rm Diag}\mathbb{P}}(x)\leq \mathbb{P}_{n}^{T}(x)$ for $x\geq p_{n}^{(n)}$ and we get the result.
The answers to Questions B and C will follow from our next result, which is of independent interest.
\[AsympDiagP\]
1. Let $n\in\mathbb{N}$. Then $$\begin{aligned}
\mathbb{P}_{n}^{T}(x)\sim\frac{\log x}{\log\log x}.
\end{aligned}$$
2. We have $$\begin{aligned}
{{\rm Diag}\mathbb{P}}(x)\sim \frac{\log x}{\log\log x}.
\end{aligned}$$
In view of Theorem \[AsympEqThm\], it is enough to show the statement for the function ${{\rm Diag}\mathbb{P}}(x)$. Let us fix an arbitrarily small number $\varepsilon >0$ and take a sufficiently large real number $x$ and find $k$ such that $p_{k}^{(k)}\leq x<p_{k+1}^{(k+1)}$. Then ${{\rm Diag}\mathbb{P}}(x)=k$ and by Lemmas \[lemUP\] and \[lemDOWN\] we have $$\begin{aligned}
k^{k}<x<k^{(1+\varepsilon )k}.\end{aligned}$$ Let us write $x=e^{y}$. Then $$\begin{aligned}
k\log k< y<(1+\varepsilon )k\log k.\end{aligned}$$ If $y$ is sufficiently large, this implies $$\begin{aligned}
\label{ineqyk}
(1-\varepsilon )\frac{y}{\log y}< k<(1+\varepsilon )\frac{y}{\log y}.\end{aligned}$$ Indeed, if $k\leq (1-\varepsilon )\frac{y}{\log y}$, then $$\begin{aligned}
y<(1+\varepsilon)k\log k\leq (1-\varepsilon^{2})\frac{y}{\log y}\log\left((1-\varepsilon )\frac{y}{\log y}\right)<(1-\varepsilon^{2})\left(1-\frac{\log\log y}{\log y}\right)y,\end{aligned}$$ which is impossible. Similarly, if $k\geq (1+\varepsilon )\frac{y}{\log y}$, then $$\begin{aligned}
y>k\log k\geq (1+\varepsilon )\frac{y}{\log y}\log\left((1+\varepsilon )\frac{y}{\log y}\right)>(1+\varepsilon )\left(1-\frac{\log\log y}{\log y}\right)y.\end{aligned}$$ The above inequality cannot hold if $y$ is sufficiently large.
If we go back to $k={{\rm Diag}\mathbb{P}}(x)$ and $y=\log x$ in (\[ineqyk\]), we get $$\begin{aligned}
(1-\varepsilon )\frac{\log x}{\log\log x}<{{\rm Diag}\mathbb{P}}(x)<(1+\varepsilon )\frac{\log x}{\log\log x}.\end{aligned}$$ The number $\varepsilon >0$ was arbitrary, so the result follows.
\[CoroBC\] There do not exist constants $c>0$ and $\beta$ such that $$\begin{aligned}
\exp\mathbb{P}_{n}^{T}(x)\sim cx(\log x)^{\beta}\end{aligned}$$ for some $n$, or $$\begin{aligned}
\exp{{\rm Diag}\mathbb{P}}(x)\sim cx(\log x)^{\beta}.\end{aligned}$$
We prove the result only for ${{\rm Diag}\mathbb{P}}(x)$. The case of $\mathbb{P}_{n}^{T}(x)$ is analogous.
Assume to the contrary, that $\exp{{\rm Diag}\mathbb{P}}(x)\sim cx(\log x)^{\beta}$ for some $c>0$ and $\beta$. Then $\exp{{\rm Diag}\mathbb{P}}(x)=(1+o(1))cx(\log x)^{\beta}$. This, after taking logarithms on both sides, implies ${{\rm Diag}\mathbb{P}}(x)=\log x+O(\log\log x)$, contradicting Theorem \[AsympDiagP\].
Acknowledgement {#acknowledgement .unnumbered}
===============
I would like to express my gratitude to Piotr Miska, who informed me about the problem and helped to improve the quality of the paper. I would also like to thank Carlo Sanna, whose comments allowed me to simplify a part of the paper.
[20]{} K. Broughan, R. Barnett, [*On the subsequence of primes having prime subscripts*]{}, Journal of Integer Sequences **12**, article 09.2.3, 2019. J. Bayless, D. Klyve, T. Oliveira e Silva, [*New bounds and computations on prime-indexed primes*]{}, Integers **13**: A43:1–-A43:21, 2013. P. Miska, J. Tóth, [*On interesting subsequences of the sequence of primes*]{}, preprint: arXiv:1908.10421. J. B. Rosser, L. Schoenfeld, [*Approximate formulas for some functions of prime numbers*]{}, Illinois J. Math. **6**, no. 1 , 64–94, 1962. doi:10.1215/ijm/1255631807.
[^1]: During the preparation of the work, the author was a scholarship holder of the Kartezjusz program funded by the Polish National Center for Research and Development.
| ArXiv |
---
abstract: |
For each ${\small b\in\left( 0,\,\infty\right) }$ we intend to generate a decreasing sequence of subsets $\left( \mathcal{Y}_{b}^{\left( n\right)
}\right) \subset Y_{\mathrm{conc}}$ depending on $b$ such that whenever $n\in\mathbb{N}$, then $\mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right) }%
$ is dense in $\mathcal{Y}_{b}^{\left( n\right) }$ and the following four sets $\mathcal{Y}_{b}^{\left( n\right) }$, $\mathcal{Y}_{b}^{\left(
n\right) }\backslash\left( \mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right)
}\right) $, $\mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right) }$ and $\mathcal{Y}_{\mathrm{conc}}$ are pairwise equinumerous. Among others we also show that if $f$ is any measurable function on a measure space $\left(
\Omega,\mathcal{F},\lambda\right) $ and $p\in\left[ 1,\infty\right) $ is an arbitrary number then the quantities $\left\Vert f\right\Vert _{L^{p}}$ and $\sup_{\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}}\left( \Phi\left(
1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert
\right\Vert _{L^{p}}$ are equivalent, in the sense that they are both either finite or infinite at the same time.
address: |
Institute of Mathematics\
University of Miskolc\
H-3515 Miskolc–Egyetemváros\
Hungary
author:
- 'N. K. Agbeko'
date: 'March 24th, 2006'
title: 'Bijections and metric spaces induced by some collective properties of concave Young-functions'
---
Introduction
============
We know that concave functions play major roles in many branches of mathematics for instance probability theory ([@BURK1973], [@GARS1973], [@MOGY1981], say), interpolation theory (cf. [@TRIEB1978], say), weighted norm inequalities (cf. [@GARFRAN1985], say), and functions spaces (cf. [@SINN2002], say), as well as in many other branches of sciences. In the line of [@BURK1973], [@GARS1973] and [@MOGY1981], the present author also obtained in martingale theory some results in connection with certain collective properties or behaviors of concave Young-functions (cf. [@AGB1986], [@AGB1989]). The study presented in [@AGB2005] was mainly motivated by the question why strictly concave functions possess so many properties, worth to be characterized using appropriate tools that await to be discovered.
We say that a function $\Phi:\left[ 0,\,\infty\right) \rightarrow\left[
0,\,\infty\right) $ belongs to the set $\mathcal{Y}_{\mathrm{conc}}$ (and is referred to as a concave Young-function) if and only if it admits the integral representation$$\Phi\left( x\right) =\int\nolimits_{0}^{x}\varphi\left( t\right) dt,
\label{id1}%$$ (where $\varphi:\left( 0,\,\infty\right) \rightarrow\left( 0,\,\infty
\right) $ is a right-continuous and decreasing function such that it is integrable on every finite interval $\left( 0,\,x\right) $) and $\Phi\left(
\infty\right) =\infty$. It is worth to note that every function in $\mathcal{Y}_{\mathrm{conc}}$ is strictly concave.
We will remind some results obtained so far in [@AGB2005].
We shall say that a concave Young-function $\Phi$ satisfies the *density-level property* if $A_{\Phi}\left( \infty\right) <\infty$, where $A_{\Phi}\left( \infty\right) :=\int\nolimits_{1}^{\infty}%
\frac{\varphi\left( t\right) }{t}dt$. All the concave Young-functions possessing the density-level property will be grouped in a set $\mathcal{A}$.
In Theorems \[theo1\] and \[theo2\] (cf. [@AGB2005]), we showed that the composition of any two concave Young-functions satisfies the density-level property if and only if at least one of them satisfies it. These two theorems show that concave Young-functions with the density-level property behave like left and right ideal with respect to the composition operation.
We also proved ([@AGB2005], Lemma 5, page 12) that if $\Phi\in
\mathcal{Y}_{\mathrm{conc}}$, then there are constants $C_{\Phi}>0$ and $B_{\Phi}\geq0$ such that$$A_{\Phi}\left( \infty\right) -B_{\Phi}\leq%
%TCIMACRO{\dint _{0}^{\infty}}%
%BeginExpansion
{\displaystyle\int_{0}^{\infty}}
%EndExpansion
\frac{\Phi\left( t\right) }{\left( t+1\right) ^{2}}dt\leq C_{\Phi}%
+A_{\Phi}\left( \infty\right) .$$ This led us to the idea to search for a Lebesgue measure (described here below) with respect to which every concave Young-function turns out to be square integrable ([@AGB2005], Lemma 6, page 13), i.e. $\mathcal{Y}%
_{\mathrm{conc}}\subset L^{2}:=L^{2}\left( \left[ 0,~\infty\right)
,~\mathcal{M},~\mu\right) $, where $\mathcal{M}$ is a $\sigma$-algebra (of $\left[ 0,~\infty\right) $) containing the Borel sets and $\mu
:\mathcal{M}\rightarrow\left[ 0,~\infty\right) $ is a Lebesgue measure defined by $\mu\left( \left[ 0,~x\right) \right) =\frac{1}{3}\left(
1-\frac{1}{\left( x+1\right) ^{3}}\right) $ for all $x\in\left[
0,~\infty\right) $. The mapping $d:L^{2}\times L^{2}\rightarrow\left[
0,~\infty\right) $, defined by$$\operatorname*{d}\left( f,g\right) =\sqrt{%
%TCIMACRO{\dint _{\left[ 0,~\infty\right) }}%
%BeginExpansion
{\displaystyle\int_{\left[ 0,~\infty\right) }}
%EndExpansion
\left( f-g\right) ^{2}d\mu}=\sqrt{%
%TCIMACRO{\dint _{0}^{\infty}}%
%BeginExpansion
{\displaystyle\int_{0}^{\infty}}
%EndExpansion
\frac{\left( f\left( x\right) -g\left( x\right) \right) ^{2}}{\left(
x+1\right) ^{4}}dx}, \label{dist}%$$ is known to be a semi-metric.
Further on, we proved in ([@AGB2005], Theorem 8, page 16) that $\mathcal{A}$ is a dense set in $\mathcal{Y}_{\mathrm{conc}}$.
Throughout this communication $\Phi_{\operatorname{id}}$ will denote the identity function defined on the half line $\left[ 0,\text{ }\infty\right) $ and we write $\left\Vert \Phi\right\Vert :=\sqrt{%
%TCIMACRO{\dint _{\left[ 0,\text{ }\infty\right) }}%
%BeginExpansion
{\displaystyle\int_{\left[ 0,\text{ }\infty\right) }}
%EndExpansion
\Phi^{2}d\mu}$ whenever $\Phi\in\mathcal{Y}_{\mathrm{conc}}$.
We intend to generate a decreasing sequence of subsets $\left( \mathcal{Y}%
_{b}^{\left( n\right) }\right) \subset\mathcal{Y}_{\mathrm{conc}}$ depending on $b$ such that whenever $n\in\mathbb{N}$, then $\mathcal{A}%
\cap\mathcal{Y}_{b}^{\left( n\right) }$ is dense in $\mathcal{Y}%
_{b}^{\left( n\right) }$ and the following four sets $\mathcal{Y}%
_{b}^{\left( n\right) }$, $\mathcal{Y}_{b}^{\left( n\right) }%
\backslash\left( \mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right) }\right)
$, $\mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right) }$ and $\mathcal{Y}%
_{\mathrm{conc}}$ are pairwise equinumerous. We shall also prove that the two pairs $\left( \mathcal{Z}^{\ast\left( n\right) },\operatorname*{dist}%
\right) $ and $\left( \mathcal{Z}^{\left( n\right) },\operatorname*{dist}%
\right) $ are metric spaces, where $\mathcal{Z}^{\ast\left( n\right)
}=\left\{ \mathcal{Y}_{b}^{\left( n\right) }:{\small b\in\left(
0,\,\infty\right) }\right\} $ and $\mathcal{Z}^{\left( n\right) }=\left\{
\mathcal{A}_{b}^{\left( n\right) }:{\small b\in\left( 0,\,\infty\right)
}\right\} $ for each $n\in\mathbb{N}$ and the distance between any two sets $\mathcal{F}$ and $\mathcal{G}$ in $\mathcal{Y}_{\mathrm{conc}}$ being defined by $$\begin{aligned}
\operatorname*{dist}\left( \mathcal{F},\mathcal{G}\right) & :=\sup\left\{
\inf\left\{ \operatorname{d}\left( \Phi,\Psi\right) :\Psi\in\mathcal{G}%
\right\} :\Phi\in\mathcal{F}\right\} \\
& =\sup\left\{ \inf\left\{ \operatorname{d}\left( \Phi,\Psi\right)
:\Phi\in\mathcal{F}\right\} :\Psi\in\mathcal{G}\right\} .\end{aligned}$$ We show in the last section that if $f$ is any measurable function on a measure space $\left( \Omega,\mathcal{F},\lambda\right) $ and $p\in\left[
1,\infty\right) $ is an arbitrary number then the quantities $\left\Vert
f\right\Vert _{L^{p}}$ and $\sup_{\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}%
}}}\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert
f\right\vert \right\Vert _{L^{p}}$ are equivalent, in the sense that they are both either finite or infinite at the same time, where $\widetilde
{\mathcal{Y}_{\mathrm{conc}}}$ is a proper subset of $\mathcal{Y}%
_{\mathrm{conc}}$.We then use this subset to express the value of $\left\Vert
f\right\Vert _{L^{p}}$ whenever $\left\Vert f\right\Vert _{L^{p}}<\infty$.
Bijections between subsets of $\mathcal{Y}_{\mathrm{conc}}$
===========================================================
We first anticipate that there are as many elements in each of the sets $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$ as there exist in $\mathcal{Y}_{\mathrm{conc}}$, showing how broad the set of concave Young-functions possessing the density-level property and its complement really are.
\[theo1\]The sets $\mathcal{A}$, $\mathcal{Y}_{\mathrm{conc}}$ and $\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$ are pairwise equinumerous.
We first show that there is a bijection between $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$. In fact, since $\mathcal{A}$ is a proper subset of $\mathcal{Y}_{\mathrm{conc}}$ there is an injection from $\mathcal{A}$ to $\mathcal{Y}_{\mathrm{conc}}$, as a matter of fact, the identity mapping from $\mathcal{A}$ into $\mathcal{Y}_{\mathrm{conc}}$ will do. Fix any number $\alpha\in\left( 0,\,1\right) $ and define the mapping $S_{\alpha
}:\mathcal{Y}_{\mathrm{conc}}\rightarrow\mathcal{A}$ by $S_{\alpha}\left(
\Phi\right) =\Phi^{\alpha}$. We point out that this mapping exists in virtue of Theorem 2 in [@AGB2005]. It is not hard to see that $S_{\alpha}$ is an injection. Then the Schröder-Bernstein theorem entails that there exists a bijection between $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$. To complete the proof it is enough to show that there is a bijection between $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$. In fact, fix arbitrarily some $\Phi\in\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$ and define the function $h_{\Phi}:\mathcal{A}\rightarrow\mathcal{Y}_{\mathrm{conc}%
}\backslash\mathcal{A}$ by $h_{\Phi}\left( \Delta\right) =\Delta+\Phi$. Obviously, $h_{\Phi}$ is an injection. Now, fix any $\Delta\in\mathcal{A}$ and define the function $f_{\Delta}:\mathcal{Y}_{\mathrm{conc}}\backslash
\mathcal{A}\rightarrow\mathcal{A}$ by $f_{\Delta}\left( \Phi\right)
=\Delta\circ\Phi$. We point out that this function always exists due to Theorem \[theo2\] in [@AGB2005]. It is not difficult to show that $f_{\Delta}$ is an injection if we take into account that $\Delta$ is an invertible function. Consequently, the Schröder-Bernstein theorem guarantees the existence of a bijection between $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$. Therefore, we can conclude on the validity of the argument.
Write $\mathcal{A}_{b}:=\left\{ \Phi\in\mathcal{A}:\Phi\left( b\right)
=b\right\} $ and $\mathcal{Y}_{b}:=\left\{ \Phi\in\mathcal{Y}_{\mathrm{conc}%
}:\Phi\left( b\right) =b\right\} $ for every number $b\in\left(
0,\,\infty\right) $.
Let us denote by $\mathcal{Z}:=\left\{ \mathcal{A}_{b}:b\in\left(
0,\,\infty\right) \right\} $ and $\mathcal{Z}^{\ast}:=\left\{
\mathcal{Y}_{b}:b\in\left( 0,\,\infty\right) \right\} $.
It is obvious that $\mathcal{A}_{b}\subset\mathcal{Y}_{b}$ for every number $b\in\left( 0,\,\infty\right) $ and $\mathcal{Z}\cap\mathcal{Z}^{\ast
}=\varnothing$.
\[lem1\]For every number $b\in\left( 0,\,\infty\right) $ the identities $\mathcal{A}_{b}=\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi
\in\mathcal{A}\right\} $ and $\mathcal{Y}_{b}=\left\{ \frac{b\Phi}%
{\Phi\left( b\right) }:\Phi\in\mathcal{Y}_{\mathrm{conc}}\right\} $ hold true.
Pick any function $\Psi\in\mathcal{A}_{b}$. Then $\Psi\in\mathcal{A}$ and $\Psi\left( b\right) =b$, so that $\Psi=\frac{b\Psi}{\Psi\left( b\right)
}\in\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi\in\mathcal{A}\right\}
$, i.e. $\mathcal{A}_{b}\subset\left\{ \frac{b\Phi}{\Phi\left( b\right)
}:\Phi\in\mathcal{A}\right\} $. To show the reverse inclusion consider any function $\Psi\in\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi
\in\mathcal{A}\right\} $. Then necessarily there must exist some $\Phi
\in\mathcal{A}$ such that $\Psi=\frac{b\Phi}{\Phi\left( b\right) }$. It is obvious that $\Psi\in\mathcal{A}$ and $\Psi\left( b\right) =b$, i.e. $\Psi\in\mathcal{A}_{b}$. Hence, $\left\{ \frac{b\Phi}{\Phi\left( b\right)
}:\Phi\in\mathcal{A}\right\} \subset\mathcal{A}_{b}$. These two inclusions yield that $\mathcal{A}_{b}=\left\{ \frac{b\Phi}{\Phi\left( b\right) }%
:\Phi\in\mathcal{A}\right\} $. The proof of identity $\mathcal{Y}%
_{b}=\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi\in\mathcal{Y}%
_{\mathrm{conc}}\right\} $ can be similarly carried out.
\[def1\]A proper subset $\mathcal{G}$ of $\mathcal{A}$ is said to be maximally bounded if each of the sets $\mathcal{G}$ and $\mathcal{A}%
\backslash\mathcal{G}$ is equinumerous with $\mathcal{A}$, i.e. there is a bijection between $\mathcal{A}$ and $\mathcal{G}$, and $\operatorname*{diam}%
(\mathcal{G})<\infty$, where $\operatorname*{diam}(\mathcal{G}):=\sup\left\{
\operatorname*{d}\left( \Phi_{1},\Phi_{2}\right) :\Phi_{1},~\Phi_{2}%
\in\mathcal{G}\right\} $ is the diameter of $\mathcal{G}$.
We note that Definition \[def1\] makes sense for the two reasons here below.
On the one hand we assert that $\operatorname*{diam}(\mathcal{A})=\sup\left\{
\operatorname*{d}\left( \Phi_{1},\Phi_{2}\right) :\Phi_{1},~\Phi_{2}%
\in\mathcal{A}\right\} =\infty$. In fact, fix some $\Phi\in\mathcal{A}$ and define a sequence $\left( \Phi_{n}\right) \subset\mathcal{Y}%
_{\mathrm{conc}}$ by $\Phi_{2n}=4n\Phi$ and $\Phi_{2n-1}=\left( 2n-1\right)
\Phi$, $n\in\mathbb{N}$. It is clear that $\left( \Phi_{n}\right)
\subset\mathcal{A}$ and $\operatorname*{d}\left( \Phi_{2n},\Phi
_{2n-1}\right) =\left( 2n+1\right) \left\Vert \Phi\right\Vert $, $n\in\mathbb{N}$. Hence, $\operatorname*{diam}(\mathcal{A})=\infty$.
On the other hand the set $\left\{ \left( \Phi\left( 1\right) \right)
^{-1}\Phi:\Phi\in\mathcal{Y}_{\mathrm{conc}}\right\} $ is of finite diameter. In fact for any $\Phi$, $\Psi\in\mathcal{Y}_{\mathrm{conc}}$ we have, via Lemma \[lem3\] in [@AGB2005], that $$\operatorname*{d}\left( \left( \Phi\left( 1\right) \right) ^{-1}%
\Phi,\left( \Psi\left( 1\right) \right) ^{-1}\Psi\right) \leq\left\Vert
\left( \Phi\left( 1\right) \right) ^{-1}\Phi\right\Vert +\left\Vert
\left( \Psi\left( 1\right) \right) ^{-1}\Psi\right\Vert \leq2\left\Vert
S\right\Vert <\infty.$$
Let us define two relations $\mathrm{\bot}\subset\mathcal{A}\times\mathcal{A}%
$ and $\mathrm{\bot}^{\ast}\subset\mathcal{Y}_{\mathrm{conc}}\times
\mathcal{Y}_{\mathrm{conc}}$ as follows:
1. We say that $\left( \Phi,\Psi\right) \in\mathrm{\bot}$, where $\left(
\Phi,\Psi\right) \in\mathcal{A}\times\mathcal{A}$, (and write $\Phi
\mathrm{\bot}\Psi$) if and only if there is some constant $c\in\left(
0,\,\infty\right) $ such that $\Psi\left( x\right) =c\Phi\left( x\right)
$ for all $x\in\left( 0,\,\infty\right) $.
2. We say that $\left( \Phi,\Psi\right) \in\mathrm{\bot}^{\ast}$, where $\left( \Phi,\Psi\right) \in\mathcal{Y}_{\mathrm{conc}}\times\mathcal{Y}%
_{\mathrm{conc}}$, (and write $\Phi\mathrm{\bot}^{\ast}\Psi$) if and only if there is some constant $c\in\left( 0,\,\infty\right) $ such that $\Psi\left( x\right) =c\Phi\left( x\right) $ for all $x\in\left(
0,\,\infty\right) $.
It is not hard to see that $\mathrm{\bot}$ and $\mathrm{\bot}^{\ast}$ are equivalence relations on $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$ respectively, i.e. they are reflexive, symmetric and transitive. Their corresponding equivalence classes are respectively$$\begin{aligned}
p_{\mathrm{\bot}}\left( \Psi\right) & :=\left\{ \Phi:\Phi\in
\mathcal{A}\text{ and }\Phi\mathrm{\bot}\Psi\right\} ,\text{ }\Psi
\in\mathcal{A}\\
p_{\mathrm{\bot}^{\ast}}\left( \Delta\right) & :=\left\{ \Phi:\Phi
\in\mathcal{Y}_{\mathrm{conc}}\text{ and }\Phi\mathrm{\bot}^{\ast}%
\Delta\right\} ,\text{ }\Delta\in\mathcal{Y}_{\mathrm{conc}}%\end{aligned}$$ and their respective induced factor (or quotient) sets can be given by$$\begin{aligned}
\mathcal{A}/\mathrm{\bot} & :=\left\{ \mathcal{C}:\mathcal{C}%
\subset\mathcal{A}\text{ and }\mathcal{C}=p_{\mathrm{\bot}}\left(
\Psi\right) \text{ for some }\Psi\in\mathcal{A}\right\} ,\\
\mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot}^{\ast} & :=\left\{ \mathcal{C}%
:\mathcal{C}\subset\mathcal{Y}_{\mathrm{conc}}\text{ and }\mathcal{C}%
=p_{\mathrm{\bot}^{\ast}}\left( \Delta\right) \text{ for some }\Delta
\in\mathcal{Y}_{\mathrm{conc}}\right\}\end{aligned}$$
One can easily verify that for all $\Psi\in\mathcal{A}$ and $\Delta
\in\mathcal{Y}_{\mathrm{conc}}$ the equivalence classes $p_{\mathrm{\bot}%
}\left( \Psi\right) $ and $p_{\mathrm{\bot}^{\ast}}\left( \Delta\right)
$ are of continuum size or magnitude.
\[theo2\]Let $b\in\left( 0,\,\infty\right) $ be any fixed number.**Part I.** Define the mapping $f:\mathcal{A}%
\rightarrow\mathcal{A}_{b}$ by $f\left( \Phi\right) =\frac{b}{\Phi\left(
b\right) }\Phi$. Then there is a unique mapping $g:\mathcal{A}/\mathrm{\bot
}\rightarrow\mathcal{A}_{b}$ for which the diagram$$\xymatrix{\mathcal{A} \ar[r]^{p_{\mathrm{\bot}}} \ar[dr]_f & \mathcal{A}/\mathrm{\bot} \ar[d]^g\\ ~ & \mathbf{b} }
\label{com3}%$$ commutes *(*i.e. $f=g\circ p_{\mathrm{\bot}}$*)* and moreover, the mapping $g$ is a bijection.**Part II.** Define the mapping $f^{\ast}:\mathcal{Y}_{\mathrm{conc}}\rightarrow\mathcal{Y}_{b}$ by $f^{\ast
}\left( \Delta\right) =\frac{b}{\Delta\left( b\right) }\Delta$. Then there is a unique mapping $g^{\ast}:\mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot}%
^{\ast}\rightarrow\mathcal{Y}_{b}$ for which the diagram$$\xymatrix{\mathcal{Y}_{\mathrm{conc}} \ar[r]^{p_{\mathrm{\bot}^{\ast}}} \ar[dr]_{f^{\ast}} & \mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot}^{\ast} \ar[d]^{g^{\ast}}\\ ~ & \mathbf{b}^{\ast} }
\label{com4}%$$ commutes *(*i.e. $f^{\ast}=g^{\ast}\circ p_{\mathrm{\bot}^{\ast}}%
$*)* and moreover, the mapping $g^{\ast}$ is a bijection.
We point out that the proof of Theorem \[theo2\] is obvious.
\[prop1\]Let $b\in\left( 0,\,\infty\right) $ be an arbitrarily fixed number.**Part I.** There is a bijection between $\mathcal{Y}_{b}$ and $\mathcal{Y}_{\mathrm{conc}}$.**Part II.** There is a bijection between $\mathcal{A}_{b}$ and $\mathcal{A}$.
We shall only show the first part because the other case can be similarly proved. To this end, write $\mathcal{Y}_{bb}:=\left\{ b\Phi:\Phi
\in\mathcal{Y}_{\mathrm{conc}}\right\} $. We note that $\mathcal{Y}_{bb}$ and $\mathcal{Y}_{\mathrm{conc}}$ are equinumerous for the reasons that $\mathcal{Y}_{bb}\subset\mathcal{Y}_{\mathrm{conc}}$ and the function $F:\mathcal{Y}_{\mathrm{conc}}\rightarrow\mathcal{Y}_{bb}$, defined by $F\left( \Phi\right) =b\Phi$, can be easily shown to be an injection. Thus it will be enough to prove that $\mathcal{Y}_{bb}$ and $\mathcal{Y}_{b}$ are equinumerous. In fact, consider the function $Q:\mathcal{Y}_{b}\rightarrow
\mathcal{Y}_{bb}$ defined by $Q\left( \frac{b}{\Phi\left( b\right) }%
\Phi\right) =b\Phi$. We shall just point out that function $Q$ can be easily shown to be a bijection, which ends the proof.
\[cor1\]Let $b\in\left( 0,\,\infty\right) $ be arbitrary. Then the following six sets $\mathcal{A}$, $\mathcal{A}_{b}$, $\mathcal{Y}_{b}$, $\mathcal{A}/\mathrm{\bot}$, $\mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot}^{\ast
}$and $\mathcal{Y}_{\mathrm{conc}}$ are equinumerous.
We note that $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$ are equinumerous (by Theorem \[theo1\]) and, by Theorem 2, $\mathcal{A}/\mathrm{\bot}$ and $\mathcal{A}_{b}$ are equinumerous. On the other hand $\mathcal{A}$ and $\mathcal{A}_{b}$ are equinumerous as well as $\mathcal{Y}_{b}$ and $\mathcal{Y}_{\mathrm{conc}}$ are (by Proposition \[prop1\]). Thus $\mathcal{A}_{b}$ and $\mathcal{Y}_{\mathrm{conc}}$ are equinumerous. Therefore, as $\mathcal{Y}_{b}$ and $\mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot
}^{\ast}$ are equinumerous (by Theorem 2), we can conclude on the validity of the argument.
\[rem1\]Let $b_{1}$ and $b_{2}\in\left( 0,\,\infty\right) $ be two arbitrary distinct numbers. Then $\mathcal{A}_{b_{1}}\cap\mathcal{A}_{b_{2}}$ and $\mathcal{Y}_{b_{1}}\cap\mathcal{Y}_{b_{2}}$ are empty sets.
\[rem2\]Let $b_{1}$ and $b_{2}\in\left( 0,\,\infty\right) $ be two arbitrary distinct numbers. Then $\mathcal{A}_{b_{1}}\cup\mathcal{A}_{b_{2}%
}\notin\mathcal{Z}$ and $\mathcal{Y}_{b_{1}}\cup\mathcal{Y}_{b_{2}}%
\notin\mathcal{Z}^{\ast}$.
\[rem3\]Fix arbitrarily a number $b\in\left( 0,\,\infty\right) $. Then it is easily seen that the function $h_{b}:\left[ 0,~\infty\right)
\rightarrow\left[ 0,~\infty\right) $, defined by $h_{b}\left( x\right)
=x+b$, is square integrable with respect to measure $\mu$ and, moreover, $C_{b}:=%
%TCIMACRO{\dint _{0}^{\infty}}%
%BeginExpansion
{\displaystyle\int_{0}^{\infty}}
%EndExpansion
\frac{\left( h_{b}\left( x\right) \right) ^{2}}{\left( x+1\right) ^{4}%
}dx=\frac{1}{3}\left( b^{2}+b+1\right) <\infty$.
\[rem4\]If $\Phi\in\mathcal{Y}_{b}$, then $\Phi\left( x\right) \leq
h_{b}\left( x\right) $ for all $x\in\left[ 0,\,\infty\right) $.
Fix any $\Phi\in\mathcal{Y}_{b}$. As $\Phi$ is a concave function its graph must lie below the tangent of equation $y=\varphi\left( b\right) \left(
x-b\right) +b$ at point $\left( b,b\right) $ since $\Phi\left( b\right)
=b$. Consequently, for all $x\in\left[ 0,\,\infty\right) $ we have:$$\begin{aligned}
\Phi\left( x\right) & \leq\varphi\left( b\right) \left( x-b\right)
+b\leq\varphi\left( b\right) x+b=b\varphi\left( b\right) \frac{x}{b}+b\\
& \leq\Phi\left( b\right) \frac{x}{b}+b=h_{b}\left( x\right) .\end{aligned}$$
\[prop2\]Let $b\in\left( 0,\,\infty\right) $ be any number. Then $\mathcal{Y}_{b}$ is of finite diameter.
Let $b\in\left( 0,\,\infty\right) $ be the source of $\mathcal{Y}_{b}%
\in\mathcal{Z}^{\ast}$. We need to prove that $\mathcal{Y}_{b}$ has a finite diameter. In fact, consider two arbitrary functions $\Phi_{1}$, $\Phi_{2}%
\in\mathcal{Y}_{b}$. Then$$\operatorname{d}\left( \Phi_{1},\Phi_{2}\right) =\left\Vert \Phi_{1}%
-\Phi_{2}\right\Vert \leq\left\Vert \Phi_{1}\right\Vert +\left\Vert \Phi
_{2}\right\Vert \leq\sqrt{2C_{b}}\text{,}%$$ via Remarks \[rem4\] and \[rem3\]. Therefore,$$\operatorname*{diam}\left( \mathcal{Y}_{b}\right) :=\sup\left\{
\operatorname{d}\left( \Phi_{1},\Phi_{2}\right) :\Phi_{1},~\Phi_{2}%
\in\mathcal{Y}_{b}\right\} \leq\sqrt{2C_{b}}<\infty.$$
\[theo3\]Let $b\in\left( 0,\,\infty\right) $ be any number. Then $\mathcal{Y}_{b}$ is maximally bounded.
We just point out that the proof follows from the conjunction of both Propositions \[prop2\] and \[prop1\].
In the sequel $H_{\left[ 0,~1\right] }$ will stand for the collection of all finite sequences $\left( t_{1},~\ldots,~t_{k}\right) \subset\left[
0,~1\right] $ such that $t_{1}+\ldots+t_{k}=1$.
For any fixed $b\in\left( 0,\,\infty\right) $ and every counting number $n\in\mathbb{N}$ write $\operatorname*{X}\limits_{i=1}^{n}\mathcal{A}_{b}$ (resp. $\operatorname*{X}\limits_{i=1}^{n}\mathcal{Y}_{b}$) for the $n$-fold Descartes product of $\mathcal{A}_{b}$ (resp. $\mathcal{Y}_{b}$).
For $n=1$ let us set $\mathcal{A}_{b}^{\left( 1\right) }=\mathcal{A}_{b}$, $\mathcal{Y}_{b}^{\left( 1\right) }=\mathcal{Y}_{b}$ and whenever $n\geq2$, write $\mathcal{Y}_{b}^{CO\left( n\right) }=\left\{ \Delta_{1}\circ
\Delta_{2}\circ\ldots\circ\Delta_{n}:\left( \Delta_{1},\Delta_{2}%
,\ldots,\Delta_{n}\right) \in\operatorname*{X}\limits_{i=1}^{n}%
\mathcal{Y}_{b}\right\} $, $$\mathcal{A}_{b}^{CO\left( n\right) }=\left\{ \Phi_{1}\circ\ldots\circ
\Phi_{n}:\left( \Phi_{1},\ldots,\Phi_{n}\right) \in\operatorname*{X}%
\limits_{i=1}^{n}\mathcal{Y}_{b}\text{ and }\Phi_{j}\in\mathcal{A}_{b}\text{
for some index }j\right\} ,$$ $\mathcal{Y}_{b}^{\left( n\right) }=\left\{ \sum_{i=1}^{k}t_{i}\Delta
_{i}:\Delta_{1},~\Delta_{2},~\ldots,~\Delta_{k}\in\mathcal{Y}_{b}^{CO\left(
n\right) },~\left( t_{1},~\ldots~t_{k}\right) \in H_{\left[ 0,~1\right]
}\right\} $, $\mathcal{A}_{b}^{\left( n\right) }=\left\{ \sum_{i=1}%
^{k}t_{i}\Phi_{i}:\Phi_{1},~\Phi_{2},~\ldots,~\Phi_{k}\in\mathcal{A}%
_{b}^{CO\left( n\right) },~\left( t_{1},~\ldots~t_{k}\right) \in
H_{\left[ 0,~1\right] }\right\} $.
Further, for $n=1$ write $\mathcal{Z}^{\left( 1\right)
}=\mathcal{Z}$, $\mathcal{Z}^{\ast\left( 1\right) }=\mathcal{Z}^{\ast}$ and, for $n\in\mathbb{N}\backslash\left\{ 1\right\} $ write $\mathcal{Z}^{\left(
n\right) }:=\left\{ \mathcal{A}_{b}^{\left( n\right) }:b\in\left(
0,\,\infty\right) \right\} $ and $\mathcal{Z}^{\ast\left( n\right)
}:=\left\{ \mathcal{Y}_{b}^{\left( n\right) }:b\in\left( 0,\,\infty
\right) \right\} $.
\[rem5\]For any pair of numbers $n\in\mathbb{N}$ and $b\in\left(
0,\,\infty\right) $ the set $\mathcal{A}_{b}^{\left( n\right) }$ is a proper subset of $\mathcal{Y}_{b}^{\left( n\right) }$.
\[rem6\]For any pair of numbers $n\in\mathbb{N}$ and $b\in\left(
0,\,\infty\right) $ we have $\mathcal{A}_{b}^{\left( n\right) }%
\subset\mathcal{A}_{b}^{\left( 1\right) }=\mathcal{A}_{b}$.
We point out that Remark \[rem6\] is a direct consequent of Theorem 2 in [@AGB2005], page 6.
\[rem7\]Let $b\in\left( 0,\,\infty\right) $, $n\in\mathbb{N}$ and $k\geq
n$ be arbitrary numbers. Then *(1)* $\Phi_{1}\circ\Phi_{2}%
\circ\ldots\circ\Phi_{k}\in\mathcal{A}_{b}^{CO\left( n\right) }$ whenever $\Phi_{1},~\Phi_{2},~\ldots,~\Phi_{k}\in\mathcal{Y}_{b}^{\left( 1\right) }$ and $\Phi_{j}\in\mathcal{A}_{b}^{\left( 1\right) }$ for some index $j\in\left\{ 1,~\ldots,~k\right\} $*(2)* $\Delta_{1}\circ
\Delta_{2}\circ\ldots\circ\Delta_{k}\in\mathcal{Y}_{b}^{CO\left( n\right) }$ whenever $\Delta_{1},~\Delta_{2},~\ldots,~\Delta_{k}\in\mathcal{Y}%
_{b}^{\left( 1\right) }$.
Note that $\Phi_{1}\circ\Phi_{2}\circ\ldots\circ\Phi_{k}=\Phi_{1}\circ\Phi
_{2}\circ\ldots\circ\Phi_{n-1}\circ\Psi_{1}$ and $\Delta_{1}\circ\Delta
_{2}\circ\ldots\circ\Delta_{n-1}\circ\Psi_{2}$, where $\Psi_{1}=\Phi_{n}%
\circ\Phi_{n+1}\circ\ldots\circ\Phi_{k}$ and $\Psi_{2}=\Delta_{n}\circ
\Delta_{n+1}\circ\ldots\circ\Delta_{k}$. From this simple observation the result easily follows.
From Remark \[rem7\] the following result can be easily derived, since it implies that $\mathcal{A}_{b}^{CO\left( n+1\right) }$ is a proper subset of $\mathcal{A}_{b}^{CO\left( n\right) }$ and, $\mathcal{Y}_{b}^{CO\left(
n+1\right) }$ is also a proper subset of $\mathcal{Y}_{b}^{CO\left(
n\right) }$.
\[lem2\]Let $b\in\left( 0,\,\infty\right) $ and $n\in\mathbb{N}$ be arbitrary numbers. Then the following two assertions are valid.*(1)* The set $\mathcal{A}_{b}^{\left( n+1\right) }$ is a proper subset of $\mathcal{A}_{b}^{\left( n\right) }$.*(2)* The set $\mathcal{Y}_{b}^{\left( n+1\right) }$ is a proper subset of $\mathcal{Y}%
_{b}^{\left( n\right) }$.
\[theo4\]For any fixed pair of numbers $n\in\mathbb{N}$ and $b\in\left(
0,\,\infty\right) $, the two sets $\mathcal{A}_{b}^{\left( n\right) }$ and $\mathcal{A}_{b}$ are equinumerous.
Throughout the proof we shall fix any counting number $n\in\mathbb{N}$. We first note that the identity function $I_{\operatorname{id}}:\mathcal{A}%
_{b}^{\left( n\right) }\rightarrow\mathcal{A}_{b}$ is an injection, since $\mathcal{A}_{b}^{\left( n\right) }\subset\mathcal{A}_{b}$. Next, pick any $\Delta\in\mathcal{A}_{b}$ and define the function $f_{\Delta}:\mathcal{A}%
_{b}\rightarrow\mathcal{A}_{b}^{\left( n\right) }$ by $f_{\Delta}\left(
\Phi\right) =\underset{\left( n-1\right) \text{-fold}}{\underbrace
{\Delta\circ\ldots\circ\Delta}}\circ\Phi$. We show that $f_{\Delta}$ is an injection. In fact, let $\Phi_{1}$, $\Phi_{2}\in\mathcal{A}_{b}$ be arbitrary and assume that $f_{\Delta}\left( \Phi_{1}\right) =f_{\Delta}\left(
\Phi_{2}\right) $. Then taking into account that $\Delta$ is an invertible function we can easily deduce that $\Phi_{1}=\Phi_{2}$, i.e. $f_{\Delta}$ is an injection. Therefore, the Schröder-Bernstein theorem entails that there is a bijection between $\mathcal{A}_{b}$ and $\mathcal{A}_{b}^{\left(
n\right) }$. This was to be proved.
\[prop3\]For any pair of numbers $n\in\mathbb{N}$ and $b\in\left(
0,~\infty\right) $ the sets $\mathcal{A}_{b}^{\left( n\right) }$ and $\mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{A}_{b}^{\left(
n\right) }$ are equinumerous.
Let $\Phi\in\mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{A}%
_{b}^{\left( n\right) }$ and $\left( \alpha,\beta\right) \in H_{\left[
0,~1\right] }$ be arbitrarily fixed. Define the function $h_{\Phi}^{\left(
\alpha,\beta\right) }:\mathcal{A}_{b}^{\left( n\right) }\rightarrow
\mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{A}_{b}^{\left(
n\right) }$ by $h_{\Phi}^{\left( \alpha,\beta\right) }\left(
\Delta\right) =\alpha\Delta+\beta\Phi$. It is clear that $h_{\Phi}^{\left(
\alpha,\beta\right) }$ is actually an injection. Now, fix any $\Delta
\in\mathcal{A}_{b}^{\left( n\right) }$ and define the function $f_{\Delta
}:\mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{A}_{b}^{\left(
n\right) }\rightarrow\mathcal{A}_{b}^{\left( n\right) }$ by $f_{\Delta
}\left( \Phi\right) =\Delta\circ\Phi$. We note that this function always exists because of the inclusion $\mathcal{A}_{b}^{\left( n\right) }%
\subset\mathcal{A}$ and Theorem \[theo2\] in [@AGB2005]. Here too we can easily check that $f_{\Delta}$ is an injection. Therefore, The Schröder-Bernstein theorem yields the result to be proven.
For any pair of numbers $n\in\mathbb{N}$ and $b\in\left( 0,~\infty\right) $ the following five sets $\mathcal{Y}_{b}^{\left( n\right) }$, $\mathcal{A}%
_{b}^{\left( n\right) }$, $\mathcal{Y}_{b}^{\left( n\right) }%
\backslash\mathcal{A}_{b}^{\left( n\right) }$, $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$ are pairwise equinumerous.
The metrization of sets $\mathcal{Z}^{\left( n\right) }$ and $\mathcal{Z}^{\ast\left( n\right) }$
=====================================================================================================
We shall only deal with the metrization of sets $\mathcal{Z}$ and $\mathcal{Z}^{\ast}$ since all the results in this section can be easily extended to the sets $\mathcal{Z}^{\left( n\right) }$ and $\mathcal{Z}%
^{\ast\left( n\right) }$.
Whenever $\Phi\in\mathcal{Y}_{\mathrm{conc}}$ write $G_{\Phi}:=\left\{
\left( x,\Phi\left( x\right) \right) :x\in\left( 0,~\infty\right)
\right\} $ for the graph of $\Phi$ on $\left( 0,~\infty\right) $ and $G_{\Phi}^{a||b}:=\left\{ \left( x,\Phi\left( x\right) \right)
:x\in\left[ a,~b\right] \right\} $ for the graph of $\Phi$ on the interval $\left[ a,~b\right] $ where $a<b$ are any non-negative numbers.
\[rem8\]Let $b_{1}$ and $b_{2}\in\left( 0,\,\infty\right) $ be two arbitrary distinct numbers. If $b_{1}<b_{2}$, then the following two assertions hold true:*(1)* For all $\Phi_{1}\in\mathcal{A}%
_{b_{1}}$ and $\Phi_{2}\in\mathcal{A}_{b_{2}}$ the inequality $\Phi
_{1}\left( b_{2}\right) <\Phi_{2}\left( b_{1}\right) $ holds.$\emph{(2)}$ For all $\Phi_{1}\in\mathcal{Y}_{b_{1}}$ and $\Phi_{2}%
\in\mathcal{Y}_{b_{2}}$ the inequality $\Phi_{1}\left( b_{2}\right)
<\Phi_{2}\left( b_{1}\right) $ holds.
Suppose that $b_{1}<b_{2}$ and fix arbitrarily two functions $\Phi_{1}%
\in\mathcal{Y}_{b_{1}}$ and $\Phi_{2}\in\mathcal{Y}_{b_{2}}$. Obviously, $\Phi_{1}$ must hit $\Phi_{\operatorname{id}}$ *prior to* $\Phi_{2}$. Hence, $G_{\Phi_{1}}^{b_{1}||b_{2}}$ lies below $G_{\Phi_{2}}^{b_{1}||b_{2}}$. But since $G_{\Phi_{1}}^{b_{1}||\infty}$ lies above the graph of the line of equation $y=b_{1}$ in the interval $\left( b_{1},\,\infty\right) $, we have as an aftermath that $\Phi_{1}\left( b_{1}\right) <\Phi_{1}\left(
b_{2}\right) <\Phi_{2}\left( b_{1}\right) $. To end the proof we note that assertion (2) can be similarly shown.
The binary relations $\prec$ and $\preceq$ , defined on $\mathcal{Z}$ respectively by $\mathcal{A}_{b_{1}}\prec\mathcal{A}_{b_{2}}$ if and only if $\Phi_{1}\left( b_{2}\right) <\Phi_{2}\left( b_{1}\right) $ for all pairs $\left( \Phi_{1},\Phi_{2}\right) \in\mathcal{A}_{b_{1}}\times\mathcal{A}%
_{b_{2}}$, and by $\mathcal{A}_{b_{1}}\preceq\mathcal{A}_{b_{2}}$ if and only if $\mathcal{A}_{b_{1}}\prec\mathcal{A}_{b_{2}}$ or $\mathcal{A}_{b_{1}%
}=\mathcal{A}_{b_{2}}$. We point out that The binary relations $\prec$ and $\preceq$ can be similarly defined on $\mathcal{Z}^{\ast}$.
We point out that the law of trichotomy is valid on $\left( \mathcal{Z}%
,\preceq\right) $ and $\left( \mathcal{Z}^{\ast},\preceq\right) $, i.e. whenever $\left( \mathcal{A}_{b_{1}},\mathcal{A}_{b_{2}}\right)
\in\mathcal{Z}\times\mathcal{Z}$ or $\left( \mathcal{A}_{b_{1}}%
,\mathcal{A}_{b_{2}}\right) \in\mathcal{Z}^{\ast}\times\mathcal{Z}^{\ast}$, then precisely one of the following holds: $\mathcal{A}_{b_{1}}=\mathcal{A}%
_{b_{2}}$, $\mathcal{A}_{b_{1}}\prec\mathcal{A}_{b_{2}}$, $\mathcal{A}_{b_{2}%
}\prec\mathcal{A}_{b_{1}}$. Hence, we can easily check that $\left(
\mathcal{Z},\preceq\right) $ and $\left( \mathcal{Z}^{\ast},\preceq\right)
$ are chains, i.e. they are totally ordered sets.
\[theo5\]The functions $f_{1}:\left( 0,\,\infty\right) \rightarrow
\mathcal{Z}$ and $f_{2}:\left( 0,\,\infty\right) \rightarrow\mathcal{Z}%
^{\ast}$, defined respectively by $f_{1}\left( p\right) =\mathcal{A}_{p}$ and $f_{2}\left( p\right) =\mathcal{Y}_{p}$, are order preserving bijections.
We show that the function $f_{1}:\left( 0,\,\infty\right) \rightarrow
\mathcal{Z}$, $f_{1}\left( p\right) =\mathcal{A}_{p}$, is an order preserving bijection. In fact, it is not hard to see via Remark \[rem1\] that $f_{1}$ is an injection. Now pick any element $\mathcal{C}\in\mathcal{Z}%
$. Obviously, there must exist some number $p\in\left( 0,\,\infty\right) $ such that $\mathcal{C}=\mathcal{A}_{p}=f_{1}\left( p\right) $, i.e. $f_{1}$ is a surjection. Consequently, $f_{1}$ is a bijection. To end the proof of this part we simply point out that the bijection $f_{1}$ is order preserving in virtue of Remark \[rem5111\]. Finally, we note that we can similarly prove that $f_{2}$ is also an order preserving bijection.
Since the sets $\left( \mathcal{Z},\preceq\right) $ and $\left(
\mathcal{Z}^{\ast},\preceq\right) $ are chains it is natural to look for a metric on them. We shall do this in the following two results. But before that let us recall the definitions of some distances known in the literature (cf. [@KUR1966], say). If $\Phi\in\mathcal{Y}_{\mathrm{conc}}$ is any function and $\mathcal{F}$, $\mathcal{G}\subset\mathcal{Y}_{\mathrm{conc}}$ are arbitrary non-empty subsets, then we define the distance from the point $\Phi$ to the set $\mathcal{G}$ by $\rho\left( \Phi,\mathcal{G}\right)
:=\inf\left\{ \operatorname{d}\left( \Phi,\Psi\right) :\Psi\in
\mathcal{G}\right\} =\inf\left\{ \operatorname{d}\left( \Psi,\Phi\right)
:\Psi\in\mathcal{G}\right\} =\rho\left( \mathcal{G},\Phi\right) $ and the distance between the two sets $\mathcal{F}$ and $\mathcal{G}$ by $$\begin{aligned}
\operatorname*{dist}\left( \mathcal{F},\mathcal{G}\right) & :=\sup\left\{
\inf\left\{ \operatorname{d}\left( \Phi,\Psi\right) :\Psi\in\mathcal{G}%
\right\} :\Phi\in\mathcal{F}\right\} \\
& =\sup\left\{ \inf\left\{ \operatorname{d}\left( \Phi,\Psi\right)
:\Phi\in\mathcal{F}\right\} :\Psi\in\mathcal{G}\right\} .\end{aligned}$$
First we find sufficient conditions for which the distance from a point to a subset (both in $\mathcal{Y}_{\mathrm{conc}}$) should be positive, in order to guarantee that the distance between two sets in $\mathcal{Y}_{\mathrm{conc}}$ have sense.
\[lem3\]Let $b_{1}$ and $b_{2}\in\left( 0,\,\infty\right) $ be two arbitrary distinct numbers. Then $\rho\left( \mathcal{Y}_{b_{1}},\Phi
_{2}\right) >0$ and $\rho\left( \mathcal{A}_{b_{1}},\Phi_{2}\right)
>0$ whenever $\Phi_{2}\in\mathcal{Y}_{b_{2}}$.
It is enough to show that $\rho\left( \mathcal{A}_{b_{1}},\Phi_{2}\right)
>0$ whenever $\Phi_{2}\in\mathcal{Y}_{b_{2}}$. In fact, suppose in the contrary that $\rho\left( \mathcal{A}_{b_{1}},\Phi_{2}\right) =0$ for some $\Phi_{2}\in\mathcal{Y}_{b_{2}}$. Then there can be extracted some sequence $\left( \Delta_{n}\right) \subset\mathcal{A}_{b_{1}}$ such that $\operatorname{d}\left( \Delta_{n},\Phi_{2}\right) >\operatorname{d}\left(
\Delta_{n+1},\Phi_{2}\right) $, $n\in\mathbb{N}$, and $\lim_{n\rightarrow
\infty}\operatorname{d}\left( \Delta_{n},\Phi_{2}\right) =\rho\left(
\mathcal{A}_{b_{1}},\Phi_{2}\right) =0$. We point out that this can be done because of the definition of the infimum. For each $n\in\mathbb{N}$ let us set $\Gamma_{n}:=\inf_{k\geq n}\left( \Delta_{k}-\Phi_{2}\right) ^{2}%
$. Clearly, $\left( \Gamma_{n}\right) $ is a non-decreasing sequence of measurable functions with its corresponding sequence of integrals $\left(
\int_{\left[ 0,\text{ }\infty\right) }\Gamma_{n}d\mu\right) $ been bounded above by $C_{b_{1}}+C_{b_{2}}<\infty$, see Remark \[rem3\]. Then by the Beppo Levi’s Theorem we can derive that sequence $\left( \Gamma_{n}\right) $ converges almost everywhere to some integrable measurable function $\Gamma$ and $\int_{\left[ 0,\text{ }\infty\right) }\Gamma d\mu=\lim_{n\rightarrow
\infty}\int_{\left[ 0,\text{ }\infty\right) }\Gamma_{n}d\mu\leq
\lim_{n\rightarrow\infty}\operatorname{d}\left( \Delta_{n},\Phi_{2}\right)
=0$, meaning that $\lim_{n\rightarrow\infty}\inf_{k\geq n}\Delta_{k}=\Phi_{2}$ almost everywhere. There are two cases to be clarified. First assume that $b_{1}<b_{2}$. Obviously, $\mu\left( \left( b_{1},~b_{2}\right) \right)
>0$, so that there must be at least one point $x_{0}\in\left( b_{1}%
,~b_{2}\right) $ such that $\lim_{n\rightarrow\infty}\inf_{k\geq n}\Delta
_{k}\left( x_{0}\right) =\Phi_{2}\left( x_{0}\right) $. But since $b_{1}<b_{2}$ the concave property implies that the graph of $\Phi_{2}%
$ (resp. the graph of each function $\inf_{k\geq n}\Delta_{k}$) lies above (resp. below) the graph of the line of equation $y=x$ in the interval $\left(
b_{1},~b_{2}\right) $. Consequently, $\lim_{n\rightarrow\infty}\inf_{k\geq
n}\Delta_{k}\left( x_{0}\right) \leq x_{0}<\Phi_{2}\left( x_{0}\right) $. This, however, is absurd since $\lim_{n\rightarrow\infty}\inf_{k\geq n}%
\Delta_{k}\left( x_{0}\right) =\Phi_{2}\left( x_{0}\right) $. Considering the second case when $b_{1}>b_{2}$ we can similarly get into a contradiction. Therefore, the statement is valid.
\[lem4\]Let $b$ and $c\in\left( 0,\,\infty\right) $ be two arbitrary numbers. Then the following assertions are equivalent:*(1)* The equality $b=c$ holds.*(2)* The sets $\mathcal{Y}_{b}$ and $\mathcal{Y}_{c}$ are equal.*(3)* The equality $\operatorname*{dist}\left( \mathcal{Y}_{b},\mathcal{Y}_{c}\right) =0$ holds.
We first note that the chain of implications (1) $\rightarrow$ (2) $\rightarrow$ (3) is obviously true. Thus we need only show the conditional (3) $\rightarrow$ (1). In fact, assume that $\operatorname*{dist}\left(
\mathcal{Y}_{b},\mathcal{Y}_{c}\right) =0$ but $b\neq c$. Then $\rho\left(
\mathcal{Y}_{b},\Delta\right) =0$ for all $\Delta\in\mathcal{Y}_{c}$. Nevertheless, this contradicts Lemma \[lem3\], since $b\neq c$. Therefore, the argument is valid.
We can similarly prove that:
\[lem5\]Let $b$ and $c\in\left( 0,\,\infty\right) $ be two arbitrary numbers. Then the following assertions are equivalent*:(1)* The equality $b=c$ holds.*(2)* The sets $\mathcal{A}%
_{b}$ and $\mathcal{A}_{c}$ are equal.*(3)* The equality $\operatorname*{dist}\left( \mathcal{A}_{b},\mathcal{A}_{c}\right) =0$ holds.
\[theo6\]Let $b$ and $c\in\left( 0,\,\infty\right) $ be two arbitrary numbers. Then the quantities $\operatorname*{dist}\left( \mathcal{A}%
_{b},\mathcal{A}_{c}\right) $ and $\operatorname*{dist}\left(
\mathcal{Y}_{b},\mathcal{Y}_{c}\right) $ define metrics on $\mathcal{Z}$ and $\mathcal{Z}^{\ast}$ respectively. Hence, the couples $\left( \mathcal{Z}%
,\operatorname*{dist}\right) $ and $\left( \mathcal{Z}^{\ast}%
,\operatorname*{dist}\right) $ are metric spaces.
We need only show that $\operatorname*{dist}\left( \mathcal{Y}_{b}%
,\mathcal{Y}_{c}\right) $ is a metric on the set $\mathcal{Z}^{\ast}$, because the other case can be similarly proved. In fact, we first point out that the condition $\operatorname*{dist}\left( \mathcal{Y}_{b},\mathcal{Y}%
_{c}\right) \geq0$ is obvious and, by Lemma \[lem4\] the equality holds if and only if $\mathcal{Y}_{b}=\mathcal{Y}_{c}$. We also note that the symmetry property trivially holds true. We are now left with the proof of the triangle inequality. In fact, let $\mathcal{Y}_{b_{j}}\in\mathcal{Z}^{\ast}$ and $\Phi_{j}\in\mathcal{Y}_{b_{j}}$ ($j\in\left\{ 1,~2,~3\right\} $) be arbitrary. Then by Proposition 5 (cf. [@AGB2005], page 15) we have that $\operatorname{d}\left( \Phi_{1},\Phi_{3}\right) \leq\operatorname{d}\left(
\Phi_{1},\Phi_{2}\right) +\operatorname{d}\left( \Phi_{2},\Phi_{3}\right)
$. Next, by taking the infimum over $\Phi_{3}\in\mathcal{Y}_{b_{3}}$ it follows that $$\rho\left( \Phi_{1},\mathcal{Y}_{b_{3}}\right) \leq\operatorname{d}\left(
\Phi_{1},\Phi_{2}\right) +\rho\left( \Phi_{2},\mathcal{Y}_{b_{3}}\right)
\leq\operatorname{d}\left( \Phi_{1},\Phi_{2}\right) +\operatorname*{dist}%
\left( \mathcal{Y}_{b_{2}},\mathcal{Y}_{b_{3}}\right) ,$$ i.e. $\rho\left( \Phi_{1},\mathcal{Y}_{b_{3}}\right) \leq\operatorname{d}%
\left( \Phi_{1},\Phi_{2}\right) +\operatorname*{dist}\left( \mathcal{Y}%
_{b_{2}},\mathcal{Y}_{b_{3}}\right) $. Finally, taking the infimum over $\Phi_{2}\in\mathcal{Y}_{b_{2}}$ yields $\rho\left( \Phi_{1},\mathcal{Y}%
_{b_{3}}\right) \leq\rho\left( \Phi_{1},\mathcal{Y}_{b_{2}}\right)
+\operatorname*{dist}\left( \mathcal{Y}_{b_{2}},\mathcal{Y}_{b_{3}}\right)
$, so that $$\operatorname*{dist}\left( \mathcal{Y}_{b_{1}},\mathcal{Y}_{b_{3}}\right)
\leq\operatorname*{dist}\left( \mathcal{Y}_{b_{1}},\mathcal{Y}_{b_{2}%
}\right) +\operatorname*{dist}\left( \mathcal{Y}_{b_{2}},\mathcal{Y}_{b_{3}%
}\right) .$$ This was to be proven.
By the law of trichotomy it is not hard to see that $\left( \mathcal{Z}%
,\preceq\right) $ and $\left( \mathcal{Z}^{\ast},\preceq\right) $ are lattices. Here too, the supremum and infimum binary operations on the lattices $\left( \mathcal{Z},\preceq\right) $ and $\left( \mathcal{Z}^{\ast}%
,\preceq\right) $ will be denoted by the usual symbols $\vee$ and $\wedge$ respectively. We also point out that $\left( \mathcal{Z},\preceq\right) $ and $\left( \mathcal{Z}^{\ast},\preceq\right) $ are infinite graphs. Between two vertices $\mathcal{A}_{b_{1}}$, $\mathcal{A}_{b_{2}}\in\mathcal{Z}$ we can define the edge in two different ways: one by $e=\operatorname*{dist}\left(
\mathcal{A}_{b_{1}},\mathcal{A}_{b_{2}}\right) \in\left( 0,~\infty\right) $ and the other one by $\mathcal{A}_{e}\in\mathcal{Z}$ where $e=\operatorname*{dist}\left( \mathcal{A}_{b_{1}},\mathcal{A}_{b_{2}}\right)
$. These two edges can apply for the vertices of $\mathcal{Z}^{\ast}$ as well.
Dense subsets in $\mathcal{Y}_{b}^{\left( n\right) }$
=======================================================
\[theo7\]Let $b\in\left( 0,\,\infty\right) $ be an arbitrary number. Then $\mathcal{A}_{b}$ is a dense set in $\mathcal{Y}_{b}$.
Fix arbitrarily any function $\Psi\in\mathcal{Y}_{b}$. Then there is some $\Phi\in\mathcal{Y}_{\mathrm{conc}}$ such that $\Psi=\frac{b\Phi}{\Phi\left(
b\right) }$ (by Lemma \[lem1\]). Define $\Psi_{n}\left( x\right)
=\frac{b\left( \Phi\left( x\right) \right) ^{1-1/(n+1)}}{\left(
\Phi\left( b\right) \right) ^{1-1/(n+1)}}$, for all $x\in\left[
0,\,\infty\right) $ and $n\in\mathbb{N}$. As we know from Theorem \[theo2\] (cf. [@AGB2005], page 6) function $\Phi^{1-1/(n+1)}\in\mathcal{A}$ for all $\Phi\in\mathcal{Y}_{\mathrm{conc}}$, $n\in\mathbb{N}$. Then $\left( \Psi
_{n}\right) \subset\mathcal{A}$ (via Lemma \[lem1\], [@AGB2005], page 5). Hence, $\left( \Psi_{n}\right) \subset\mathcal{A}_{b}$, since $\Psi
_{n}\left( b\right) =b$ for all $n\in\mathbb{N}$. We can easily show that $\left( \Psi_{n}\right) $ converges pointwise to $\Psi$. By Remark \[rem4\] it ensues that $\Psi\left( x\right) \leq h_{b}\left( x\right) $ and $\Psi_{n}\left( x\right) \leq h_{b}\left( x\right) $ for all $x\in\left[ 0,\,\infty\right) $ and $n\in\mathbb{N}$, where $h_{b}\left(
x\right) =x+b$, $x\in\left[ 0,\,\infty\right) $. We know via Remark \[rem3\] that function $h_{b}$ is square integrable. Then by applying twice the Dominated Convergence Theorem one can verify that$$\lim_{n\rightarrow\infty}%
%TCIMACRO{\dint _{\left[ 0,\,\infty\right) }}%
%BeginExpansion
{\displaystyle\int_{\left[ 0,\,\infty\right) }}
%EndExpansion
\Psi_{n}^{2}d\mu=%
%TCIMACRO{\dint _{\left[ 0,\,\infty\right) }}%
%BeginExpansion
{\displaystyle\int_{\left[ 0,\,\infty\right) }}
%EndExpansion
\Psi^{2}d\mu\text{ \ and }\lim_{n\rightarrow\infty}%
%TCIMACRO{\dint _{\left[ 0,\,\infty\right) }}%
%BeginExpansion
{\displaystyle\int_{\left[ 0,\,\infty\right) }}
%EndExpansion
\Psi_{n}\Psi d\mu=%
%TCIMACRO{\dint _{\left[ 0,\,\infty\right) }}%
%BeginExpansion
{\displaystyle\int_{\left[ 0,\,\infty\right) }}
%EndExpansion
\Psi^{2}d\mu,$$ so that $\lim_{n\rightarrow\infty}\operatorname{d}\left( \Psi,\Psi
_{n}\right) =0$, because $\Psi\left( x\right) \Psi_{n}\left( x\right)
\leq\left( h_{b}\left( x\right) \right) ^{2}$ for all $x\in\left[
0,\,\infty\right) $ and $n\in\mathbb{N}$ (by Remark \[rem4\]). This was to be proven.
\[theo8\]Fix any pair of numbers $n\in\mathbb{N}\backslash\left\{
1\right\} $ and $b\in\left( 0,~\infty\right) $. Then $\mathcal{A}%
_{b}^{\left( n\right) }$ is dense in $\mathcal{Y}_{b}^{\left( n\right) }$.
Pick arbitrarily some $\Delta\in\mathcal{Y}_{b}^{\left( n\right) }$. Since obviously $\mathcal{Y}_{b}^{CO\left( n\right) }$ is a proper subset of $\mathcal{Y}_{b}^{\left( n\right) }$, we will have two cases to take into consideration. First assume that $\Delta\in\mathcal{Y}_{b}^{CO\left(
n\right) }$. This means that there can be found a counting number $k\geq n$ and a finite sequence $\Phi_{1},~\ldots,~\Phi_{k}\in\mathcal{Y}_{b}^{\left(
1\right) }=\mathcal{Y}_{b}$ such that $\Delta=\Phi_{1}\circ\ldots\circ
\Phi_{k}$. Fix any integer $j\in\mathbb{N}$ and write $\Delta_{j}=\Psi
_{j}\circ\Delta$, where $\Psi_{j}\left( x\right) =\left( b^{1/j}x\right)
^{j/(j+1)}$, $x\in\left[ 0,~\infty\right) $. Clearly, $\Psi_{j}%
\in\mathcal{A}_{b}^{\left( 1\right) }$ for all $j\in\mathbb{N}$. Then applying Theorem \[theo2\] in [@AGB2005] and via the structure of set $\mathcal{A}_{b}^{CO\left( n\right) }$, we can deduce that $\Delta_{j}%
\in\mathcal{A}_{b}^{CO\left( n\right) }$ for all $j\in\mathbb{N}$. It is not difficult to see that sequence $\left( \Delta_{j}\right) $ converge pointwise to $\Delta$. By Remark \[rem4\] we observe that $\Delta\leq h_{b}%
$, $\Delta_{j}\leq h_{b}$ and hence, $\Delta\Delta_{j}\leq\left(
h_{b}\right) ^{2}$ on $\left[ 0,~\infty\right) $. Then recalling twice the Dominated Convergence Theorem we can easily verify that $$\lim_{j\rightarrow\infty}\int_{\left[ 0,~\infty\right) }\left( \Delta
_{j}\right) ^{2}d\mu=\int_{\left[ 0,~\infty\right) }\Delta^{2}d\mu
=\lim_{j\rightarrow\infty}\int_{\left[ 0,~\infty\right) }\Delta\Delta
_{j}d\mu.$$ Consequently, $\lim_{j\rightarrow\infty}\operatorname{d}\left( \Delta
,\Delta_{j}\right) =0$. In the second case we can suppose that $\Delta
\in\mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{Y}_{b}^{CO\left(
n\right) }$. Then without loss of generality we may choose $\Phi_{1}%
,~\ldots,~\Phi_{k}\in\mathcal{Y}_{b}^{CO\left( n\right) }$, whose graphs are pairwise distinct, and some finite sequence $\left( t_{1},~\ldots
~t_{k}\right) \in H_{\left[ 0,~1\right] }$ with $\left( t_{1}%
,~\ldots~t_{k}\right) \subset\left( 0,~1\right) $ such that $\Delta
=\sum_{i=1}^{k}t_{i}\Phi_{i}$. Consider $\Delta_{j}=\sum_{i=1}^{k}t_{i}\left(
\Psi_{j}\circ\Phi_{i}\right) $, where $\Psi_{j}\left( x\right) =\left(
b^{1/j}x\right) ^{j/(j+1)}$, $x\in\left[ 0,~\infty\right) $, $j\in
\mathbb{N}$. Clearly, on the one hand we have that $\left( \Delta_{j}\right)
\subset\mathcal{A}_{b}^{\left( n\right) }$ because $\left( \Psi_{j}%
\circ\Phi_{i}\right) \subset\mathcal{A}_{b}^{CO\left( n\right) }$ for every fixed index $i\in\left\{ 1,~\ldots~k\right\} $ and on the other hand $\lim_{j\rightarrow\infty}\operatorname{d}\left( \Phi_{i},\Psi_{j}\circ
\Phi_{i}\right) =0$, $i\in\left\{ 1,~\ldots~k\right\} $, because of the first part of this proof. Consequently, by the Minkowski inequality we can observe that $\lim_{j\rightarrow\infty}\operatorname{d}\left( \Delta
,\Delta_{j}\right) \leq\sum_{i=1}^{k}t_{i}\lim_{j\rightarrow\infty
}\operatorname{d}\left( \Phi_{i},\Psi_{j}\circ\Phi_{i}\right) =0$. This completes the proof.
Some criterium on the $L^{p}$-norm
==================================
The result here below is worth being mentioned, which is an answer to the second open problem in [@AGB2005].
\[theo9\]Let $\Phi\in\mathcal{Y}_{\mathrm{conc}}$ be arbitrary. Then the following assertions are equivalent.*(1)* $\lim_{t\rightarrow
\infty}\frac{\Phi\left( t\right) }{t}=\lim_{t\rightarrow\infty}%
\varphi\left( t\right) \in\left( 0,\,\infty\right) $.*(2)* There is some constant $c\in\left[ 1,\,\infty\right) $ such that $c\Phi
>\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $.*(3)* There is some constant $c\in\left[ 1,\,\infty\right) $ and some strictly concave function $\Delta:\left[ 0,\,\infty\right) \rightarrow\left[
0,\,\infty\right) $, differentiable on $\left( 0,\,\infty\right) $ and vanishing at the origin such that $c\Phi=\Phi_{\operatorname{id}}+\Delta$ on $\left[ 0,\,\infty\right) $.
We first prove the conditional (1)$\rightarrow$(2). In fact, assume that $\lim_{t\rightarrow\infty}\frac{\Phi\left( t\right) }{t}\in\left(
0,\,\infty\right) $ but in the contrary for every counting number $k\in\mathbb{N}$ there is some $x_{k}\in\left( 0,\,\infty\right) $ for which $k\Phi\left( x_{k}\right) \leq x_{k}$. Obviously, $\limsup
\limits_{k\rightarrow\infty}\frac{\Phi\left( x_{k}\right) }{x_{k}}\leq
\lim_{k\rightarrow\infty}k^{-1}=0$ which is absurd since $\limsup
\limits_{k\rightarrow\infty}\frac{\Phi\left( x_{k}\right) }{x_{k}}\in\left(
0,\,\infty\right) $ by the assumption. Next we show the implication (2)$\rightarrow$(3). In fact, assume that there is some constant $c\in\left[
1,\,\infty\right) $ such that $c\Phi>\Phi_{\operatorname{id}}$ on $\left(
0,\,\infty\right) $ and write $\Delta:=c\Phi-\Phi_{\operatorname{id}}$. Clearly, $\Delta:\left[ 0,\,\infty\right) \rightarrow\left[ 0,\,\infty
\right) $ is a function such that $\Delta\left( 0\right) =0$ and $\Delta$ is positive on $\left( 0,\,\infty\right) $. We also note that $\Delta$ is differentiable on $\left( 0,\,\infty\right) $. Writing $\delta$ for the derivative of $\Delta$, we can observe that $\delta=c\varphi-1$ on $\left(
0,\,\infty\right) $. To show that $\Delta$ is strictly concave it is enough if we prove that $$\left( y-x\right) \delta\left( y-0\right) <\Delta\left( y\right)
-\Delta\left( x\right) <\left( y-x\right) \delta\left( x+0\right)
=\left( y-x\right) \delta\left( x\right)$$ for all $x$, $y\in\left( 0,\,\infty\right) $ with $x<y$ (where, $\delta\left( t-0\right) $ respectively is the left derivative and $\delta\left( t+0\right) $ the right derivative of $\Delta$ at point $t$). In fact, fix arbitrarily two numbers $x$, $y\in\left( 0,\,\infty\right) $ such that $x<y$. But since $\Phi$ is strictly concave we have that$$\left( y-x\right) \varphi\left( y-0\right) <\Phi\left( y\right)
-\Phi\left( x\right) <\left( y-x\right) \varphi\left( x+0\right)
=\left( y-x\right) \varphi\left( x\right)$$ which easily leads to$$c\varphi\left( y-0\right) <\frac{c\Phi\left( y\right) -c\Phi\left(
x\right) }{y-x}<c\varphi\left( x+0\right) =c\varphi\left( x\right) .$$ Hence,$$c\varphi\left( y-0\right) -1<\frac{c\Phi\left( y\right) -c\Phi\left(
x\right) }{y-x}-1<c\varphi\left( x\right) -1,$$ i.e. $$\left( y-x\right) \delta\left( y-0\right) <\Delta\left( y\right)
-\Delta\left( x\right) <\left( y-x\right) \delta\left( x\right) .$$ This ends the proof of the implication (2)$\rightarrow$(3). In the last step, we just point out that the conditional (3)$\rightarrow$(1) is obvious. Therefore, we can conclude on the validity of the argument.
Denote $\widetilde{\mathcal{Y}_{\mathrm{conc}}}:=\left\{ \Phi\in
\mathcal{Y}_{\mathrm{conc}}:\lim_{t\rightarrow\infty}\frac{\Phi\left(
t\right) }{t}>0\right\} $. It is not difficult to check that $\widetilde
{\mathcal{Y}_{\mathrm{conc}}}=\left\{ \Delta\in\mathcal{Y}_{\mathrm{conc}%
}:c\Delta>\Phi_{\operatorname{id}}\text{ on }\left( 0,\,\infty\right) \text{
for some }c\in\left[ 1,\,\infty\right) \right\} $. Write $T_{\Delta
}=\left\{ c\in\left[ 1,\,\infty\right) :c\Delta>\Phi_{\operatorname{id}%
}\text{ on }\left( 0,\,\infty\right) \right\} $, $\Delta\in\widetilde
{\mathcal{Y}_{\mathrm{conc}}}$.
Some few words about set $\widetilde{\mathcal{Y}_{\mathrm{conc}}}$.
\[rem9\]Let $\alpha\in\left( 0,\,\infty\right) $ be arbitrary. Then $\alpha\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ provided that $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$.
Whenever $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ we can choose a corresponding $c\in T_{\Delta}$ such that $c\Delta>\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $. Now choose a constant $t_{0}\in\left(
1,\,\infty\right) $ such that $\alpha t_{0}\geq c$. Hence, $t_{0}\left(
\alpha\Delta\right) \geq c\Delta>\Phi_{\operatorname{id}}$ on $\left(
0,\,\infty\right) $, i.e. $\alpha\Delta\in\widetilde{\mathcal{Y}%
_{\mathrm{conc}}}$.
\[rem10\]Every function $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ can be written as the sum of a finite number of elements of $\widetilde
{\mathcal{Y}_{\mathrm{conc}}}$. Conversely, the sum of a finite number of elements of $\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ also belongs to $\widetilde{\mathcal{Y}_{\mathrm{conc}}}$.
Next, we show that the quantities $\left\Vert f\right\Vert _{L^{p}}$ and $\sup_{\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}}\left( \Phi\left(
1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert
\right\Vert _{L^{p}}$ are equivalent, in the sense that they are both either finite or infinite at the same time. This provides a kind of criterium for a measurable function to belong to $L^{p}$.
\[theo10\]Let $f$ be any measurable function on an arbitrarily fixed measure space $\left( \Omega,\mathcal{F},\lambda\right) $ and $p\in\left[
1,\infty\right) $ be any number. Then$$\left\Vert f\right\Vert _{L^{p}}\leq\sup_{\Phi\in\widetilde{\mathcal{Y}%
_{\mathrm{conc}}}}\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert
\Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}}\leq\left\Vert
f\right\Vert _{L^{p}}+\lambda\left( \Omega\right) .$$
Pick any function $\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$. Then$$\int_{\Omega}\left( \left( \Phi\left( 1\right) \right) ^{-1}\Phi
\circ\left\vert f\right\vert \right) ^{p}d\lambda\leq\int_{\Omega}\left(
\left\vert f\right\vert +1\right) ^{p}d\lambda$$ because $\Delta\leq\left( \Phi_{\operatorname{id}}+1\right) \Delta\left(
1\right) $ for all $\Delta\in\mathcal{Y}_{\mathrm{conc}}$. Consequently, via the Minkowski inequality, it follows that $\left( \Phi\left( 1\right)
\right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}%
}\leq\left\Vert f\right\Vert _{L^{p}}+\lambda\left( \Omega\right) $, which proves the inequality on the right hand-side of the above chain. To show the left side inequality fix any $\Delta\in\mathcal{Y}_{\mathrm{conc}}$ and write $\Delta_{n}=n^{-1}\Delta$, $n\in\mathbb{N}$. Clearly, $\left( \Delta
_{n}\right) \subset\mathcal{Y}_{\mathrm{conc}}$. It is also evident that $\Phi_{\operatorname{id}}+\Delta_{n}\in\widetilde{\mathcal{Y}_{\mathrm{conc}}%
}$, $n\in\mathbb{N}$. Then $$\begin{aligned}
\sup_{\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}}\left( \Phi\left(
1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert
\right\Vert _{L^{p}} & \geq\left( 1+n^{-1}\right) ^{-1}\left\Vert \left(
\Phi_{\operatorname{id}}+\Delta_{n}\right) \circ\left\vert f\right\vert
\right\Vert _{L^{p}}\\
& =\left( 1+n^{-1}\right) ^{-1}\left\Vert \left\vert f\right\vert
+n^{-1}\left\vert f\right\vert \right\Vert _{L^{p}}\geq\left( 1+n^{-1}%
\right) ^{-1}\left\Vert f\right\Vert _{L^{p}}.\end{aligned}$$ Passing to the limit yields $\sup_{\Phi\in\widetilde{\mathcal{Y}%
_{\mathrm{conc}}}}\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert
\Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}}\geq\left\Vert
f\right\Vert _{L^{p}}$. Therefore, we have obtained a valid argument.
\[theo11\]Let $\left( \Omega,\mathcal{F},\lambda\right) $ be any measure space and on it let $f$ be any measurable function. Then$$\lambda\left( \left\vert f\right\vert \geq\varepsilon\right) =\inf\left\{
\inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert \geq\varepsilon
c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in\widetilde{\mathcal{Y}%
_{\mathrm{conc}}}\right\}$$ for every number $\varepsilon\in\left[ 0,\,\infty\right) $.
Throughout the proof $\varepsilon\in\left[ 0,\,\infty\right) $ will be any fixed number. We first note that the assertion is trivial when $\left( \left\vert f\right\vert =\infty\right)
\neq\varnothing$. We shall then prove it when $\left( \left\vert
f\right\vert <\infty\right) \neq\varnothing$. Pick some $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ and $c\in
T_{\Delta}$ such that $c\Delta>\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $. It is not hard to see that $\left(
\left\vert f\right\vert \geq\varepsilon\right) =\left(
\Delta\circ\left\vert f\right\vert \geq\Delta\left(
\varepsilon\right) \right) \subset\left(
\Delta\circ\left\vert f\right\vert \geq\varepsilon c^{-1}\right) $ and thus$$\lambda\left( \left\vert f\right\vert \geq\varepsilon\right) =\lambda\left(
\Delta\circ\left\vert f\right\vert \geq\Delta\left( \varepsilon\right)
\right) \leq\lambda\left( \Delta\circ\left\vert f\right\vert \geq\varepsilon
c^{-1}\right) .$$ Consequently, $$\lambda\left( \left\vert f\right\vert \geq\varepsilon\right) \leq
\inf\left\{ \inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert
\geq\varepsilon c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in
\widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} .$$ To prove the converse statement, we need show that $$\lambda\left( \left\vert f\right\vert \geq\varepsilon\right) \geq
\inf\left\{ \inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert
\geq\varepsilon c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in
\widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} .$$ In fact, for any $n\in\mathbb{N}$ set $\Delta_{n}=\Phi_{\operatorname{id}%
}+n^{-1}\left( 1-e^{-\Phi_{\operatorname{id}}}\right) $. It is not difficult to see that $\Delta_{n}\in\mathcal{Y}_{\mathrm{conc}}$ and $\Delta_{n}%
>\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $, $n\in\mathbb{N}$. This means that $\left( \Delta_{n}\right) \subset\widetilde{\mathcal{Y}%
_{\mathrm{conc}}}$ and moreover, $1\in T_{\Delta_{n}}$, $n\in\mathbb{N}$. Consequently, $$\begin{aligned}
\inf\left\{ \inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert
\geq\varepsilon c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in
\widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} & \leq\lambda\left(
\Delta_{n}\circ\left\vert f\right\vert \geq\varepsilon\right) \\
& =\lambda\left( \left\vert f\right\vert +n^{-1}\left( 1-e^{-\left\vert
f\right\vert }\right) \geq\varepsilon\right) \text{.}%\end{aligned}$$ However, as $\left( \Delta_{n}\right) $ is a decreasing sequence it is obvious that $\left( \Delta_{n+1}\circ\left\vert f\right\vert \geq
\varepsilon\right) \subset\left( \Delta_{n}\circ\left\vert f\right\vert
\geq\varepsilon\right) $, $n\in\mathbb{N}$. Thus having passed to the limit we can observe that $$\inf\left\{ \inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert
\geq\varepsilon c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in
\widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} \leq\lambda\left( \left\vert
f\right\vert \geq\varepsilon\right) .$$ Therefore, the proof is a valid argument.
\[theo12\]Let $f\in L^{p}\left( \Omega,\mathcal{F},\lambda\right) $, $p\geq1$, where $\left( \Omega,\mathcal{F},\lambda\right) $ is any given measure space. Then $$\left\Vert f\right\Vert _{L^{p}}=\inf\left\{ \inf\left\{ c\left\Vert
\Delta\circ\left\vert f\right\vert \right\Vert _{L^{p}}:c\in T_{\Delta
}\right\} :\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} .$$
Pick arbitrarily some $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ and $c\in T_{\Delta}$ such that $c\Delta>\Phi_{\operatorname{id}}$ on $\left(
0,\,\infty\right) $. Clearly, $c\left\Vert \Delta\circ\left\vert f\right\vert
\right\Vert _{L^{p}}\geq\left\Vert f\right\Vert _{L^{p}}$. We can then easily observe that $$\inf\left\{ \inf\left\{ c\left\Vert \Delta\circ\left\vert f\right\vert
\right\Vert _{L^{p}}:c\in T_{\Delta}\right\} :\Delta\in\widetilde
{\mathcal{Y}_{\mathrm{conc}}}\right\} \geq\left\Vert f\right\Vert _{L^{p}}.$$ To prove the converse of this inequality consider the sequence $\left(
\Delta_{n}\right) \subset\widetilde{\mathcal{Y}_{\mathrm{conc}}}$, where $\Delta_{n}=\Phi_{\operatorname{id}}+n^{-1}\left( 1-e^{-\Phi
_{\operatorname{id}}}\right) >\Phi_{\operatorname{id}}$ on $\left(
0,\,\infty\right) $, $n\in\mathbb{N}$. Then as $1\in T_{\Delta_{n}}$, $n\in\mathbb{N}$, we have$$\inf\left\{ \inf\left\{ c\left\Vert \Delta\circ\left\vert f\right\vert
\right\Vert _{L^{p}}:c\in T_{\Delta}\right\} :\Delta\in\widetilde
{\mathcal{Y}_{\mathrm{conc}}}\right\} \leq\left\Vert \Delta_{n}%
\circ\left\vert f\right\vert \right\Vert _{L^{p}}.$$ Since $\left( \Delta_{n}\right) $ is a decreasing sequence it ensues that $\left( \Delta_{n}\circ\left\vert f\right\vert \right) $ is also a decreasing sequence which tends to $\left\vert f\right\vert $. As every member of sequence $\left( \Delta_{n}\circ\left\vert f\right\vert \right) $ is dominated by $\Delta_{1}\circ\left\vert f\right\vert \in L^{p}$, then by applying the Dominated Convergence Theorem it will entail that $$\inf\left\{ \inf\left\{ c\left\Vert \Delta\circ\left\vert f\right\vert
\right\Vert _{L^{p}}:c\in T_{\Delta}\right\} :\Delta\in\widetilde
{\mathcal{Y}_{\mathrm{conc}}}\right\} \leq\left\Vert f\right\Vert _{L^{p}}.$$ This completes the proof.
\[cor3\]Suppose that $h:\mathbb{R\rightarrow R}$ is a continuous function. Then $\left\vert h\right\vert =\frac{1}{\sqrt[p]{\lambda\left( \Omega\right)
}}\inf\left\{ \inf\left\{ \left( \Delta\circ\left\vert h\right\vert
\right) c:c\in T_{\Delta}\right\} :\Delta\in\widetilde{\mathcal{Y}%
_{\mathrm{conc}}}\right\} $.
Fix any number $x\in\mathbb{R}$ and let $f\in L^{p}\left( \Omega
,\mathcal{F},\lambda\right) $ be the constant function defined by $f\equiv
h\left( x\right) $ on $\Omega$. Then by applying Theorem \[theo10\] we can easily deduce the result.
Given any number $k\in\mathbb{N}$ characterize all pairs of functions $\Phi$ and $\Delta\in\mathcal{Y}_{\mathrm{conc}}$ such that $\left\vert \left\{
x\in\left( 0,~\infty\right) :\Phi\left( x\right) =\Delta\left( x\right)
\right\} \right\vert =k$.
Characterize all pairs of functions $\Phi$ and $\Delta\in\mathcal{Y}%
_{\mathrm{conc}}$ such that the sets $\left( 0,~\infty\right) $ and $\left\{ x\in\left( 0,~\infty\right) :\Phi\left( x\right) =\Delta\left(
x\right) \right\} $ should be equinumerous.
[99]{}
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<span style="font-variant:small-caps;">Kuratowski, K.</span>: *Topology*, vol. 1, Academic Press, New York, etc., 1966.
<span style="font-variant:small-caps;">MacLane, S. and Birkhoff G.</span>: *Algèbre*, Tome 1, *Structures fondamentales*, Gauthier-Villars, 1971.
<span style="font-variant:small-caps;">Mogyoródi, J.</span>: On a concave function inequality for martingales, Annales Univ. Sci. Bud. Sect. Math. **24**(1981), 255 - 271.
<span style="font-variant:small-caps;">Reed, M. C.</span>: *Fundamental ideas of analysis*, John Wiley & Sons, New York ..., 1998.
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| ArXiv |
---
author:
- 'Andrzej Dbrowski, Nursena Günhan and Gökhan Soydan'
title: 'On a class of Lebesgue-Ljunggren-Nagell type equations'
---
[*Abstract*]{}. Given odd, coprime integers $a$, $b$ ($a>0$), we consider the Diophantine equation $ax^2+b^{2l}=4y^n$, $x, y\in\Bbb Z$, $l \in \Bbb N$, $n$ odd prime, $\gcd(x,y)=1$. We completely solve the above Diophantine equation for $a\in\{7,11,19,43,67,163\}$, and $b$ a power of an odd prime, under the conditions $2^{n-1}b^l\not\equiv \pm 1(\mod a)$ and $\gcd(n,b)=1$. For other square-free integers $a>3$ and $b$ a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers $x$, $y$ with ($\gcd(x,y)=1$), $l\in\mathbb{N}$ and all odd primes $n>3$, satisfying $2^{n-1}b^l\not\equiv \pm 1(\mod a)$, $\gcd(n,b)=1$, and $\gcd(n,h(-a))=1$, where $h(-a)$ denotes the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-a})$.
Key words: Diophantine equation, Lehmer number, Fibonacci number, class number, modular form, elliptic curve
2010 Mathematics Subject Classification: 11D61, 11B39
Introduction
============
The Diophantine equation $x^2+C=y^n$ ($x\geq 1$, $y\geq 1$, $n\geq 3$) has a rich history. Lebesgue proved that this equation has no solution when $C=1$, and Cohn solved the equation for several values of $1\leq C\leq 100$. The remaining values of $C$ in the above range were covered by Mignotte and de Weger, and finally by Bugeaud, Mignotte and Siksek. Barros in his PhD thesis considered the range $-100\leq C\leq -1$. Also, several authors (Abu Muriefah, Arif, Dbrowski, Le, Luca, Pink, Soydan, Togbé, Ulas,...) became interested in the case where only the prime factors of $C$ are specified. Surveys of these and many others topics can be found in [@AB] and [@BP]. Some people studied the more general equation $ax^2+C=2^iy^n$, $a>0$ and $i\leq 2$.
Given odd, coprime integers $a$, $b$ ($a>0$), we consider the Diophantine equation $$\label{maineq}
ax^2+b^{2l}=4y^n, \quad x, y\in \Bbb Z,\,\, l, n \in \Bbb N, \, n \, odd \, \, prime, \gcd(x,y)=1.$$ If $a \equiv 1 \, \mod 4$, then reducing modulo $4$ we trivially obtain that the equation has no solution.
It is known (due to Ljunggren [@Lj]) that the Diophantine equation $ax^2+1=4y^n$, $n\geq 3$, has no positive solution with $y>1$ such that $a \equiv 3 (\mod 4)$ and the class number of the quadratic field $\mathbb Q(\sqrt{-a})$ is not divisible by $n$. When $a=3$, then $3x^2+1=4y^n$ has the only positive solution $(x,y)=(1,1)$.
As our first result, we completely solve the equation for $a\in\{7,11,19,$ $43,67,163\}$, under the conditions $2^{n-1}b^l\not\equiv \pm 1(\mod a)$ and $\gcd(n,b)=1$.
\[thm.1\] Fix $p\in\{7,11,19,43,67,163\}$ and $b= \pm q^r$, with $q$ an odd prime different from $p$ and $r\geq 1$.
$(i)$ The Diophantine equation $$\label{sec.eq}
px^2+b^{2l}=4y^n,\, l\in\mathbb{N},\, \gcd(x,y)=1$$ has no solutions $(p,x,y,b,l,n)$ with integers $x$, $y$ and primes $n>3$, satisfying the conditions $2^{n-1}b^l\not\equiv \pm 1(\mod p)$ and $\gcd(n,b)=1$.
$(ii)$ If $n=3$ and $p\not= 7$, then the equation has no solutions $(p,x,y,b,l,3)$ satisfying the conditions $4b^l\not\equiv \pm 1(\mod p)$ and $\gcd(3,b)=1$.
$(iii)$ If $n=3$ and $p=7$, then the equation leads to $6$ infinite families of solutions, corresponding to solutions of Pell-type equations , , , , and satisfying the conditions $4b^l\not\equiv \pm 1(\mod 7)$ and $\gcd(3,b)=1$.
[**Remarks.**]{} (i) The Diophantine equation has many solutions (infinitely many ?) satisfying the conditions $2^{n-1}b^l\equiv \pm 1(\mod p)$ and $\gcd(n,b)=1$. Examples include
$(p,x,y,b,l,n) \in\{ (7,\pm 1,$ $2,\pm 11,1,5),(11,\pm 1,3,\pm 31,1,5)$, $(7,\pm 7,2,\pm 13,1,7),$ $(19,\pm 1,5,\pm 559,1,7)$, $(11, \pm 253, 3, \pm 67, 1,11), (19, \pm 2531, 5, \pm 8579, 1, 11)$,\
$(7,\pm 1,2,\pm 181,1,13), (11, \pm 1801, 3, \pm 21929, 1,17 ),
(7,\pm 457, 2, \pm 797, 1, 19), \\
(7, \pm 967, 2, \pm 5197, 1, 23)\}$.
\(ii) If $b$ is divisible by at least two different odd primes, then the Diophantine equation may have solutions satisfying the conditions $2^{n-1}b^l\not\equiv \pm 1(\mod p)$. Examples include
$(p,x,y,b,l,n) \in
\{ (7,103820535541,4,10341108537,1,37)$,\
$(7,4865,46,1320267,1,7)$, $(19,315003,49,909715,1,7)$,\
$(19,581072253,49,3037108805,1,11) \}$.
\(iii) Write the equation as $px^2+b^{2l}=4y(y^{(n-1)/2})^2$ (compare [@Lj p.116]). Now using $4y=u^2+pv^2$, taking $u=\pm 1$, and multiplying the equation by $p$, we arrive at the equation
$$\label{Ljunggren}
X^2 - p(1+pv^2)Y^2 = -pb^{2l}.$$
If $b=\pm 1$, we obtain the equation (7’) in [@Lj]. Ljunggren used an old result by Mahler to deduce that, if $p>3$, then has no solution with $Y>1$ such that any prime divisor of $Y$ divides $p(1+pv^2)$ as well.
\(iv) Question: may we extend Ljunggren’s idea to prove non-existence of solutions of our equation for some $b^l$ ?
For a family of positive square-free integers $a$ with $h(-a)>1$ we can prove the following result (a variant of the results by Bugeaud [@Bu] and Arif and Al-Ali [@AA] in a case of the equation $ax^2+b^{2l+1}=4y^n$). Let $h(-a)$ denote the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-a})$.
\[thm.2\] Fix a positive square-free integer $a$, different from $3$, $7$, $11$, $19$, $43$, $67$, $163$, and $b= \pm q^r$, with $q$ an odd prime not dividing $a$ and $r\geq 1$. Then the Diophantine equation has no solutions $(a,x,y,b,l,n)$, with integers $x$, $y$ and primes $n>3$ satisfying the conditions $\gcd(n,h(-a))=1$, $2^{n-1}b^l\not\equiv \pm 1(\mod a)$, and $\gcd(n,b)=1$.
[**Remarks.**]{} (i) There are a lot of positive square-free integers $a$ with $rad(h(-a)) | 6$ (hypothetically, infinitely many): $18$ values of $a$ with $h(-a)=2$, $54$ values of $a$ with $h(-a)=4$, $31$ values of $a$ with $h(-a)=6$, etc. Here $rad(m)$ denotes the radical of a positive integer $m$, i.e. the product of all prime divisors of $m$.
\(ii) For fixed $a$ and $b$ we can (in some cases) use MAGMA [@Magma] to solve the Diophantine equation $ax^2+b^{2l}=4y^3$ (applying `SIntegralPoints` subroutine of MAGMA to associated families of elliptic curves). In a general case, one can try to prove a variant of Dahmen’s result [@Da] saying that the above equation has no solution for a positive proportion of $l$’s, not divisible by $3$.
\(iii) The following variant of a result by Laradji, Mignotte and Tzanakis (see [@LMT Theorem 2.3]) follows immediately from out Theorem 2 (note that always $h(-p) < p$). Let $p$, $q$ be odd primes with $p\equiv 3 (\mod 8)$ and $p>3$. Then the Diophantine equation $px^2+q^{2l}=4y^p$ has no solution $(x,y,l)$ with positive integers $x$, $y$, $l$ satisfying $\gcd(x,y)=1$.
\(iv) Dieulefait and Urroz [@DU] used the method of Galois representations attached to $\mathbb Q$-curves to solve the Diophantine equation $3x^2+y^4=z^n$. The authors suggest that their method can be applied to solve this type of equations with $3$ replaced by other values of $a$. We expect that their method can be extended to the case $ax^2+y^4=4z^n$ with small $a$ as well.
\(v) We can solve the Diophantine $ax^2+b^{2l}=4y^n$ for relatively small values of $a>0$ (at least) in positive integers $x,y,l,n$, $\gcd(x,y)=1$, $n\geq 7$ a prime dividing $l$, by using the Bennett-Skinner strategy [@BS]. We treat some examples in Section 3. Let us also mention that the smallest positive integer $a$ with $h(-a)=7$ is $71$, and one needs to consider newforms of weight $2$ and level $10082$.
\(vi) Pink [@P] used estimates for linear forms in two logarithms in the complex and the $p$-adic case, to give an explicit bound for the number of solutions of the Diophantine equation $x^2+ (p_1^{\alpha_1}\cdots p_s^{\alpha_s})^2 =2y^n$ in terms of $s$ and $\max\{p_1, \cdots, p_s\}$. We can prove analogous result concerning the equations $px^2 + (p_1^{\alpha_1}\cdots p_s^{\alpha_s})^2 = 4y^n$, with $p\in\{7,11,19,43,67,163\}$.
Proofs of Theorems \[thm.1\] and \[thm.2\]
==========================================
[*Proof of Theorem \[thm.1\]*]{}.
Below in the proof, $b$ is a power of an odd prime $q \not= p$.
As the class number of $\mathbb Q(\sqrt{-p})$ with $p\in\{7,11,19,43,67,163\}$ is $1$, we have the following factorization
$${b^l+x\sqrt{-p}\over 2} \cdot {b^l-x\sqrt{-p}\over 2} = y^n.$$ Now we have
$${b^l+x\sqrt{-p}\over 2} = \left({u+v\sqrt{-p}\over 2}\right)^n,$$ where $u$, $v$ are odd rational integers. Note that necessarily $\gcd(u,v)=1$. Equating real parts we get
$$2^{n-1}b^l = u\sum_{r=0}^{(n-1)/2}\binom{n}{2r}u^{n-2r-1}(-p)^rv^{2r}.$$ As $u$ is odd, its possible values are among divisors of $b^l$. Here, we assume that $2^{n-1}b^l\not\equiv \pm 1 (\mod p)$.
\(i) If $u=\pm 1$, then $2^{n-1}b^l = \sum_{r=0}^{(n-1)/2}\binom{n}{2r}(-p)^rv^{2r}$, and in particular $2^{n-1}b^l\equiv \pm 1 (\mod p)$, a contradiction.
\(ii) If $u \not= \pm 1, \pm b^l$, then $q$ divides $pvn$. Since $\gcd(u,v) = \gcd(p,b) =1$, then $q$ divides $n$, a contradiction.
\(iii) Assume $u=\pm b^l$. Put $\alpha = {v\sqrt{p}+b^li\over 2}$. Then $(\alpha + \overline{\alpha})^2 = v^2p$, $\alpha \overline{\alpha} = {1\over 4}(v^2p + b^{2l})$, and $\alpha/\overline{\alpha}$ is not a root of unity. Hence $(\alpha,\overline{\alpha})$ is a Lehmer pair. Note that ${\alpha^n - \overline{\alpha}^n \over \alpha - \overline{\alpha}}=\pm 1$. On the other hand, using [@BHV-M] we obtain that ${\alpha^n - \overline{\alpha}^n \over \alpha - \overline{\alpha}}$ has primitive divisors for $n=11$ and all primes $n>13$, and hence our equation has no solution for $n=11$ and for primes $n>13$. Let us consider the cases $n\in\{3,5,7,13\}$ separately. Let us stress that the data in [@BHV-M] are given for equivalence classes of $n$-detective Lehmer pairs: two Lehmer pairs $(\alpha_1,\beta_1)$, $(\alpha_2,\beta_2)$ are equivalent (we write $(\alpha_1,\beta_1) \sim(\alpha_2,\beta_2)$) if $\alpha_1/\alpha_2 = \beta_1/\beta_2 \in\{\pm 1,\pm i\}$.
$\underline{n=3}$. According to [@BHV-M], we have two possibilities: (a) ${v\sqrt{p}+ui \over 2} \sim {\sqrt{1+\lambda} + \sqrt{1-3\lambda} \over 2}$, $\lambda\not=1$, or (b) ${v\sqrt{p}+ui \over 2} \sim {\sqrt{3^k+\lambda} + \sqrt{3^k-3\lambda} \over 2}$, $k>0$, $3\nmid \lambda$.
In the case (a) we have four subcases: (i) $1+\lambda=v^2p$ and $1-3\lambda=-u^2$ or (ii) $1+\lambda=-v^2p$ and $1-3\lambda=u^2$ or (iii) $1+\lambda=-u^2$ and $1-3\lambda=v^2p$ or (iv) $1+\lambda=u^2$ and $1-3\lambda=-v^2p$ .
In the subcase (i) we obtain a contradiction reducing the second equation modulo $3$.
In the subcase (ii) we obtain relation $u^2=3pv^2+4$. If $p\not =7$, then reducing this equation modulo $8$, we obtain $1\equiv 5(\mod 8)$, a contradiction. Now the case $p=7$ leads to Pell-type equation $$\label{Pell.1}
u^2-21v^2=4.$$ Using the assumption $u$ and $v$ are odd for our equation, any solution to is given by $$\dfrac{u_t+v_t\sqrt{21}}{2}=\left(\dfrac{u_0+v_0\sqrt{21}}{2}\right)^t,$$ where $(u_0,v_0)=(5,1)$ is minimal solution and $3\nmid t.$ Thus an infinite family of solutions of equation is given by $$(x_t,y_t,b_t^l,n)=
\left(\dfrac{3u_t^2v_t-7v_t^3}{4},\dfrac{7v_t^2+u_t^2}{4},\dfrac{u_t^3-21u_tv_t^2}{4},3\right).$$ (see [@Coh Proposition 6.3.16] for the details about the equation ).\
In the subcase (iii), note that $4+3u^2=v^2p$, and hence $7\equiv 3(\mod 8)$ if $p\not=7$, a contradiction. If $p=7$, then we need to consider the Diophantine equation $$\label{Pell.2}
7v^2-3u^2=4.$$ Such an equation has $3$ infinite families of solutions $(v,u) \in \{(s+3r,s+7r), (-s+3r,s-7r), (4s+18r,6s+28r)\}$, where $s^2-21r^2=1$. But since $u$ and $v$ are odd, one gets $2$ infinite families of solutions $(v,u) \in \{(s+3r,s+7r), (-s+3r,s-7r)\}$ for the equation . Any solution to the equation $s^2-21r^2=1$ is given by $$s_t+r_t\sqrt{21}=(s_0+r_0\sqrt{21})^t$$ where $(s_0,r_0)=(55,12)$ is minimal solution. Thus 2 infinite families of solutions of equation are given by
$$\begin{aligned}
x_t&=&-3r_ts_t^2+63r_t^3+21s_tr_t^2-s_t^3\\
y_t&=&2s_t^2+14s_tr_t+28r_t^2\\
b_t^l&=&-5s_t^3-63s_t^3r_t-231s_tr_t^2-245r_t^3\end{aligned}$$
or $$\begin{aligned}
x_t&=&-3s_t^2r_t+s_t^3-21s_tr_t^2+63r_t^3\\
y_t&=&2s_t^2-14s_tr_t+28r_t^2\\
b_t^l&=&-5s_t^3+63s_t^2r_t-231s_tr_t^2+245r_t^3\end{aligned}$$ with $n=3$. (see [@AnAn Theorems 4.5.1, 4.5.2] for details about the equation ).
In the subcase (iv) note that $4+v^2p=3u^2$, and hence $7\equiv 3(\mod 8)$ if $p\not=7$, a contradiction. If $p=7$, then reducing $4+7v^2=3u^2$ modulo $7$ we obtain $\square = -\square$, a contradiction.
In the case (b) we have four subcases: (i) $3^k+\lambda=v^2p$ and $3^k-3\lambda=-u^2$ or (ii) $3^k+\lambda=-v^2p$ and $3^k-3\lambda=u^2$ or (iii) $3^k+\lambda=-u^2$ and $3^k-3\lambda=v^2p$ or (iv) $3^k+\lambda=u^2$ and $3^k-3\lambda=-v^2p$.
In the subcase (i) note that $3v^2p=u^2+4\cdot 3^k$, hence necessarily $k=1$ (otherwise $3 | \gcd(u,v)$). Therefore $v^2p=3t^2+4$, where $u=3t$. If $p\not =7$, then reducing this equation modulo $8$, we obtain $3\equiv 7(\mod 8)$, a contradiction. If $p=7$, then we need to consider the Diophantine equation $$\label{Pell.3}
7v^2-3t^2=4.$$ As in subcase (iii) of part (a) Such an equation has $3$ infinite families of solutions $(v,u) \in \{(s+3r,3(s+7r)), (-s+3r,3(s-7r)), (4s+18r,3(6s+28r))\}$, where $s^2-21r^2=1$. But since $u$ and $v$ are odd, one obtains $2$ infinite families of solutions $(v,u) \in \{(s+3r,3(s+7r)), (-s+3r,3(s-7r))$ for the equation . Any solution to the equation $s^2-21r^2=1$ is given by $$s_t+r_t\sqrt{21}=(s_0+r_0\sqrt{21})^t$$ where $(s_0,r_0)=(55,12)$ is minimal solution. Thus 2 infinite families of solutions of equation are given by $$\begin{aligned}
x_t&=&567s_tr_t^2+99r_ts_t^2+5s_t^3+945r_t^3\\
y_t&=&4s_t^2+42s_tr_t+126r_t^2\\
b_t^l&=&-9s_t^3-63s_t^2r_t+189s_tr_t^2+1323r_t^3\end{aligned}$$ or $$\begin{aligned}
x_t&=&-567s_tr_t^2+99r_ts_t^2-5s_t^3+945r_t^3\\
y_t&=&4s_t^2-42s_tr_t+126r_t^2\\
b_t^l&=&-9s_t^3+63s_t^2r_t+189s_tr_t^2-1323r_t^3\end{aligned}$$ with $n=3$.
In the subcase (ii) note that $4\cdot 3^k+3pv^2=u^2$, hence necessarily $k=1$ (otherwise $3 | \gcd(u,v)$). Therefore $4=3t^2-pv^2$, where $u=3t$. If $p\not =7$, then reducing this equation modulo $8$, we obtain $4\equiv 0(\mod 8)$, a contradiction. Now reducing $4=3t^2-7v^2$ modulo $7$, we obtain $\square = -\square$, a contradiction again.
In the subcase (iii) note that $3t^2p=u^2+4$, where $v=3t$. Now reducing modulo $3$, we obtain a contradiction.
In the subcase (iv) note that $u^2=3pt^2+4$, where $v=3t$. If $p\not =7$, then reducing this equation modulo $8$, we obtain $1\equiv 5(\mod 8)$, a contradiction. Now the case $p=7$ leads to Pell-type equation $$\label{Pell.4}
u^2-21t^2=4.$$ Since $u$ and $v$ are odd, one gets that $t$ is odd for the equation . So, any solution to is given by $$\dfrac{u_m+t_m\sqrt{21}}{2}=\left(\dfrac{u_0+t_0\sqrt{21}}{2}\right)^m,$$ where $(u_0,t_0)=(5,1)$ is minimal solution and $3\nmid m.$ Thus an infinite family of solutions of equation is given by $$(x_m,y_m,b_m^l,n)=
\left(\dfrac{3u_m^2v_m-7v_m^3}{4},\dfrac{7v_m^2+u_m^2}{4},\dfrac{u_m^3-21u_mv_m^2}{4},3\right),$$ with $v_m=3t_m$.
$\underline{n=5}$. According to [@BHV-M], we have two possibilities to consider: (a) ${v\sqrt{p}+ui \over 2} \sim {\sqrt{F_{k-2\epsilon}} + \sqrt{F_{k-2\epsilon}-4F_k} \over 2}$, $k\geq 3$, $\epsilon=\pm 1$, or (b) ${v\sqrt{p}+ui \over 2} \sim {\sqrt{L_{k-2\epsilon}} + \sqrt{L_{k-2\epsilon}-4F_k} \over 2}$, $k\not=1$, $\epsilon=\pm 1$. Here $F_m$ and $L_m$ denote $m$-th Fibonacci and Lucas number respectively.
In the case (a) we have four subcases: (i) $v^2p=F_{k-2\epsilon}-4F_k$ and $-u^2=F_{k-2\epsilon}$ or (ii) $-v^2p=F_{k-2\epsilon}-4F_k$ and $u^2=F_{k-2\epsilon}$ or (iii) $v^2p=F_{k-2\epsilon}$ and $-u^2=F_{k-2\epsilon}-4F_k$ or (iv) $-v^2p=F_{k-2\epsilon}$ and $u^2=F_{k-2\epsilon}-4F_k$.
In the subcase (i) we obtain $F_{k-2\epsilon}=-u^2<0$, a contradiction.
In the subcase (ii), due to the fundamental work by Ljunggren [@Lj0] [@Lj2] (see also [@BMS2 Section 2]) we can find all solutions to the equation $u^2=F_{k-2\epsilon}$ ($k\geq 3$, $\epsilon=\pm 1$). Ljunggren has proved that the only squares in the Fibonacci sequence are $F_0=0$, $F_1=F_2=1$, and $F_{12}=144$.
The case $k-2\epsilon=1$ gives $k=3$, $\epsilon=1$, $u^2=1$, hence using the first equation from (ii) we obtain $-v^2p=-7$, i.e. $p=7$, $v^2=1$. This case gives the solution $(p,x,y,b^l,n) = (7,\pm 1,2,\pm 11,5)$ which contradicts with $2^{n-1}b^l\not\equiv \pm (\mod p)$.
The case $k-2\epsilon=2$, gives $k=4$, $\epsilon=1$, $u^2=1$, hence using the first equation from (ii) we obtain $-v^2p=-11$, i.e. $p=11$, $v^2=1$. This case gives the solution $(p,x,y,b^l,n) = (11,\pm 1,3,\pm 31,5)$, which is impossible since $2^{n-1}b^l\equiv \pm 1$ $(\mod p)$.
The case $k-2\epsilon=12$, gives $k=14$, $\epsilon=1$, $u^2=144$ or $k=10$, $\epsilon=-1$, $u^2=144$. The first possibility gives $-v^2p=-2^2 \cdot 11 \cdot 31$, i.e. $p=11\cdot 31$, $v^2=4$. The second possibility gives $-v^2p=-2^2\cdot 19$, i.e. $p=19$, $v^2=4$. This case gives no solution satisfying $\gcd(x,b)=1$ (note that both $u$ and $v$ are even).
In the subcase (iii), let us note that $u^2 = -(F_{k-2\epsilon}-4F_k) = F_k+F_{k+2\epsilon}$. Now a short look at the paper by Luca and Patel (see [@LP Theorem 1], and their calculations in Section 5) shows that $k=4$, $\epsilon=-1$ is the only possibility. But then $F_6 = 8 = v^2p$, a contradiction.
In the subcase (iv) we obtain $F_{k-2\epsilon}=-v^2p<0$, a contradiction.
In the case (b) we have four subcases: (i) $v^2p=L_{k-2\epsilon}-4L_k$ and $-u^2=L_{k-2\epsilon}$ or (ii) $-v^2p=L_{k-2\epsilon}-4L_k$ and $u^2=L_{k-2\epsilon}$ or (iii) $v^2p=L_{k-2\epsilon}$ and $-u^2=L_{k-2\epsilon}-4L_k$ or (iv) $-v^2p=L_{k-2\epsilon}$ and $u^2=L_{k-2\epsilon}-4L_k$.
In the subcase (i) we obtain $L_{k-2\epsilon}=-u^2<0$, a contradiction.
In the subcase (ii) we can find all solutions to the equation $u^2=L_{k-2\epsilon}$, ($k\not= 1$, $\epsilon=\pm 1$). By the work by Cohn [@Cohn] we know all solutions: $L_1=1^2$ and $L_3=2^2$.
The case $k-2\epsilon=1$, gives $k=3$, $\epsilon=1$, $u^2=1$, hence using the first equation from (ii) we obtain $-v^2p=-13$, i.e. $p=13$, $v^2=1$. The case $k-2\epsilon=3$, gives $k=5$, $\epsilon=1$, $u^2=4$, hence using the first equation from (ii) we obtain $-v^2p=-40$, i.e. $p=10$, $v^2=4$. None of these two cases lead to solution of our Diophantine equation ($13$ is congruent to $1$ modulo $4$, while $10$ is even).
In the subcase (iii), let us note that $L_k+L_{k-2\epsilon} = 5F_{k-\epsilon}$. Therefore we need to determine all $k$ such that $5F_{k-\epsilon}$ is a square. Again, the paper by Bugeaud, Mignotte and Siksek [@BMS] shows that the only possibility is $5F_5 = 5^2$. But then $v^2p=L_4=7$ in case $\epsilon=1$ or $v^2p=L_6=18$ in case $\epsilon=-1$. In the first case we obtain $p=7$, but then $u^2=65$, a contradiction. The second case gives a contradiction by trivial observation.
In the subcase (iv) we obtain $L_{k-2\epsilon}=-v^2p<0$, a contradiction.
$\underline{n=7}$. According to [@BHV-M] we have six equivalence classes of $7$-defective Lehmer pairs. Two of them, $({1-\sqrt{-7}\over 2}, {1+\sqrt{-7}\over 2})$ and $({1-\sqrt{-19}\over 2}, {1+\sqrt{-19}\over 2})$, come from our Lehmer pairs, giving $(p,x,y,b^l,n)\in \{(7,\pm 7,2,\pm 13,7), (19,\pm 1,5,$ $\pm 559,7)\}$, which are impossible since $2^{n-1}b^l\equiv \pm 1$ $(\mod p)$.
$\underline{n=13}$. The unique $13$-detective equivalence class $({1-\sqrt{-7}\over 2}, {1+\sqrt{-7}\over 2})$ leads to the solution $(p,x,y,b^l,n) = (7,\pm 1,2,\pm 181,13)$, which contradicts with $2^{n-1}b^l\not\equiv \pm 1 (\mod p)$.
.
In this case, thanks to [@Bilu Lemma 1], we can follow the same lines as in the proof of Theorem 1 for $n>3$.
Method via Galois representations and modular forms
===================================================
We will consider the Diophantine equation $ax^2+b^{2k}=4y^n$, for $a\in\{3, 7, 11, 15\}$ in positive integers $x, y, k, n$, $\gcd(x,y)=1$, $n\geq 7$ a prime dividing $k$. We will apply the Bennett-Skinner strategy [@BS], in particular we will use the results we need from [@BS]. We can compute systems of Hecke eigenvalues for conjugacy classes of newforms using MAGMA (or use Stein’s Modular Forms Database provided the level is $\leq 7248$).
[**Remarks.**]{} (a) If $a \equiv 3 \, \mod 8$, then $y$ is necessarily odd: if $y$ is even, then reducing modulo $8$ we obtain that the left hand side is congruent to $4$ modulo $8$, while the right hand side is congruent to $0$ modulo $8$, a contradiction.
\(b) If $a \equiv 7 \, \mod 8$, then $y$ is necessarily even.
\(i) The Diophantine equation $3x^2+b^{2k}=4y^n$ has no solution $(x,y,k,n)$, $xy\not=1$, $n\geq 7$ prime dividing $k$.
We will consider a more general Diophantine equation $X^n+4Y^n=3Z^2$ ($n\geq 7$ a prime) and use [@BS]. We are in case (iii) of [@BS p.26], hence $\alpha\in\{1, 2\}$. From Lemma 3.2 it follows, that we need to consider the newforms of weight $2$ and levels $N\in\{36, 72\}$.
a\) There is only one newform of weight $2$ and level $36$, corresponding to an elliptic curve $E$ of conductor $36$ with complex multiplication by $\mathbb Q(\sqrt{-3})$. Here we will apply [@BS Subsection 4.4], to prove that $ab=\pm 1$. Assume (a.a) that $ab\not= \pm 1$. Then (using Prop. 4.6 (b)) holds, hence if $n=7$ or $13$, $n$ splits in $K=\mathbb Q (\sqrt{-3})$ and $E(K)$ is infinite for all elliptic curves of conductor $2n$. One checks that both primes $7$ and $13$ split in $K$. Now using Cremona’s online tables we check, that all elliptic curves of conductor $126=2\times 7 \times 3^2$ have rank zero, and all elliptic curves of conductor $234=2 \times 13 \times 3^2$, which are quadratic twists by $3$ of quadratic curves of conductor $26$, have rank zero too.
b\) There is only one newform of weight $2$ and level $72$, corresponding to isogeny class of elliptic curves of conductor $72$, with $j$-invariant $u/3^v$, with $v>0$ and $u$ some non-zero integer prime to $3$. To eliminate such an elliptic curve we use [@BS Prop. 4.4].
\(ii) The Diophantine equation $7x^2+b^{2k}=4y^n$ has no solution $(x,y,k,n)$, $n\geq 7$ prime dividing $k$.
We need to consider the newforms of weight $2$ and level $N=98$.
There are two Galois conjugacy classes of forms of weight $2$ and level $98$. We will use numbering as in Stein’s tables: we have $a_3(f_2)=\pm \sqrt{2}$ and we can use [@BS Prop. 4.3] to eliminate $f_2$. On the other hand, the form $f_1$ corresponds to an elliptic curve of conductor $98$, with $j$-invariant $u/7^v$, with $v>0$ and $u$ some non-zero integer prime to $7$. To eliminate such an elliptic curves we use [@BS Prop. 4.4].
\(iii) The Diophantine equation $11x^2+b^{2k}=4y^n$ has no solution $(x,y,k,n)$, $n\geq 7$, $n\not=11, 13$ prime dividing $k$.
We need to consider the newforms of weight $2$ and levels $N\in\{484, 968\}$.
a\) There are five Galois conjugacy classes of forms of weight $2$ and level $484$. We have $a_3(f_1)=1$, $a_3(f_4)={1\pm \sqrt{33}\over 2}$, and we can use [@BS Prop. 4.3] to eliminate $f_1$ and $f_4$. To eliminate $f_2$ and $f_3$ we need to consider coefficients $a_3$ and $a_5$: we have $a_3(f_2)=a_3(f_3)={-3\pm \sqrt{5}\over 2}$ and $a_5(f_2)=a_5(f_3)={-1\pm \sqrt{5}\over 2}$ (we cannot avoid $n=29$ when considering only $a_3$). Finally, $a_7(f_5)=\pm 2\sqrt{3}$, and we can use [@BS Prop. 4.3] to eliminate $f_5$ when $n\geq 7$ and $n\not=13$.
b\) There are fourteen Galois conjugacy classes of forms of weight $2$ and level $968$. We have $a_3(f_1)=-3$, $a_5(f_2)=a_5(f_3)=3$, $a_3(f_4)=a_3(f_5)=1$, $a_3(f_6)=a_3(f_7)=\pm 2\sqrt{5}$, $a_3(f_{10})={1\pm\sqrt{17}\over 2}$, and we can easily use [@BS Prop. 4.3] to eliminate $f_1,...,f_7$, and $f_{10}$. Now considering $a_3$ and $a_{13}$ for both newforms $f_8$, $f_9$, and $a_3$ and $a_5$ for both newforms $f_{13}$, $f_{14}$, and using [@BS Prop. 4.3], we can eliminate these four forms when $n\geq 7$ and $n\not= 11$.
\(iv) The Diophantine equation $15x^2+b^{2k}=4y^n$ has no solution $(x,y,k,n)$, $n\geq 7$ prime dividing $k$.
We need to consider the newforms of weight $2$ and level $N=450$.
There are seven Galois conjugacy classes of forms of weight $2$ and level $450$. We have $a_{11}(f_3)=a_{11}(f_7)=3$, and we can use [@BS Prop. 4.3] to eliminate $f_3$ and $f_7$. The forms $f_2$ and $f_6$ correspond to elliptic curves of conductor $450$ (named $C$ and $A$ respectively in Cremona’s tables), with $j$-invariants $u/3^v$, with $v>0$ and $u$ some non-zero integer prime to $3$; the forms $f_1$ and $f_5$ correspond to elliptic curves of conductor $450$ (named $F$ and $E$ respectively in Cremona’s tables), with $j$-invariants $u/5^v$, with $v>0$ and $u$ some non-zero integer prime to $5$; the form $f_4$ corresponds to elliptic curve $G$ of conductor $450$, with $j$-invariant $u/(3^v5^w)$, with $v,w>0$ and $u$ some non-zero integer prime to $15$. To eliminate all these elliptic curves we use [@BS Prop. 4.4]
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Professor Mike Bennett for providing us with some references. Andrzej Dbrowski would like to thank Professor Gökhan Soydan for inviting him to the [*Friendly Workshop on Diophantine Equations and Related Problems*]{} (Bursa Uludağ University, Bursa, July 6 - 8, 2019), where this work started. This workshop was partially supported by TÜBİTAK (the Scientific and Technological Research Council of Turkey) under Project No: 117F287.
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Andrzej Dbrowski, Institute of Mathematics, University of Szczecin, 70-451 Szczecin, Poland, E-mail: [email protected] and [email protected]
Nursena Günhan, Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey, E-mail: [email protected]
Gökhan Soydan, Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey, E-mail: [email protected]
| ArXiv |
---
abstract: 'We show that conservation laws in quantum mechanics naturally lead to metric spaces for the set of related physical quantities. All such metric spaces have an “onion-shell” geometry. We demonstrate the power of this approach by considering many-body systems immersed in a magnetic field, with a finite ground state current. In the associated metric spaces we find regions of allowed and forbidden distances, a “band structure” in metric space directly arising from the conservation of the $z$ component of the angular momentum.'
author:
- 'P. M. Sharp and I. D’Amico'
bibliography:
- 'References.bib'
title: Metric Space Formulation of Quantum Mechanical Conservation Laws
---
Introduction
============
Conservation laws are a central tenet of our understanding of the physical world. Their tight relationship to natural symmetries was demonstrated by Noether in 1918 [@Noether1918] and has since been a fundamental tool for developing theoretical physics. In this paper we demonstrate how these laws induce appropriate “natural” metrics on the related physical quantities. Conservation laws are central to the behavior of physical systems and we show how this relevant physics is translated into the metric analysis. We argue that this alternative picture provides a new powerful tool to study certain properties of many-body systems, which are often complex and hardly tractable when considered within the usual coordinate space-based analysis, while may become much simpler when analyzed within metric spaces. We exemplify this concept by considering functional relationships fundamental to current density functional theory (CDFT) [@Vignale1987; @Vignale1988].
We will first introduce a way to derive appropriate “natural” metrics from a system’s conservation laws. Second, as an example application of the approach, we will explicitly consider an important class of systems – systems with applied external magnetic fields. In contrast with those to which standard density functional theory (DFT) [@Dreizler1990] can be applied, systems subject to external magnetic fields are not simply characterized by their particle densities as even their ground states may display a finite current [@Vignale1987; @Vignale1988]. These systems are of great importance, e.g., due to the emerging quantum technologies of spintronics and quantum information where, for example, few electrons in nano- or microstructures immersed in magnetic fields are proposed as hardware units [@Takahashi2010; @daSilva2009; @Brandner2013; @Amaha2013; @Castellanos-Beltran2013].
To analyze systems immersed in a magnetic field, we will introduce a metric associated with the paramagnetic current density, which can be associated with the angular momentum components. We will show that, at least for systems which preserve the $z$ component of the angular momentum, the paramagnetic current density metric space displays an “onion-shell” geometry, directly descending from the related conservation law. In recent work [@D'Amico2011; @Artacho2011; @D'Amico2011b] appropriate metrics for characterizing wavefunctions and particle densities within quantum mechanics were introduced. It was shown that wavefunctions and their particle densities both form metric spaces with an “onion-shell” structure [@D'Amico2011]. We will show that, within the same general procedure used for the paramagnetic current, these metrics descend from the respective conservation laws. We will then focus on ground states and characterize them not only through the mapping between wavefunctions and particle densities, but importantly through mappings involving the paramagnetic current density. In fact, for systems with an applied magnetic field, ground state wavefunctions are characterized uniquely only by knowledge of both particle *and* paramagnetic current densities (and vice versa), as demonstrated within CDFT [@Vignale1987; @Vignale1988].
The rest of this paper is organized as follows: In Sec. \[metric\] we introduce our general approach to derive metric spaces from conservation laws. We demonstrate the application of this approach to wavefunctions, particle densities, and paramagnetic current densities in Sec. \[apply\]. We consider systems subject to magnetic fields in Sec. \[cdft\]. Here we use the metrics derived from our approach to study the fundamental theorem of CDFT. We present our conclusions in Sec. \[conclusion\].
Derivation of Metric Spaces from Conservation Laws {#metric}
==================================================
A metric or distance function $D$ over a set $X$ satisfies the following axioms for all $x,y,z \in X$ [@Megginson1998; @Sutherland2009]: $$\begin{aligned}
D(x,y) &\geqslant 0\ \text{and}\ D(x,y)=0 \iff x=y, \label{axiom1}\\
D(x,y) &= D(y,x), \label{axiom2}\\
D(x,y) &\leqslant D(x,z)+D(z,y), \label{axiom3}\end{aligned}$$ with (\[axiom3\]) known as the triangle inequality. The set $X$ with the metric $D$ forms the metric space $(X,D)$. It can be seen from the axioms (\[axiom1\]) - (\[axiom3\]) that many metrics could be devised for the same set, some trivial. Here we introduce “natural” metrics associated to conservation laws: this will avoid arbitrariness and in turn will ensure that the proposed metrics stem from core characteristics of the systems analyzed and contain the related physics.
In quantum mechanics, many conservation laws take the form $$\label{conservation}
\int {\left|f(x)\right|}^{p} dx = c$$ for $0<c<\infty$. For each value of $1\leqslant p<\infty$, the entire set of functions that satisfy (\[conservation\]) belong to the $L^p$ vector space, where the standard norm is the $p$ norm [@Megginson1998] $$\label{lp_norm}
{\left|\left|f(x)\right|\right|}_p =\left[\int {\left|f(x)\right|}^{p} dx \right]^{\frac{1}{p}}.$$ From any norm a metric can be introduced in a standard way as $D(x,y)={\left|\left|x-y\right|\right|}$ so that with $p$ norms we get $$\label{lp_metric}
D_{f}(f_1,f_2):={\left|\left|f_1-f_2\right|\right|}_p.$$ However before assuming this metric for the physical functions related to the conservation laws, an important consideration must be made: Eq. (\[lp\_metric\]) has been derived assuming the ensemble $\{f\}$ to be a vector space; this is in fact necessary to introduce a norm. If we want to retain the metric (\[lp\_metric\]), but restrict it to the ensemble of *physical* functions satisfying (\[conservation\]), which does not necessarily form a vector space, we must show that (\[lp\_metric\]) is a metric for this restricted function set. This can be done using the general theory of metric spaces: given a metric space $(X,D)$ and $S$ a non empty subset of $X$, $(S,D)$ is itself a metric space with the metric $D$ inherited from $(X,D)$. The metric axioms (\[axiom1\]) - (\[axiom3\]) automatically hold for $(S,D)$ because they hold for $(X,D)$ [@Megginson1998; @Sutherland2009]. Hence, we have a metric for the functions of interest, as their sets are non empty subsets of the respective $L^p$ sets.
The metric (\[lp\_metric\]) is then the one that *directly descends* from the conservation law (\[conservation\]). Conversely any conservation law which can be recast as (\[conservation\]) (for example conservation of quantum numbers) can be interpreted as inducing a metric on the appropriate, physically relevant, subset of $L^{p}$ functions. This provides a general procedure to derive “natural” metrics from physical conservation laws.
Applications of the Metric Space Approach {#apply}
=========================================
We now consider specific quantum mechanical functions and conservation laws. Following Ref. [@D'Amico2011] we use a convention where wavefunctions are normalized to the particle number $N$ [^1]. Then the particle density of an $N$-particle system and its paramagnetic current density are defined as $$\begin{aligned}
\rho({\mathbf{r}})&=\int {\left|\psi\left({\mathbf{r}},{\mathbf{r}}_{2},\ldots,{\mathbf{r}}_{N}\right)\right|}^{2} d{\mathbf{r}}_{2}\ldots d{\mathbf{r}}_{N},\label{density}\\
{\mathbf{j}}_{p}({\mathbf{r}})&=-\frac{i}{2}\int \left(\psi^{\ast}\nabla\psi - \psi\nabla\psi^{\ast}\right) d{\mathbf{r}}_{2}\ldots d{\mathbf{r}}_{N}.\label{current}\end{aligned}$$ First of all we note that $\psi\left({\mathbf{r}}_1,{\mathbf{r}}_{2},\ldots ,{\mathbf{r}}_{N}\right)$ and $\rho({\mathbf{r}})$ are subject to the following conservation laws (wavefunction norm and particle conservation): $$\begin{aligned}
&\int{\left|\frac{\psi\left({\mathbf{r}}_1,{\mathbf{r}}_{2},\ldots ,{\mathbf{r}}_{N}\right)}{\sqrt{N}}\right|}^{2}d{\mathbf{r}}_{1}\ldots d{\mathbf{r}}_{N} = 1,\label{psi_cons}\\
&\int\rho({\mathbf{r}}) d{\mathbf{r}} = N.\label{rho_cons}\end{aligned}$$ Similarly the paramagnetic current density ${\mathbf{j}}_{p}({\mathbf{r}})$ obeys $$\label{Lz}
\int \left[{\mathbf{r}}\times{\mathbf{j}}_{p}({\mathbf{r}})\right]_z d{\mathbf{r}} = \langle\psi|\hat{L}_z|\psi\rangle.$$ For eigenstates of systems for which the $z$ component of the angular momentum is preserved we then have $\langle\hat{L}_z\rangle=m$, with $m$ an integer, and (\[Lz\]) can be recast as $$\label{j_p_cons}
\int {\left|\left[{\mathbf{r}}\times{\mathbf{j}}_{p}({\mathbf{r}})\right]_z\right|} d{\mathbf{r}} = {\left|m\right|}.$$ For wavefunctions and particle densities our procedure leads to the metrics introduced in Ref. [@D'Amico2011] ($N$ fixed) [@Artacho2011; @D'Amico2011b] $$\begin{aligned}
D_{\psi}(\psi_{1},\psi_{2})=&\left[\int \left({\left|\psi_{1}\right|}^{2}+{\left|\psi_{2}\right|}^{2}\right)d{\mathbf{r}}_1\ldots d{\mathbf{r}}_{N}\right.\nonumber\\
&- \left. 2{\left|\int\psi_{1}^{*}\psi_{2}d{\mathbf{r}}_1\ldots d{\mathbf{r}}_{N}\right|}\right]^{\frac{1}{2}},\label{dpsi}\\
D_{\rho}(\rho_{1},\rho_{2})=&\int{\left|\rho_{1}({\mathbf{r}})-\rho_{2}({\mathbf{r}})\right|} d{\mathbf{r}}; \label{drho}\end{aligned}$$ for the paramagnetic current density, our procedure introduces the following metric: $$\label{dj_p}
D_{{\mathbf{j}}_{p_{\perp}}}({\mathbf{j}}_{p,1},{\mathbf{j}}_{p,2})=\int{\left|\left\{{\mathbf{r}}\times\left[{\mathbf{j}}_{p,1}({\mathbf{r}})-{\mathbf{j}}_{p,2}({\mathbf{r}})\right]\right\}_{z}\right|} d{\mathbf{r}}.$$ We note that $D_{{\mathbf{j}}_{p_{\perp}}}$ will be a distance between equivalence classes of paramagnetic currents, each class characterized by current densities having the same transverse component ${\mathbf{j}}_{p_{\perp}}\equiv(j_{p,x},j_{p,y})$. $D_{{\mathbf{j}}_{p_{\perp}}}$ is gauge invariant provided that ${\mathbf{j}}_{p,1}$ and ${\mathbf{j}}_{p,2}$ are within the same gauge and $[\hat{L_{z}},\hat{H}]=0$.
Next we show that conservation laws naturally build within the related metric spaces a hierarchy of concentric spheres, or “onion-shell” geometry. If we set as the center of each sphere the zero function $f^{(0)}(x)\equiv 0$, and consider the distance between it and any other element in the metric space, we recover the $p$-norm expressions (\[lp\_norm\]) directly descending from the related conservation laws. This procedure induces in the related metric spaces a structure of concentric spheres with radii, in the cases considered here, of natural numbers to the power of $1/p$: all functions corresponding to the same value of a certain conserved quantity will lay on the surface of the same sphere. Specifically, for systems of $N$ particles, wavefunctions lie on spheres of radius $\sqrt{N}$, and particle densities on spheres of radius $N$; for the metric space of paramagnetic current densities, all paramagnetic current densities with a $z$ component of the angular momentum equal to $\pm m$ lie on spheres of radius ${\left|m\right|}$.
The first axiom of a metric (\[axiom1\]) guarantees that the minimum value for all distances is $0$, and that this value is attained for two identical states. The onion-shell geometry guarantees that, for functions on the surface of the same sphere, i.e., which satisfy a certain conservation law with the same value, there is also an upper limit for their distance associated with the diameter of the sphere. From (\[dj\_p\]) we see that for paramagnetic current densities this upper limit is achieved in the limit of currents which do not spatially overlap. This is also the case for particle densities, as seen in (\[drho\]).
![(Color online) For the ISI system energy is plotted against the confinement frequency for several values of the angular momentum quantum number $m$ (as labeled), and with constant cyclotron frequency and interaction strength. Arrows indicate where the value of $m$ for the ground state changes.[]{data-label="energy"}](energy.pdf){width="\columnwidth"}
Interestingly, and in contrast to wavefunctions and particle densities [@D'Amico2011], even when considering systems with the same number of particles it may be necessary to consider paramagnetic current densities with different values of $m$; in terms of their metric space geometry, current densities that have different values of ${\left|m\right|}$ lie on different spheres. Therefore, the maximum value for the distance between paramagnetic current densities of a system of $N$ particles is related to the upper limit of the number of spheres in the onion-shell geometry. Using the triangle inequality we have in fact $$\begin{aligned}
D_{{\mathbf{j}}_{p_{\perp}}}({\mathbf{j}}_{p,m_{1}},{\mathbf{j}}_{p,m_{2}})&\leqslant D_{{\mathbf{j}}_{p_{\perp}}}({\mathbf{j}}_{p,m_{1}},{\mathbf{j}}^{(0)}_{p})+D_{{\mathbf{j}}_{p_{\perp}}}({\mathbf{j}}^{(0)}_{p},{\mathbf{j}}_{p,m_{2}})\nonumber\\
&={\left|m_1\right|}+{\left|m_2\right|}\leqslant l_{1}+l_{2},\end{aligned}$$ where $l_i$ is the quantum number related to the total angular momentum of system $i$.
Study of Model Systems {#cdft}
======================
We now concentrate on the sets of ground state wavefunctions, related particle densities, and related paramagnetic current densities. Since ground states are non empty subsets of all states, ground-state-related functions form metric spaces with the metrics (\[dpsi\]), (\[drho\]), and (\[dj\_p\]). The importance of characterizing ground states and their properties has been highlighted by the huge success of DFT (in all its flavors) as a method to predict devices’ and material properties [@Dreizler1990; @Ullrich2013]. Standard DFT is built on the Hohenberg-Kohn (DFT-HK) theorem [@HK1964], which demonstrates a one-to-one mapping between ground state wavefunctions and their particle densities. This theorem is highly complex and nonlinear in coordinate space. However, Ref. [@D'Amico2011] showed that the DFT-HK theorem is a mapping between metric spaces, and may be very simple when described in these terms, becoming monotonic and almost linear for a wide range of parameters and for the systems there analyzed. CDFT is a formulation of DFT for systems in the presence of an external magnetic field. In CDFT [@Vignale1987; @Vignale1988] the original HK mapping is extended (CDFT-HK theorem) to demonstrate that $\psi$ is uniquely determined only by knowledge of both $\rho({\mathbf{r}})$ and ${\mathbf{j}}_p({\mathbf{r}})$ (and vice versa). This is the theorem we will consider in this section.
To further our analysis, we now explicitly examine two model systems with applied magnetic fields. They both consist of two electrons parabolically confined that interact via different potentials, Coulomb (magnetic Hooke’s atom) [@Taut2009] and inverse square interaction (ISI) [@Quiroga1993], respectively. Both systems may be used to model electrons confined in quantum dots. The Hamiltonians for the magnetic Hooke’s atom and the ISI system are $$\begin{aligned}
\hat{H}_{HA}&=\sum_{i=1}^{2}\left\{\frac{1}{2}\left[\hat{{\mathbf{p}}}_{i}+{\mathbf{A}}\left({\mathbf{r}}_{i}\right)\right]^{2}+\frac{1}{2}\omega_{0}^{2}r_{i}^{2}\right\}+\frac{1}{{\left|{\mathbf{r}}_{2}-{\mathbf{r}}_{1}\right|}}, \label{Hooke_H}\\
\hat{H}_{ISI}&=\sum_{i=1}^{2}\left\{\frac{1}{2}\left[\hat{{\mathbf{p}}}_{i}+{\mathbf{A}}\left({\mathbf{r}}_{i}\right)\right]^{2}+\frac{1}{2}\omega_{0}^{2}r_{i}^{2}\right\}+\frac{\alpha}{\left({\mathbf{r}}_{1}-{\mathbf{r}}_{2}\right)^{2}}, \label{ISI_H}\end{aligned}$$ (atomic units, $\hbar=m_e=e=1$). Here $\alpha$ is a positive constant, ${\mathbf{A}}=\frac{1}{2}{\mathbf{B}}\times{\mathbf{r}}$ (symmetric gauge), and ${\mathbf{B}}=\omega_{c}c{\mathbf{\hat{z}}}$ is a homogeneous, time-independent external magnetic field. For these systems $\langle\hat{L}_z\rangle$ is a conserved quantity. Following Refs. [@Vignale1987; @Taut2009] we disregard spin to concentrate on the features of the orbital currents. For Hooke’s atom, we obtain highly precise numerical solutions following the method in Ref. [@Coe2008]. The ISI system is solved exactly [@Quiroga1993].
{width="\textwidth"}
To produce families of ground states, for each system we systematically vary the value of $\omega_0$ (while keeping all other parameters constant), and for each value we calculate the ground state wavefunction, particle density, and paramagnetic current density. A reference state is determined by choosing a specific $\omega_0$ value, and the appropriate metric is then used to calculate the distances between it and each member of the family. To ensure that we select ground states, varying $\omega_0$ may require varying the quantum number $m$ [@Taut2009; @Quiroga1993]. This is shown for the ISI system in Fig. \[energy\]. Here, as $\omega_{0}$ increases, we must decrease the value of ${\left|m\right|}$ in order to remain in the ground state. As a result of this property, within each family of ground states, paramagnetic current densities will “jump” from one sphere of the onion-shell geometry to another \[see Fig. \[spheres\](a), where the reference state is the ‘north pole’ of its sphere\]. To obtain ground states with nonzero paramagnetic currents, we must use $\omega_0$ values corresponding to $m<0$ [@Taut2009; @Quiroga1993].
![(Color online) (a) Sketch of the onion-shell geometry of the metric space for paramagnetic current densities, where ${\left|m_q\right|}>{\left|m_r\right|}>{\left|m_{ref}\right|}$ (left) and ${\left|m_{ref}\right|}>{\left|m_s\right|}>{\left|m_t\right|}$ (right). The reference state is at the north pole on the reference sphere. The dark gray areas denote the regions where ground state currents are located (‘bands’), with dashed lines indicating their widths. (b) Results of the angular displacement of ground state currents for the ISI system. Lines are a guide to the eye. Inset: Definition of relevant angles.[]{data-label="spheres"}](sphere_fig.pdf){width="\columnwidth"}
In Fig. \[results\], we plot each pair of distances for the two systems. The reference states have been chosen so that most of the available distance range can be explored both for the case of increasing and for the case of decreasing values of $\omega_0$. When considering the relationship between ground state wavefunctions and related particle densities, Figs. \[results\](a) and \[results\](b), our results confirm the findings in Ref. [@D'Amico2011]: a monotonic mapping, linear for low to intermediate distances, and where vicinities are mapped onto vicinities; also curves for increasing and decreasing $\omega_0$ collapse onto each other. However closer inspection reveals a fundamental difference with Ref. [@D'Amico2011], the presence of a “band structure.” By this we mean regions of allowed (“bands”) and forbidden (“gaps”) distances, whose widths depend, for the systems considered here, on the value of ${\left|m\right|}$. This structure is due to the changes in the value of the quantum number $m$, which result in a substantial modification of the ground state wavefunction (and therefore density) and a subsequent large increase in the related distances.
When we focus on the plots of paramagnetic current densities’ against wavefunctions’ distances, Figs. \[results\](c) and \[results\](d), we find that the “band structure” dominates the behavior. Here the change in ${\left|m\right|}$ has an even stronger effect, in that $dD_{{\mathbf{j}}_{p_{\perp}}}/dD_{\psi}$ is noticeably discontinuous when moving from one sphere to the next in ${\mathbf{j}}_p$ metric space. This discontinuity is more pronounced for the path ${\left|m\right|}<{\left|m_{ref}\right|}$ than for the path ${\left|m\right|}>{\left|m_{ref}\right|}$. Similarly to Figs. \[results\](a) and \[results\](b), the mapping of $D_{\psi}$ onto $D_{{\mathbf{j}}_{p_{\perp}}}$ maps vicinities onto vicinities and remains monotonic, but for small and intermediate distances it is only piecewise linear. In contrast with $D_{\rho}$ vs $D_{\psi}$, curves corresponding to increasing and decreasing $\omega_0$ do not collapse onto each other.
Figures \[results\](e) and \[results\](f) show the mapping between particle and paramagnetic current density distances: this has characteristics similar to the one between $D_{\psi}$ and $D_{{\mathbf{j}}_{p_{\perp}}}$, but remains piecewise linear even at large distances.
We will now concentrate on the ${\mathbf{j}}_{p}$ metric space to characterize the “band structure” observed in Fig. \[results\]. Within the metric space geometry, we consider the polar angle $\theta$ between the reference ${\mathbf{j}}_{p,ref}$ and the paramagnetic current density ${\mathbf{j}}_{p}$ of angular momentum ${\left|m\right|}$. Using the law of cosines, $\theta$ is given by $$\label{angle}
\cos{\theta}=\frac{m_{ref}^2+m^2-D_{{\mathbf{j}}_{p_{\perp}}}^2({\mathbf{j}}_{p,ref},{\mathbf{j}}_{p}) }{2{\left|m_{ref}\right|}{\left|m\right|}}.$$ We define the polar angles corresponding to the two extremes of a given band as $\theta_{min}$ and $\theta_{max}$ (inset of Fig. \[spheres\]). The width of each band is then ${\Delta}{\theta}={\theta}_{max}-{\theta}_{min}$, and its position defined by $\theta_{min}$. Now we can calculate the bands’ widths and positions by sweeping, for each ${\left|m\right|}$, the values of $\omega_0$ corresponding to ground states (Fig. \[spheres\]).
For both systems under study, we find that as ${\left|m\right|}$ increases from ${\left|m_{ref}\right|}$, both ${\theta}_{max}$ and ${\theta}_{min}$ increase. This has the effect of the bands moving from the north pole to the south pole as we move away from the reference. Additionally, we find that the bandwidth ${\Delta}{\theta}$ decreases as ${\left|m\right|}$ increases \[sketched in Fig. \[spheres\](a), left\]. As ${\left|m\right|}$ decreases from ${\left|m_{ref}\right|}$, we again find that both ${\theta}_{max}$ and ${\theta}_{min}$ increase, with the bands moving from the north pole to the south pole. However, this time, as ${\left|m\right|}$ decreases, ${\Delta}{\theta}$ increases, meaning that the bands get wider as we move away from the reference \[sketched in Fig. \[spheres\](a), right\].
Quantitative results for the ISI system are shown in Fig. \[spheres\](b). We obtain similar results for Hooke’s atom (not shown). The band on the surface of each sphere indicates where all ground state paramagnetic current densities lie within that sphere. In contrast with particle densities or wavefunctions, we find that, at least for the systems at hand, ground state currents populate a well-defined, limited region of each sphere, whose size and position display monotonic behavior with respect to the quantum number $m$. This regular behavior is not at all expected, as the CDFT-HK theorem does not guarantee monotonicity in metric space, and not even that the mapping of $D_{\psi}$ to $D_{{\mathbf{j}}_{p_{\perp}}}$ is single valued. In the CDFT-HK theorem ground state wavefunctions are uniquely determined only by particle and paramagnetic current densities *together*. In this sense we can look at the panels in Fig. \[results\] as projections on the axis planes of a 3-dimensional $D_{\psi}D_{\rho}D_{{\mathbf{j}}_{p_{\perp}}}$ relation. The complexity of the mapping due to the application of a magnetic field – the changes in quantum number $m$ – is fully captured by $D_{{\mathbf{j}}_{p_{\perp}}}$ only, as this is related to the relevant conservation law. However the mapping from $D_{\rho}$ to $D_{\psi}$ inherits the “band structure,” showing that the two mappings $D_{{\mathbf{j}}_{p_{\perp}}}$ to $D_{\psi}$ and $D_{\rho}$ to $D_{\psi}$ are not independent.
Conclusion
==========
In conclusion we showed that conservation laws induce related metric spaces with an “onion-shell” geometry and that they may induce a “band structure” in ground state metric spaces, a signature of the enhanced constraints due to the system conservation laws on the relation between wavefunctions and the relevant physical quantities.
The method proposed may help with understanding extended HK theorems, such as, in the case at hand, the CDFT-HK theorem. In this respect we find that in metric spaces and for the systems considered, the relevant mappings display distinctive signatures, including (piecewise) linearity at short and medium distances, the mapping between ground state $\psi$ and ${\mathbf{j}}_{p}$ resembling the one between $\rho$ and ${\mathbf{j}}_{p}$, and the mapping between ground state $\psi$ and ${\mathbf{j}}_{p}$ showing different trajectories for increasing or decreasing Hamiltonian parameters, in contrast with the mapping between $\psi$ and $\rho$. Features like this could be used to build or test (single-particle) approximate solutions to many-body problems, e.g., within DFT schemes.
Our results show that using conservation laws to derive metrics makes these metrics a powerful tool to study many-body systems governed by integral conservation laws.
We thank M. Taut, K. Capelle, and C. Verdozzi for helpful discussions. P. M. S. acknowledges EPSRC for financial support. I. D. and P. M. S. gratefully acknowledge support from a University of York - FAPESP combined grant.
[^1]: This allows the description of Fock space as a set of concentric spheres
| ArXiv |
---
author:
- |
Jon A. Bailey, Sunkyu Lee,\
Lattice Gauge Theory Research Center, CTP, and FPRD,\
Department of Physics and Astronomy,\
Seoul National University, Seoul 08826, South Korea\
E-mail:
- |
Yong-Chull Jang\
Physics Department, Brookhaven National Laboratory, Upton, NY11973, USA
- |
Jaehoon Leem\
School of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, South Korea
- |
Sungwoo Park\
Los Alamos National Laboratory, Theoretical Division T-2, Los Alamos, NM87545, USA
- SWME Collaboration
bibliography:
- 'refs.bib'
title: '2018 Update on $\epsK$ with lattice QCD inputs'
---
Introduction
============
This paper is a brief summary of our previous paper [@Bailey:2018feb]. This paper is also an update of our previous papers [@Jang:2017ieg; @Bailey:2015tba; @Bailey:2015frw].
Input parameters: $\Vcb$ and $\xi_0$ {#sec:Vcb}
====================================
In Table \[tab:Vcb\], we present updated results for both exclusive $\Vcb$ and inclusive $\Vcb$. Recently, HFLAV reported them in Ref. [@Amhis:2016xyh]. The results for exclusive $\Vcb$ are obtained using lattice QCD results for the semileptonic form factors of Refs. [@Bailey2014:PhysRevD.89.114504; @Lattice:2015rga; @Detmold:2015aaa]. Here, we use the combined results (ex-combined) for exclusive $\Vcb$ and the results of the $1S$ scheme for inclusive $\Vcb$ to evaluate $\epsK$. For more details on $\Vcb$ and the related caveats, refer to Ref. [@Bailey:2018feb].
The absorptive part of long distance effects on $\epsK$ is parametrized into $\xi_0$. $$\begin{aligned}
\xi_0 &= \frac{\Im A_0}{\Re A_0}, \qquad
\xi_2 = \frac{\Im A_2}{\Re A_2}, \qquad
\Re \left(\frac{\eps'}{\eps} \right) =
\frac{\omega}{\sqrt{2} |\eps_K|} (\xi_2 - \xi_0) \,.
\label{eq:e'/e:xi0}\end{aligned}$$ There are two independent methods to determine $\xi_0$ in lattice QCD: one is the indirect method and the other is the direct method. In the indirect method, one can determine $\xi_0$ using Eq. with lattice QCD input $\xi_2$ and with experimental results for $\eps'/\eps$, $\epsK$, and $\omega$. In the direct method, one can determine $\xi_0$ directly using lattice QCD results for $\Im A_0$ combined with experimental results for $\Re A_0$. In Table \[tab:xi0+d0\](), we summarize results for $\xi_0$ calculated by RBC-UKQCD using the indirect and direct methods. Here, we use the results of the indirect method for $\xi_0$ to evaluate $\epsK$.
In Ref. [@Bai:2015nea], RBC-UKQCD also reported the S-wave scattering phase shift for the $I=0$ channel: $\delta_0 =
23.8(49)(12)$, which is different from those of the dispersion relations [@Colangelo:2001df; @GarciaMartin:2011cn] by $\approx 3
\sigma$. In Ref. [@Wang:2018Latt], they have accumulated higher statistics to obtain $\delta_0 = 19.1(25)(12)$, which is about $5\sigma$ different from those of the dispersion analyses. They introduce a $\sigma$ operator and make all possible combinations with the $\sigma$ and $\pi-\pi$ operators. Then, RBC-UKQCD has obtained $\delta_0 = 32.8(12)(30)$ which is consistent with those of the dispersion relations. These results are presented in Table \[tab:xi0+d0\]() and Figure \[tab:xi0+d0\]().
Input parameters: Wolfenstein parameters, $\BK$, $\xi_\text{LD}$, and others
============================================================================
In Table \[tab:input-WP-eta\](), we summarize the Wolfenstein parameters on the market. The CKMfitter and UTfit collaboration provide the Wolfenstein parameters determined by the global unitarity triangle (UT) fit. Unfortunately, $\epsK$, $\BK$, and $\Vcb$ are used as inputs to the global UT fit, which leads to unwanted correlation with $\epsK$. We want to avoid this correlation, and so take another input set from the angle-only fit (AOF) suggested in Ref. [@Bevan2013:npps241.89]. The AOF does not use $\epsK$, $\BK$, and $\Vcb$ as input to determine the UT apex $(\bar{\rho}, \bar{\eta})$. Here the $\lambda$ parameter is determined from $\Vus$ which is obtained from the $K_{\ell 2}$ and $K_{\ell 3}$ decays using lattice QCD results for the form factors and decay constants. The $A$ parameter is determined from $\Vcb$.
In the FLAG review [@Aoki:2016frl], they present lattice QCD results for $\BK$ with $N_f=2$, $N_f=2+1$, and $N_f= 2+1+1$. Here, we use the results for $\BK$ with $N_f=2+1$, which is obtained by taking a global average over the four data points from BMW 11 [@Durr:2011ap], Laiho 11 [@Laiho:2011np], RBC-UKQCD 14 [@Blum:2014tka], and SWME 15 [@Jang:2015sla]. In Table \[tab:input-BK-other\](), we present the FLAG 17 result for $\BK$ with $N_f = 2+1$, which is used to evaluate $\epsK$.
The dispersive long distance (LD) effect is defined as $$\begin{aligned}
\xi_\text{LD} &= \frac{m^\prime_\text{LD}}{\sqrt{2} \Delta M_K} \,,
\qquad
m^\prime_\text{LD}
= -\Im \left[ \mathcal{P}\sum_{C}
\frac{\mate{\wbar{K}^0}{H_\text{w}}{C} \mate{C}{H_\text{w}}{K^0}}
{m_{K^0}-E_{C}} \right]
\label{eq:xi-LD}\end{aligned}$$ If the CPT invariance is well respected, the overall contribution of the $\xi_\text{LD}$ to $\epsK$ is about $\pm 2\%$.
Lattice QCD tools to calculate $\xi_\text{LD}$ are well established in Refs. [@Christ2012:PhysRevD.88.014508; @Bai:2014cva; @Christ:2015pwa]. In addition, there have been a number of attempts to calculate $\xi_\text{LD}$ on the lattice [@Christ:2015phf; @Bai:2016gzv]. In them, RBC-UKQCD used a pion mass of 329 MeV and a kaon mass of 591 MeV, and so the energy of the 2 pion and 3 pion states are heavier than the kaon mass. Hence, the sign of the denominator in Eq. \[eq:xi-LD\] is opposite to that of the physical contribution. Therefore, this work belongs to the category of exploratory study rather than to that of precision measurement.
In Ref. [@Buras2010:PhysLettB.688.309], they use chiral perturbation theory to estimate the size of $\xi_\text{LD}$ and claim that $$\begin{aligned}
\xi_\text{LD} &= -0.4(3) \times \frac{\xi_0}{ \sqrt{2} }
\label{eq:xiLD:bgi}\end{aligned}$$ where we use the indirect results for $\xi_0$ and its error. Here, we call this method the BGI estimate for $\xi_\text{LD}$. In Refs. [@Christ2012:PhysRevD.88.014508; @Christ:2014qwa], RBC-UKQCD provides another estimate for $\xi_\text{LD}$: $$\begin{aligned}
\xi_\text{LD} &= (0 \pm 1.6)\%.
\label{eq:xiLD:rbc}\end{aligned}$$ Here, we call this method the RBC-UKQCD estimate for $\xi_\text{LD}$.
In Table \[tab:input-WP-eta\](), we present higher order QCD corrections: $\eta_{ij}$ with $i,j = t,c$. In Table \[tab:input-BK-other\](), we present other input parameters needed to evaluate $\epsK$. Since Lattice 2017, three parameters: $m_t(m_t)$, $m_{K^{0}}$, $F_K$ have been updated. The $m_t(m_t)$ parameter is the scale-invariant (SI) top quark mass renormalized in the $\MSb$ scheme. The pole mass of top quarks comes from Ref. [@Patrignani:2016xqp]: $ M_t = 173.5 \pm 1.1 \GeV$. We convert the top quark pole mass into the SI top quark mass using the four-loop perturbation formula. For more details, refer to Ref. [@Bailey:2018feb].
Results for $\epsK$
===================
In Fig. \[fig:epsK:cmp:rbc\], we present results for $|\epsK|$ evaluated directly from the standard model (SM) with lattice QCD inputs given in the previous sections. In Fig. \[fig:epsK:cmp:rbc\](), the blue curve represents the theoretical evaluation of $|\epsK|$ using the FLAG-2017 $\BK$, AOF for Wolfenstein parameters, and exclusive $\Vcb$, and the RBC-UKQCD estimate for $\xi_\text{LD}$. The red curve in Fig. \[fig:epsK:cmp:rbc\] represents the experimental value of $|\epsK|$. In Fig. \[fig:epsK:cmp:rbc\](), the blue curve represents the same as in Fig. \[fig:epsK:cmp:rbc\]() except for using the inclusive $\Vcb$.
Our results for $|\epsK|$ are summarized in Table \[tab:epsK\]. Here, the superscript ${}^\text{SM}$ means that it is obtained directly from the standard model, the subscript ${}_\text{excl}$ (${}_\text{incl}$) means that it is obtained using exclusive (inclusive) $\Vcb$, and the superscript ${}^\text{Exp}$ represents the experimental value. Results in Table \[tab:epsK\]() are obtained using the RBC-UKQCD estimate for $\xi_\text{LD}$ and those in Table \[tab:epsK\]() are obtained using the BGI estimate for $\xi_\text{LD}$. In Table \[tab:epsK\](), we find that the theoretical evaluation of $|\epsK|$ with lattice QCD inputs (with exclusive $\Vcb$) $|\epsK|^\text{SM}_\text{excl}$ has $4.2\sigma$ tension with the experimental result $|\epsK|^\text{Exp}$, while there is no tension with inclusive $\Vcb$ (heavy quark expansion with QCD sum rules).
In Fig. \[fig:depsK:sum:rbc:his\](), we plot the $\Delta \epsK \equiv |\epsK|^\text{Exp} -
|\epsK|^\text{SM}_\text{excl}$ in units of $\sigma$ (the total error) as a function of time starting from 2012. In 2012, $\Delta \epsK$ was $2.5\sigma$, but now it is $4.2\sigma$. In Fig. \[fig:depsK:sum:rbc:his\](), we plot the history of the average $\Delta \epsK$ and the error $\sigma_{\Delta \epsK}$. We find that the average has increased with some fluctuations by 27% during the period of 2012-2018, and its error has decreased monotonically by 25% in the same period.
In Table \[tab:err-bud+his-DepsK\](), we present the error budget for $|\epsK|^\text{SM}_\text{excl}$. Here, we find that the largest error comes from $\Vcb$. Hence, it is essential to reduce the error in $\Vcb$ significantly.
In Table \[tab:err-bud+his-DepsK\](), we present how the values of $\Delta\epsK$ have changed from 2015 [@Bailey:2015tba] to 2018 [@Bailey:2018feb]. Here, we find that the positive shift of $\Delta \epsK$ is about the same for the inclusive and exclusive $\Vcb$. This reflects the changes in other parameters since 2015.
We thank Shoji Hashimoto and Takashi Kaneko for helpful discussion on $\Vcb$. The research of W. Lee is supported by the Creative Research Initiatives Program (No. 2017013332) of the NRF grant funded by the Korean government (MEST). J.A.B. is supported by the Basic Science Research Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2015024974). W. Lee would like to acknowledge the support from the KISTI supercomputing center through the strategic support program for the supercomputing application research (No. KSC-2016-C3-0072). Computations were carried out on the DAVID GPU clusters at Seoul National University.
| ArXiv |
---
abstract: |
In the past years, analyzers have been introduced to detect classes of non-terminating queries for definite logic programs. Although these non-termination analyzers have shown to be rather precise, their applicability on real-life Prolog programs is limited because most Prolog programs use non-logical features. As a first step towards the analysis of Prolog programs, this paper presents a non-termination condition for Logic Programs containing integer arithmetics. The analyzer is based on our non-termination analyzer presented at ICLP 2009. The analysis starts from a class of queries and infers a subclass of non-terminating ones. In a first phase, we ignore the outcome (success or failure) of the arithmetic operations, assuming success of all arithmetic calls. In a second phase, we characterize successful arithmetic calls as a constraint problem, the solution of which determines the non-terminating queries.
Keywords: non-termination analysis, numerical computation, constraint-based approach
author:
- |
Dean Voets[^1] $~~~~~$ Danny De Schreye\
Department of Computer Science, K.U.Leuven, Belgium\
Celestijnenlaan 200A, 3001 Heverlee\
{Dean.Voets, Danny.DeSchreye }@cs.kuleuven.be
bibliography:
- 'prolog.bib'
title: 'Non-termination Analysis of Logic Programs with integer arithmetics'
---
**Note:** This article has been published in *Theory and Practice of Logic Programming, volume 11, issue 4-5, pages 521-536, 2011*.
Introduction
============
The problem of proving termination has been studied extensively in Logic Programming. Since the early works on termination analysis in Logic Programming, see e.g. [@DBLP:journals/jlp/SchreyeD94], there has been a continued interest from the community for the topic. Lots of in-language and transformational tools have been developed, e.g. [@Giesl06aprove1.2] and [@DBLP:journals/corr/abs-0912-4360], and since 2004, there is an annual Termination Competition[^2] to compare the current analyzers on the basis of an extensive database of logic programs. In contrast with termination analysis, the dual problem, to detect non-terminating classes of queries, is a fairly new topic. The development of the first and most well-known non-termination analyzer, $NTI$ [@nti_06], was motivated by difficulties in obtaining precision results for termination analyzers. Since the halting problem is undecidable, one way of demonstrating the precision of a termination analyzer is with a non-termination analyzer. For $NTI$ it was already shown that for many examples one can partition queries in terminating and non-terminating. $NTI$ compares the consecutive calls in the program using binary unfoldings and proves non-termination by comparing the head and body of these binary clauses with a special more general relation.
Recently, in joined work with Yi-Dong Shen, we integrated loop checking into termination analysis, yielding a very accurate technique to predict the termination behavior for classes of queries described using modes [@term_prediction]. Classes of queries are represented as *moded queries*. A moded query consists of a query and a label, input or output, for each variable in the query. These moded queries are then evaluated with a *moded SLD-tree* obtained by applying clauses to the partially instantiated query and propagating the labels. To guarantee a finite analysis, this moded SLD-tree is constructed using a complete loop check. After evaluating the moded query, the analysis predicts the termination behavior of the program for the considered queries based on the labels and substitutions in the moded SLD-tree.
Motivated by the elegance of this approach and the accuracy of the predictions, our research focused on defining a non-termination condition based on these moded queries. In [@DBLP:conf/iclp/VoetsS09], we introduced a non-termination condition identifying paths in a moded SLD-tree that can be repeated infinitely often. This approach was implemented in a system called $P2P$, which proved more accurate than $NTI$ on the benchmark of the termination competition. An evaluation of the classes of queries not handled by current approaches lead to considerable improvements in our non-termination analysis. These improvements were presented in [@VDS10] and implemented in the analyzer $pTNT$. Both termination and non-termination analyzers have been rather successful in analyzing the termination behavior of definite logic programs, but only a few termination analyzers, e.g. [@DBLP:conf/lpar/SerebrenikS01a], and none of the non-termination analyzers handle non-logical features such as arithmetics or cuts, typically used in practical Prolog programs. In this paper, we introduce a technique for proving non-termination of logic programs containing a subset of the built-in predicates for integer arithmetic, commonly found in Prolog implementations.
Given a program, containing integer arithmetics, and a class of queries, described using modes, we infer a subset of these queries for which we prove existential non-termination (i.e. the derivation tree for these queries contains an infinite path). The inference and proof are done in two phases. In the first phase, non-termination of the logic part of the program is proven by assuming that all comparisons between integer expressions succeed. We will show that only a minor adaption of our technique presented in [@DBLP:conf/iclp/VoetsS09] is needed to achieve this. In the second phase, given the moded query, integer arguments are identified and constraints over these arguments are formulated, such that solutions for these constraints correspond to non-terminating queries.
The paper is structured as follows. In the next section, we introduce some preliminaries concerning logic programs, integer arithmetics and we present the symbolic derivation trees used to abstract the computation. In Section 3, we introduce our non-termination condition for programs containing integer arithmetics. In Section 4, we describe our prototype analyzer and some results. Finally, we conclude in Section 5.
Preliminaries
=============
Logic Programming
-----------------
We assume the reader is familiar with standard terminology of logic programs, in particular with SLD-resolution as described in [@Lloyd_foundations]. Variables are denoted by strings beginning with a capital letter. Predicates, functions and constant symbols are denoted by strings beginning with a lower case letter. We denote the set of terms constructible from a program $P$ by $Term_P$. Two atoms are called *variants* if they are equal up to variable renaming. An atom $A$ is *more general* than an atom $B$ and $B$ is an *instance* of $A$ if there exists a substitution $\theta$ such that $A\theta = B$.
We restrict our attention to definite logic programs. A logic program $P$ is a finite set of clauses of the form $H\leftarrow A_1,..., A_n$, where $H$ and each $A_i$ are atoms. A goal $G_i$ is a headless clause $\leftarrow A_1,..., A_n$. A top goal is also called the query. Without loss of generality, we assume that a query contains only one atom.
Let $P$ be a logic program and $G_0$ a goal. $G_0$ is evaluated by building a *generalized SLD-tree* as defined in [@term_prediction], in which each node is represented by $N_i:G_i$ where $N_i$ is the name of the node and $G_i$ is a goal attached to the node. Throughout the paper, we choose to use the best-known *depth-first, left-most* control strategy, as is used in Prolog, to select goals and atoms. So by the *selected atom* in each node $N_i:\leftarrow A_1,..., A_n$, we refer to the left-most atom $A_1$. For any node $N_i:G_i$, we use $A_i^1$ to refer to the selected atom in $G_i$. Let $A_i^1$ and $A_j^1$ be the selected atoms at two nodes $N_i$ and $N_j$, respectively. $A_i^1$ is an *ancestor* of $A_j^1$ if the proof of $A_i^1$ goes through the proof of $A_j^1$.
A derivation step is denoted by $N_i:G_i\Longrightarrow_{C} N_{i+1}:G_{i+1}$, meaning that applying a clause $C$ to $G_i$ produces $N_{i+1}:G_{i+1}$. Any path of such derivation steps starting at the root node $N_0:G_0$ is called a *generalized SLD-derivation*.
Integer arithmetics
-------------------
Prolog implementations contain special purpose predicates for handling integer arithmetics. Examples are $is/2, \geq/2, =:=/2,\ldots$
\[integer\_expressions\] An expression $Expr$ is an *integer expression* if it can be constructed by the following recursive definition.
- $Expr = z \in {{\mathbb{Z}}}\mid -Expr \mid Expr+Expr \mid Expr-Expr \mid Expr*Expr$ $\hfill \square$
An atom `"V is Expr"`, with $V$ a free variable and $Expr$ an integer expression, is called an *integer constructor*. An atom $Expr1 \circ Expr2$ is called an *integer condition* if $Expr1$ and $Expr2$ are integer expressions and $\circ \in \lbrace$`>,>=,=<,<,=:=,=/=`$\rbrace$.
Moded SLD-trees and loop checking
---------------------------------
In [@DBLP:conf/iclp/VoetsS09], classes of queries are represented as *moded queries*. Moded queries are partially instantiated queries, in which variables can be labeled as *input*. Variables labeled input are called *input variables* and represent arbitrary ground terms. To indicate that a variable is labeled as input, the name of the variable is underlined. A query in which no variable is labeled as input is called a *concrete query*. The set of concrete queries represented by a moded query $Q$ is called the *denotation* of $Q$.
\[def:denotation\] Let $Q$ be a query and $\lbrace \underline{I_1},\ldots,\underline{I_n} \rbrace$ its set of input variables. The *denotation* of $Q$, $Den(Q)$, is defined as:
- $Den(Q) = \left\lbrace Q\lbrace \underline{I_1} \setminus t_1,\ldots,\underline{I_n} \setminus t_n \rbrace \mid
t_i \in Term_P, t_i~is~ground, 1\leq i \leq n \right\rbrace $. $\hfill \square$
Note that the denotation of a concrete query is a singleton containing the query itself. Denotations of moded goals and atoms are defined similarly.
A moded query $\leftarrow Q$ is evaluated by constructing a *moded SLD-tree*, representing the derivations of the queries in $Den(\leftarrow Q)$. This moded SLD-tree is constructed by applying SLD-resolution to the query and propagating the labels. An input variable $\underline{I}$ can be unified with any term $t \in Term_P$. After unifying $\underline{I}$ and $t$, all variables of $t$ will be considered input as well.
\[example:moded\_sld\] Figure \[fig:eq\_plus\_symbolic\] shows the moded SLD-tree of the program $eq\_plus$ for the moded query $\leftarrow eq\_plus(\underline{I},\underline{J},\underline{P})$. This program is non-terminating for any query in $Den(\leftarrow eq\_plus(\underline{I},\underline{I},0))$ and fails for all other queries in $Den(\leftarrow eq\_plus(\underline{I},\underline{J},\underline{P}))$. A query fails if its derivation tree is finite, with no path ending with the empty goal.
eq_plus(I,J,P):- eq(I,J), plus(P,I,In), eq_plus(In,J,P).
eq(A,A). plus(0,B,B). plus(s(A),B,s(C)):- plus(A,B,C).
![Moded SLD-tree $eq\_plus$[]{data-label="fig:eq_plus_symbolic"}](figs/eq_plus_symbolic.pdf){width="70ex"}
Substitutions on input variables express conditions for the clause to be applicable. The edge from node $N_2$ to $N_3$ shows that clause two is applicable if the concrete term denoted by $\underline{P}$ can be unified with $0$. The substitution, $I1 \setminus \underline{I}$, shows that applying this clause unifies $I1$ with the term corresponding to $\underline{I}$.
Every derivation in a moded SLD-tree for a query $\leftarrow Q$ corresponds to a concrete derivation for a subclass of $Den(\leftarrow Q)$. The subclass of queries for which a derivation to node $N_i$ is applicable is obtained by applying all substitutions on input variables from $N_0$ to $N_i$. Our condition of [@DBLP:conf/iclp/VoetsS09] proves non-termination for every query for which the derivation to $N_3$ is applicable. The substitutions on input variables in the derivation to $N_3$ are $\underline{J}\setminus \underline{I}$ and $\underline{P} \setminus 0$. Applying these to the query proves non-termination for the queries in $Den(\leftarrow eq\_plus(\underline{I},\underline{I},0))$. $\hfill \square$
As in the example, moded SLD-trees are usually infinite. To obtain a finite analysis, a complete loop check is applied during the construction of the tree. As in our previous works, [@DBLP:conf/iclp/VoetsS09] [@VDS10], we use the complete loop check *LP-check*, [@shen_dynamic_approach]. Without proof, we state that this loop check can also be used for moded SLD-trees and refer to [@shen_dynamic_approach] for more information.
In Figure \[fig:eq\_plus\_symbolic\], LP-check cuts clause 4 at node $N_6$ and clause 3 at node $N_7$. $\hfill \square$
Combined with the loop check, a moded SLD-tree can be considered a light-weight alternative to an abstract interpretation for mode analysis.
Non-termination analysis for programs with integer arithmetics
==============================================================
In this section, we introduce a non-termination condition for programs containing integer arithmetics. To abstract the computations for the considered queries, the moded SLD-tree of [@term_prediction] is used, with some modifications to handle integer constructors and integer conditions. LP-check ensures finiteness of the tree and detects paths that may correspond to infinite loops. For every such path, two analyses are combined to identify classes of non-terminating queries.
In the first phase, an adaption of the non-termination condition of [@DBLP:conf/iclp/VoetsS09] detects a class of queries such that each query is non-terminating or fails due to the evaluation of an integer condition such as $>/2$. This class of queries is a moded query with an additional integer label for variables representing unknown integers. In the second phase, the class of queries is restricted to a class of non-terminating queries by formulating additional constraints on the integer variables of the moded query. To prove that this class of non-terminating queries is not empty, these constraints over unknown integers are transformed to constraints over the natural numbers and solved by applying well-known techniques from termination analysis. Then we try to solve these constraints by transforming them to constraints over the natural numbers and applying well-known techniques on them.
Moded SLD-tree for programs with integer arithmetics
----------------------------------------------------
The first step of the extension is rather straightforward. The extensions to the moded SLD-tree of [@term_prediction] are limited to the introduction of the label *integer variable* and additional transitions to handle integer constructors and integer conditions. Integer variables are also input variables and will also be represented by underlining the name of the variable. An integer constructor, i.e. $is/2$, is applicable if the first argument is a free variable and the second argument is an integer expression. The application of an integer constructor labels the free variable as an integer variable. An integer condition, e.g. $\geq/2$, is applicable if both arguments are integer expressions. Since integer variables denote unknown integers, integer expressions are allowed to contain integer variables. Applications of integer constructors and integer conditions in the moded SLD-tree are denoted by derivation steps $N_i:G_i\Longrightarrow_{cons} N_{i+1}:G_{i+1}$ and $N_i:G_i\Longrightarrow_{cond} N_{i+1}:G_{i+1}$, respectively.
\[example:count\_to\] The following program, $count\_to$, is a faulty implementation of a predicate generating the list starting from 0 up to a given number. The considered class of queries is represented by the moded query $\leftarrow count\_to(\underline{N},L)$ with $\underline{N}$ an integer variable.
count_to(N,L):- count(0,N,L). count(N,N,[N]).
count(M,N,[M|L]):- M > N, M1 is M+1, count(M1,N,L).
In the last clause, the integer condition should be `M < N` instead of `M > N`. Due to this error, the program:
- fails for the queries for which $\underline{N}>0$ holds,
- succeeds for $\leftarrow count\_to(0,L)$,
- loops for the queries for which $\underline{N} < 0$ holds.
![Moded SLD-tree $count\_to$[]{data-label="fig:count_to"}](figs/count_to.pdf){width="60ex"}
Figure \[fig:count\_to\] shows the moded SLD-tree for the considered query, constructed using LP-check. LP-check cuts clause 3 at node $N_9$. $\hfill \square$
Note that by ignoring the possible values for the integer variables when constructing the tree, some derivations in it may not be applicable to any considered query. For example the refutations at nodes $N_6$ and $N_{10}$ in the previous example cannot be reached by the considered queries.
Adapting the non-termination condition
--------------------------------------
In [@DBLP:conf/iclp/VoetsS09], programs are shown to be non-terminating for a moded query, by proving that a path in the moded SLD-tree can be repeated infinitely often. Such a path, from a node $N_b$ to a node $N_e$, is identified based on three properties. The path should be applicable, independent from the concrete terms represented by the input variables. Therefore, the first property states that no substitutions on input variables may occur between $N_b$ and $N_e$. The second property states that the selected atom of $N_b$ – i.e. $A_b^1$ – has to be an ancestor of $A_e^1$. These two properties prove that the sequence of clauses in the path from $N_b$ to $N_e$ is applicable to any goal with a selected atom from $Den(A_b^1)$. Therefore, non-termination is proven by requiring that $Den(A_e^1)$ is a subset of $Den(A_b^1)$. This property can be relaxed by requiring that each atom in $Den(A_e^1)$ is more general than some atom in $Den(A_b^1)$. If this is the case, $A_e^1$ is called *moded more general* than $A_b^1$. For definite logic programs, these three properties imply non-termination.
Let $A$ and $B$ be moded atoms. $A$ is *moded more general* than $B$ if
- $ \forall I \in Den(A),~ \exists J \in Den(B): I \textit{ is more general than } J$.$\hfill \square$
In Figure \[fig:eq\_plus\_symbolic\], the path from $N_3$ to $N_6$ satisfies these properties. The ancestor relation holds. There are no substitutions on input variables in the path. Finally, the selected atoms are identical and therefore denote the same concrete atoms. $\hfill \square$
The following proposition provides a practical sufficient condition to verify whether the moded more general relation holds.
\[prop:mmg\] Let $A$ and $B$ be moded atoms. Let $A_1$ and $B_1$ be renamings of these atoms such that they do not share variables. $A$ is moded more general than $B$ if $A_1$ and $B_1$ are unifiable with most general unifier $\lbrace V_1\setminus t_1,\ldots,
V_n \setminus t_n \rbrace$, $t_i \in Term_P$, $1 \leq i, \leq n$, such that for each binding $V_i \setminus t_i$, either:
- $V_i \in Var(B_1)$ and $V_i$ is labeled as input, or
- $V_i \in Var(A_1)$, $V_i$ is not labeled as input and no variable of $Var(t_i)$ is labeled as input. $\hfill \square$
As stated, we want to prove that every query in the denotation of the considered moded query is either non-terminating or terminates due to the evaluation of an integer condition. To achieve this, we need to guarantee that integer constructors are repeatedly evaluated with a free variable and an integer expression as arguments and that integer conditions are repeatedly evaluated with integer expressions as arguments. Proposition \[prop:mmg\] already implies that the first argument of all integer constructors are free variables in the subsequent iterations of the loop.
To prove the repeated behavior on integer constructors and integer expressions stated above, the *integer-similar to* relation is defined. Intuitively, given some loop in the computation, if an atom at the end of the loop is integer-similar to an atom at the start of the loop, then it will provide the required integer expressions to the first atom. First, we introduce positions to identify subterms and a function to obtain a subterm from a given position.
\[def:func\_subterm\] Let $L$ be a list of natural numbers, called a *position*, and $A$ a moded atom or term. The function *subterm(L,A)* returns the subterm obtained by:
- if $L = [I]$ and $A=f(A_1,\ldots,A_I,A_{I+1},\ldots,A_n)$ then $subterm(L,A) = A_I$
- else if $L=[I|T]$ and $A=f(A_1,\ldots,A_I,A_{I+1},\ldots,A_n)$ then $subterm(L,A) = subterm(T,A_I)$ $\hfill \square$
An atom $A$ is integer-similar to an atom $B$ if it has integer expressions on all positions corresponding to integer expressions in $B$.
Let $A$ and $B$ be moded atoms. $A$ is *integer-similar to* $B$ if for every integer expression $t_B$ of $B$, with $subterm(L,B) = t_B$, there exists an integer expression $t_A$ of $A$, with $subterm(L,A) = t_A$. $\hfill \square$
- $count(0,\underline{N},L)$ is integer-similar to $count(\underline{M},\underline{N},L)$
- $count(\underline{M},\underline{N},L)$ is integer-similar to $count(0,\underline{N},L)$
- $count(\underline{M} + 1,\underline{N},L)$ is integer-similar to $count(\underline{M},\underline{N},L)$
- $count(\underline{M},\underline{N},L)$ is not integer-similar to $count(\underline{M}+1,\underline{N},L)$
Note that the last one is a counterexample because $count(\underline{M}+1,\underline{N},L)$ has integer expressions on $[1,1]$ and $[1,2]$, while $count(\underline{M},\underline{N},L)$ does not have any subterms on these positions. $\hfill \square$
\[th:analysis1\] Let $N_b$ and $N_e$ be nodes in a moded SLD-tree for a moded query $Q$. Let $Q'$ be the moded atom obtained by applying to $Q$ all substitutions on input variables from $N_0$ to $N_b$. Every query in $Den(Q')$ is either non-terminating or terminates due to the evaluation of an integer condition if the following properties hold:
- $A_b^1$ is an ancestor of $A_e^1$
- no substitutions on input variables occur from $N_b$ to $N_e$
- $A_e^1$ is moded more general than $A_b^1$
- $A_e^1$ is integer-similar to $A_b^1$ $\hfill \square$
\[example:mmg\_adaption\] The path between nodes $N_5$ and $N_9$ in Figure \[fig:count\_to\] satisfies the conditions of Theorem \[th:analysis1\]. There are no substitutions on input variables from $N_0$ to $N_5$ and thus, every query in $Den(\leftarrow count\_to(\underline{N},L))$ is either non-terminating or fails due to the evaluation of an integer condition. Note that although $\leftarrow count\_to(0,L)$ has a succeeding derivation to $N_2$, its derivation to $N_9$ fails due to the integer condition $0 > \underline{N}$. $\hfill \square$
To verify the last property automatically, we strengthen Proposition \[prop:mmg\] to imply both the moded more general relation and the integer-similar to relation.
\[prop:mmg\_int\_ins\] Let $A$ and $B$ be moded atoms. Let $A_1$ and $B_1$ be renamings of these atoms such that they do not share variables. $A$ is moded more general than $B$ and $A$ is integer-similar to $B$, if $A_1$ and $B_1$ are unifiable with most general unifier $\lbrace V_1\setminus t_1,\ldots,
V_n \setminus t_n \rbrace$, such that for each binding $V_i \setminus t_i$, $1\leq i \leq n$, either:
- $V_i \in Var(B_1)$ and $V_i$ is labeled as integer and $t_i$ is an integer expression, or
- $V_i \in Var(B_1)$ and $V_i$ is labeled as input but not as integer variable, or
- $V_i \in Var(A_1)$, $V_i$ is not labeled as input, no variable of $Var(t_i)$ is labeled as input and $t_i$ does not contain integers. $\hfill \square$
Since the selected atoms of nodes $N_5$ and $N_9$ in Figure \[fig:count\_to\] are variants, Proposition \[prop:mmg\_int\_ins\] holds. $\hfill \square$
Generating the constraints on the integers of the query
-------------------------------------------------------
In this subsection, we introduce the constraints on the integer variables of the moded query, identifying values for which all integer conditions in the considered derivations succeed. These constraints consist of reachability constraints, identifying queries for which the derivation up till the last node is applicable, and an implication proving that the integer conditions will also succeed in the following iterations.
\[example:count\_to\_int\_cons\] As a first example, we introduce the constraints for the path between $N_5$ and $N_9$ in the moded SLD-tree of $count\_to$ in Figure \[fig:count\_to\]. For this path, Theorem \[th:analysis1\] holds and thus every query denoted by $\leftarrow count\_to(\underline{N},L)$ is either non-terminating or terminates due to an integer condition.
To restrict the class of considered queries to those for which the derivation to $N_9$ is applicable, all integer conditions in the derivation are expressed in terms of the integers of the query, yielding $0 > \underline{N}$ and $0 + 1 > \underline{N}$. For this program and considered class of queries, the condition $0 > \underline{N}$ implies that the derivation is applicable until node $N_9$. The following implication states that if the condition of node $N_7$ holds for any two values $M$ and $N$, then it also holds for the values of the next iteration. $$\forall M,N \in {{\mathbb{Z}}}: M > N \Longrightarrow M+1>N$$ This implication is correct and thus proves non-termination for the considered queries if the precondition holds in the first iteration. This is the case for all queries in $Den(\leftarrow count\_to(\underline{N},L))$ with $0 > \underline{N}$ since the value corresponding to $M$ in the first iteration is $0$ and the value corresponding to $N$ is $\underline{N}$. This proves non-termination of all considered queries for which $0 > \underline{N}$. $\hfill \square$
In the following example, applicability of the derivation does not imply non-termination. To detect a class of non-terminating queries, a domain constraint is added to the pre- and postcondition of the implication.
\[example:constants\_nt\_cond\]
constants(I,J):- I =:= 2, In is J*2, Jn is I-J, constants(In,Jn).
The clause in *constants* is applicable to any goal with $constants(2,\underline{J})$ as selected atom, with $\underline{J}$ an integer variable. Since the first argument in the next iteration is the value corresponding to $\underline{J}*2$, only goals with the selected atom $constants(2,1)$ are non-terminating for this program.
Since applicability of the derivation does not imply non-termination, a similar implication as in the previous example is false, $\forall I,J \in {{\mathbb{Z}}}: I=2 \Longrightarrow J*2 = 2$. To overcome this, a constraint is added to the pre- and post-condition of this implication, restricting the considered values of $\underline{J}$ to an unknown set of integers, called its *domain*. $$\exists Dom_j \subset {{\mathbb{Z}}}, \forall I,J \in {{\mathbb{Z}}}: I=2, J \in Dom_j \Longrightarrow J*2 = 2, I-J \in Dom_j$$ The resulting implication is true for $Dom_j = \lbrace 1 \rbrace$. By requiring that the considered moded query satisfies both the reachability constraint and the additional constraint in the pre-condition, the non-terminating query $\leftarrow constants(2,1)$ is obtained. $\hfill \square$
All information needed to construct these constraints can be obtained from the moded SLD-tree.
Let $C$ be an integer condition or expression and $N_i$ and $N_j$ two nodes in a moded SLD-tree $D$. Let $Cons$ be the set of all integer constructors occurring as selected atom in a node $N_p~(i \leq p \leq j)$ in $D$.
The function *$apply\_cons(C,N_i,N_j)$* returns the integer condition or expression obtained by exhaustively applying $\underline{I}\setminus Expr$ to $C$, for any $\underline{I} ~is~ Expr \in Cons$. $\hfill \square$
The constraints guaranteeing a derivation to $N_j$ to be applicable, can be obtained using $apply\_cons(Cond,N_0,N_i)$ for any integer condition $Cond$ in a node $N_i$ in the considered derivation. For a path from $N_b$ to $N_e$, the precondition of the implication is obtained using $apply\_cons(Cond,N_b,N_i)$, for each condition $Cond$ in a node $N_i$ between nodes $N_b$ to $N_e$ and universally quantifying the integer variables of $N_b$.
\[example:apply\_cons\] The derivation to $N_9$ in Figure \[fig:count\_to\], contains integer conditions in nodes $N_3$ and $N_7$. These are expressed on the integer variable of the query, $\underline{N}$, using $apply\_cons$.
- $apply\_cons(0>\underline{N},N_0,N_3) = 0 > \underline{N}$
- $apply\_cons(\underline{M1}>\underline{N},N_0,N_7) = 0 + 1 > \underline{N}$
To obtain the precondition of the implication, the integer condition in $N_7$ is expressed in terms of the integer variables of $N_5$.
- $apply\_cons(\underline{M1}>\underline{N},N_5,N_7) = \underline{M1} > \underline{N}$
Universally quantifying these variables yields the precondition. $\hfill \square$
To obtain the consequence of the implication for a path from $N_b$ to $N_e$, one first replaces the integer variables of $N_b$ in the precondition by the corresponding integer variables of $N_e$. Then, $apply\_cons$ is used to express the consequence in terms of the values in the previous iteration.
Let $LHS$ be the precondition of an implication, consisting of integer conditions and constraints of the form $I \in Dom_I$. Let $N_i$ and $N_j$ be two nodes in a moded SLD-derivation such that all integer variables in $LHS$ are in $A_i^1$ and let $\underline{I_1},\ldots,\underline{I_n}$ be all integer variables of $A_i^1$.
If there exist subterms of $A_j^1$, $t_1,\ldots,t_n$, such that $\forall L: subterm(L,A_i^1)=\underline{I_p} \Longrightarrow
subterm(L,A_j^1)=t_p, 1 \leq p \leq n$, then *$replace(LHS,N_i,N_j)$* is obtained by applying $\lbrace \underline{I_1} \setminus t_1, \ldots, \underline{I_n} \setminus t_n\rbrace$ to all constraints in $LHS$. $\hfill \square$
In Example \[example:apply\_cons\], we generated the precondition of the implication, $\underline{M1} > \underline{N}$. To obtain the consequence, $replace(\underline{M1} > \underline{N},N_5,N_9)$ is applied, yielding $\underline{M2} > \underline{N}$. Then, the integer variable of $N_9$, $\underline{M_2}$, is expressed in terms of the integer variables of $N_5$ using $apply\_cons(\underline{M2} > \underline{N},N_5,N_9)=\underline{M_1}+1 > \underline{N}$.
Adding the domains to the pre- and postcondition yields the desired implication: $\exists Dom_N, Dom_{M1} \subset {{\mathbb{Z}}}, \forall N,M1 \in {{\mathbb{Z}}}: M1 > N,~N \in Dom_N,~M1 \in Dom_{M1} \Longrightarrow$\
$~~~~~~~~~M1+1 > N,~N \in Dom_N,~M1+1 \in Dom_M$ $\hfill \square$
Adding these constraints to the class of queries detected by Theorem \[th:analysis1\], yields a class of non-terminating queries.
Proving that the constraints on integers are solvable
-----------------------------------------------------
The previous subsection introduced constraints, implying that all integer conditions in a considered derivation succeed. In this subsection, we introduce a technique to check if these constraints have solutions, using a constraint-based approach. Symbolic coefficients represent values for the integers in the query and domains in the implication, for which the considered path is a loop. After these coefficients are introduced, the implication is transformed into a set of equivalent implications over natural numbers. These implications can then be solved automatically in the constraint-based approach, based on Proposition 3 of [@DBLP:journals/corr/abs-0912-4360].
\[prop:rem\_imp\] Let $prem$ be a polynomial over $n$ variables and $conc$ a polynomial over 1 variable, both with natural coefficients, where $conc$ is not a constant. Moreover, let $p_1,\ldots,p_{n+1},q_1,\ldots,q_{n+1}$ be arbitrary polynomials with integer coefficients[^3] over the variables $\overline{X}$. If $$\forall \overline{X} \in {{\mathbb{N}}}: conc(p_{n+1})-conc(q_{n+1})-prem(p_1,\ldots,p_n)+
prem(q_1,\ldots,q_n) \geq 0$$ is valid, then $\forall \overline{X} \in {{\mathbb{N}}}: p_1 \geq q_1, \ldots,p_n\geq q_n \Longrightarrow p_{n+1}\geq q_{n+1}$ is also valid. $\hfill \square$
### Introducing the symbolic coefficients.
To represent half-open domains in the implication by symbolic coefficients, the domains are described by two symbolic coefficients, one upper or lower limit and one for the direction. Constraints of the form $Exp \in Dom_I$ in the implication, are replaced by constraints of the form $d_I * Exp \geq d_I* c_I$ with $d_I$ either $1$ or $-1$, describing the domain $\lbrace c_I, c_I-1, \ldots\rbrace$ for $d_I=-1$ and $\lbrace c_I, c_I+1, \ldots\rbrace$ for $d=1$. The values to be inferred for the integers of the query should satisfy the precondition of the implication. Off course, the symbolic coefficients $c_I$ should also be consistent with the values of the integers in the query.
In Example \[example:count\_to\_int\_cons\], we introduced constraints on the integer variable $\underline{N}$, $0 > \underline{N}$ and $0 + 1 > \underline{N}$, proving non-termination for queries in $Den(\leftarrow count\_to(\underline{N},L))$. By convention, we denote the symbolic coefficients as constants. For the integer variable $\underline{N}$, we introduce the symbolic coefficient $n$.
The implication introduced in Example \[example:count\_to\_int\_cons\], for the path from $N_5$ to $N_9$ in Figure \[fig:count\_to\], does not contain constraints on the domains. When adding these constraints to the pre- and postcondition, we obtain the following implication.
- $\forall M,N \in {{\mathbb{Z}}}: ~M > N, ~N \in Dom_N, ~M \in Dom_M \Longrightarrow $\
$~~~~~~~~~~~M+1>N, ~N \in Dom_N, ~M+1 \in Dom_M$
Representing these domains by symbolic coefficients yields the following implication.
- $\forall M,N \in {{\mathbb{Z}}}: ~M > N, ~d_N * N \geq d_N * c_N, ~d_M * M \geq d_M * c_M \Longrightarrow $\
$~~~~~~~~~~~M+1>N, ~d_N * N \geq d_N * c_N, ~d_M * (M+1) \geq d_M * c_M$
To guarantee that the precondition succeeds for the considered derivation, $c_M$ and $c_N$ are required to be the values for $\underline{M}$ and $\underline{N}$ in node $N_5$. Combining these constraints implies non-termination for the query $\leftarrow count\_to(n,L)$, for which the following constraints are satisfied with some unknown integers $c_N,c_M,d_N$ and $d_M$.
- $0>n,~0+1>n$ to guarantee applicability of the derivation
- $c_N = n, ~c_M = 0+1$ to guarantee that the precondition holds
- $d_N = 1 \lor d_N = -1, ~d_M = 1 \lor d_M = -1$,
- $\forall M,N \in {{\mathbb{Z}}}: M > N, d_N * N \geq d_N * c_N, d_M * M \geq d_M * c_M \Longrightarrow $\
$~~~~~~~~~~~M+1>N, d_N * N \geq d_N * c_N, d_M * (M+1) \geq d_M * c_M$ to prove that the condition succeeds infinitely often.
Due to the implication, $d_M$ has to be $1$. $d_N$ can be either $1$ or $-1$. $\hfill \square$
To be able to infer singleton domains, we allow the constant describing the direction of the interval to be $0$. If in such a constant $d_I$ is zero, the constraints on the domain are satisfied trivially because they simplify to $0 \geq 0$. To guarantee that the domain is indeed a singleton when $d_I$ is inferred to be zero, a constraint of the form $(1-d_I^2)Exp=(1-d_I^2)*c_I$ is added to the postcondition for every constraint $d_I * I \geq d_I * c_I$. This constraint is trivially satisfied for half-open domains and proves that $\lbrace c_I \rbrace$ is the domain in the case that $d_I = 0$.
In Example \[example:constants\_nt\_cond\], we introduced constraints on the integer variables $\underline{I}$ and $\underline{J}$, proving non-termination for queries in $Den(\leftarrow constants(\underline{I},\underline{J}))$. Introducing symbolic coefficient $i$ and $j$ for the integers of the query and for the domains of $\underline{I}$ and $\underline{J}$, yields the following constraints.
1. $i = 2$ to guarantee applicability of the derivation
2. $c_I = i, ~c_J = j$ to guarantee that the precondition holds
3. $d_I \leq 1, ~d_I \geq -1, ~d_J \leq 1, ~d_J \geq -1$,
4. $\forall I,J \in {{\mathbb{Z}}}: I=2, ~d_I * I \geq d_I * c_I, ~d_J * J \geq d_J * c_J \Longrightarrow $\
$~~~~~J*2=2, ~d_I * (J*2) \geq d_I * c_I, (1-d_I^2)*(J*2) = (1-d_I^2)*c_I, $\
$~~~~~d_J * (I-J) \geq d_J * c_J, (1-d_J^2)*(I-J) = (1-d_J^2)*c_J$
The implication in $(4)$ can only be satisfied with $d_J$ equal to zero. $\hfill \square$
### To implications over the natural numbers
The symbolic coefficients to be inferred which represent the domains, allow to transform the implication over ${{\mathbb{Z}}}$ to an equivalent implication over ${{\mathbb{N}}}$.
- for $d_I = 1$, any integer in $\lbrace c_I,~c_I+1,~\ldots\rbrace$ that satisfies the precondition is in $\lbrace c_I+d_I*N \mid N \in {{\mathbb{N}}}\rbrace$
- for $d_I = -1$, any integer in $\lbrace c_I,~c_I-1,~\ldots\rbrace$ that satisfies the precondition is in $\lbrace c_I+d_I*N \mid N \in {{\mathbb{N}}}\rbrace$
- for $d_I = 0$, any integer in $\lbrace c_I \rbrace$ that satisfies the precondition is in $\lbrace c_I+d_I*N \mid N \in {{\mathbb{N}}}\rbrace$
Therefore, we obtain an equivalent implication over the natural numbers by replacing each integer $I$ by its corresponding expression $c_I+d_I*N$ and replacing the universal quantifier over $I$ by a quantifier over $N$.
### Automation by a translation to diophantine constraints
To solve the resulting constraints, we use the approach of [@DBLP:journals/corr/abs-0912-4360]. Constraints of the form $A =:= B$ in the implication, are replaced by the conjunction $A\geq B,~B\geq A$. Constraints of the form $A =/= B$, yield two disjunctive cases. One obtained by replacing the $=/=$ in the pre- and postcondition by $>$ and one obtained by replacing it by $<$. The other conditions – i.e. $>,<$ and $\leq$ – are transformed into $\geq$-constraints in the obvious way. Implications with only one consequence are obtained by creating one implication for each consequence, with the pre-condition of the original implication.
The resulting implications allow to apply Proposition \[prop:rem\_imp\]. These inequalities of the form, $p\geq0$, are then transformed into a set of *diophantine constraints*, i.e. constraints without universally quantified variables, by requiring that all coefficients of $p$ are non-negative. As proposed in [@DBLP:journals/corr/abs-0912-4360], the resulting diophantine constraints are then transformed into a SAT-problem. The constraints are then proven to have solutions by a SAT solver by inferring one possible solution.
Evaluation
==========
We have implemented our analysis and integrated it within our existing non-termination analyzer $pTNT$. The analyzer can be downloaded from\
http://www.cs.kuleuven.be/\~dean/iclp2011.html. We tested our analysis on a benchmark of 16 programs similar to those in the paper. These programs are also available online. To solve the resulting SAT-Problem, MiniSat [@ES03] is used.
$ $ linear-class, 3 bits linear-class, 4 bits max2-class, 3 bits max2-class, 4 bits
----------- ---------------------- ---------------------- -------------------- --------------------
count\_to $+$ $+$ $+$ $+$
constants $+$ $+$ $+$ $OS$
int1 $+$ $+$ $+$ $+$
int2 $+$ $+$ $+$ $+$
int3 $+$ $+$ $+$ $OS$
int4 $+$ $+$ $+$ $OS$
int5 $+$ $+$ $+$ $OS$
int6 $+$ $+$ $+$ $OS$
int7 $+$ $+$ $OS$ $OS$
int8 $+$ $+$ $OS$ $OS$
int9 $-$ $+$ $OS$ $OS$
int10 $-$ $-$ $+$ $OS$
int11 $-$ $+$ $-$ $OS$
int12 $-$ $+$ $-$ $OS$
int13 $+$ $+$ $+$ $+$
int14 $+$ $+$ $+$ $OS$
: An overview of the experiments[]{data-label="Table:evaluation"}
We experimented with different bit-sizes in the translation to SAT and different classes of functions for the $prem$ functions in Proposition \[prop:rem\_imp\]. As $conc$ functions, the identity function was used. Table \[Table:evaluation\] shows the results for the considered settings, $+$ denotes that non-termination is proven successfully, $-$ denotes that non-termination could not be proven and $OS$ denotes that the computation went out of stack. The considered settings are 3 and 4 as bit-sizes and $linear$ and $max2$ as forms for the symbolic $prem$-functions. The $linear$ class is a weighted sum of each argument. The $max2$ class contains a weighted term for each multiplication of two arguments. The analysis time is between $1$ and $20$ seconds for all programs and settings.
Table \[Table:evaluation\] shows non-termination can be proven for any program of the benchmark when choosing the right combination of parameters, but no setting succeeds in proving non-termination for all programs. Programs $int9$ and $int12$ require a constant that cannot be represented with bit-size 3. Linear prem-functions cannot prove non-termination for $int10$. However, the setting with 4 as a bit-size and $max2$ as class of $prem$-function usually fails, because these settings cause an exponential increase in memory use during the translation to SAT.
Conclusion
==========
In this paper we introduced a technique to detect classes of non-terminating queries for logic programs with integer arithmetic. The analysis starts with a given program and class queries, specified using modes, and detects subclasses of non-terminating queries. First, the derivations for the given class of queries are abstracted by building a moded SLD-tree [@term_prediction] with additional transitions to handle integer arithmetic. Then, this moded SLD-tree is used to detect subclasses of non-terminating queries in two phases. In the first phase, we ignore the conditions over integers, e.g. $>/2$, and detect paths in the moded SLD-tree that correspond to infinite derivations if all conditions on integers in those derivations succeed. For every such path, the corresponding subclass of queries is generated. In the second phase, the obtained classes of queries are restricted to classes of non-terminating queries, by formulating constraints implying that all conditions on integers will succeed. These constraints are then solved by transforming them into a SAT problem.
We implemented this approach in our non-termination analyzer $pTNT$ and evaluated it on small benchmark of non-terminating Prolog programs with integer arithmetic. The evaluation shows that the proposed technique is rather powerful, but also that the parameters in the transformation to SAT must be chosen carefully to avoid excessive memory use. For future work, we plan to improve the efficiency by using SMT solvers.
#### Acknowledgment
We thank the referees for their useful and constructive comments.
[^1]: Supported by the Fund for Scientific Research - FWO-project G0561-08
[^2]: Results are available at http://termcomp.uibk.ac.at/
[^3]: Proposition 3 in [@DBLP:journals/corr/abs-0912-4360] uses natural coefficients, but the proposition also holds for polynomials with integer coefficients.
| ArXiv |
Precise knowledge of the spin susceptibility $\chi({\bf q}, \omega)$ of the cuprates is essential for understanding their unusual normal state properties. The imaginary part, $\chi^{\prime \prime}({\bf q},\omega)$ can be probed either by inelastic neutron scattering (INS) [@Kei; @Bou; @Hay; @Mook97], or in the low frequency limit by NMR measurements of the spin-lattice relaxation rate $1/T_1$ [@T1]. In contrast, one knows little about the real part of the susceptibility, $\chi^\prime({\bf q})$, since information can, so far, only be extracted from the NMR observation of the Gaussian component of the transverse relaxation time, $T_{\rm 2G}$, of planar Cu [@PS91; @Curro97]. In particular, the analysis of INS and NMR experiments has not yet led to a consensus on the shape of $\chi({\bf q},\omega)$ in momentum space and the temperature ($T$) dependence of the antiferromagnetic correlation length, $\xi$. In this communication we present new insight into this issue based on experiments by Bobroff [*et al.*]{} [@BAY97]. Our principal conclusions are that $\xi$ in YBa$_2$Cu$_3$O$_{6+\delta}$ is $T$-dependent and that the Lorentzian form of $\chi^\prime(\bf q)$ provides a completely consistent description of the data, whereas the Gaussian form can be ruled out.
Bobroff [*et al.*]{} [@BAY97] recently presented a novel approach to the measurement of $\chi^\prime({\bf q})$ using Ni impurities in YBa$_2$(Cu$_{1-x}$Ni$_x$)$_3$O$_{6+\delta}$. These impurities induce a spin polarization at the planar Cu sites via $\chi^\prime({\bf q})$. The hyperfine coupling between Cu and O induces a spatially varying polarization and an additional broadening $$\Delta \nu_{\rm imp} = \Delta \nu-\Delta \nu_0=\alpha f(\xi)/T
\label{dnu}$$ of the planar $^{17}$O NMR, where $\Delta \nu$ and $\Delta \nu_0$ are the total and $x=0$ line width, respectively. In Eq. \[dnu\] $\alpha$ is the overall amplitude of $\chi^\prime({\bf q})$ and $f(\xi)$ characterizes the dependency of $\Delta \nu$ on $\xi$ ($\alpha=4\pi \chi^*$ in the notation of Ref. [@BAY97] and[@Morr97]). Finally, the factor $1/T$ is caused by the Curie behavior of the Ni impurities in YBa$_2$Cu$_3$O$_{6+\delta}$[@Mah94; @Men94] with effective moment $p_{\rm eff}\approx 1.9 \mu_{\rm B}$ ($1.59 \mu_{\rm B}$) for $\delta=0.6
(\delta=1)$. Bobroff [*et al.*]{} found that $T \Delta \nu(T)$ strongly depends on temperature and the Ni concentration $x$ in the sample. Furthermore they observed a much stronger broadening in the underdoped, $\delta=0.6$, sample than in the overdoped one with $\delta=1$. Performing numerical simulations of the NMR line shape by assuming a Gaussian form for $\chi^\prime({\bf q})$, they found that $f(\xi)$ is basically constant for all physically reasonable values of $\xi$. Combining these results with $T_{\rm2G}$ data by Takigawa [@Tak94], they concluded that $\xi$ is $T$-independent for the underdoped samples. On the other hand, in every scenario of cuprate superconductors in which the anomalous low-energy behavior is driven by spin fluctuations one would expect the correlation length $\xi$ to be $T$-dependent (for recent reviews, see: [@Pines; @Scal]). Thus their result has important implications about the mechanism of superconductivity. We recently pointed out [@Morr97], that our simulations using a Lorentzian form of $\chi^\prime({\bf q})$ yield a different result and are actually compatible with a $T$-dependent $\xi$.
Before going into the details of our calculations, it is important to notice that the fact that $\xi$ must be $T$-dependent can be deduced even without a detailed model from the very experimental data by Bobroff [*et al.*]{} [@BAY97] for $\Delta \nu(T)$ and Takigawa [@Tak94] for $T_{\rm 2G}$. To show this, we need to recognize that we can always express $T_{\rm 2G}$ as a product of $\alpha$ and a function of $\xi$, namely $$T_{\rm 2G}^{-1} = \alpha g(\xi) \ .
\label{t2g}$$ We can then eliminate $\alpha$ by forming the product $$T \Delta \nu_{\rm imp} T_{\rm 2G} = { f(\xi) \over g(\xi)}
\label{product}$$ which depends solely on $\xi$. In Fig. \[prod\], we plot the product $T_{\rm 2G} T \Delta \nu_{\rm imp}$ as a function of $T$[@T1corr].
\[t\]
=7.5cm
We see that this product is strongly $T$-dependent, dropping by more than a factor of 2 between $100 \, K$ and $200 \, K$. Therefore $\xi$ must have a substantial $T$-dependence.
To have a more quantitative insight into the $T$-dependence of $T \Delta \nu(T)$ of Ref. [@BAY97], we must go into details. We present in the following a theoretical analysis of the $^{17}$O line shape using a method first applied by Bobroff [*et al.*]{}, to simulate their experimental data.
To simulate the $^{17}$O line shape numerically, we distribute Ni impurities on a $(100 \times 100)$ lattice with concentration $\frac{3}{2}x$ randomly at positions ${\bf r}_j$ on a two dimensional lattice [@com2]. We consider the Ni impurities as foreign atoms embedded in the pure material, which is characterized by a non-local spin-susceptibility $\chi'({\bf q})$. In the following ${\bf s}_i$ characterizes the spin dynamics of the pure material, ${\bf S}_i$ the difference at the Ni site brought about by the Ni. These Ni spins polarize the spin ${\bf
s}_j$ of the itinerant strongly correlated electrons. To calculate the induced moments we need to know how the Ni impurities couple to these spins. Without discussing the microscopic origin of the effective Ni spin ${\bf S}_j$, we assume that it obeys a Curie law, and that the coupling to the spin ${\bf s}_j$ occurs via an on-site interaction described by $${ \cal H}_{int} = -J \sum_{j} {\bf s}_j \cdot {\bf S}_j \ .
\label{Hint}$$ The coupling constant $J$ is an unknown parameter of the theory and will be estimated below. Furthermore, we will assume like Bobroff [*et al.*]{} that the Ni impurities do [*not*]{} change the magnetic correlation length or the magnitude of the spin susceptibility.
For the NMR experiments we consider an external magnetic field $B_0$ along the $z$-direction. The Ni spins have a non-zero average value obeying $\langle S^z_j \rangle = C_{\rm Curie} B_0/T $ with Curie constant $C_{\rm Curie}=p_{\rm eff}/(2\sqrt{3} k_{\rm B})$ [@Mah94; @Men94].
Adopting a mean field picture, the induced polarization for the electron spins at the Cu sites ${\bf r}_i$ is given by $$\langle s^z_i \rangle = \frac{J}{(g\mu_{\rm B})^2} \sum_j
\chi'({\bf r}_i-{\bf r}_j) \langle S^z_j \rangle\, .
\label{Sind}$$ Here, $\chi'({\bf r})$ is the real space Fourier transform of $\chi'({\bf
q})$. In the following we consider two different forms of the spin susceptibility [@mp92]. For the commensurate case, there is only one peak, whereas in the incommensurate case, one has to sum over four peaks. The Gaussian form of $\chi'({\bf q})$ is given by $$\chi_{\rm G}'({\bf q})=\alpha \xi^2 \exp\left(-({\bf q}-{\bf Q})^2
\xi^2\right)
\label{chig}$$ and the Lorentzian form by $$\chi_{\rm L}'({\bf q})=\alpha \xi^2/(1+({\bf q}-{\bf Q})^2 \xi^2) \ .
\label{chil}$$ Since the question whether there exist incommensurate peaks in YBa$_2$Cu$_3$O$_{6+\delta}$ has not been settled yet [@Mook97], we will consider below both cases, a commensurate wavevector ${\bf Q}=(\pm \pi, \pm \pi)$, and an incommensurate one with ${\bf Q}=\delta_{\rm i} (\pm \pi, \pm \pi)$. The calculation of the real space Fourier transform finally yields $$\begin{aligned}
\chi_{\rm G}'({\bf r}) &=&\frac{\alpha}{4\pi} F({\bf Q}) \exp
\Big( - { {\bf r}^2 \over 4 \xi^2} \Big) \, ,\nonumber \\
\chi_{\rm L}'({\bf r}) &=& \frac{\alpha}{4\pi}F({\bf Q}) K_0 \Big( { r \over \xi } \Big)\, ,\end{aligned}$$ where $K_0$ is the modified Bessel function, and $ F({\bf Q}) = \cos(Q_x r_x)\cos(Q_y r_y) \ . $
Having determined the Ni induced Cu spin polarization $\langle s^z_i \rangle$, it is straightforward to investigate the $^{17}$O NMR lineshape, determined by the coupling of the $I=\frac{5}{2}$ nuclear spins $^{17}{\bf I}_l$ to the Cu electron spins ${\bf s}_i$ with spatially varying mean value $\langle s^z_i \rangle$. The hyperfine Hamiltonian is $${\cal H}_{hf} = \hbar^2 \gamma_n \gamma_e
\sum_{l,i} C_{i,l} \, {\bf s}_i\cdot ^{17}{\bf I}_l \, ,
\label{HMR}$$ where $\gamma_n,\gamma_e$ are the gyromagnetic ratios for the $^{17}$O nucleus and the electron, respectively. The hyperfine coupling constants $C_{i,l}$ is dominated by a nearest neighbor hyperfine coupling $C\approx 3.3 {\rm T}/\mu_B$ [@Zha96]. However, it was recently argued that a next-nearest neighbor hyperfine coupling $C^\prime\approx 0.25C$ is relevant for the explanation of the spin-lattice relaxation rate [@Zha96] in La$_{ 2-x}$Sr$_x$CuO$_4$. We will therefore also consider its effects on the $^{17}$O NMR line. Using a mean field description of this hyperfine coupling by replacing ${\bf s}_i$ by $\langle s^z_i \rangle$ of Eq. \[Sind\], we finally obtain for the shift of the resonance at a given $^{17}$O site ${\bf r}_l$ $$\begin{aligned}
\nu_l &=& \frac{A}{T} \sum_{i,j} C_{i,l} \chi'({\bf r}_{i}-{\bf r}_{j})\, .
\label{ox_shift}\end{aligned}$$ Here, the sum over $i$ runs over the Cu spin sites, coupled to the $^{17}$O nuclear spin, whereas the $j$-summation goes over all Ni-sites. Furthermore, the constant prefactor $A$ is given by $
\frac{5}{2}\gamma_n \gamma_e J \hbar C_{Curie} B_0 /( g \mu_B)^2$. Note, $\nu_l$ as given in Eq. \[ox\_shift\] is the shift of the $^{17}$O resonance with respect to the case without Ni impurities.
To obtain the $^{17}$O NMR line shape, we create a histogram $I_o(\nu)=\sum_l\delta(\nu-\nu_l)$ counting the number of nuclei with shift $\nu$. Since we want to compare the resulting distribution with the experimental data where the line has a finite width even in the absence of impurities, we convolute $I_o(\nu)$ with a Gaussian distribution $\exp\left(-\nu^2/(2\sigma^2)\right)/\sqrt{2 \pi \sigma^2}$, yielding the lineshape $I(\nu)$. By comparison with the experiments of Ref. [@BAY97] we expect that $\Delta \nu_0=\sqrt{2 \log 2} \sigma $ should be of the order of the high temperature (i.e. $\xi < 1$) Ni-impurity induced linewidth. In the following calculation we therefore choose $\sigma=20$kHz for both the Lorentzian and Gaussian $\chi'({\bf q})$. Finally, we define the resulting $\Delta \nu$ by half the width of the peak at half maximum.
\[t\]
=7.5cm
In Fig. \[shape\] we present the lineshape of the $^{17}$O NMR signal, calculated with the Lorentzian form $\chi_{\rm L}'({\bf q})$ for two different values of $\xi$. We clearly observe that the line becomes broader as we increase $\xi$. From a comparison of Eq.(\[ox\_shift\]) with the experimentally measured broadening we can extract the value of the interaction $J$ in Eq.(\[Hint\]). For $C'=0$ and $\xi(200{\rm K})=4$ ($\xi(200{\rm K})=3$) we obtain $J \approx 25 \
{\rm meV}$ ($43 \ {\rm meV}$). These values are accompanied by some uncertainties, but enable us to estimate the effects of a Cu spin mediated Ni-Ni spin (RKKY-type) interaction. We find within a self consistent mean field treatment of this interaction that the effect of the Ni-Ni interaction changes $\Delta \nu$ only within a few percent, consistent with the fact that no significant deviation from a Curie law was found in susceptibility measurements [@Mah94; @Men94].
In Fig. \[comp\] we present a comparison of $\Delta \nu(\xi)$ for the Gaussian $\chi_{\rm G}'({\bf q})$ (open diamonds) and the Lorentzian $\chi_{\rm
L}'({\bf q})$ (filled squares).
\[t\]
=7.5cm
Here we chose a Ni concentration of $x=2\%$, $C^\prime=0$, and ${\bf Q}=
\delta_{\rm i}
(\pm \pi, \pm \pi)$ to be incommensurate with $\delta_{\rm i}=0.94$ [@Mook97]. We also compute $\Delta \nu $ for the commensurate case, and find that in general $\Delta \nu$ decreases. However, since the incommensurability, $1-\delta_{\rm i}$, in YBa$_2$Cu$_3$O$_{6.6}$, if present at all, is rather small, differences are negligible for $\xi < 8$. In Fig. \[comp\], we clearly see that the effect of $\chi_{\rm G}'({\bf
q})$ and $\chi_{\rm L}'({\bf q})$ on the behavior of the line width is [*qualitatively*]{} different. In agreement with the results by Bobroff [*et al.*]{} we find using $\chi_{\rm G}'({\bf q})$ that $\Delta \nu$ is basically independent of $\xi$ for all physically reasonable values $2<\xi < 5$. The Lorentzian form $\chi_{\rm
L}'({\bf q})$, however, yields a much stronger increase in $\Delta \nu$ between $\xi=2$ and $\xi=5$ than the Gaussian. This result immediately implies that a temperature dependent $\xi$ is clearly compatible with the experimental results by Bobroff [*et al*]{}. Furthermore, we find that the function $f(\xi)$ of Eq. \[dnu\] behaves like $f(\xi) \sim \xi^{3/2} $ in the Lorentzian case and $f(\xi) \sim const.$ in the Gaussian case. This qualitatively different behavior of $ \Delta
\nu(\xi)$ for $\chi_{\rm G}'({\bf q})$ and $\chi_{\rm L}'({\bf q})$ makes this experiment extremely sensitive to details of the momentum dependence of $\chi'({\bf q})$.
Next we discuss the $\xi$ dependence of $\Delta \nu$ for different values of the Ni concentration $x$. We present our results for a Ni concentration of $x=0.5
\%, 2 \%$ and $4 \%$ and for $C'=0.25C$ in Fig. \[conc\].
\[t\]
=7.5cm
In agreement with the experimental results we find that $\Delta \nu$ for a given $\xi$ increases with $x$. We believe that the results in Figs. \[comp\] and \[conc\] also provide an explanation for the different behavior of $\Delta \nu$ in the underdoped (YBa$_2$Cu$_3$O$_{6.6}$) and overdoped (YBa$_2$Cu$_3$O$_{7}$) samples. Bobroff [*et al.*]{} obtained that for the overdoped sample the variation of $\Delta \nu$ with $T$ is much weaker than for the underdoped sample. As far as $\chi'({\bf q})$ is concerned, the main difference between these two regimes consist in the value of $\xi$, namely $\xi=1..2$ for the overdoped and $\xi=2..4$ for the underdoped sample. We see from Figs. \[comp\] and \[conc\] that the $\xi$ variation of $\Delta \nu$ for the overdoped sample is much weaker than for the underdoped one, in agreement with the experimental results.
Finally, we can use our numerical results to investigate in more detail the consequences of the $T$-dependence of $T \Delta \nu_{\rm imp} T_{\rm 2G}$ shown in Fig. \[prod\]. Using $g(\xi)\propto \xi$ [@Curro97; @Pines] and the above results for $f(\xi)$, it follows from Eq. \[product\] for the Gaussian case ($f(\xi)\sim const.$) that $T \Delta \nu_{\rm imp} T_{\rm 2G} \propto \xi^{-1}$, i.e. $\xi$ has to increase with increasing $T$. This result seems to be unphysical and thus strongly suggests that the Gaussian form $\chi_{\rm G}'({\bf q})$ is inappropriate for the description of the spin susceptibility. On the other hand, for the Lorentzian case, $ f(\xi) \sim \xi^{3/2}$ and it follows $T \Delta \nu_{\rm imp} T_{\rm 2G} \propto \xi^{1/2}$, i.e. $\xi$ decreases as $T$ increases, as we would expect. One can also solve Eqs.(\[dnu\]) and (\[t2g\]) to obtain $\alpha$ as a function of $T$. However, our result possesses error bars which are quite large. The conclusion that $\alpha$ is independent of $T$ is acceptable within those errorbars, however, a weak $T$ dependence cannot be excluded.
It is important to contrast our findings with the observations of INS experiments. In YBa$_2$Cu$_3$O$_{6+\delta}$, INS observes a $T$-independent broad peak around $(\pi,\pi)$, above $T_c$, resembling a Gaussian form of $\chi^{\prime \prime}({\bf q}, \omega)$ [@Kei; @Bou]. However, strong indications for incommensurate peaks with Lorentzian like shape in YBa$_2$Cu$_3$O$_{6.6}$ [@Mook97] suggest that the broad structure around $(\pi,\pi)$ is only a superposition of incommensurate peaks. Its width is therefore dominated by the merely $T$-independent incommensuration instead of $\xi^{-1}$. This is consistent with the recent analysis by Pines [@Pines] that the overall magnitude of $\chi''({\bf q},\omega)$ in YBa$_2$Cu$_3$O$_{6+\delta}$, as obtained from NMR experiments, necessitates a considerable improvement of the experimental resolution of INS experiments to resolve the incommensurate peaks in the normal state.
In conclusion we obtain from the analysis of the $^{17}$O NMR data by Bobroff [*et al.*]{} and the $T_{\rm 2G}$ data by Takigawa that the correlation length $\xi$ must have a substantial temperature dependence. A detailed analysis shows that the Gaussian form $\chi_{\rm G}'({\bf q})$ of the spin susceptibility can be excluded as an appropriate description of the spin dynamics in the doped cuprates. A more correct description is provided by a Lorentzian-type form $\chi_{\rm
L}'({\bf q})$, which is fully compatible with the experimental data and a temperature dependent $\xi$. Though the resolution of the experiment does not allow us yet to determine the precise $T$-dependence of $\xi$, our analysis shows that $\xi$ considerably decreases with increasing temperature.
This work has been supported by STCS under NSF Grant No. DMR91-20000, the U.S. DOE Division of Materials Research under Grant No. DEFG02-91ER45439 (C.P.S., R.S.) and the Deutsche Forschungsgemeinschaft (J.S.) We would like to thank H. Alloul, J. Bobroff, A. Chubukov, D. Pines and M. Takigawa for valuable discussions.
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C.H. Pennington and C.P. Slichter, Phys. Rev. Lett. [**66**]{}, 381 (1991). N. Curro, T. Imai, C. P. Slichter, and B. Dabrowski, Phys. Rev. B [**56**]{}, 877 (1997) J. Bobroff , H. Alloul, Y. Yoshinari, A. Keven. P. Mendels, N. Blanchard, G. Collin, J. F. Marucco, Phys. Rev. Lett. [**79**]{}, 2117 (1997). A.V. Mahajan, H. Alloul, G. Collin, J. F. Marucco, Phys. Rev. Lett. [**72**]{}, 3100 (1994). D.K. Morr, J. Schmalian, R. Stern, and C.P. Slichter, preprint cond-mat/9710257. P. Mendels, H. Alloul, G. Collin, N. Blanchard, J. F. Marucco, J. Bobroff, Physica C [**235-240**]{}, 1595 (1994). D. Pines, Z. Phys. B [**103**]{}, 129 (1997); Proc. of the NATO ASI on [*The Gap Symmetry and Fluctuations in High-T$_c$ Superconductors*]{}, J. Bok and G. Deutscher, eds., Plenum Pub. (1998), and references therein. D. J. Scalapino, Phys. Rep. [**250**]{}, 329 (1995). M. Takigawa, Phys. Rev. B [**49**]{}, 4158 (1994). We thank M. Takigawa for reanalyzing his original $T_{2{\rm G}}$ data in Ref.[@Tak94] using the T$_1$ corrections by Curro [*et al.*]{} [@Curro97]. The concentration of Ni impurities in the CuO$_2$ planes is given by $\frac{3}{2}x$, since predominantely Cu sites in the planes are substituted by Ni. A. J. Millis and H, Monien, Phys. Rev. B [**45**]{}, 3059 (1992). Y. Zha, V. Barzykin and D. Pines, Phys. Rev. B [**54**]{}, 7561 (1996).
| ArXiv |
---
abstract: |
0.2truecm
Classical strings coupled to a metric, a dilaton and an axion, as conceived by superstring theory, suffer from ultraviolet divergences due to self-interactions. Consequently, as in the case of radiating charged particles, the corresponding effective string dynamics can not be derived from an action principle. We propose a [*fundamental principle*]{} to build this dynamics, based on local energy-momentum conservation in terms of a well-defined distribution-valued energy-momentum tensor. Its continuity equation implies a finite equation of motion for self-interacting strings. The construction is carried out explicitly for strings in uniform motion in arbitrary space-time dimensions, where we establish cancelations of ultraviolet divergences which parallel superstring non-renormalization theorems. The uniqueness properties of the resulting dynamics are analyzed.
---
Preprint DFPD/2016/TH05\
April 2016\
**Dynamics of self-interacting strings and energy-momentum conservation** 0.3truecm
Kurt Lechner
1truecm
*Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Italy*
*and*
*INFN, Sezione di Padova,*
*Via F. Marzolo, 8, 35131 Padova, Italy*
2.0truecm Keywords: classical strings, self-interaction, renormalization, energy-momentum conservation, distribution theory. PACS: 11.25.-w, 11.30.-j, 11.10.Kk, 11.10.Gh, 02.30.Sa.
Introduction
============
In the same way as charged particles in four space-time dimensions are subject to divergent electromagnetic self-interactions, generic charged extended objects, $p$-branes, in $D$ space-time dimensions are subject to infinite self-interactions. The reason for this is that the fields created by a brane become singular on the brane world-volume, meaning that the [*self-fields*]{}, and hence the [*self-forces*]{}, are infinite. A - in a certain sense dramatic - consequence of these ultraviolet divergences is that the theory of self-interacting branes can not be derived from a variational principle: while the original fundamental equations of motion for fields and branes follow of course from an action principle, once one substitutes the fields resolving the formers in the equations of motion of the latter, the resulting equations are divergent. If one isolates and subtracts - adapting whatever prescription - the infinities, the resulting non-local equations of motion of the brane do no longer follow from an action principle. This in turn implies that the conservation laws, in particular energy-momentum conservation, can not be derived from Nöther’s theorem, see [*e.g.*]{} [@R; @LM; @KL0] for the case of self-interacting charged particles and dyons in $D=4$. Within this approach one looses thus the control over energy-momentum conservation.
More precisely ultraviolet divergences show up in brane theory in two, a priori, unrelated physical quantities: $i)$ in the [*self-force*]{} of the brane, [*i.e.*]{} the force exerted by the field generated by the brane on the brane itself, as explained above, and $ii)$ in the $D$-momentum contained in a volume $V$ enclosing (a portion of) the brane. Although the origins of the divergences appearing in these two quantities - the self-force and the $D$-momentum - are the same, [*i.e.*]{} the bad ultraviolet behavior of the field in the vicinity of the brane, their cures require actually two distinct unrelated procedures [@KL1].
To cure the divergent self-force one may proceed, as anticipated above, regularizing the field produced by the brane in some way, evaluating it on the brane and trying then to isolate and subtract the divergent terms.
The cure of the infinite $D$-momentum requires instead the construction of a well-defined [*distribution-valued*]{} energy-momentum tensor and offers - at the same time - a strategy for the derivation of the self-force, that is alternative to the approach described above and overcomes its main drawback, [*i.e.*]{} the missing control over energy-momentum conservation. It works as follows.
Generically the standard total energy-momentum tensor has the structure \^=\^\_[field]{}+ \_[kin]{}\^, \_[kin]{}\^=Ml\^\^D(x-y())d\^2, where $\tau_{\rm kin}^{\m\n}$ is the free [*kinetic*]{} energy-momentum tensor of the brane (with $M$ the brane tension and $y^\m(\s)$ the brane coordinates, see sections \[aad\] and \[eom\] for the notations) and $\tau^{\m\n}_{\rm field}$ is the [*bare*]{} energy-momentum tensor produced by the fields[^1]: while the fields - solutions of linear d’Alembert equations - are by definition distributions, the tensor $\tau^{\m\n}_{\rm field}$ - a product of the fields - is [*not*]{} a distribution. Consequently, $i)$ the $D$-momentum of the field $$P^\m_V=\int_V \tau^{0\m}_{\rm field}\, d^3x$$ contained in a volume $V$ is in general divergent and, $ii)$ it makes no sense to evaluate the divergence $\pa_\m \tau^{\m\n}_{\rm field}$ to analyze the conservation properties of $\tau^{\m\n}$. The cure of these pathologies requires the construction of a [*renormalized*]{} distribution-valued energy-momentum tensor $T^{\m\n}_{\rm field}$, out of $\tau^{\m\n}_{\rm field}$. A - in principle standard - way to do this consists in the introduction of a regularization - preserving possibly Lorentz- as well as reparameterization-invariance - and the subsequent subtraction from the regularized energy-momentum tensor $(\tau^{\m\n}_{\rm field})_{reg}$ of [*divergent local counterterms*]{}, [*i.e.*]{} of counterterms supported on the brane that do not converge to distributions as the regularization is removed. By construction the resulting energy-momentum tensor $T^{\m\n}_{\rm field}$ is a distribution and admits hence a well-defined divergence, supported on the word-volume, \[prel0\] \_T\^\_[field]{} =-\^\^D(x-y()) d\^p, where the vector ${\cal S}^\n$ is going to become the [*finite*]{} self-force of the brane. In fact, for the divergence of the renormalized total energy-momentum tensor $T^{\m\n}=T^{\m\n}_{\rm field}+ \tau_{\rm kin}^{\m\n}$ one obtains now \[prel\] \_T\^ =(M\_iU\^[i]{}- [S]{}\^)\^D(x-y())d\^p, where the quantity $\D_iU^{\n i}$ represents the generalized acceleration of the brane. Upon requiring local energy-momentum conservation one derives then the equation of motion for the brane coordinates \[fam\] M \_iU\^[i]{}= [S]{}\^.
This strategy to derive the self-force may however encounter an obstacle: it can happen that the vector ${\cal S}^\n$ in [(\[prel0\])]{} is not a pure [*multiplication*]{} operator but contains also terms involving derivatives acting on the $\delta$-function, as for example ${\cal S}^\n \sim \pa^\n$. In this case there would be no equation of motion for the brane ensuring the vanishing of $\pa_\m T^{\m\n}$. This obstacle can be faced through the [*finite-counterterm ambiguity*]{} inherent in any renormalization process in physics - in the present case the fact that after the subtraction of [*divergent*]{} local counterterms, the renormalized energy-momentum tensor is defined only modulo [*finite*]{} local counterterms.
The general strategy just described has been envisaged in [@KL1], where a $p$-brane interacting minimally with a $(p+1)$-form potential in $D$ dimensions has been considered, based on previous work facing the analogous problem for massive [@LM] as well as massless [@AL1; @AL2; @KL2] point-charges in four dimensions. The present paper represents the first step in the application of this method to the physically more interesting case of the low energy effective superstring theory, compactified to dimensions $D<10$, where the string couples to the metric $g_{\m\n}$, the dilaton $\Phi$ and the axion field $B_{\m\n}$. Particular attention will be paid to four-dimensional space-time. We will actually consider two prototype models: a) the [*general model*]{}, where a certain set of free parameters, or coupling constants, assume generic values, and b) the [*fundamental string model*]{}, where these parameters are tied by the special relations [(\[fundpar\])]{} predicted by ten-dimensional superstring theory.
The problem of ultraviolet divergences and self-interactions of strings moving in a space-time of dimension $D\ge4$ has a long history, especially w.r.t. the problem of tension renormalization and the related finiteness/divergence properties of the self-force and the self-energy. A far from exhaustive literature with this respect is [@DH; @CHH; @DQ; @DGHRR; @BS; @BC1; @C; @BC2; @BD12; @BD12bis; @CBU; @BCM]; for some recent results on the same problem for point-particles see [*e.g.*]{} [@FZ1; @FZ2; @HFT]. As observed above, by-hand subtractions of divergences from the self-force or from the self-energy - as the ones performed in these references - in general do not ensure energy-momentum conservation. On the contrary the core of our approach is a systematic renormalization of the energy-momentum tensor, comprising $i)$ a covariant separation of the - in the sense of distributions - [*divergent counterterms*]{}, $ii)$ the identification of possible finite counterterms and eventually, $iii)$ the implementation of energy-momentum conservation and the consequent derivation of the self-force. In the present paper the implementation of this program will be carried out explicitly for [*flat*]{} strings, [*i.e.*]{} for strings in uniform motion, already a non-trivial task, although in this case the self-force is expected to vanish. Being based essentially on the criteria of [*finiteness*]{} and [*energy-momentum conservation*]{}, we regard our approach as a [*fundamental principle*]{} for the determination of the dynamics of self-interacting extended objects. A particularly powerful aspect of the method - that supports its universality further - is that it is able to control even strong ultraviolet singularities, as for example the violent divergences generated by growing space-time dimensions or the [*a priori*]{} uncontrollable divergences generated by objects moving at the speed of light [@KL2].
Since with this respect the contribution of the gravitational self-energy is of fundamental importance, we have to face the problem of which gravitational energy-momentum [*pseudo-tensor*]{}, and hence which [*total*]{} energy-momentum pseudo-tensor, we choose. To test the “stability” of our construction against different choices we resort to three [*frameworks*]{}: a) in the, in a certain sense [*hybrid*]{}, [*Dirac*]{} framework the gravitational pseudo-tensor [@PAM] is based on the Nöther procedure, while the matter tensor is the [*symmetric*]{} one; b) in the [*Landau-Lifshitz*]{} framework both the gravitational pseudo-tensor [@LL] and the matter tensor are the symmetric ones; c) in the [*canonical*]{} framework both tensors are based on the Nöther procedure and correspondingly the total energy-momentum pseudo-tensor is neither symmetric nor gauge-invariant.
In the spirit of the above references we will analyze the the dynamics of the theory at the linearized level, see [*e.g.*]{} [@BCM], which corresponds to a perturbative treatment at first order in Newton’s constant $G$. In this setting the [*on-shell*]{} divergent parts of the self-force of the string turn, however, out to be of order $G^2$ [@BD12bis]. Consequently there is an intrinsic ambiguity in the tension renormalization, inherent in standard self-force approaches, in that at first order in $G$ the divergences simply drop out. These on-shell ambiguities are absent in our approach, since we do not impose any [*a-priori*]{} equation of motion on the string.
With respect to the case of a string interacting minimally with a two-form potential $B_{\m\n}$, the coupling to a metric and to a dilaton introduces additional ultraviolet singularities, due to the presence in the energy-momentum tensor of interaction-terms between the string and the fields, that are [*localized*]{} on the string world-sheet, see [(\[tint\])]{}. These divergences have a different origin w.r.t. the bulk-divergences of the energy-momentum tensor discussed above, and our approach entails the further advantage of separating them cleanly from the formers. This distinction is completely lost if one considers only the divergences of the total energy [@DH; @CHH] or of the total effective action [@BD12] - a feature that in the past has led to conflicting results concerning tension renormalization: these contradictions are clarified and solved by our approach.
Considering gravity, as well as the exponential interactions of the dilaton, at a full non-linear level leads in the presence of distributional sources, like strings, to further problems, that we will not face, see [*e.g.*]{} [@GT].
In the next section we present the action describing the microscopic dynamics which gives rise to self-interacting strings in $D$ space-time dimensions, and present the relevant gravitational energy-momentum pseudo-tensors. In section \[ld\] we linearize the dynamics, restricting correspondingly the energy-momentum tensors of the fields to their quadratic expressions, and present the solutions of the linearized equations of motion in terms of Green functions. In section \[cra\] we introduce a [*universal*]{} covariant ultraviolet regularization, preserving all fundamental symmetries, and present our general approach for the derivation of the self-force.
In section \[siu\] we apply this approach to strings in uniform motion, constructing a regularized energy-momentum tensor and performing its renormalization via subtraction of divergent counterterms, relying on the Dirac framework. Particular attention will be paid to the cancelation of ultraviolet divergences in the [*fundamental string*]{} model, that comprises the non-renormalization of the string tension. This latter property, in turn, is directly related to the non-renormalization theorems of superstring amplitudes [@DH], that are supposed to hold at all orders of perturbation theory. We find that, while in the Landau-Lifshitz and canonical frameworks for all $D\ge 4$ all divergences cancel, so that in particular the string tension gets not renormalized, in the Dirac framework these cancelations occur only for $D=4$. This may signal a conflict between this classical framework and the postulates of superstring theory. The subsection \[emc\] is devoted specifically to the energy-momentum-conservation paradigm and the role of finite counterterms in establishing the correct self-force - which for strings in uniform motion must vanish.
Sections \[llf\] and \[cfw\] are devoted respectively to the analogous analysis in the Landau-Lifshitz and canonical frameworks: while, as anticipated above, the actual cancelation of divergences depends on the choice of the framework, our general renormalization approach applies of course independently of the occurrence of those cancelations. In these sections we establish also the relations between our approach and the energy-divergences analysis of [@DH; @CHH] and the effective-action approach of [@BD12]. In section \[tgc\] we outline the steps to be carried out in the future to derive the dynamics of self-interacting strings in arbitrary motion and discuss the uniqueness properties of our approach. This more ambitious program of using our approach to compute the self-force explicitly and compare it, where possible, with known results, may shed new light on classical-string radiation reaction, on the causality issue and, may be more marginally, on the viability of cosmic string dynamics. The final section \[cr\] contains a concise summary of our results and of possible future developments.
Classical string dynamics {#aad}
=========================
We consider a classical string theory in $D$ space-time dimensions whose microscopic dynamics is determined by the action \[fands\] I=I\_f+I\_s, where the field-action $I_f$ and the string-action $I_s$ are given respectively by $$\begin{aligned}
\label{ib}
I_f &=\frac{1}{G}\int \left(-R+\frac{1}{12}\,e^{-2\a\Phi}H^{\m\n\rho}H_{\m\n\rho}+\frac12\, g^{\m\n}\pa_\m\Phi\, \pa_\n\Phi\right)\sqrt{g}\,d^Dx,\\
\label{is}I_s &=-M \int e^{\bt \Phi} \sqrt{\G}\,d^2\s -\frac{\La}{2} \int W^{\m\n}B_{\m\n} \sqrt{\G}\,d^2\s.\end{aligned}$$ We use indices $\m,\n=0,\cdots,D-1$ for the [*bulk*]{} space-time coordinates $x^\m$, with a mostly minus lorentzian signature, and indices $i,j=0,1$ for the [*world-sheet*]{} coordinates $\s^i$. The action $I$ is inspired by superstring theory in that it corresponds to the bosonic part of the low energy effective action of ten-dimensional $N=1$ supergravity, compactified to $D$ dimensions, in the Einstein frame [@CFMP; @GS; @LTZ]. Correspondingly the space-time fields to which the string couples are the dilaton $\Phi(x)$, the axion $B_{\m\n}(x)$ and the $D$-dimensional metric $g_{\m\n}(x)$. In [(\[ib\])]{} $R$ is the scalar curvature associated to $g_{\m\n}$ and $H_{\m\n\rho}=3\pa_{[\m}B_{\n\rho]}$ is the field strength of the axion. $G$ is related to Newton’s constant through $G_N=G/16\pi$.
In the string-action [(\[is\])]{} - that describes the string propagation as well as its interaction with the bulk fields - we introduced the string coordinates $y^\m(\s)$, with tangent vectors $U^\m_i(\s)=\pa_i y^\m(\s)$, and the induced world-sheet metric \[wsm\] \_[ij]{}=U\^\_i U\^\_j g\_, with inverse $\G^{ij}$. We introduced also the antisymmetric world-sheet tensor \[w2\] W\^=U\^\_i U\^\_j, =-det (\_[ij]{}). On the world-sheet the space-time metric can be decomposed in parallel and orthogonal projectors \[gdec\] g\^=L\^+K\^, L\^=U\^\_i U\^\_j \^[ij]{}. The parallel projector $L^{\m\n}$ is sometimes referred to as the [*first fundamental tensor*]{}. Bulk indices and world-sheet indices are raised and lowered respectively with the metrics $g_{\m\n}$ and $\G_{ij}$ and their inverses.
By definition, the dimensionless parameters $\a$ and $\bt$ and the dimension-one parameters $M$ and $\La$, respectively the [*tension*]{} and the [*charge*]{} of the string, are arbitrary in the [*general*]{} model. As we anticipated in the introduction, we will pay particular attention to the [*fundamental string*]{} model where they assume the values [@CFMP] \[fundpar\] ==, M=. This will allow us on one hand to probe the non-renormalization properties of a superstring-inspired model [@DH; @CHH; @BC2; @BD12], and on the other to analyze the consistency and renormalizability properties of a generic self-interacting classical string model.
Inspired by superstring theory we will assume that the dilaton takes generically a non-vanishing vacuum expectation value $\langle\Phi\rangle \equiv \Psi $, so that, denoting its fluctuation by $\vp$, we have \[dil\] =+, =0.
Equations of motion
-------------------
The equations of motion for $\Phi$, $B_{\m\n}$, $g_{\m\n}$ and the string coordinates $y^\m$ arising from the action $I=I_f+I_s$ are $$\begin{aligned}
g^{\m\n}D_\m \pa_\n \Phi+\frac{\a}{6}\,e^{-2\a\Phi}H^{\m\n\rho}H_{\m\n\rho}&= -GM\bt \int e^{\bt \Phi}\,\frac{\dl^D(x-y)}{\sqrt{g}}\,\sqrt{\G}\,d^2\s,\label{eqfi}\\
D_\m\left(e^{-2\a\Phi}H^{\m\n\rho}\right)&=-G\La\int W^{\n\rho}\, \frac{\dl^D(x-y)}{\sqrt{g}}\sqrt{\G}\,d^2\s,\label{eqax}\\
G^{\m\n}\equiv R^{\m\n}-\frac{1}{2}\,g^{\m\n}R&=\frac{G}{2}\, \Theta^{\m\n},\label{eqg}\\
Me^{\bt\Phi}\left(D_iU^{\m i}-\bt K^{\m\n}\pa_\n \Phi\right)&=\frac{\La}{2}\,H^{\m\n\rho}W_{\n\rho}.
\label{eqst}\end{aligned}$$ $G^{\m\n}$ is the Einstein tensor built with the metric $g_{\m\n}$ and the generalized acceleration $D_iU^{\m i}$ of the string coordinates $y^\m$ is given by \[acc\] D\_iU\^[i]{}=\_i(\^[ij]{}U\_j\^)+\^\_ L\^, where $\G^\m_{\nu\rho}$ is the affine connection built with $g_{\m\n}$.
The [*matter*]{} energy-momentum tensor $\Theta^{\m\n}$ decomposes into a bulk contribution, due to the fields $\Phi$ and $B$, and a string contribution, supported on the world-sheet, \[sumsb\] \^=\^\_b+\_s\^, given by $$\begin{aligned}
\Theta_b^{\m\n}&=\frac{1}{G}\left(D^\m\Phi D^\n\Phi-\frac{1}{2}\,g^{\m\n}D^\rho\Phi D_\rho\Phi +\frac{1}{2}\,e^{-2\a\Phi}\left(H^{\m\rho\s}H^\n{}_{\rho\s}-\frac{1}{6}\,g^{\m\n}
H^{\rho\s\la}H_{\rho\s\la}\right)\right)\label{enfb},\\
\Theta^{\m\n}_s&=M\int \label{ts} e^{\bt\Phi}L^{\m\n}\,\frac{\dl^D(x-y)}{\sqrt{g}}\sqrt{\G}\,d^2\s.\end{aligned}$$ Obviously in $\Theta^{\m\n}_s $ there is no contribution from the axion field $B_{\m\n}$ because its minimal coupling to the string in [(\[is\])]{}, being topological, does not contain the metric.
Computing [*mechanically*]{} the covariant divergence of $\Theta^{\m\n}$ one obtains the identity
\[consform\] D\_\^=&H\^\_D\_(e\^[-2]{}H\^) +(g\^D\_\_+ e\^[-2]{}H\^H\_)D\^\
&+Me\^(D\_iU\^[i]{}+L\^\_) d\^2,
and if one uses the equations [(\[eqfi\])]{}, [(\[eqax\])]{} and [(\[eqst\])]{} one gets obviously $D_\m\Theta^{\m\n}=0$. As stressed in the introduction, the operations leading to [(\[consform\])]{} have however only [*formal*]{} validity, in that $\Theta^{\m\n}$ is not a distribution - it diverges too violently in the vicinity of the string - and hence its $D$-divergence “$\pa_\m \Theta^{\m\n}$” is meaningless. [*A fortiori*]{} one is not allowed to resort to the Leibnitz-rule $\pa_\mu(f_1f_2)=\pa_\m f _1 f_2+f_1\pa_\m f_2$, that has been used thoughtless to derive [(\[consform\])]{}.
Gravitational energy-momentum pseudo-tensors
--------------------------------------------
Since the implementation of $D$-momentum conservation requires an energy-momentum tensor that satisfies a [*standard*]{} continuity equation, before attacking the renormalization issue we must recast the formal equation $D_\m\Theta^{\m\n}=0$ in a (still formal) equation of the type $\pa_\m \tau^{\m\n}=0$, for some pseudo-tensor $\tau^{\m\n}$. A standard continuity equation is also in line with our distributional framework in that the $D$-divergence of a distribution - as $\tau^{\m\n}$ should eventually be - is [*always*]{} a distribution, while on the contrary an object like $D_\m \Theta^{\m\n} \sim \pa \Theta +\G \Theta$ - involving [*products*]{} between distributions - would not be so.
To attack this problem we must face first the issue of the - non unique - [*gravitational*]{} energy-momentum pseudo-tensor. We resort to three different choices, giving rise to the three different conservation [*frameworks*]{} described in the introduction.
### Dirac’s energy-momentum pseudo-tensor
The distinguished feature of Dirac’s gravitational energy-momentum pseudo-tensor $\Sigma^{\m}{}_\n$ [@PAM] is that it descends canonically from Nöther’s theorem, applied to the Einstein-Hilbert action. It carries one upper and one lower index and reads \[TD\] \^\_= ((\^\_-\^\_\_\^)\_( g\^) -\^\_(\^\_-\^\_\_\^)\_( g\^)). Notice that $\Sigma^{\m}{}_\n$ is quadratic in the first derivatives of the metric. The term multiplying $\dl^\m{}_\n$ is related to the Einstein-Hilbert action through the identity $$-\frac{1}{G} \int\! R\sqrt{g}\,d^Dx=\frac{1}{2G}\int\! \left(\G^\g_{\a\bt}-\delta^\g_\a\G_{\bt\la}^\la\right)\pa_\g\left(\sqrt{g}
g^{\a\bt}\right) d^Dx,$$ [*i.e.*]{} it differs from $R\sqrt{g}$ by total derivatives, and represents thus an equivalent quadratic lagrangian.
As shown by Dirac, $\Sigma^{\m}{}_\n$ is tied to the Einstein tensor $G^{\m\n}=R^{\m\n}-\frac{1}{2}\,g^{\m\n}R$ through the identity \[idenD\] \_(\^\_+G\^\_)=0. Introducing the [*total*]{} energy-momentum tensor - actually a pseudo-tensor, too - with one upper and one lower index \[tott\] \^\_\^\_+\^\_, from [(\[idenD\])]{} and $D_\m G^\m{}_\n=0$ we derive that it satisfies the identity \[iden0\] \_\^\_= D\_\^\_- (G\^- \^)\_g\_. Since the matter energy-momentum tensor $\Theta^\m{}_\n$ satisfies the identity [(\[consform\])]{}, $\tau^\m{}_\n$ obeys the continuity equation $\pa_\m \tau^\m{}_\n=0$, if all fields satisfy their equations of motion [(\[eqfi\])]{}-[(\[eqst\])]{}.
Since [(\[idenD\])]{} is an algebraic identity we infer the existence of a tensor $P^{\rho\m}{}_\nu$ - antisymmetric in $\rho$ and $\m$ and built only with $g_{\m\n}$ - such that \[d1\] G\^\_=(\_P\^\_-\^\_). A direct calculation gives[^2] \[wex\] P\^\_= \_(gg\^)g\_. Actually equations [(\[d1\])]{} and [(\[wex\])]{} could be taken equivalently as the defining equations for $\Sigma^\m{}_\n$.
### The Landau-Lifshitz energy-momentum pseudo-tensor
In analogy to [(\[d1\])]{} and [(\[wex\])]{}, the Landau-Lifshitz gravitational energy-momentum pseudo-tensor $\w\Sigma^{\m\n}$ [@LL] - a [*symmetric*]{} tensor with two upper indices - is defined through the relations \[ll\] gG\^=(\_P\^-\^), where the tensor $\w P^{\rho\m\n}$, antisymmetric in $\rho$ and $\m$, is by definition [@LL] \[z\] P\^= g P\^\_g\^= \_(gg\^). Like $\Sigma^\m{}_\n$ also $\w\Sigma^{\m\n}$ can be seen to be quadratic in the first derivatives of the metric, and from [(\[ll\])]{} follows the identity $$\pa_\m\left(\w\Sigma^{\m\n}+\frac{2}{G}\,gG^{\m\n}\right)=0,$$ analogous to [(\[idenD\])]{}. In this framework the [*total*]{} energy-momentum tensor, with two upper indices, is defined by \[ttilde\] \^g\^+\^, and thanks to $D_\m G^{\m\n}=0$ this time one arrives at \[iden1\] \_\^= g D\_\^+(G\^- \^)(\^\_-\_\^\_), counterpart of [(\[iden0\])]{}. The r.h.s. vanishes again if the equations [(\[eqfi\])]{}-[(\[eqst\])]{} hold.
With the help of [(\[z\])]{} we can establish an explicit link between $\tau^\m{}_\n$ and $\w \tau^{\m\n}$. Equating the r.h.s. of [(\[ll\])]{} with $\sqrt{g}$ times the r.h.s. of [(\[d1\])]{} with the index $\n$ raised, we establish first the link between the pseudo-tensors $\w\Sigma^{\m\n}$ and $\Sigma^\m{}_\n$ \[dll\] \^=\^\_ g\^+ P\^\_\_(g\^), that is consistent with the fact that both tensors are quadratic in $ \pa g$. [(\[dll\])]{} implies then the relation between the total energy-momentum tensors [(\[tott\])]{} and [(\[ttilde\])]{} \[tot12\] \^= \^\_ g\^+ P\^\_\_(g\^). From this relation, using again [(\[tott\])]{} and [(\[d1\])]{}, one finds eventually that the two total energy-momentum tensors are connected through a three-tensor $L^{\rho\m\n}$, antisymmetric in its first two indices, modulo equations of motion, as it should be: \[t12\] \^=\^\_\^+\_L\^-( G\^\_-\^\_)(g\^-\^), where \[l3\] L\^=P\^\_(g\^-\^). The analysis of this paper will be performed primarily in the Dirac framework, based on the energy-momentum tensor $\tau^\m{}_\n$. Equations [(\[tot12\])]{} and [(\[t12\])]{} will then be used to translate this analysis to the Landau-Lifshitz framework, based on $\w\tau^{\m\n}$
### The canonical energy-momentum pseudo-tensor
By definition the [*canonical*]{} total energy-momentum tensor $\wh\tau^\m{}_\n$ follows from Nöther’s theorem applied to the whole action [(\[fands\])]{}. Consequently it differs from Dirac’s choice [(\[tott\])]{} by the divergence of a three-tensor, antisymmetric in its first two indices, modulo the equations of motion of the axion[^3]: \[cano\] \^\_\^\_+\_S\^\_+(D\_(e\^[-2]{}H\^)+G W\^ d\^2)B\_. The tensor $S^{\rho\m}{}_\n$ is quadratic in the axion and reads \[sss\] S\^\_=e\^[-2]{}H\^B\_= -S\^\_. The major shortcoming of the tensor [(\[cano\])]{} is that it is no longer gauge-invariant under $\dl B_{\m\n} =\pa_\m\La_\n -\pa_\n\La_\m$. Inserting [(\[sss\])]{} in [(\[cano\])]{} we obtain the relation \[cano1\] \^\_= \^\_- e\^[-2]{}H\^\_B\_-W\^B\_\^D(x-y)d\^2. The second term at its r.h.s. amounts in [(\[enfb\])]{} to the replacement $$H^{\m\rho\s}H_{\n\rho\s}\quad\rightarrow\quad
H^{\m\rho\s}\pa_\n B_{\rho\s},$$ while the third term represents a modification of the world-sheet term [(\[ts\])]{}, corresponding to the replacement $$Me^{\bt\Phi}L^\m{}_\n\quad\ra\quad
Me^{\bt\Phi}L^\m{}_\n-\La W^{\m\s}B_{\s\n}.$$ From [(\[iden0\])]{} and [(\[cano\])]{} we deduce that $\pa_\m\wh\tau^\m{}_\n=0$, if the fields satisfy [(\[eqfi\])]{}-[(\[eqst\])]{}.
Linearized dynamics {#ld}
===================
As stated previously we consider the theory in the linear regime, which corresponds to an analysis at first order in Newton’s constant $G$. This amounts to keep in the field-action [(\[ib\])]{} the terms quadratic in the fields, and in the string-action [(\[is\])]{} the terms linear in the fields. Correspondingly in the equations of motion [(\[eqfi\])]{}-[(\[eqst\])]{} we must keep only the terms linear in the fields.
We write the dilaton as in [(\[dil\])]{} as its constant vacuum expectation value $\Psi$ plus a fluctuation $\vp$, [*i.e.*]{} $\Phi(x)=\Psi +\vp(x)$. Moreover, to simplify the formalism, we parameterize the fluctuation of the space-time metric in terms of a symmetric field $F^{\m\n}$, specified by \[gf\] g\_=\_+h\_\_+F\_-\_F,FF\_\^, $F_{\m\n}=h_{\m\n}-\frac{1}{2}\,\eta_{\m\n}h^\a{}_\a$ and $F=\left(1-\frac{D}{2}\right)h^\a{}_\a$. This choice is convenient in that we have, in any dimension $D\ge4$, \[ghat\] g\^=\^-F\^+o(F\^2). In particular, the harmonic gauge for diffeomorphisms $\pa_\m(\sqrt{g}g^{\m\n})=0$, that we will use throughout the rest of the paper, assumes then the simple linearized form \[hamonic\] \_F\^=0. For the axion we use the Lorenz-gauge $\pa_\m B^{\m\n}=0$. Henceforth all indices will be raised and lowered with the flat metric $\eta_{\m\n}$. In conclusion, the linearization will be in the fields $f=\{\vp, B_{\m\n}, F_{\m\n}\}$.
Equations of motion {#eom}
-------------------
The linearized equations of motion [(\[eqfi\])]{}-[(\[eqst\])]{} read $(\square =\pa_\m\pa^\m$) $$\begin{aligned}
\square\vp=& - e^{\bt\Psi}GM\bt \int\dl^D(x-y)\sqrt{\g}\,d^2\s,\label{eqfil}\\
\square B_{\m\n} &=-e^{2\a\Psi} G\La\int w_{\m\n}\, \dl^D(x-y)\sqrt{\g}\,d^2\s,\label{bmn}\\
\square F_{\m\n}=&-e^{\bt\Psi} GM\int l_{\m\n}\, \dl^D(x-y)\sqrt{\g}\,d^2\s,\label{eqgl}\\
Me^{\bt\Psi} \Delta_i U^{\m i}=&\frac{\La}{2}\,H^{\m\a\bt}w_{\a\bt}+
Me^{\bt\Psi}\left[k^{\m\n}\left(\left(\frac{1}{2}\,\pa_\n F_{\a\bt}-\pa_\a F_{\bt\n}\right)l^{\a\bt}+\frac{1}{D-2}\,\pa_\n F-\bt\pa_\n\vp\right)\right.\nn\\
&\left.l^\m{}_{\a\bt}F^{\a\bt} -k^{\m\n}F_{\n\rho}\,\Delta_i U^{\rho i}+\left(\frac{F}{D-2}-\frac{1}{2}\,l^{\a\bt}F_{\a\bt}\right)\Delta_i U^{\m i}
\right] \equiv {\cal S}^\m_{bare},
\label{ill}
\end{aligned}$$ where we introduced the flat-space counterparts of the tensors $\G_{ij}$, $L^{\m\n}$, $K^{\m\n}$ and $W^{\m\n}$ in [(\[wsm\])]{}-[(\[gdec\])]{} \[flat\] \_[ij]{}=U\_i\^U\_j\^\_, =-det \_[ij]{},l\^=U\^\_i U\^\_j\^[ij]{},k\^=\^-l\^, w\^=U\^\_i U\^\_j, obeying the relations \[proj\] l\^l\_=l\^\_,k\^k\_=k\^\_,k\^l\_=0,w\^w\^=l\^l\^-l\^l\^. Correspondingly $l^{\m}{}_{\m}=2$ and $k^\m{}_\m=D-2$. $\Delta_i$ is the covariant derivative w.r.t. the flat-target-space world-sheet metric $\g_{ij}$ and $\Delta_i U^{\m i}$ is the flat-target-space reparameterization invariant acceleration of the string \[acclin\] \_i U\^[i]{}=\_i(\^[ij]{}U\_j\^). In [(\[ill\])]{} $l^\m{}_{\a\bt}$ represents the [*second fundamental tensor*]{}, which can be expressed in different ways and entails several properties [*e.g.*]{} ($\Delta_\a\equiv U_\a^i\Delta_i$) \[2fund\] l\^\_=l\_\_l\^=k\^\_l\_,l\^\_=l\^\_, l\_l\^\_=0.
Equations [(\[eqgl\])]{} are the linearized Einstein equations. Applying $\pa_\m$ to both sides of them one arrives formally at a mismatch, since at its r.h.s. one gets a non-vanishing term: $\square \pa_\m F^{\m\n}\sim G\Delta_iU^{\n i}\neq 0$. This is a remnant of the peculiar property of Einstein’s equations to [*imply*]{} the geodesic equation of motion, in the present case the string equation of motion. At the linearized level this does however not lead to an inconsistency; in fact, since $\Delta_iU^{\n i}$ eventually equals the finite self-force - which is of order $G$ - the above mismatch is of order $o(G^2)$.
Just for the sake of completeness above we wrote out also the linearized version [(\[ill\])]{} of the string equation of motion [(\[eqst\])]{} that, contrary to the field equations [(\[eqfil\])]{}-[(\[eqgl\])]{}, is actually ill-defined. In fact, in [(\[ill\])]{} the fields $f(x)$ are evaluated at the world-sheet $x=y(\sigma)$, where they diverge, so that the bare self-force ${\cal S}^\m_{bare}$ is infinite. There is moreover an intrinsic ambiguity in this equation concerning the renormalization of the string tension $M$, due to the appearance of the acceleration $\Delta_i U^{\m i}$ also at its right hand side, where it is multiplied by the self-fields $f(y(\s))$. Since the latter are of order $G$, as is the acceleration $\Delta_i U^{\m i}$, the terms of the kind $\Delta_i U^{\m i}f(y(\s))$ are actually of order $G^2$ and should have therefore be omitted in [(\[ill\])]{} from the beginning. Similarly (the divergent parts of) the terms $\pa f(y(\s)) $ in the first line of [(\[ill\])]{} have the structure $\pa f(y(\s))\sim G\Delta_i U^{\m i}\sim G^2$ [@BC2; @BCM]. Consequently, as observed already in [@BD12bis], in a first-order setting as the present one, it appears intrinsically impossible to perform an unambiguous [*quantitative*]{} analysis of tension renormalization, [*upon renormalizing directly ${\cal S}^\m_{bare}$ at the basis of*]{} [(\[ill\])]{}.
According to our approach, our starting point to derive a finite self-force will actually not be [(\[ill\])]{}, but rather energy-momentum conservation.
Linearized total energy-momentum tensors
----------------------------------------
### Dirac framework
We present first the linearized version of the total energy-momentum tensor [(\[tott\])]{} of the Dirac framework. We write it as a sum of three terms, each term having its specific physical meaning. For notational convenience we write it with two flat [*upper*]{} indices - an operation that preserves the formal conservation law $\pa_\m \tau^{\m\n}=0$ - albeit maintaining for the linearized version the same symbol $\tau$ as for the exact one: \[123\] \^\^\_|\_[lin]{}\^ =\_f\^ +\_[int]{}\^+\_[kin]{}\^. The first term, the [*field*]{} energy-momentum tensor $\tau_f^{\m\n}$, represents the energy due solely to the fields and is supported on the bulk. It is obtained extracting from [(\[enfb\])]{} and [(\[TD\])]{} the terms quadratic in the fields $f=\{\vp,B,F\}$
\[tf\] \^\_f=& (\_b\^g\_+\^\_)|\_[f\^2]{} \^\
=&{\^\^-\^\^\_+e\^[-2]{}(H\^H\^\_ -\^ H\^H\_).\
+&\^F\^\^F\_-\^F\^\^F\_ -\^F\^F\
-&.\^(\^F\^\_F\_-\^F\^\_F\_ -\^F\_F)}.
The gravitational contribution of $\tau^{\m\n}_f$ in the last two rows is not symmetric in its indices - a characteristic feature of the Dirac tensor [(\[TD\])]{}.
The [*interaction*]{} energy-momentum tensor $\tau^{\m\n}_{int}$ arises from the interaction between the fields and the string and is hence supported on the world-sheet. It is obtained extracting from $\sqrt{g}\,\Theta^{\m\bt}_sg_{\bt\rho}$, see [(\[ts\])]{}, the terms linear in the fields:
\^\_[int]{}&=(\^\_sg\_) |\_f \^\
&=Me\^((l\^F\_- +)l\^-l\^l\^F\_+l\^ F\_\^)\^D(x-y)d\^2.\[tint\]
Only the gravitational field and the dilaton contribute to $\tau^{\m\n}_{int}$, but not the axion, for the reasons explained above.
The term $\tau_{\rm kin}^{\m\n}$ represents the free [*kinetic*]{} energy-momentum tensor of the string and is obtained from $\Theta_s^{\m\n}$ [(\[ts\])]{} setting all fields $f=\{\vp,B,F\}$ to their background values, [*i.e.*]{} zero, \[tslin\] \_[kin]{}\^=(\_s\^g\_) |\_[f=0]{}\^=Me\^l\^\^D(x-y)d\^2.
### Landau-Lifshitz framework
In the framework of Landau and Lifshitz the linearized equations of motion [(\[eqfil\])]{}-[(\[eqgl\])]{} remain clearly the same, what changes is the form of the energy-momentum tensor $\w \tau^{\m\n}$ in [(\[ttilde\])]{}. The most simplest way to write it down is to use its relation to the Dirac-tensor [(\[tot12\])]{}. Setting as in [(\[123\])]{} - from now on with the symbol $\w\tau^{\m\n}$ we understand its linearized version - \[123ll\] \^=\_f\^+\_[int]{}\^+\_[kin]{}\^, from [(\[tot12\])]{} we see that $\w\tau_f^{\m\n}$ receives additional contributions from the gravitational field and that, due to the presence of the factor $\sqrt{g}g^{\rho\n}$ in the first term of [(\[tot12\])]{}, also the interaction-term changes: $$\begin{aligned}
\label{fdll}
\w\tau^{\m\n}_f&=\tau_f^{\m\n}+ \frac{1}{G}\left(\pa_\a F_\bt{}^\m \pa^\a
F^{\bt\nu}+\pa_\a F^{\a\bt}\pa_\bt F^{\m\n}-\pa_\a F^{\a\m}\pa_\bt F^{\bt\n}-\pa^\a F^{\bt\n}\pa^\m F_{\a\bt} \right),\\
\w\tau^{\m\n}_{int}&=\tau^{\m\n}_{int}-
Me^{\bt\Psi}\int l^{\m\a} F_\a{}^\n\,\dl^D(x-y)\sqrt{\g}\,d^2\s.\label{intll}\end{aligned}$$ To derive [(\[fdll\])]{} we used in particular the linearized version of [(\[wex\])]{} \[wexl\] P\^\_=(\^[\[]{}F\^[\]]{}\_+\_F\^), following from [(\[ghat\])]{}. Obviously $\tau_{\rm kin}^{\m\n}$, the free energy-momentum tensor of the string, remains the same. It is easily seen that the tensor $\w \tau^{\m\n}$ given by [(\[123ll\])]{}-[(\[intll\])]{} is symmetric.
### Canonical framework {#cf}
Writing also the linearized version of the canonical total energy-momentum tensor in the form \[canlin\] \^=\_f\^+\_[int]{}\^+\_[kin]{}\^, from [(\[cano1\])]{} we obtain $$\begin{aligned}
\label{fdcan}
\wh\tau^{\m\n}_f&=\tau_f^{\m\n}-\frac{1}{G}\,e^{-2\a\Psi}H^{\m\rho\s}\pa_\rho B_\s{}^\n,
\\
\wh\tau^{\m\n}_{int}&=\tau^{\m\n}_{int}-\La
\int w^{\m\s}B_\s{}^\n\,\dl^D(x-y)\sqrt{\g}\,d^2\s.\label{intcan}\end{aligned}$$ Thanks to (the linearized versions of) [(\[iden0\])]{}, [(\[iden1\])]{} and [(\[cano\])]{}, the [*formal*]{} conservation laws \[conslin\] \_\^=0, \_\^=0, \_\^=0 hold, if the fields and the string coordinates satisfy the equations of motion [(\[eqfil\])]{}-[(\[ill\])]{}. Since singularities do arise only on the world-sheet, and the tensors $\tau_{int}^{\m\n}$, $\w\tau_{int}^{\m\n}$, $\wh\tau_{int}^{\m\n}$, as well as $\tau_{\rm kin}^{\m\n}$, are supported on the world-sheet, too, the formal equations [(\[conslin\])]{} imply that, if only the fields satisfy their equations of motion [(\[eqfil\])]{}-[(\[eqgl\])]{}, in the [*complement of the world-sheet*]{} the field tensors satisfies the [*true*]{} conservation laws \[fieldt\] \_\^\_f=0, \_\^\_f=0, \_\^\_f=0. This property will become crucial later one.
Regularized field solutions and renormalization {#cra}
===============================================
We address now the solutions of the equations [(\[eqfil\])]{}-[(\[eqgl\])]{} obeyed by the fields $f=\{\vp,B,F\}$. They are all of the d’Alembert-type \[square\] f(x)=j()\^D(x-y())d\^2and admit thus the solution \[fx\] f(x)=(x-y()) j()d\^2, where ${\cal G}(x)$ - the retarded Green function in $D$ space-time dimensions - satisfies the equation $\square {\cal G}(x)=\dl^D(x)$; for explicit expressions see [(\[greene\])]{} with $\ve=0$.
Singularities and distributions
-------------------------------
Denoting generically the $D-2$ coordinates orthogonal to the string world-sheet with $x_\perp$, in the vicinity of the string, that is for $x_{\perp}\ra 0$, the fields [(\[fx\])]{} diverge schematically as (see for example [(\[fe1\])]{}-[(\[fe3\])]{} with $\ve=0$) \[fxdiv\] f(x)\~
,&D>4,\
x\_,&D=4.
The behaviors [(\[fxdiv\])]{} represent [*distributional*]{} types of singularity: the fields $f(x)$ in [(\[fx\])]{} [*are*]{} indeed distributions. In contrast the [*bare*]{} field energy-momentum tensor [(\[tf\])]{} diverges for $x_{\perp}\ra 0$ as \[tmndiv\] \^\_f(x)\~f(x)f(x)\~, a behavior that is [*not*]{} of the distributional type[^4], unless $D<4$. Said in other words, for $D\ge 4$ the functions $\tau^{\m\n}_f(x)$ are not distributions.
Similarly, also the interaction energy-momentum tensor $\tau_{int}^{\m\n}$ [(\[tint\])]{} is ill-defined, because the self-fields $f(y(\s))=f(x)|_{x_\perp =0}$ appearing therein are infinite. Contrary to the singularities of $\tau^{\m\n}_f$, the singularities of $\tau_{int}^{\m\n}$ are hence [*strongly*]{} local, [*i.e.*]{} they are localized on the world-sheet as is the whole $\tau_{int}^{\m\n}$. Correspondingly their subtraction encounters no technical difficulty, so that the finite part of $\tau_{int}^{\m\n}$ gives rise directly to a [*renormalized*]{} interaction energy-momentum tensor $T^{\m\n}_{int}$, see [(\[tmnint\])]{} below.
By contrast the construction of a renormalized field energy-momentum tensor $T^{\m\n}_f$ out of $\tau_f^{\m\n}$ is more involved. We impose on $T^{\m\n}_f$ the [*minimal*]{} conditions:
- $T^{\m\n}_f(x)$ is a tempered distribution, [*i.e.*]{} an element of ${\cal S}'(\mathbb{R}^D)$;
- $T^{\m\n}_f(x)=\tau^{\m\n}_f(x)$, if $x$ belongs to the complement of the world-sheet.
Condition a) is a necessary [*pre*]{}-consistency condition for local energy-momentum conservation: the distributional divergence $\pa_\m T^{\m\n}_f$ of a distribution is indeed always a distribution. Condition b) says instead that we want to modify $\tau_f^{\m\n}$ “as little as possible”, [*i.e.*]{} we do not want to change its values in the complement of the world-sheet, since there it is regular. This condition represents a cornerstone of our approach.
By construction conditions a) and b) determine $T^{\m\n}_f$ modulo terms supported on the world-sheet: this is the aforementioned [*finite-counterterm-ambiguity*]{}, that we have to take into account in the following.
Covariant regularization and renormalized energy-momentum tensor {#rem}
----------------------------------------------------------------
To construct out of the (ill-defined) tensor $\tau_f^{\m\n}$ a tensor $T^{\m\n}_f$ satisfying the above conditions a) and b), we need first of all a set of [*regular*]{} fields $f_\ve(x)$, that for $\ve\ra0$ tend (pointwise and in the sense of distributions) to the fields ${(\ref{fx})}$. In what follows $\ve$ is a positive regularization parameter with the dimension of a length. A convenient covariant regularization consists in replacing in the solutions [(\[fx\])]{} the Green functions ${\cal G}(x)$ with the regularized - but still Lorentz-invariant - Green functions (for $D=4$ see [@LR]) \_(x)=
[ ([ddx\^2]{})\^N(x\^2-\^2)]{},&D=2N+4,\
&\[greene\]\
[ ([ddx\^2]{})\^N[H(x\^2-\^2)]{}]{},&D=2N+3,
where $H(\,\cdot\,)$ is the Heaviside function and $x^2=x_\m x^\m$. In practical applications of these formulae it may be useful to replace the derivative $d/dx^2$ with $-d/d\ve^2$. The smoothed fields \[fxe\] f\_(x)=\_(x-y()) j()d\^2are now [*regular*]{} distributions and on the world-sheet one has in particular the small-$\ve$ behaviors, see below,
f\_(y())\~
, & D>4,\
, & D=4.
The main virtue of the regularization [(\[greene\])]{} is that it preserves manifest Lorentz- as well as reparameterization-invariance. Consequently the regularized field energy-momentum tensor $\tau_{f\ve}^{\m\n}$ - obtained from [(\[tf\])]{} replacing the fields $f$ with $f_\ve$ - are distributions, too, and they are covariant [*tensors*]{}. However, while in the complement of the world-sheet one has the [*point-wise*]{} limit $$\lim_{\ve\ra 0}\tau_{f\ve}^{\m\n}(x)=\tau_{f}^{\m\n}(x),$$ the [*distributional*]{} limit $${\cal S}'-\lim_{\ve\ra0}\tau_{f\ve}^{\m\n}$$ does not exist. Indeed, before taking this limit one must isolate from $\tau_{f\ve}^{\m\n}$ the term $\tau_{f\ve}^{\m\n}\big|_{div}$ that diverges as $\ve\ra0$ and that, in turn, must be supported on the world-sheet. The renormalized energy-momentum tensor $T^{\m\n}_f$ can then be defined subtracting this [*divergent counterterm*]{} and performing then the distributional limit \[tmnf\] T\^\_f’-\_[0]{}(\_[f]{}\^-\_[f]{}\^|\_[div]{}). By construction this tensor satisfies the above conditions a) and b).
Similarly one introduces a regularized interaction energy-momentum tensor $\tau_{int\,\ve}^{\m\n}$, replacing in [(\[tint\])]{} the fields $f$ with $f_\ve$, and subtracts then its divergent part obtaining the renormalized interaction energy-momentum tensor \[tmnint\] T\^\_[int]{}’-\_[0]{}(\_[int]{}\^-\_[int]{}\^|\_[div]{}). Although the formulae [(\[tmnf\])]{} and [(\[tmnint\])]{} are formally identical, in [(\[tmnint\])]{} the subtraction of the divergent counterterm, as anticipated above, will be a conceptually [*trivial*]{} operation, while the analogous process in [(\[tmnf\])]{} will require the whole apparatus of distribution theory.
Obviously the kinetic energy-momentum tensor $\tau_{\rm kin}^{\m\n}$ [(\[tslin\])]{} is well-defined by itself and needs no renormalization. Eventually we define then the [*total*]{} renormalized energy-momentum tensor as \[tmnr\] T\^=T\^\_f +T\^\_[int]{} + \_[kin]{}\^, that by construction is a distribution and coincides in the complement of the world-sheet with the original - bare - energy-momentum [(\[123\])]{}.
Energy-momentum conservation and self-force {#emca}
-------------------------------------------
Both properties $a)$ and $b)$ play an essential role in the implementation of energy-momentum conservation and in the derivation of the self-force. At the end of section \[cf\] we saw that the bare field energy-momentum tensor has the property $$\pa_\m \tau^{\m\n}_f(x)=0, \quad\mbox{if $x$ belongs to the complement of the world-sheet,}$$ thanks to the fact that the fields satisfy the linearized equations of motion [(\[eqfil\])]{}-[(\[eqgl\])]{}. But since by construction - see condition b) above - the tensor $T^{\m\n}_f$ [(\[tmnf\])]{} equals $\tau^{\m\n}_f$ in the complement of the world-sheet, it follows that the [*distributional*]{} divergence $\pa_\m T^{\m\n}_f$ is supported on the world-sheet. Since also $T^{\m\n}_{int}$ is supported on the world-sheet, and our whole construction preserves Lorentz- as well reparameterization-invariance, we derive the distributional relation \[ident1\] \_(T\^\_f + T\^\_[int]{})=-\^\^D(x-y())d\^2, where ${\cal S}^\n$ is some covariant vector defined on the world-sheet. Since the kinetic energy-momentum tensor of the string [(\[tslin\])]{} satisfies the identity, see [(\[acclin\])]{}, \_\_[kin]{}\^= Me\^\_i U\^[i]{} \^D(x-y())d\^2, imposing on the tensor [(\[tmnr\])]{} total energy-momentum conservation we obtain \[iden2\] \_T\^= (Me\^\_i U\^[i]{} - [S]{}\^) \^D(x-y())d\^2= 0. In this way we deduce the equation of motion for the self-interacting string \[eqself\] Me\^\_i U\^[i]{} = [S]{}\^, replacing the ill-defined equation [(\[ill\])]{}. Equation [(\[eqself\])]{} identifies the vector ${\cal S}^\m$ showing up in [(\[ident1\])]{} as the [*self-force*]{}.
As anticipated in the introduction, it could happen that ${\cal S}^\m$ is not a multiplicative vector, but contains also derivative operators, like \[sder\] [S]{}\^\~\^+A\^\_\^+. In this case [(\[iden2\])]{} could no longer be made to vanish upon imposing [(\[eqself\])]{}. As we will see, even in the most simplest case of a string in uniform motion, ${\cal S}^\m$ will actually contain terms like [(\[sder\])]{}, but those terms can always be eliminated thanks to the finite-counterterm-ambiguity.
Concerning this strategy to derive the self-force we insist on the fact that, in presence of singularities, there is no longer a [*fundamental*]{} principle - as the action principle - allowing to derive the dynamics of a theory, in particular the self-force. As we observed already, the alternative strategy based on the direct renormalization of the bare self-force [(\[ill\])]{}, as done [*e.g.*]{} in [@BD12bis; @BCM], entails no control on energy-momentum conservation: if the singularities are [*too strong*]{}, this strategy may even turn out to violate energy-momentum conservation, in which case it must be rejected; for a concrete example - regarding massless charges in four dimensions - see [@KL2]. The physical meaning of this potential conflict between different procedures to derive the dynamics of self-interacting objects in [*extremal*]{} cases, is still an open problem, to be investigated further. Its origin is however clear: the failure of the action principle to describe self-interactions.
Strings in uniform motion {#siu}
=========================
In this section we apply the procedure outlined in sections \[rem\] and \[emca\] to a flat string moving uniformly - in which case the entire program can be carried out analytically - thereby illustrating its internal consistency in a simple, although non-trivial, physical situation. In this case we expect of course to gain ${\cal S}^\m=0$. As above, in the following we will work out the details in the Dirac framework, relegating the differences that occur in the other two frameworks to sections \[llf\] and \[cfw\].
The world-sheet swept out by a string in uniform motion has the form \[sflat\] y\^()=U\^\_i\^i and correspondingly the tangent vectors $U^\m_i=\pa_i y^\m(\s)$ are [*constant*]{}, as are the geometric objects in [(\[flat\])]{}.
Regularized fields and energy-momentum tensors {#rfa}
----------------------------------------------
For a configuration like [(\[sflat\])]{} the regularized fields [(\[fxe\])]{} can computed analytically, upon reading the [*currents*]{} $j(\s)$ from [(\[square\])]{} and [(\[eqfil\])]{}-[(\[eqgl\])]{} and inserting the regularized Green functions [(\[greene\])]{}. The integral over the $\s^i$ in [(\[fxe\])]{} can be carried out explicitly for even as well as for odd $D$ - see [*e.g.*]{} the appendix in reference [@KL1] - and the regularized fields $f_\ve$ have the same analytical form for all $D>4$: $$\begin{aligned}
\label{fe1}
\vp_\ve(x)&=\displaystyle\frac{\bt G M e^{\bt\Psi}}{(4-D)\Omega_{D-2}
(-k_{\a\bt}x^\a x^\bt+\ve^2)^{\frac{D}{2}-2}},\\
B_\ve^{\m\n}(x)&=\displaystyle\frac{G\La e^{2\a\Psi}}{(4-D)\Omega_{D-2}
(-k_{\a\bt}x^\a x^\bt+\ve^2)^{\frac{D}{2}-2}}\,w^{\m\n},\\
F_\ve^{\m\n}(x)&=\displaystyle\frac{GM e^{\bt\Psi}}{(4-D)\Omega_{D-2}
(-k_{\a\bt}x^\a x^\bt+\ve^2)^{\frac{D}{2}-2}}\,l^{\m\n},\label{fe3}\end{aligned}$$ where we introduced the $(D-2)$-dimensional solid angle $$\Omega_{D-2}=\frac{2\pi^{\frac{D-2}{2}}}{\G\left(\frac{D-2}{2}\right)}.$$ For $D=4$ the integrals [(\[fxe\])]{} are infrared divergent due to the infinite spatial extension of a flat string. This is merely an artifact of the Green-function method, that for infinitely extended strings in $D=4$ does not work properly[^5]. In this case it is however easy to solve the equations [(\[square\])]{} from scratch[^6], and regularized solutions can be obtained upon replacing $k_{\a\bt}x^\a x^\bt \ra k_{\a\bt}x^\a x^\bt-\ve^2$: $$\begin{aligned}
\label{fe14}
\vp_\ve(x)&=\displaystyle\frac{\bt G M e^{\bt\Psi}}{4\pi}
\ln\left(\frac{-k_{\a\bt}x^\a x^\bt+\ve^2}{\la^2}\right),\\
B_\ve^{\m\n}(x)&=\displaystyle\frac{G\La e^{2\a\Psi} w^{\m\n}}{4\pi}\ln\left(\frac{-k_{\a\bt}x^\a x^\bt+\ve^2}{\la^2}\right),\\
F_\ve^{\m\n}(x)&=\displaystyle\frac{GM e^{\bt\Psi}l^{\m\n}}{4\pi}
\ln\left(\frac{-k_{\a\bt}x^\a x^\bt+\ve^2}{\la^2}\right).\label{fe34}\end{aligned}$$ For dimensional reasons we are obliged to introduce a parameter $\la$ with the dimension of length - in principle a new constant of the theory. When computing the field-strengths $\pa_\m f_\ve(x)$, appearing in the the regularized field energy-momentum tensor $\tau^{\m\n}_{f\ve}$, the constant $\la$ drops out. It will however survive in the regularized interaction energy-momentum tensor $\tau^{\m\n}_{int\,\ve}$, see [(\[tint\])]{}, where the fields [(\[fe14\])]{}-[(\[fe34\])]{} are evaluated on the world-sheet.
The regularized fields [(\[fe1\])]{}-[(\[fe34\])]{} depend in a simple way on $x^\m$ through the factor $-k_{\a\bt}x^\a x^\bt+\ve^2$, that is positive definite since the orthogonal projector to the string $k_{\a\bt}$ is negative definite. They depend in particular only on the $D-2$ orthogonal coordinates $k^{\m\n} x_\n$. The fields [(\[fe1\])]{}-[(\[fe34\])]{} are regular on the world-sheet: for $x^\m= y^\m(\s)=U^\m_i\s^i$ we in fact have $-k_{\a\bt}x^\a x^\bt+\ve^2=\ve^2\neq 0$. These fields are actually of class $C^\infty$ in whole $\mathbb{R}^D$ for all $D\ge 4$. For $\ve=0$, near the string they exhibit the singular behavior anticipated in [(\[fxdiv\])]{}.
Inserting the (derivatives of the) fields [(\[fe1\])]{}-[(\[fe34\])]{} in [(\[tf\])]{} we obtain a single analytic expression for the regularized field energy-momentum tensor, valid for all $D\ge 4$: \[tmnfe\] \^\_[f]{}=(C( k\^k\^x\_x\_-\^k\_x\^x\^)-\^2 e\^[2]{}l\^k\_x\^x\^). The coefficient $C$ has the expression \[c\] C= M\^2e\^[2]{}(\^2+)-\^2e\^[2]{}, which in the [*fundamental string*]{} model [(\[fundpar\])]{} is zero for all $D\ge4$. In [(\[tmnfe\])]{} the contributions from the scalar field are those proportional to $M^2\bt^2$, those from the gravitational field are the ones proportional to $M^2$, and the ones from the axion are proportional to $\La^2$.
For what concerns the regularized interaction energy-momentum tensor, substituting [(\[fe1\])]{}-[(\[fe3\])]{} in [(\[tint\])]{} for $D>4$ we obtain \[tinte\] \^\_[int]{}=(\^2 +)l\^\^D(x-y)d\^2, while for $D=4$ from [(\[fe14\])]{}-[(\[fe34\])]{} we get[^7] \[tinte4\] \^\_[int]{}= l\^\^4(x-y)d\^2. For strings in uniform motion these tensors have thus purely a divergent part, $$\tau_{int\,\ve}^{\m\n}\big|_{div}= \tau_{int\,\ve}^{\m\n},$$ so that the renormalized interaction energy-momentum tensors [(\[tmnint\])]{} vanish for all $D\ge4$, \[tzero\] T\^\_[int]{}=0. We see that the divergent counterterm $\tau_{int\,\ve}^{\m\n}\big|_{div}$ is non-vanishing for all $D\ge4$, for both our string-models: in the [*general*]{} model the parameters are arbitrary, and in the [*fundamental string*]{} model [(\[fundpar\])]{} we have $\bt^2=\frac{2}{D-2}$. Notice, however, that in $D=4$ the gravitational field - even in the [*general*]{} model - does not contribute to $\tau_{int\,\ve}^{\m\n}$: in [(\[tinte4\])]{} there is in fact no term proportional to $M^2$, but only a term proportional to $\bt^2M^2$ coming from the dilaton.
As anticipated, the renormalization of the field energy-momentum tensors [(\[tmnfe\])]{} is more involved since its support is the bulk $\mathbb{R}^D$; we face it in the next sections.
Renormalization: an example
---------------------------
As $\ve$ tends to zero point-wise in [(\[tmnfe\])]{}, we obtain a function $\tau^{\m\n}_{f}(x)$ - the bare energy-momentum tensor - that is regular for $k^{\m\n} x_\n\neq 0$, [*i.e.*]{} in the complement of the world-sheet. In the vicinity of the world-sheet $\tau^{\m\n}_{f}(x)$ behaves, however, as in [(\[tmndiv\])]{} and is thus not a distribution. To isolate the divergent counterterm $\tau^{\m\n}_{f\ve}\big|_{div}$ of $\tau^{\m\n}_{f\ve}$, that diverges as $\ve\ra 0$ in the distributional sense, we use a technique that we illustrate first in a simple example. The results of the actual calculation of $\tau^{\m\n}_{f\ve}\big|_{div}$, first for $D=4$ and then for $D>4$, will be given subsequently.
Consider the functions of a single variable $${\cal T}_\ve(x)=\frac{1}{(x^2+\ve^2)^2},$$ depending on a positive real parameter $\ve$ with the dimension of a length. For every $\ve>0$ these functions represent distributions ${\cal T}_\ve\in {\cal S}'(\mathbb R)$. More precisely, if we apply them to a test function $\vp\in {\cal S}(\mathbb R)$ the resulting integrals \[fef\] [T]{}\_()= dx are convergent[^8] for every $\vp$. The [*point-wise*]{} limit for $x\neq 0$ \[point\] \_[0]{}[T]{}\_(x)= does however not represent a distribution, because the integrals dx diverge due to the non-integrable singularity at $x=0$.
We want to overcome this difficulty at the price of modifying ${\cal T}_\ve(x)$ as little as possible, [*i.e.*]{} only at $x=0$. To this order we isolate the singularity at $x=0$ in [(\[fef\])]{} writing \[fef1\]\_()&=& dx+ (0) +”(0)\
&= &dx + (0)+”(0).\[exam\] Since the first integral in [(\[exam\])]{} converges now as $\ve\ra0$ for every $\vp\in {\cal S}(\mathbb R)$, we read off the “divergent part” of ${\cal T}_\ve$ as \[divf\] [T]{}\_|\_[div]{}= |\_[div]{}= (x)+”(x). ${\cal T}_\ve\big|_{div}$ contains a leading divergence, supported in $x=0$, proportional to $1/\ve^3$, and a sub-leading one - yet supported in $x=0$ - proportional to $1/\ve$: the general lesson is that the stronger the divergences (higher inverse powers of $x$) present in ${\cal T}_\ve$, the more terms proportional to higher derivatives of the $\dl$-function (higher inverse powers of $\ve$) are present in ${\cal T}_\ve\big|_{div}$.
Subtracting the “divergent counterterm” we conclude then that the distributional limit \[renf\] [S]{}’-\_[0]{}([T]{}\_- [T]{}\_|\_[div]{}) exists and defines the renormalized version of the function [(\[point\])]{}. The explicit expression of ${\cal T}$ is $${\cal T}(\vp)=\int \frac{\vp(x)-\vp(0)-\frac{x^2}{2}\,\vp''(0)}{x^4}\,dx.$$ We have thus achieved our goal: from [(\[exam\])]{}, [(\[divf\])]{} and [(\[renf\])]{} we deduce that ${\cal T}$ is a [*distribution*]{}, that in $\mathbb{R}\setminus\{0\}$ coincides with $1/x^4$, [*i.e.*]{} with the point-wise limit [(\[point\])]{}[^9].
### Subtraction schemes and finite counterterms
In choosing the divergent part [(\[divf\])]{} we tacitly “resolved” an indeterminacy regarding the [*finite*]{} part of ${\cal T}$ - relying on what in quantum field theory would be called a [*minimal subtraction scheme*]{}. In fact, the “renormalized” distribution ${\cal T}$ is determined only modulo the [*finite local counterterms*]{} $${\cal T}\ra {\cal T}+a\,\dl(x)+b\,\dl''(x),$$ where we omitted odd derivatives of the $\dl$-function to preserve the invariance under parity of ${\cal T}_\ve$. In the present case the choice [(\[divf\])]{} might be justified because the coefficients $a$ and $b$ must be dimensionful, [*i.e.*]{} of length dimension respectively $1/L^3$ and $1/L$. If no fundamental constants with inverse length-dimensions show up in the theory, then $a$ and $b$ must actually vanish.
Consider with this respect the further example $${\cal U}_\ve(x)=\frac{1}{|x|+\ve},$$ whose divergent part is $${\cal U}_\ve\big|_{div}=-2\ln(\ve/L)\,\dl(x).$$ In this case, for dimensional reasons the separation of the divergent part required the introduction of an arbitrary parameter $L$ with the dimension of a length. This leads in the renormalized distribution $${\cal U} ={\cal S}'-\lim_{\ve\ra0}\big({\cal U}_\ve+2\ln(\ve/L)\,\dl(x)\big),$$ to an indeterminacy of the type $${\cal U}\ra {\cal U}+a\,\dl(x),$$ where $a$ is a dimensionless parameter, that [*a priori*]{} can not be set to zero. This is a simple example of the [*finite-counterterm-ambiguity*]{}, that will play a significant role in sections \[fc\] and \[ufc\].
Renormalization of the field energy-momentum tensor in $D=4$
------------------------------------------------------------
The determination of the divergent counterterm of the tensors [(\[tmnfe\])]{} relies on a straightforward generalization of the above example. Due to its obvious relevance we analyze first the four-dimensional case.
For $D=4$ [(\[tmnfe\])]{} reduces to \[tmnfe4\] \^\_[f]{}=(C\_4( k\^k\^x\_x\_-\^k\_x\^x\^)-\^2 e\^[2]{}l\^k\_x\^x\^), where $$C_4=M^2\bt^2e^{2\bt\Psi}-\La^2e^{2\a\Psi}.$$ The formula analogous to [(\[divf\])]{} we need is \[form4\] [k\^k\^x\_x\_(-k\_x\^x\^+\^2)\^2]{}|\_[div]{}= (/L) (\^-l\^)\^4(x-y)d\^2. There is only a logarithmic divergence, since near the world-sheet for $\ve=0$ the l.h.s. of [(\[form4\])]{} diverges as $x_\perp^2$, and the orthogonal space is tow-dimensional. For dimensional reasons we are obliged to introduce an arbitrary length scale $L$, that reflects the subtraction-scheme ambiguity discussed above.
Applying [(\[form4\])]{} to [(\[tmnfe4\])]{} we obtain the divergent counterterm \[div4\] \^\_[f]{}|\_[div]{}=-(M\^2\^2e\^[2]{}+\^2 e\^[2]{}) l\^\^4(x-y)d\^2. As in the case of the interaction energy-momentum tensor [(\[tinte4\])]{}, also in [(\[div4\])]{} there is no divergent contribution from the gravitational field. Given [(\[tmnfe4\])]{} and [(\[div4\])]{}, the distributional limit \[tmnf4\] T\^\_f’-\_[0]{}(\_[f]{}\^-\_[f]{}\^|\_[div]{}) exists now and defines the renormalized field energy-momentum tensor.
### Cancelation of divergences {#nrt}
Within our approach the energy-momentum tensors are always “renormalizable” - in the sense that the divergent counterterms are localized on the world-sheet - so that the vanishing of the divergent counterterms is actually not of central importance. Nevertheless it is instructive, also for the comparison with known results in the literature, to see if there are models for which the divergences cancel out. To make this analysis comparative we anticipate some results from later sections.
The divergent counterterms [(\[tinte4\])]{} and [(\[div4\])]{} are non-vanishing in the [*general*]{} model as well as in the [*fundamental string*]{} model, unless $\La=\bt=0$. The situation is different for what concerns the [*total*]{} counterterm (we omit finite terms) \^\_[int]{}|\_[div]{} + \^\_[f]{}|\_[div]{} = (/L) (M\^2\^2e\^[2]{}-\^2e\^[2]{})l\^\^4(x-y)d\^2. \[tot4\] For the [*general*]{} model this is still divergent, while for the [*fundamental string*]{} model [(\[fundpar\])]{} the divergences actually cancel. The cancelation occurs between the dilaton $(M^2\bt^2)$ and the axion $(\La^2)$, while, as we observed above, the divergences of the gravitational field $(M^2)$ just drop out, even in the [*general*]{} model. This result proves in particular, for $D=4$, the compensation between field-divergences, originating from the bulk, and interaction-divergences, genuinely localized on the world-sheet, conjectured in the effective-action approach [@BD12]. As we will see in section \[rot\], for $D>4$ this compensation will no longer occur, neither in the Dirac framework that we are applying here, nor in the Landau-Lifshitz and canonical frameworks. Nevertheless in the last two frameworks the field-divergences and interaction-divergences will cancel [*separately*]{} for all $D\ge4$, see sections \[llf\] and \[cfw\].
The cancelation of [*gravitational*]{} divergences, noticed previously, is special to $D=4$ and occurs - even in the [*general*]{} model - [*separately*]{} in $\tau^{\m\n}_{int\,\ve}|_{div}$ [(\[tinte4\])]{} and $\tau^{\m\n}_{f\ve}|_{div}$ [(\[div4\])]{}. This separate cancelation occurs in the Dirac framework and, as we will see, in the canonical framework, while in the Landau-Lifshitz framework the gravitational field-divergences will cancel against the gravitational interaction-divergences. In general the pattern of cancelation of divergences, even in $D=4$, is thus [*framework-dependent*]{}.
A characteristic feature of the four-dimensional total counterterm [(\[tot4\])]{} is that, being proportional to $l^{\m\n}$, it [*could*]{} be eliminated via the string-tension redefinition (see [(\[tslin\])]{}) \[redef\] MM’= M + (M\^2\^2e\^-\^2e\^[(2-)]{})(/L). In contrast, in dimensions $D>4$ there will be several different tensorial structures showing up in the divergent counterterms, whose cancelation could not be achieved renormalizing the parameters of the original theory: in the general case, [*by-hand*]{} subtractions of divergences, as in [(\[tmnf4\])]{}, represent thus a basic ingredient of our approach.
From the presence of the $(\ln L)$-term in [(\[tot4\])]{} we conclude that in $D=4$ the finite-counterterm-ambiguity amounts simply to a redefinition of the string tension.
Renormalization of the field energy-momentum tensor in $D>4$ {#rot}
------------------------------------------------------------
To determine the divergent counterterm of the field energy-momentum tensor [(\[tmnfe\])]{} in a generic space-time, we need the generalization of [(\[form4\])]{} to a generic $D\ge4$ (see reference [@KL1]) \[formD\] [k\^k\^x\_x\_(-k\_x\^x\^+\^2)\^[D-2]{}]{}|\_[div]{}= \_[j=0]{}\^[D-4]{}’A\_j((l\^-\^)-j \^\^)\^[j/2-1]{}\^D(x-y)d\^2. The [*prime*]{} indicates that the sum extends only over [*even*]{} $j$ and the coefficients $A_j$ (divergent for $\ve\ra0$) are given by A\_j=
, & j< D-4,\
&\
\[boh\] (/L), & j=D-4.
Applying [(\[formD\])]{} to [(\[tmnfe\])]{} we obtain for its divergent counterterm the expression, valid for all $D\ge4$, $$\begin{aligned}
\tau^{\m\n}_{f\ve}\big|_{div}=\frac{G}{\Omega_{D-2}^2}\sum_{j=0}^{D-4}{}'A_j\int&\left(
\big(C+(D+j-2)\La^2e^{2\a\Psi}\big)l^{\m\n}\,\square+\frac{1}{2}\,C\big(D+j-4\big)\eta^{\m\n}
\,\square\right.
\nn\\
&-j\,C\pa^\m\pa^\n\bigg)\square^{j/2-1}\,\dl^D(x-y)\sqrt{\g}\,d^2\s.\label{divD}\end{aligned}$$ Contrary to the four-dimensional case, this counterterm exhibits a sum of derivatives of $\dl$-functions $\pa^j\dl^D(x-y)$, multiplied by the divergent factor $1/\ve^{D-j-4}$. The leading divergence is $1/\ve^{D-4}$ and corresponds to $j=0$. The terms with $j>0$ represent an entire series of [*subleading*]{} divergences - absent in $D=4$ - and none of them could be eliminated through the redefinition of the tension, or some other coupling constants, like in [(\[redef\])]{}. Notice also the appearance of [*gravitational*]{} divergences, [*i.e.*]{} the terms proportional to $(D-4)M^2$ in the coefficient $C$ in [(\[c\])]{}, that were absent in $D=4$.
Let us analyze more closer the leading divergence in [(\[divD\])]{}, that has the form
\^\_[f]{}|\_[div]{}\^[lead]{}= [G\^[1/2]{}()2\^[D-1]{}\_[D-2]{}()\^[D-4]{}]{} &((C+(D-2)\^2e\^[2]{})l\^\
&. +C(D-4)\^ ) \^D(x-y)d\^2.\[divDlead\]
For $D>4$ it contains hence the two tensorial structures $l^{\m\n}$ and $\eta^{\m\n}$. Correspondingly, for $D>4$ in the [*general*]{} model there is no way to cancel even this leading field-divergence against the interaction-divergence [(\[tinte\])]{}, which contains only $l^{\m\n}$.
But even in the [*fundamental string*]{} model, where $C=0$ and the tensor $\eta^{\m\n}$ drops out, the numerical coefficients of $l^{\m\n}$ in [(\[tinte\])]{} and [(\[divDlead\])]{} do not match. We conclude thus that in the Dirac framework in the [*fundamental string*]{} model the total divergences cancel in $D=4$, but not for $D>4$. In particular in this framework for $D>4$ the string tension suffers a non-vanishing renormalization - a feature that would not be expected at the basis of the non-renormalization theorems of superstring amplitudes [@DH]. This occurrence may disfavor the Dirac-framework w.r.t. the other two frameworks, although we were not able to find a physical reason for this.
In the [*general*]{} model we define the renormalized field energy-momentum tensor as in [(\[tmnf4\])]{}, with $\tau^{\m\n}_{f\ve}$ and $\tau^{\m\n}_{f\ve}\big|_{div}$ given respectively in [(\[tmnfe\])]{} and [(\[divD\])]{}, \[tmnfD\] T\^\_[f(0)]{}’-\_[0]{}(\_[f]{}\^-\_[f]{}\^|\_[div]{}). We put a $(0)$ in the definition of $ T^{\m\n}_f$, due to the finite-counterterm-ambiguity that we will encounter in the next section.
We emphasize that the tensor [(\[tmnfD\])]{} is not a merely abstract object in that, being regular in whole space, [*i.e.*]{} being a distribution, it can be used to compute concretely the finite energy and momenta in arbitrary finite volumes - even if these volumes intersect the world-sheet. $T^{\m\n}_{f(0)}$ shares this property with the renormalized energy-momentum tensor of the electromagnetic field of a charged point-particle in four dimensions, whose integrals over a volume [*enclosing*]{} the particle always converge, giving rise to finite four-momenta [@LM].
Energy-momentum conservation {#emc}
----------------------------
By construction [(\[tmnfD\])]{} is a distribution and so its divergence $\pa_\m T^{\m\n}_{f(0)}$ is perfectly well-defined. From the general analysis of section \[rem\] we know furthermore that $\pa_\m T^{\m\n}_{f(0)}$ is supported on the word-sheet. To evaluate it explicitly we use that derivatives are [*continuous*]{} operations in distribution space, so that in [(\[tmnfD\])]{} we can freely interchange the derivatives with the limit. Moreover, since $\tau_{f\ve}^{\m\n}$ is a regular distribution, its derivatives can be computed in the usual way. From [(\[tmnfe\])]{} and [(\[divD\])]{} we get
\_(\_[f]{}\^-\_[f]{}\^|\_[div]{}) =\^&(.\
&- \_[j=0]{}\^[D-4]{}’(D-j-4)A\_j\^[j/2]{}\^D(x-y)d\^2),\[cons1\]
where, for convenience, we factorized out a derivative. The first term between parentheses at the right hand side, coming from the divergence of $\tau_{f\ve}^{\m\n}$, multiplies a factor of $\ve^2$. This means that, when taking $\ve\ra0$, this term is entirely supported on the world-sheet, as it must be. Applying this term to a test function $\vp$ and performing the expansion in powers of $\ve$, one gets [@KL1] \[factor\] = \_[j=0]{}\^[D-4]{}’B\_j\^[j/2]{}\^D(x-y)d\^2+o(), where \[bj\] B\_j=
(D-j-4)A\_j,&j<D-4,\
,&j=D-4,( D ,
with $A_j$ given in [(\[boh\])]{}. In [(\[factor\])]{} with $o(\ve)$ we understood terms that converge to zero as $\ve\ra0$ in the distributional sense. We see that all divergences in [(\[cons1\])]{} cancel out, as they must by construction. However, for $D$ [*even*]{} the expansion [(\[factor\])]{} contains also a non-vanishing [*finite*]{} term, the one corresponding to $j=D-4$. Consequently, for the divergence of the energy-momentum tensor [(\[tmnfD\])]{} we get
[\_T\^\_[f(0)]{}=]{} 0, & for $D$ odd,\[odd\]\
\^\^\^D(x-y)d\^2,&for $D$ even.\[even\]
For the four-dimensional string we have for example $$\pa_\m T^{\m\n}_{f(0)}=\frac{GC}{8\pi} \int\pa^\n\dl^4(x-y)\sqrt{\g}\,d^2\s.$$
### Finite counterterms {#fc}
In principle, according to our approach the [*anomaly*]{} encountered in [(\[even\])]{} for $D$ even - a non-vanishing $D$-divergence for the otherwise well-behaved distribution $T^{\m\n}_{f(0)}$ - determines the self-force ${\cal S}^\m$. Recalling that the renormalized interaction energy-momentum tensor [(\[tzero\])]{} is zero, from [(\[ident1\])]{} and [(\[even\])]{} we would then get an ${\cal S}^\m$ that is a derivative operator, and not a simply a vector. There would thus exist no string-equation of motion ensuring total energy-momentum conservation.
On the other hand it is a basic fact in any renormalization process, in quantum as well as in classical theory, that once we subtract divergent terms from a physical quantity, this quantity remains by itself determined only modulo finite terms of the [*same structure*]{} as the divergent ones. This offers a way out thanks to the fact that the anomaly in [(\[even\])]{} is a [*trivial*]{} anomaly, in that in can - and must - be eliminated by subtracting a finite counterterm, in very much the same way as one eliminates trivial anomalies in quantum field theory, once one has introduced a regularization that breaks a classical symmetry.
In the present case the appropriate finite counterterm is \[finc\] T\^|\_[fin]{}= \^ \^\^D(x-y)d\^2, which in $D = 4$ becomes \[finitec4\] T\^|\_[fin]{}= \^\^4(x-y)d\^2. The final renormalized field energy-momentum tensor \[finitec\] T\^\_f= T\^\_[f(0)]{}-T\^|\_[fin]{} satisfies in turn \_T\^\_f=0. Together with [(\[tzero\])]{} equation [(\[ident1\])]{} gives then rise to a vanishing self-force, ${\cal S}^\m=0$, as is of course in line with our string moving freely in space-time.
Landau-Lifshitz framework {#llf}
=========================
In this section we display the main changes that arise w.r.t. the preceding analysis, when we use for the gravitational field the Landau-Lifshitz pseudo-tensor $\w\Sigma^{\m\n}$ ([(\[ll\])]{} and [(\[z\])]{}) in place of the Dirac pseudo-tensor $\Sigma^\m{}_\n$ [(\[TD\])]{}. As we saw, the use of $\w\Sigma^{\m\n}$ instead of $\Sigma^\m{}_\n$ induces in the field and interaction energy-momentum tensors the modifications [(\[fdll\])]{} and [(\[intll\])]{}, so that it is easy to extract from those relations and our previous results [(\[tmnfe\])]{} and [(\[tinte\])]{}, using still [(\[fe1\])]{}-[(\[fe34\])]{}, the new regularized tensors for a generic $D\ge4$[^10] $$\begin{aligned}
\label{tmnfell}
\w\tau^{\m\n}_{f\ve}&=\frac{G}{\Omega^2_{D-2}(-k_{\a\bt}x^\a x^\bt+\ve^2)^{D-2}}\left(C\left(
k^{\m\a}k^{\n\bt}x_\a x_\bt-\frac{1}{2}\,\eta^{\m\n}k_{\a\bt}x^\a x^\bt\right)+K l^{\m\n}k_{\a\bt}x^\a x^\bt\right),\\
\label{tintell}
\w\tau^{\m\n}_{int\,\ve}&=\frac{GM^2e^{2\bt\Psi}N}{(4-D)\Omega_{D-2}\,\ve^{D-4}}\int l^{\m\n}\dl^D(x-y)\sqrt{\g}\,d^2\s= \w\tau^{\m\n}_{int\,\ve}\big|_{div}.\end{aligned}$$ We introduced the coefficients ($C$ is the same as in [(\[c\])]{}) $$\begin{aligned}
\label{c1}
C&=\dis M^2e^{2\bt\Psi}\left(\bt^2+\frac{D-4}{D-2}\right)-\La^2e^{2\a\Psi},\\
K&=M^2e^{2\bt\Psi}-\La^2 e^{2\a\Psi},\label{k1}\\
N&=\bt^2-\frac{2}{D-2}\label{n1}.\end{aligned}$$ Notice that w.r.t. [(\[tmnfe\])]{} in [(\[tmnfell\])]{} only the coefficient of the last term changed. The divergent counterterm of [(\[tmnfell\])]{} has correspondingly a structure very similar to [(\[divD\])]{}
\^\_[f]{}|\_[div]{}=\_[j=0]{}\^[D-4]{}’A\_j&( (C-(D+j-2)K)l\^+C(D+j-4)\^ .\
&-jC\^\^)\^[j/2-1]{}\^D(x-y)d\^2.\[divDll\]
$D=4$
-----
Specializing the above formulae to $D=4$ we obtain $$\begin{aligned}
\label{div4ll}
\w\tau^{\m\n}_{f\ve}\big|_{div}&=-\frac{G\ln(\ve/L)}{4\pi}\left((\bt^2-2)M^2
e^{2\bt\Psi}+\La^2 e^{2\a\Psi}\right) \int l^{\m\n}\dl^4(x-y)\sqrt{\g}\,d^2\s,
\\
\label{tinte4ll}
\w\tau^{\m\n}_{int\,\ve}\big|_{div}&=\frac{G\ln(\ve/\la)}{2\pi}\,(\bt^2-1)M^2e^{2\bt\Psi}
\int l^{\m\n}\dl^4(x-y)\sqrt{\g}\,d^2\s.\end{aligned}$$ For the total counterterm, disregarding finite terms, we get then \[tot4ll\] \^\_[int]{}|\_[div]{} + \^\_[f]{}|\_[div]{} = (/L) (M\^2\^2e\^[2]{}-\^2e\^[2]{})l\^\^4(x-y)d\^2. Comparing with the Dirac-framework results we notice first of all that the total divergent counterterm [(\[tot4ll\])]{} matches exactly [(\[tot4\])]{}. The main difference is, however, the appearance of [*gravitational*]{} divergences in [(\[div4ll\])]{} as well as in [(\[tinte4ll\])]{}, proportional respectively to $M^2e^{2\bt\Psi}/2\pi$ and $-M^2e^{2\bt\Psi}/2\pi$, which are absent in [(\[div4\])]{} and [(\[tinte4\])]{}. In the sum [(\[tot4ll\])]{} they cancel therefore out.
In the [*general*]{} model there are again no cancelations, while in the [*fundamental string*]{} model [(\[fundpar\])]{} - a further main difference w.r.t. the Dirac framework - the field-divergences and the interaction-divergences cancel now [*separately*]{} \[vanis\] \^\_[f]{}|\_[div]{}=0= \^\_[int]{}|\_[div]{}. These results support in particular the hypothesis formulated in [@BD12] to explain the apparently contradictory results of the analysis of [@DH; @CHH], concerned with field-energy-divergencies of static strings in $D=4$. The authors of [@DH; @CHH] found indeed that the total field-divergences cancel, whilst the gravitational field-divergences alone did not. Since the authors of [@BD12] - within their effective-action approach - found that in $D=4$ there were no divergent gravitational divergences contributing to the tension renormalization, they hypothesized that the gravitational field-divergences revealed in [@DH; @CHH] should cancel against gravitational interaction-divergences. Yet the total divergences had to cancel. All these expectations are precisely confirmed by our formulae [(\[div4ll\])]{}-[(\[vanis\])]{}[^11].
$D>4$
-----
Coming back to generic dimensions $D>4$, we notice that in the [*fundamental string*]{} model the coefficients $C$, $K$ and $N$ in [(\[c1\])]{}-[(\[n1\])]{} are all zero. Given [(\[tintell\])]{} and [(\[divDll\])]{} this implies that in this model the identities [(\[vanis\])]{} hold for all dimensions $D\ge4$, meaning that all [*leading and subleading*]{} field-divergences and interaction-divergences cancel separately. However, for $D>4$ there is no compensation between these two types of divergences: in particular $\w\tau^{\m\n}_{int\,\ve}\big|_{div}$ [(\[tintell\])]{} contains only the leading divergence $1/\ve^{D-4}$, while $\w\tau^{\m\n}_{f\ve}\big|_{div}$ [(\[divDll\])]{} contains also the subleading divergences $1/\ve^{D-4-j}$ for all even $0<j\le D-4$.
In the [*fundamental string*]{} model it happens actually that the regularized tensors $\w\tau^{\m\n}_{f\ve}$ [(\[tmnfell\])]{} and $\w\tau^{\m\n}_{int\,\ve}$ [(\[tintell\])]{} vanish identically: this feature is characteristic for strings in uniform motion, while for accelerated strings these tensors will obviously be different from zero, see section \[tgc\].
In the [*general*]{} model the divergent counterterms are non-vanishing and must be subtracted, as in [(\[tmnfD\])]{}. Since w.r.t. the Dirac framework the divergent counterterms changed only by terms proportional to $l^{\m\n}$ - see [(\[tinte\])]{} [*versus*]{} [(\[tintell\])]{} and [(\[divD\])]{} [*versus*]{} [(\[divDll\])]{} - the (distributional limit of the) divergence $\pa_\m\left(\w\tau_{f\ve}^{\m\n}-\w\tau_{f\ve}^{\m\n}\big|_{div}\right)$ is the same as in the Dirac framework. This implies that also the finite counterterm [(\[finc\])]{} to be subtracted remains the same. The renormalized field energy-momentum tensor in the Landau-Lifshitz framework is therefore \[tfinll\] T\^\_f =[S]{}’-\_[0]{} (\_[f]{}\^-\_[f]{}\^|\_[div]{}) - T\^|\_[fin]{},\_T\^\_f=0. Similarly the renormalized interaction energy-momentum tensor is again zero, $\w T^{\m\n}_{int} ={\cal S}'-\lim_{\ve\ra0} \left(\w\tau_{int\,\ve}^{\m\n}-\w\tau_{int\,\ve}^{\m\n}\big|_{div}\right)=0$, as is the self-force.
Canonical framework {#cfw}
===================
From [(\[fdcan\])]{} and [(\[intcan\])]{} - proceeding as above - in the canonical framework we obtain $$\begin{aligned}
\label{tmnfec}
\wh\tau^{\m\n}_{f\ve}&=\frac{GC}{\Omega^2_{D-2}(-k_{\a\bt}x^\a x^\bt+\ve^2)^{D-2}}\left(
k^{\m\a}k^{\n\bt}x_\a x_\bt-\frac{1}{2}\,\eta^{\m\n}k_{\a\bt}x^\a x^\bt\right),\\
\label{tintec}
\wh\tau^{\m\n}_{int\,\ve}&=\frac{GC}{(4-D)\Omega_{D-2}\,\ve^{D-4}}\int l^{\m\n}\dl^D(x-y)\sqrt{\g}\,d^2\s= \wh\tau^{\m\n}_{int\,\ve}\big|_{div}.\end{aligned}$$ Contrary to the Dirac and Landau-Lifshitz frameworks, in the canonical framework the axion contributes now also to the interaction tensor $\wh \tau_{int\,\ve}^{\m\n}$.
The divergent counterterm of the field energy-momentum tensor becomes now \^\_[f]{}|\_[div]{}=\_[j=0]{}\^[D-4]{}’A\_j (l\^+(D+j-4)\^ -j\^\^)\^[j/2-1]{}\^D(x-y)d\^2.\[divDc\] The expressions [(\[tintec\])]{}, [(\[divDc\])]{} of the counterterms are simpler than the corresponding expressions [(\[tinte\])]{}, [(\[divD\])]{} and [(\[tintell\])]{}, [(\[divDll\])]{} of the other two frameworks. In particular the string coupling constants enter only through the single constant $C$ [(\[c\])]{} which, we recall, vanishes in the [*fundamental string*]{} model. In this model we have therefore for all $D\ge4$ \[cancall\] \^\_[f]{}|\_[div]{}=0= \^\_[int]{}|\_[div]{}, as in the Landau-Lifshitz framework.
In a certain sense the canonical framework “maximizes” the cancelation of ultraviolet divergences in the [*fundamental string*]{} model: for all $D\ge4$ the field- and interaction-divergences cancel separately - as in the Landau-Lifshitz framework - and in $D=4$, in addition, the gravitational field-divergences and interaction-divergences cancel separately - as in the Dirac framework.
In $D=4$ we obtain in particular ($C=M^2\bt^2e^{2\bt\Psi}-\La^2e^{2\a\Psi}$) \[d4can\] \^\_[int]{}|\_[div]{} + \^\_[f]{}|\_[div]{} = (/L)l\^\^4(x-y)d\^2,\^\_[f]{}|\_[div]{} =- \^\_[int]{}|\_[div]{}, so that the total counterterm coincides with the expressions [(\[tot4\])]{} and [(\[tot4ll\])]{} of the other two frameworks. For $D=4$ the [*total*]{} ultraviolet divergence appears thus to have [*universal*]{} character, in that it is framework-independent. We did not found an [*a priori*]{} reason for this “coincidence” - which does not occur for $D>4$.
For future reference we write out [(\[divDc\])]{} for $D=5$ \[divDc5\] \^\_[f]{}|\_[div]{}= (\^+l\^)\^5(x-y)d\^2, as well as for $D=6$ \^\_[f]{}|\_[div]{}= ((\^+l\^) +(/L)( (\^+l\^)-\^\^)) \^6(x-y)d\^2.\[divDc6\]
A part from the the simplifications showing up in formulae [(\[tmnfec\])]{}-[(\[divDc\])]{}, in the [*general*]{} model the divergent counterterms must again be subtracted, and the renormalized field energy-momentum tensor $\wh T^{\m\n}_f$ is defined exactly in the same way as in [(\[tfinll\])]{}, with the same finite counterterm [(\[finc\])]{}; it satisfies still $\pa_\m\wh T^{\m\n}_f=0$. Also in the canonical framework we have of course $\wh T^{\m\n}_{int} ={\cal S}'-\lim_{\ve\ra0} \left(\wh\tau_{int\,\ve}^{\m\n}-\wh\tau_{int\,\ve}^{\m\n}\big|_{div}\right)=0$, so that the self-force vanishes, as in the other frameworks.
In the [*general*]{} model, by construction the renormalized field energy-momentum tensors of the three frameworks $T^{\m\n}_f$ [(\[finitec\])]{}, $\w T^{\m\n}_f$ [(\[tfinll\])]{} and $\wh T^{\m\n}_f$ - being all divergence-less distributions - differ from each other by the distributional divergence $\pa_\rho C^{\rho\m\n}$ of an antisymmetric tensor: this means hat for strings in uniform motion these frameworks are physically equivalent.
Comparison with the effective-action approach
---------------------------------------------
It seems not straightforward to establish a direct link between the non-renormalization property [(\[cancall\])]{} - holding in the [*fundamental string*]{} model where $C=0$ - and the results of the effective-action method of [@BD12], applied to the same model. The latter tests indeed different physical properties w.r.t. our approach, [*i.e.*]{} the ultraviolet renormalization of the string tension through a computation of the (divergent) coefficient of the kinetic action $\int \! \sqrt{\g}\,d^2\s$. This computation amounts essentially to the (gaussian) functional integral over the fields of the linearized form of the action [(\[fands\])]{}, giving rise to the “effective action”. The latter is a non-local functional of only the string coordinates $y^\m(\s)$, that contains as divergent part a term like $M_{div}\int \!\sqrt{\g}\,d^2\s$, where $M_{div}$ is a divergent coefficient. The authors of [@BD12] found the proportionality relation \[mdiv\] M\_[div]{} C, where $C$ is precisely the coefficient [(\[c1\])]{}. If we identify the effective action with the [*total energy integrated over time*]{} - although it is not clear, at least to us, whether this is the correct physical interpretation of the effective action - we may compute the (divergent part of the) former integrating the $00$ components of the canonical-framework expressions [(\[tintec\])]{} and [(\[divDc\])]{} over whole space-time: the outcome is clearly $C \int \!\sqrt{\g}\,d^2\s $, times a divergent factor, in agreement with [(\[mdiv\])]{}. In this sense the effective-action approach appears to parallel the [*canonical*]{} framework, while in the other two frameworks the total energy integrated over time produces a divergent coefficient in front of $\int \!\sqrt{\g}\,d^2\s$, that depends in a more complicated way on the string coupling constants.
General conclusions on cancelation of divergences
-------------------------------------------------
In the [*general*]{} model there are non-vanishing divergent counterterms in all three frameworks. The occurrence of the cancelation of these divergences in the [*fundamental string*]{} model depends on the choice of the total energy-momentum pseudo-tensor: in the Landau-Lifshitz and canonical frameworks the divergences cancel for all $D\ge4$, while in the Dirac framework they cancel only in $D=4$. From this point of view the canonical framework seems the most convenient one, in that all divergences are proportional to the same coefficient $C$. The energy-divergence-analysis of [@DH; @CHH] rephrases the Landau-Lihshitz framework, while the effective-action-analysis of [@BD12] rephrases the canonical one.
In general in all frameworks the cancelation of divergences requires actually only the conditions \[weaker\] Me\^= e\^, =, which are weaker than the defining relations [(\[fundpar\])]{} of the [*fundamental string*]{} model. Notice in particular that the first relation amounts to the equality between the effective string [*tension*]{} and [*charge*]{} - a property that is strictly related to the supersymmetry, more precisely $k$-symmetry, of the Green-Schwarz sigma-model action [@GSW], that in absence of fermions reduces indeed to the action [(\[is\])]{}.
One has to keep in mind that, even if the divergences cancel for an appropriate choice of the coupling constants, the energy-momentum tensor must nevertheless be [*regularized*]{}: indeed, even in this case the [*single*]{} terms of the [*bare*]{} energy-momentum tensor are not distributions, so that it would make no sense to take their $D$-divergence. Obviously, for strings in uniform motion satisfying the conditions [(\[weaker\])]{}, the regularized field- and interaction-energy-momentum tensors [*themselves*]{} vanish before [*and*]{} after regularization (in the Landau-Lifshitz and canonical frameworks), so that for all practical purposes the regularization can be removed. However, for accelerated strings, even if the conditions [(\[weaker\])]{} hold, the energy-momentum tensors will be non-vanishing and the regularization must be maintained.
Accelerated strings {#tgc}
===================
In this section we perform a preliminary analysis of the additional problems one has to face, when our approach is applied to accelerated strings, where its final more ambitious goal is the explicit determination of the self-force. The general properties of the string self-forces - highly non-local functions of the whole retarded string-history - are poorly known, and in the literature one finds typically approximated explicit expressions, see for example [@DQ; @BS; @BD12bis]. For what concerns our approach, the main implication of a non-vanishing acceleration is the appearance of new divergent counterterms of the energy-momentum tensor, which in turn bear also new finite counterterms.
New counterterms
----------------
For strings in generic accelerated motion the velocity vectors $U_i^\m(\s)$ are no longer constant, so that their multiple covariant derivatives \[multiple\] \_[j\_1]{}\_[j\_p]{}U\^\_i are generically non-vanishing. The main implication of this feature is that the divergent counterterms [(\[tintec\])]{} and [(\[divDc\])]{} - to be specific from now on for simplicity we refer to the canonical framework - will receive corrections. Thanks to the manifest Lorentz- and reparameterization-invariances of our regularization, these corrections amount to additional [*tensorial*]{} structures in the integrands of [(\[tintec\])]{} and [(\[divDc\])]{}. Since the divergences arise from the small-distance behavior of the fields near the world-sheet, these new tensors must be, moreover, [*local*]{} expressions involving the generalized accelerations [(\[multiple\])]{}. This property restricts actually strongly the form of these new tensors. Similarly, the non-vanishing of [(\[multiple\])]{} allows for the appearance of new [*finite*]{} counterterms, too.
Generically, since the indices can be contracted only with the invariant tensors $U^\m_i$ and $\eta^{\m\n}$, or their combinations, the total number of derivatives appearing in the new tensors, acting on $\dl^D(x-y)$, or on $U^\m_i$ as in [(\[multiple\])]{}, must be [*even*]{}. Instead of presenting a general classification of these new structures, that would be rather cumbersome, in the following we work them out for low space-time dimensions.
[***D=4.***]{} In four dimensions we found that for a string in uniform motion the total (divergent + finite) counterterm to be subtracted from the regularized energy-momentum tensor is (see [(\[d4can\])]{} and [(\[finitec4\])]{}) \[acc4\] \^|\_[unif]{}=\^\_[int]{}|\_[div]{} + \^\_[f]{}|\_[div]{}+T\^|\_[fin]{} = ( (/L)l\^+\^)\^4(x-y)d\^2. In this case the divergence is logarithmic in $\ve$, and the tensor between parenthesis is dimensionless. Consequently, since acting with derivatives lowers the length-dimension, there is no new (divergent or finite) counterterm that can show up if the string is accelerated. We conclude thus that in $D=4$ also for accelerated strings the total counterterm is given by [(\[acc4\])]{}, [*i.e.*]{} $\wh \tau^{\m\n}\big|_{acc}=\wh \tau^{\m\n}\big|_{unif}$. According to [(\[tmnf\])]{}-[(\[tmnr\])]{} the total renormalized energy-momentum tensor is therefore \[tot44\] T\^= [S]{}’-\_[0]{}(\_[f]{}\^+\_[int]{}\^ -\^|\_[acc]{})+ \_[kin]{}\^. According to the general strategy of section \[emca\], the continuity equation $\pa_\m T^{\m\n}=0$ determines then the - this time non-vanishing - self-force ${\cal S}^\m$.
[***D=5.***]{} In five dimensions for strings in uniform motion there is no finite counterterm and hence the total counterterm - adding [(\[divDc5\])]{} and [(\[tintec\])]{} - becomes \[tot5\] \^|\_[unif]{}= (\^+ (1-) l\^)\^5(x-y)d\^2. This time the leading divergence is a simple pole $1/\ve$ and, in principle, for an accelerated string there could show up new subleading divergent and also finite counterterms, proportional to $\ln \ve$. However, for dimensional reasons the corresponding additional tensors in the integrand in [(\[tot5\])]{} would involve just [*one*]{} derivative, whereas, as we saw above, for covariance reasons these tensors must involve an [*even*]{} number of derivatives. This means that also in $D=5$ the total counterterm for accelerated strings is given by [(\[tot5\])]{}, so that the total energy-momentum tensor is still [(\[tot44\])]{} with $\wh \tau^{\m\n}\big|_{acc}=\wh \tau^{\m\n}\big|_{unif}$.
For generic [*odd*]{} dimensions $D\ge 7$, the total [*divergent*]{} counterterm of the uniform motion will, however, receive non-vanishing corrections if the string is accelerated, but, contrary to even $D$, there will be no [*finite*]{} counterterms at all. The reason is the same as in $D=5$: the finite tensors in the integrand in [(\[tot5\])]{} should have an odd number, [*i.e.*]{} $D-4$, of derivatives, but an even number, [*i.e.*]{} two, of indices, and there are no such tensors.
[***D=6.***]{} In six dimensions for strings in uniform motion the total counterterm to be subtracted from the regularized energy-momentum tensor $\wh\tau^{\m\n}_{f\ve} +\wh\tau^{\m\n}_{int\,\ve}$ is obtained adding up [(\[divDc6\])]{}, [(\[tintec\])]{} and [(\[finc\])]{} (there is now again a finite counterterm) $$\begin{aligned}
\wh\tau^{\m\n}\big|_{unif}=\frac{GC}{48\pi^2}
\int\bigg\{\frac{1}{\ve^2}\left(\eta^{\m\n}-5l^{\m\n}\right)& +\ln(\ve/L)\left(
\left(\eta^{\m\n}+\frac{1}{2}\,l^{\m\n}\right)\square-\pa^\m\pa^\n\right)\\
&-\left.\frac{1}{8}\,\eta^{\m\n}\,\square\right\}
\dl^6(x-y)\sqrt{\g}\,d^2\s.
\label{tot6uni}\end{aligned}$$ As for $D=4$ and $D=5$, for accelerated strings the leading divergence - in this case $1/\ve^2$ - can not be modified, and there can be no new divergences multiplying a subleading pole $1/\ve$. This time, however, the regularized tensor $\wh\tau^{\m\n}_{f\ve}+ \wh\tau^{\m\n}_{int\,\ve}$ can produce new subleading divergences of order $\ln\ve$, multiplying two-derivative terms of the kind [(\[multiple\])]{}. For accelerated strings the total divergent counterterm has, in fact, the structure \[tot6\] \^|\_[acc]{}=\^|\_[unif]{}+ G (/L) A\^ \^6(x-y)d\^2, where the tensor $A^{\m\n}$, involving two derivatives, is a [*finite*]{} sum of terms like
\[a\] A\^&= a\_1 \_iU\^[i]{} \_jU\^[j]{}+ a\_2 U\^[j]{}\_i\^i U\^\_j + a\_3 \_iU\^[i]{}\^\
&+a\_4 \^ \_iU\^[i]{}\_jU\_\^ j +a\_5 l\^ \_iU\^[i]{}\_jU\_\^ j +.
As the coefficient $C$ in [(\[c1\])]{}, the $a_i$ are - calculable - uniquely determined coefficients of the form $$a_i= b_iM^2+c_i\La^2,$$ where $b_i$ and $c_i$ are dimensionless numbers.
This procedure carries on in any dimension $D\ge6$, and the divergent part $\wh \tau^{\m\n}\big|_{acc}$ is always uniquely determined. From these examples we see that, as the space-time dimension grows, the number - as well as the inverse length-dimensions - of the new divergent counterterms, generalizing [(\[a\])]{}, become larger and larger and more involved. The renormalized total energy-momentum tensor [(\[tot44\])]{} is then a distribution, and the divergence of its field- and interaction-parts - for the reasons explained in section \[emca\] - gives rise to a local expression of the kind $$\begin{aligned}
\pa_\m\left({\cal S}'-\lim_{\ve\ra0}\left(\wh\tau_{f\ve}^{\m\n}+\wh\tau_{int\,\ve}^{\m\n} -\wh\tau^{\m\n}\big|_{acc}\right)\right)=&\nn\\
-\int\!\big( {\cal S}^\n+{\cal S}^{\n\a_1\cdots\a_n}&\, \pa_{\a_1}\cdots\pa_{\a_n}\big) \dl^D(x-y)\sqrt{\g}\,d^2\s,\end{aligned}$$ where we singled out the term ${\cal S}^\n$ without derivatives. The derivative terms would not give rise to a consistent self-force, but they can be eliminated performing the finite-counterterm subtraction \^|\_[acc]{}\^|\_[acc]{} -\^[\_2\_n]{} \_[\_2]{}\_[\_n]{} \^D(x-y)d\^2. According to [(\[ident1\])]{} the vector ${\cal S}^\m$ identifies then the self-force. As we saw, in $D=4$ and $D=5$ for accelerated strings no such new finite counterterms appear, but in $D=6$ the finite derivative-counterterm in [(\[tot6uni\])]{} could be modified by terms involving two derivatives, like the third term in [(\[a\])]{}. In conclusion, for any $D\ge 4$ the self-force will eventually be a multiplicative vector.
There is a last condition that ${\cal S}^\m$ must satisfy to be acceptable as a consistent self-force. The geometrical identity \[geom\] U\_[j]{} \_i U\^[i]{}=0, stating that the acceleration is orthogonal to the velocities, requires indeed that \[phys2\] U\_[j]{} [S]{}\^=0, as a consistency condition for the string equation [(\[eqself\])]{}. Though necessary for the existence of an internally consistent string dynamics compatible with energy-momentum conservation, the condition [(\[phys2\])]{} is not guaranteed [*a priori*]{}[^12]. It could very well be that in order to satisfy [(\[phys2\])]{} - in line with our strategy that fixes the energy-momentum tensor only in the complement of the world-sheet - one must subtract further finite counterterms of the form [(\[a\])]{}. We will come back to this point in the concluding section.
Uniqueness: finite counterterms {#ufc}
-------------------------------
Eventually we address the problem of whether the dynamics of self-interacting strings, codified by the self-force, derived according to the above procedure is uniquely determined. Within our approach this question is tied intimately to the uniqueness properties of the renormalized total energy-momentum tensor $T^{\m\n}$ [(\[tot44\])]{}, that is actually subject to an [*ultimate*]{} finite-counterterm-ambiguity, [*i.e.*]{} the freedom of modifying it according to \[modi\] T’\^ = T\^+ G I\^\^D(x-y)d\^2, where, for a generic $D\ge6$, the tensor $I^{\m\n}$ is made out of the higher dimensional kinematical analogues of [(\[a\])]{}. To say it again, the freedom of adding such terms arises from the fact that in the divergent counterterm [(\[tot6\])]{} the tensors [(\[a\])]{} appear multiplied by the finite coefficients $\ln L$, required for dimensional reasons, which are actually [*arbitrary*]{}.
The tensor $I^{\m\n}$ is, however, constrained by two consistency conditions. In the first place the divergence of the added term must have the structure \[imn\] \_I\^\^D(x-y)d\^2= I\^\_\^D(x-y)d\^2=- I\^ \^D(x-y)d\^2, for some [*multiplicative*]{} vector $I^\n$. Only in this case the modification [(\[modi\])]{} can indeed give rise, through $\pa_\m T'^{\m\n}=0 $, to the modified self-force (see again [(\[ident1\])]{}) $${\cal S}'^\m= {\cal S}^\m+ G I^\m,$$ and hence to a physically inequivalent dynamics. The second condition is that the vector $I^\m$ must respect the geometrical identity [(\[phys2\])]{}, [*i.e.*]{} \[geom2\] U\_[i]{}I\^=0.
There are of course a lot of tensors $I^{\m\n}$ satisfying [(\[imn\])]{} and [(\[geom2\])]{} trivially, namely derivative operators of the kind $I^{\m\n}= W^{\rho\m\n}\pa_\rho$, with $W^{\rho\m\n}=- W^{\m\rho\n}$. In these cases [(\[imn\])]{} would hold simply with $I^\m=0$, so that the self-force would remain unaltered. For a non-vanishing $I^\m$ the properties [(\[imn\])]{} and [(\[geom2\])]{} turn out to be very restrictive. For $D=4$, for example, there would be only the trivial choice $I^{\m\n}= bM^2l^{\m\n}$ - with $b$ a dimensionless constant - that corresponds merely to a redefinition of the string tension[^13]. No such tensor exists for $D=5$. For the simplest non-trivial dimension $D=6$ the tensor $I^{\m\n}$ must be a combination of terms like [(\[a\])]{}. We found actually only [*one*]{} tensor satisfying [(\[imn\])]{} and [(\[geom2\])]{}, whose construction is as follows.
Consider the functional of the string coordinates $y^\m(\s)$ \[acc2\] L\[y\]=bM\^2 \_iU\^[i]{}\_jU\^[j]{}\_d\^2, which can be seen to be the unique reparameterization- and Lorentz-invariant local functional containing [*two*]{} derivatives ($b$ is a dimensionless constant as above). Construct its [*curved*]{} counterpart $L_g[y]$, obtained from [(\[acc2\])]{} replacing everywhere the flat matric $\eta_{\m\n}$ with $g_{\m\n}$, [*e.g.*]{} $\g_{ij}\ra \G_{ij}$ etc. Define then the tensor $I^{\m\n}$ through the functional derivative \[func\] -|\_[g=]{}= I\^\^6(x-y)d\^2, and introduce the world-sheet vector \[wv\] I\_=-. Then from $D=6$ diffeomorphism invariance of $L_g[y]$ it follows that the so defined tensors $I^{\m\n}$ and $I^\m$ satisfy [(\[imn\])]{}, and from the world-sheet reparameterization invariance of $L[y]$ it follows that $I^\m$ satisfies [(\[geom2\])]{}. Given the structure of [(\[acc2\])]{}, the tensor $I^{\m\n}$ defined in [(\[func\])]{} has precisely the form [(\[a\])]{}, containing in particular two derivatives.
In conclusion, in $D=6$ after renormalization the effective string dynamics is determined modulo the self-force $GI^\m$, defined via [(\[acc2\])]{} and [(\[wv\])]{}, which introduces thus a [*new coupling constant*]{} in the theory, namely the coefficient $b$. It is clear that this new interaction amounts to the the replacement of the string kinetic action \[kinplus\] -Me\^ d\^2-Me\^ d\^2+GL\[y\], which introduces new local higher-derivative self-interactions.
We stress that the occurrence of this new interaction is an unavoidable consequence of the renormalization process itself: it is the simple mathematical statement that a divergent term is intrinsically defined modulo finite terms. While in the [*generic*]{} string model the coupling constant $b$ appears to be arbitrary, it may happen that in the [*fundamental string*]{} model - as a classical version of superstring theory - this coupling constant must vanish, or be fixed to some specific value. For higher [*even*]{} dimensions $D\ge8$, the invariant self-interaction functionals of the type [(\[acc2\])]{} contain a growing number of derivatives, and so the number of independent functionals, and thus new coupling constants, grows rapidly.
Uniqueness: framework-dependence
--------------------------------
A further source of non-uniqueness of the self-force could arise from the dependence of the whole procedure on the choice of the [*framework*]{}, that is, the freedom in the choice of the total energy-momentum pseudo-tensor. Before regularization each pair of the three total energy-momentum tensors [(\[tott\])]{}, [(\[ttilde\])]{}, [(\[cano1\])]{} - [*i.e.*]{} $\tau^{\m\n}$, $\w\tau^{\m\n}$ and $\wh\tau^{\m\n}$ - are tied by a relation of the kind (see [(\[t12\])]{} and [(\[cano\])]{}) \[12\] \_[(2)]{}\^-\_[(1)]{}\^=\_Z\^+, where $Z^{\rho\m\n}=-Z^{\m\rho\n}$. Since $\tau_{(2)}^{\m\n}$ and $\tau_{(1)}^{\m\n}$ contain as kinetic part of the string the same tensor $\tau_{\rm kin}^{\m\n}$ [(\[tslin\])]{}, the equation [(\[ident1\])]{} would give rise [*formally*]{} to the same self-force, since $Z^{\rho\m\n}$ simply drops out from that equation.
At the regularized level there is, however, a subtlety that may arise. If we consider the regularized energy-momentum tensors $\tau_{(i)\ve}^{\m\n}$, constructed with the regularized fields $f_\ve$ [(\[fxe\])]{}, the relation [(\[12\])]{} would still hold for the $\tau_{(i)\ve}^{\m\n}$, where the regularized tensor $Z^{\rho\m\n}_\ve$ is still antisymmetric in its first two indices. Consequently, since [*distributional*]{} derivatives always commute, the identity $\pa_\m\pa_\rho Z^{\rho\m\n}_\ve=0$ holds true also for finite $\ve$. But this time the field equations of motion at the r.h.s of [(\[12\])]{} would amount to [*regularized*]{} equations of motion - not identically vanishing - which as $\ve\ra 0$ could give rise in [(\[12\])]{} to (divergent and finite) contributions supported on the world-sheet. Clearly - by construction - the divergent contributions are removed by our renormalization procedure, but there could remain finite parts which in [(\[ident1\])]{} could give rise, in turn, to different self-forces. We believe actually that such discrepancies do not arise, in that we conjecture that the unique freedom of the dynamics of self-interacting strings is represented by the [*universal*]{} local self-coupling [(\[acc2\])]{}, arising directly from the renormalization process. After all, a part from this self-coupling, there should exist a unique well-defined dynamics “associated” to the linearized [*formal*]{} action [(\[fands\])]{}, as it happens [*e.g.*]{} for self-interacting charged particles in $D=4$ [@R]. Probably a definitive test of this conjecture can be provided only through a direct calculation of the self-forces in the three frameworks.
Conclusions {#cr}
===========
Due to the presence of ultraviolet divergences, the derivation of the dynamics of self-interacting strings, taking back-reaction into account, can not be founded on a variational principle - based on a [*canonical, local*]{} and [*finite*]{} action. In absence of such a principle, in this paper we proposed a universal procedure for the derivation of this dynamics in arbitrary dimensions that, $i)$ incorporates by construction energy-momentum conservation and, $ii)$ gives rise automatically to a finite self-force.
We tested three versions - frameworks - of this procedure, in the pilot program of flat strings, where its main characteristics and advantages emerge clearly: manifest reparameterization and Lorentz-invariance, separability and locality of divergences, the presence of subleading divergences, the need of local finite counterterms to derive a multiplicative self-force. All frameworks give in this case rise to a consistent total conserved and covariant energy-momentum tensor and to a vanishing self-force. In the [*fundamental string*]{} model for all $D\ge4$ we retrieved in the Landau-Lifshitz and canonical frameworks the cancelation of all ultraviolet divergences, including also the entire set of subleading divergences - cancelations that incorporate the tension non-renormalization retrieved in previous approaches and foreseen by superstring theory. In the Dirac framework this cancelation occurs only in $D=4$. The cancelation of ultraviolet divergences is thus in general framework-dependent; in particular the failure of the non-renormalization of the string tension in the Dirac framework for $D>4$ may signal that this framework is unable to furnish the correct classical counterpart of superstring theory. This feature is however not a problem for what concerns the construction of a correct [*classical*]{} theory, since in our approach a consistent dynamics of self-interacting strings can be derived independently of the values of the coupling constants.
We faced the problem of the derivation of the self-force ${\cal S}^\m$ for accelerated strings, analyzing in particular the form of the new divergent counterterms. We found that all derivative-self-forces can be eliminated through the subtraction of finite counterterms from the energy momentum tensor. The explicit check whether or not the resulting multiplicative self-forces satisfy the orthogonality condition [(\[phys2\])]{}, is tied to the future program of determining the self-forces explicitly. With this respect the simplest case is $D=4$, since there the exact total counterterm [(\[acc4\])]{} is known and receives no acceleration-induced corrections. The validity of the conjectured relation [(\[phys2\])]{} is based on a physical credo: would it not hold, there would exist no dynamics of self-interacting strings compatible with energy-momentum conservation. From this point of view the situation is the same as for a charged self-interacting particle in $D=4$: [*a priori*]{} there is no reason that the Lorentz-Dirac self-force $${\cal S}^\m=\frac{e^2}{6\pi}\,\bigg(\frac{d^2u^\m}{ds^2}+ \left(\frac{du}{ds}\right)^2 u^\m\bigg),$$ derived from the requirement of conservation of the renormalized total energy-momentum tensor [@R; @LM], is orthogonal to the four-velocity $u^\m$, but eventually it turns out to be so.
Regarding the uniqueness of our construction we revealed the appearance of a finite number of new [*local*]{} self-interactions - none of them occurs in $D=4$ and $D=5$ and only a single self-interaction term occurs in $D=6$ - that are tied intrinsically to the renormalization process. Correspondingly we believe that this is the unique source of ambiguity in the dynamics of self-interacting strings, so that the self-force is not framework-dependent. Again, this statement requires a test through an explicit computation.
#### Acknowledgments.
This work is supported in part by the [*INFN Iniziativa Specifica STEFI*]{}.
0.5truecm
[1]{}
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[^1]: Actually in a generic brane- or string-model, as the one considered in this paper, this tensor is given by a sum $\tau^{\m\n}_{\rm field} = \tau^{\m\n}_f+ \tau^{\m\n}_{int}$, where $\tau^{\m\n}_f$ depends only on the fields and is supported on the bulk, and $\tau^{\m\n}_{int}$ is a field-brane interaction-term supported on the world-volume.
[^2]: The most efficient way to perform it is to extract from $\sqrt{g}\,G^\m{}_\nu$ all terms linear in $\pa\pa g$ and to cast them in the form of a divergence of an antisymmetric tensor. As in the whole paper in [(\[wex\])]{} antisymmetrization is understood with unit weight.
[^3]: The dilaton is a scalar and so its canonical and symmetric energy-momentum tensors coincide.
[^4]: Applying [(\[tmndiv\])]{} to a test function $\vp(x)= \vp(x_\perp,x^0,x^1)$, schematically one has $$\int \tau^{\m\n}_f(x)\,\vp(x)\,d^{D-2}x_\perp dx^0dx^1 \sim \int\frac{dx_\perp}{x_\perp^{D-3}},$$ that diverges for $D\ge4$.
[^5]: This is similar to the failure of the Green-function method to solve Maxwell’s equations in $D=4$ in the case of a charged particle moving along a straight line at the speed of light [@AL2].
[^6]: Alternatively one may introduce an infrared cut-off $l$ for the coordinate $\s^1$ in [(\[fxe\])]{}, imposing $\s^1<l$, and send then $l\ra \infty$; equations [(\[fe14\])]{}-[(\[fe34\])]{} are then regained identifying $l\leftrightarrow \la$. Formally the expressions [(\[fe14\])]{}-[(\[fe34\])]{} could also be obtained performing in [(\[fe1\])]{}-[(\[fe3\])]{} the limit $D\ra 4$ and identifying $\frac{1}{D-4}\leftrightarrow\ln \la$.
[^7]: \[limit\][(\[tinte4\])]{} can be obtained from [(\[tinte\])]{} considering the limit $D\ra 4$ and identifying $\frac{1}{D-4}\leftrightarrow\ln \la$.
[^8]: Actually, for ${\cal T}_\ve$ to be elements of ${\cal S}'(\mathbb R)$ the quantities $|{\cal T}_\ve(\vp)|$ must be dominated by (a finite sum of) [*semi-norms*]{} of $\vp$.
[^9]: The precise meaning of this is that when applied to a test function $\vp(x)$ that vanishes in an arbitrarily small neighborhood of $x=0$, the function $1/x^4$ and the distribution ${\cal T}$ give the same value. In the case at hand ${\cal T}$ could actually be written as the distributional derivative of a basic distribution, [*i.e.*]{} of the [*principal part*]{} of $1/x$, namely ${\cal T}=-\frac{1}{6}(d/dx)^3P(1/x)$.
[^10]: It is understood that the expression of $\w\tau^{\m\n}_{int\,\ve}$ for $D=4$ is obtained from [(\[tintell\])]{}, taking the appropriate limit, see footnote \[limit\] in section \[rfa\].
[^11]: Actually the authors of [@DH; @CHH] do not specify which gravitational energy-momentum pseudo-tensor they use. To be precise, what we have shown above is that the Landau-Lifshitz choice is consistent with their results.
[^12]: Notice that, thanks to reparameterization invariance, the infinite bare self-force [(\[ill\])]{} would satisfy [(\[phys2\])]{} automatically.
[^13]: The choice between $M$ and $\La$ in $I^{\m\n}$ is purely conventional, since they have both the dimension of an inverse length squared.
| ArXiv |
---
abstract: 'We describe a novel technique for creating an artificial magnetic field for ultra-cold atoms using a periodically pulsed pair of counter propagating Raman lasers that drive transitions between a pair of internal atomic spin states: a multi-frequency coupling term. In conjunction with a magnetic field gradient, this dynamically generates a rectangular lattice with a non-staggered magnetic flux. For a wide range of parameters, the resulting Bloch bands have non-trivial topology, reminiscent of Landau levels, as quantified by their Chern numbers.'
author:
- Tomas Andrijauskas
- 'I. B. Spielman'
- Gediminas Juzeliūnas
title: 'Topological lattice using multi-frequency radiation'
---
Introduction
============
Ultracold atoms find wide applications in realising condensed matter phenomena [@Greiner2002; @Lewenstein2007; @Bloch2008a; @Lewenstein2012]. Since ultracold atom systems are ensembles of electrically neutral atoms, various methods have been used to simulate Lotentz-type forces, with an eye for realizing physics such as the quantum Hall effect (QHE). Lorentz forces are present in spatially rotating systems [@Matthews1999a; @Madison2001; @Abo-Shaeer2001; @Cooper2008; @Fetter2009; @Gemelke2010; @Wright13PRL] and appear in light-induced geometric potentials [@Dalibard2011; @Goldman2014]. The magnetic fluxes achieved with these methods are not sufficiently large for realizing the integer or fractional QHE. In optical lattices, larger magnetic fluxes can be created by shaking the lattice potential [@struck12; @Windpassinger2013RPP; @Jotzu2014; @Eckardt16review], combining static optical lattices along with laser-assisted spin or pseudo spin coupling [@Javanainen2003; @Jaksch2003; @Osterloh2005; @Dalibard2011; @Cooper2011PRL; @Aidelsburger:2013; @Goldman2014; @goldman16review; @miyake13harper]; current realizations of these techniques are beset with micro motion and interaction induced heating effects. Here we propose a new method that simultaneously creates large artificial magnetic fields and a lattice that may overcome these limitations.
Our technique relies on a pulsed atom-light coupling between internal atomic states along with a state-dependent gradient potential that together create a two-dimensional (2D) periodic potential with an intrinsic artificial magnetic field. With no pre-existing lattice potential, there are no a priori resonant conditions that would otherwise constrain the modulation frequency to avoid transitions between original Bloch bands [@Weinberg15PRA]. For a wide range of parameters, the ground and excited bands of our lattice are topological, with nonzero Chern number. Moreover, like Landau levels the lowest several bands can all have unit Chern number.
The manuscript is organized as follows. Firstly, we describe a representative experimental implementation of our technique directly suitable for alkali atoms. Secondly, because the pulsed atom-light coupling is time-periodic, we use Floquet methods to solve this problem. Specifically, we employ a stroboscopic technique to obtain an effective Hamiltonian. Thirdly, using the resulting band structure we obtain a phase diagram which includes a region of Landau level-like bands each with unit Chern number.
Pulsed lattice
==============
Figure 1 depicts a representative experimental realization of the proposed method. A system of ultracold atoms is subjected to a magnetic field with a strength $B(X)=B_{0}+B^{\prime}X$. This induces a position-dependent splitting $g_{F}\mu_{{\rm B}}B$ between the spin up and down states; $g_{F}$ is the Landé $g$-factor and $\mu_{{\rm B}}$ is the Bohr magneton. Additionally, the atoms are illuminated by a pair of Raman lasers counter propagating along ${\bf
e}_{y}$, i.e. perpendicular to the detuning gradient. The first beam (up-going in Fig. \[fig:schematic\](a)) is at frequency $\omega^{+}=\omega_{0}$, while the second (down-going in Fig. \[fig:schematic\](a)) contains frequency components $\omega_{n}^{-}=\omega_{0}+(-1)^{n}(\delta\omega+n\omega)$; the difference frequency between these beams contains frequency combs centered at $\pm\delta\omega$ with comb teeth spaced by $2\omega$, as shown in Fig. \[fig:schematic\](b). In our proposal, the Raman lasers are tuned to be in nominal two-photon resonance with the Zeeman splitting from the large offset field $B_{0}$ such that $g_{F}\mu_{{\rm B}}B_{0}=\hbar\delta\omega_{0}$, making the frequency difference $\omega_{n=0}^{-}-\omega^{+}$ resonant at $X=0$, where $B=B_{0}$. Intuitively, each additional frequency component $\omega_{n}^{-}$ adds a resonance condition at the regularly spaced points $X_{n}=n\hbar\omega/g_{F}\mu_{{\rm B}}B^{\prime}$, however, transitions using even-$n$ side bands give a recoil kick opposite from those using odd-$n$ side bands (see Fig. \[fig:schematic\](c)). Each of these coupling-locations locally realizes synthetic magnetic field experiment performed at NIST [@Lin2009b], arrayed in a manner to give a rectified artificial magnetic field with a non-zero average that we will show is a novel flux lattice.
![Floquet flux lattice. a. Experimental schematic depicting a cold cloud of atoms in a gradient magnetic field, illuminated by a pair of counter-propagating laser beams tuned near two-photon Raman resonance. The down-going beam includes sidebands both to the red and blue of the carrier ($\omega_{0}$) in resonance at different spatial positions along ${\bf
e}_{x}$. b. Level diagram showing even and odd side-bands linking the $\left|\uparrow\right\rangle $ and $\left|\downarrow\right\rangle $ states with differing detuning from resonance at $X=0$. c. Spatially dependent coupling. Bottom: different frequency components are in two-photon resonance in different $X$ positions. Top: the recoil kick associated with the Raman transition is along $\pm\mathbf{e}_{y}$ and thus alternates spatially depending on whether the Raman transition is driven from the red or blue sideband of the down-going laser beam.[]{data-label="fig:schematic"}](fig1)
We formally describe our system by first making the rotating wave approximation (RWA) with respect to the large offset frequency $\omega_{0}$. This situation is modeled in terms of a spin-1/2 atom of mass $M$ and wave-vector $\bm{K}$ with a Hamiltonian $$H(t)=H_{0}+V(t).\label{eq:full-Hamiltonian}$$ The first term is $$H_{0}=\frac{\hbar^{2}\bm{K}^{2}}{2M}+\frac{\Delta(X)}{2}\sigma_{3},\label{eq:Hamiltonian0}$$ where $\Delta(X)=\Delta^{\prime}X$ describes the detuning gradient along $\mathbf{e}_{x}$ axis, and $\sigma_{3}=|\!\uparrow\rangle\langle\uparrow\!|-|\!\downarrow\rangle\langle\downarrow\!|$ is a Pauli spin operator. In the RWA only near-resonant terms are retained, giving the Raman coupling described by $$V(t)=V_{0}\sum_{n}\left[{\rm e}^{{\rm i}(K_{0}Y-2n\omega t)}+{\rm e}^{{\rm
i}(-K_{0}Y-(2n+1)\omega
t)}\right]|\!\downarrow\rangle\langle\uparrow\!|+{\rm
H.\,c.}\,.\label{eq:Raman-coupling}$$ The first term describes coupling from the sidebands with even frequencies $2n\omega$, whereas the second term describes coupling from the sidebands with odd frequencies $\left(2n+1\right)\omega$. The recoil kick is aligned along $\pm\mathbf{e}_{y}$ with opposite sign for the even and odd frequency components. In writing Eq.(\[eq:Raman-coupling\]) we assumed that the coupling amplitude $V_{0}$ and the associated recoil wave number $K_{0}$ are the same for all frequency components. The coupling Hamiltonian $V(t)$ and therefore the full Hamiltonian $H(t)$ are time-periodic with period $2\pi/\omega$, and we accordingly apply Floquet techniques.
Theoretical analysis
====================
The outline of this Section is as follows. (1) We begin the analysis of the Hamiltonian given by Eq. (\[eq:full-Hamiltonian\]) by moving to dimensionless units; (2) subsequently derive an approximate effective Hamiltonian from the single-period time evolution operator; (3) provide an intuitive description in terms of adiabatic potentials; and (4) finally solve the band structure, evaluate its topology and discuss possibilities of the experimental implementation.
Dimensionless units
-------------------
For the remainder of the manuscript we will use dimensionless units. All energies will be expressed in units of $\hbar\omega$, derived from the Floquet frequency $\omega$; time will be expressed in units of inverse driving frequency $\omega^{-1}$, denoted by $\tau=\omega t$; spatial coordinates will be expressed in units of inverse recoil momentum $K_{0}^{-1}$, denoted by lowercase letters $(x,y)=K_{0}(X,Y)$. In these units, the Hamiltonian (\[eq:full-Hamiltonian\]) takes the form $$h(\tau)=\frac{H(\tau/\omega)}{\hbar\omega}=E_{\text{r}}\bm{k}^{2}
+\frac{1}{2}\boldsymbol{\Omega}(\tau)\cdot\boldsymbol{\sigma}\,,
\label{eq:dimless-Hamiltonian}$$ where $E_{\text{r}}=\hbar^{2}K_{0}^{2}/(2M\hbar\omega)$ is the dimensionless recoil energy associated with the recoil wavenumber $K_{0}$; $\bm{k}=\bm{K}/K_{0}$ is the dimensionless wavenumber. The dimensionless coupling $$\boldsymbol{\Omega}(x,y,\tau)=\left(2{\rm Re}\,u(y,\tau),\,2{\rm Im}\,u(y,\tau),\,\beta x\right)
\label{eq:full-coupling}$$ includes a combination of position-dependent detuning and Raman coupling. Here $\beta=\Delta^{\prime}/(\hbar\omega k_{0})$ describes the linearly varying detuning in dimensionless units; the function $u(y,\tau)=v_{0}\sum_{n}\left\{
\exp[{\rm i}(y-2n\tau)]+\exp[{\rm i}(-y-(2n+1)\tau)]\right\} $ is a dimensionless version of the sum in Eq. (\[eq:Raman-coupling\]) with $v_{0}=V_{0}/(\hbar\omega)$.
The vector $\boldsymbol{\Omega}(x,y,\tau)$ is spatially periodic along the $y$ direction with a period $2\pi$. This period can be halved to $\pi$ by virtue of a gauge transformation $U=\exp(-{\rm i}y\sigma_{3}/2)$. Subsequently, when exploring energy bands and their topological properties, this prevents problems arising from using a twice larger elementary cell. Following this transformation the dimensionless Hamiltonian becomes $$\tilde{h}(\tau)=E_{\text{r}}\left(\bm{k}+\sigma_{3}\bm{e}_{y}/2\right)^{2}
+\frac{1}{2}\tilde{\boldsymbol{\Omega}}(\tau)\cdot\boldsymbol{\sigma}$$ with $\tilde{\boldsymbol{\Omega}}(\tau)=U\boldsymbol{\Omega}(\tau)U^{-1}$.
In the time domain the coupling (\[eq:full-coupling\]) is $$\frac{1}{2}\tilde{\boldsymbol{\Omega}}(\tau)\cdot\boldsymbol{\sigma}=\frac{1}{2}\beta x\sigma_{3}
+\sum_{l}v_{l}(y)\delta(\tau-\pi l),
\label{eq:coupling-even-odd}$$ with $$v_{l}(y)=\pi v_{0}\left[{\rm e}^{{\rm i}2y}+(-1)^{l}\right]|\!\downarrow\rangle\langle\uparrow\!|+{\rm H.\,c.}\,.$$ In this way we separated the spatial and temporal dependencies in the coupling (\[eq:coupling-even-odd\]).
Effective Hamiltonian
---------------------
We continue our analysis by deriving an approximate Hamiltonian that describes the complete time evolution over a single period from $\tau=0-\epsilon$ to $\tau=2\pi-\epsilon$ with $\epsilon\to0$. This evolution includes a kick $v_{0}$ at the beginning of the period $\tau_{+}=0$ and a second kick $v_{1}$ in the middle of the period $\tau_{-}=\pi$; between the kicks the evolution includes the kinetic and gradient energies. In the full time period, the complete evolution operator is a product of four terms: $$U(2\pi,0)\equiv\lim_{\epsilon\to0}U(2\pi-\epsilon,0-\epsilon)=U_{0}U_{\text{kick}}^{(1)}U_{0}U_{\text{kick}}^{(0)}.
\label{eq:time-evolution}$$ Here $$U_{0}=\exp\left\{ -{\rm i}\pi\left[E_{\text{r}}\left(\bm{k}+\frac{1}{2}\sigma_{3}\bm{e}_{y}\right)^{2}
+\frac{1}{2}\sigma_{3}\beta x\right]\right\}
\label{eq:time-evolution-0}$$ is the evolution operator over the half period, generated by kinetic energy and gradient. The operator $$U_{\text{kick}}^{(l)}=\exp\left[-{\rm i}v_{l}(y)\right].\label{eq:kicks}$$ describes a kick at $\tau=l\pi$.
We obtain an effective Hamiltonian by assuming that the Floquet frequency $\omega$ greatly exceeds the recoil frequency, $1\gg E_{\text{r}}$, allowing us to ignore the commutators between the kinetic energy and functions of coordinates in eq.(\[eq:time-evolution\]). We then rearrange terms in the full time evolution operator (\[eq:time-evolution\]) and obtain $$U_{\text{eff}}=\exp\left\{ -{\rm i}2\pi\left[E_{\text{r}}\left(\bm{k}
+\sigma_{3}\bm{e}_{y}/2\right)^{2}+v_{\text{eff}}\right]\right\},
\label{eq:effective-time-evolution-op}$$ where $v_{\text{eff}}$ is an effective coupling defined by $$\exp\left(-{\rm i}2\pi v_{\text{eff}}\right)={\rm e}^{-{\rm i}\pi\sigma_{3}\beta x/2}U_{\text{kick}}^{(1)}{\rm e}^{-{\rm i}
\pi\sigma_{3}\beta x/2}U_{\text{kick}}^{(0)}.
\label{eq:V-eff-exponent}$$
The algebra of Pauli matrices allows us to write the effective coupling in a form $$v_{\text{eff}}(\bm{r})=\frac{1}{2}\bm{\Omega}_{\text{eff}}(\bm{r})\cdot\bm{\sigma},
\label{eq:effective-coupling}$$ where $\bm{\Omega}_{\text{eff}}=\left(\Omega_{\text{eff},1},\Omega_{\text{eff},2},\Omega_{\text{eff},3}\right)$ is a position-dependent effective Zeeman field which takes the analytic form $$\exp\left(-{\rm i}2\pi v_{\text{eff}}\right)=q_{0}-{\rm i}q_{1}\sigma_{1}
-{\rm i}q_{2}\sigma_{2}-{\rm i}q_{3}\sigma_{3}.
\label{eq:V-eff-exponent-in-q}$$ Here $q_{0}$, $q_{1}$, $q_{2}$ and $q_{3}$ are real functions of the coordinates $(x,y)$, allowing to express the effective Zeeman field as $$\bm{\Omega}_{\text{eff}}=\pi^{-1}\frac{\bm{q}}{||\bm{q}||}\arccos q_{0},
\label{eq:effective-magnetic-field}$$ where $\bm{q}$ is a shorthand of a three dimensional vector $(q_{1},q_{2,}q_{3})$. In general the equation (\[eq:V-eff-exponent-in-q\]) gives multiple solutions that correspond for different Floquet bands. Our choice (\[eq:effective-magnetic-field\]) picks only to the two bands that lie in the energy window from $-1/2$ to $1/2$ covering a single Floquet period.
Comparing (\[eq:V-eff-exponent\]) and (\[eq:V-eff-exponent-in-q\]) and multiplying four matrix exponents give explicit expressions $$\begin{aligned}
q_{0} & =\cos f_{1}\cos f_{2}\cos(\pi\beta x),\label{eq:function-q0}\\
q_{1} & =\sin f_{1}\cos f_{2}\cos(y+\pi\beta x)-\cos f_{1}\sin f_{2}\sin(y),\label{eq:function-q1}\\
q_{2} & =\sin f_{1}\cos f_{2}\sin(y+\pi\beta x)+\cos f_{1}\sin f_{2}\cos(y),\label{eq:function-q2}\\
q_{3} & =\cos f_{1}\cos f_{2}\sin(\pi\beta x)-\sin f_{1}\sin f_{2}\label{eq:function-q3}\end{aligned}$$ with $$\begin{aligned}
f_{1}(y) & =2\pi v_{0}\cos(y),\label{eq:function-f1}\\
f_{2}(y) & =2\pi v_{0}\sin(y).\label{eq:function-f2}\end{aligned}$$
![Coupling components (a) $\Omega_{1}(\bm{r})$, (b) $\Omega_{2}(\bm{r})$ and (c) $\Omega_{3}(\bm{r})$ for $v_{0}=0.25$ and $\beta=0.6$. The corresponding eigenvalues of the coupling $v_{\pm}(\bm{r})=\pm\Omega_{\text{eff}}/2$ are presented by the thick red solid lines in the fig. \[fig:floquet-spectrum\].[]{data-label="fig:coupling"}](fig2a "fig:"){width="33.00000%"}![Coupling components (a) $\Omega_{1}(\bm{r})$, (b) $\Omega_{2}(\bm{r})$ and (c) $\Omega_{3}(\bm{r})$ for $v_{0}=0.25$ and $\beta=0.6$. The corresponding eigenvalues of the coupling $v_{\pm}(\bm{r})=\pm\Omega_{\text{eff}}/2$ are presented by the thick red solid lines in the fig. \[fig:floquet-spectrum\].[]{data-label="fig:coupling"}](fig2b "fig:"){width="33.00000%"}![Coupling components (a) $\Omega_{1}(\bm{r})$, (b) $\Omega_{2}(\bm{r})$ and (c) $\Omega_{3}(\bm{r})$ for $v_{0}=0.25$ and $\beta=0.6$. The corresponding eigenvalues of the coupling $v_{\pm}(\bm{r})=\pm\Omega_{\text{eff}}/2$ are presented by the thick red solid lines in the fig. \[fig:floquet-spectrum\].[]{data-label="fig:coupling"}](fig2c "fig:"){width="33.00000%"}
These explicit expressions show that the resulting effective Zeeman field (\[eq:effective-magnetic-field\]) and the associated effective coupling (\[eq:effective-coupling\]) are periodic along both $\bm{e}_{x}$ and $\bm{e}_{y}$, with spatial periods $a_{x}=2/\beta$ and $a_{y}=\pi$ respectively. Therefore, although the original Hamiltonian containing the spin-dependent potential slope $\propto x\sigma_{3}$ is not periodic along the $x$ direction, the effective Floquet Hamiltonian is. The spatial dependence of the Zeeman field components $\Omega_{\text{eff},1}$, $\Omega_{\text{eff},2}$ and $\Omega_{\text{eff},3}$ is presented in the fig. \[fig:coupling\] for $\beta=0.6$ giving an approximately square unit cell. In fig. \[fig:coupling\] we select $v_{0}=0.25$ where the absolute value of the Zeeman field $\Omega_{\text{eff}}$ is almost uniform, as is apparent from the nearly flat adiabatic bands shown in fig. \[fig:floquet-spectrum\] below.
Adiabatic evolution and magnetic flux\[subsec:Adiabatic-evolution-and\]
-----------------------------------------------------------------------
Before moving further to an explicit numerical analysis of the band structure, we develop an intuitive understanding by performing an adiabatic analysis of motion governed by effective Hamiltonian $$h_{\text{eff}}(\bm{r})=E_{\text{r}}\left(\bm{k}+\sigma_{3}\bm{e}_{y}/2\right)^{2}
+\frac{1}{2}\bm{\Omega}_{\text{eff}}\cdot\bm{\sigma}\,
\label{eq:h_eff}$$ featured in the evolution operator $U_{\text{eff}}$, Eq. (\[eq:effective-time-evolution-op\]). The coupling field $\bm{\Omega}_{\text{eff}}(\bm{r})$ is parametrized by the spherical angles $\theta(\bm{r})$ and $\phi(\bm{r})$ defined by $$\begin{aligned}
\cos\theta & =\frac{\Omega_{\text{eff},3}}{\Omega_{\text{eff}}},\label{eq:spherical-cos-theta}\\
\tan\phi & =\frac{\Omega_{\text{eff},2}}{\Omega_{\text{eff},1}}.\label{eq:spherical-tan-phi}\end{aligned}$$ This gives the effective coupling [@Dalibard2011] $$\frac{1}{2}\bm{\Omega}_{\text{eff}}\cdot\bm{\sigma}=
\frac{1}{2}\Omega_{\text{eff}}\left[\begin{array}{cc}
\cos\theta & {\rm e}^{-{\rm i}\phi}\sin\theta\\
{\rm e}^{{\rm i}\phi}\sin\theta & -\cos\theta
\end{array}\right]\,,\label{eq:eff-coupling-in-spherical-coords}$$ characterized by the position-dependent eigenstates $$\left|+\right\rangle =\left(\begin{array}{c}
\cos\left(\theta/2\right)\\
{\rm e}^{{\rm i}\phi}\sin\left(\theta/2\right)
\end{array}\right)\,,\qquad\left|-\right\rangle =\left(\begin{array}{c}
-{\rm e}^{-{\rm i}\phi}\sin\left(\theta/2\right)\\
\cos\left(\theta/2\right)
\end{array}\right)\,.\label{eq:pm-states}$$ The corresponding eigenvalues $$v_{\pm}(\bm{r})=\pm\frac{1}{2}\Omega_{\text{eff}},
\label{eq:eigenvalues-of-V-eff}$$ are shown in Fig. \[fig:floquet-spectrum\] for various value of the Raman coupling $v_{0}$. As one can see in Fig. \[fig:floquet-spectrum\], for $v_{0}=0.25$ the resulting bands $v_{\pm}(\bm{r})$ (adiabatic potentials) are flat and have a considerable gap $\approx\omega/2$, a regime suitable for a description in terms of an adiabatic motion in selected bands [@Zoller2008].
![Adiabatic Floquet potentials for $\beta=0.6$. (a) Thin black dotted lines denote the spin-dependent gradient slopes without including the Raman coupling ($v_{0}=0$); (b) thin blue solid lines denote effective adiabatic potentials for weak Raman coupling ($v_{0}=0.05$) (c) red solid lines denote nearly flat adiabatic potentials that are achieved for stronger Raman coupling ($v_{0}=0.25$). All the curves are projected into $x$ plane for various $y$ values. A weak $y$ dependence of the adiabatic potentials is seen to appear in the strong coupling case (c) making the superimposed red lines thicker.[]{data-label="fig:floquet-spectrum"}](fig3){width="75.00000%"}
As in Ref. [@Juzeliunas2012], we consider the adiabatic motion of the atom in one of these flat adiabatic bands with the projection Shrodinger equation that includes a geometric vector potential $$\bm{A}_{\pm}(\bm{r})=\pm\frac{1}{2}\left(\cos\theta-1\right)\nabla\phi\,.
\label{eq:geometric-vector-potential}$$ This provides a synthetic magnetic flux density $\bm{B}_{\pm}(\bm{r})=\nabla\times\bm{A}_{\pm}(\bm{r})$. The geometric vector potential $\bm{A}_{\pm}(\bm{r})$ may contain Aharonov-Bohm type singularities, that give rise to a synthetic magnetic flux over an elementary cell $$\alpha_{\pm}=-\sum\oint_{{\rm singul}}{\rm d}\bm{r}\cdot\bm{A}_{\pm}(\bm{r}).
\label{eq:synthetic-magnetic-flux}$$ The singularities appear at points where $\theta=\pi$, where the angle $\phi$ and its gradient $\nabla\phi$ are undefined and $\cos\theta=-1$. The term $\cos\theta-1$ in (\[eq:geometric-vector-potential\]) is non zero and does not remove the undefined phase $\nabla\phi$. Our unit cell contains two such singularities located at $\bm{r}=(a_{x},3a_{y})/4$ and $\bm{r}=(3a_{x},a_{y})/4$, containing the same flux, so that they do not compensate each other, giving the synthetic magnetic flux $\pm2\pi$ in each unit cell.
![Geometric flux density $\bm{B}_{+}$ computed for $v_{0}=0.25$ and $\beta=0.6$. The overall spatial structure of this flux density does not depend on the gradient $\beta$; rather it scales with the corresponding lattice constant $a_{x}=2/\beta$. []{data-label="fig:flux-density"}](fig4){width="50.00000%"}
For a weak coupling (such as $v=0.05$) the geometric flux density $\bm{B}(\bm{r})\equiv\bm{B}_{\pm}(\bm{r})$ is concentrated around the intersection points of the gradient slopes shown in in Fig. \[fig:floquet-spectrum\] and has a very weak $y$ dependence. With increasing the coupling $v$, the flux extends beyond the intersection areas and acquires a $y$ dependence. Fig. \[fig:flux-density\] shows the geometric flux density $\bm{B}(\bm{r})\equiv\bm{B}_{+}(\bm{r})$ for the strong coupling ($v_{0}=0.25$) corresponding to the most flat adiabatic bands. In this regime the flux develops stripes in the $x$ direction and has a strong $y$ dependence. For the whole range of coupling strengths $0\le v_{0}\le 1/2$ the total synthetic magnetic flux per unit cell is $2\pi$ and is independent of the Floquet frequency $\omega$ and the gradient $\beta$.
Band structure and Chern numbers
--------------------------------
We analyze the topological properties of this Floquet flux lattice by explicitly numerically computing the band structure and associated Chern number using the effective Hamiltonian (\[eq:h\_eff\]) without making the adiabatic approximation introduced in Sec. \[subsec:Adiabatic-evolution-and\]. Again the gradient of the original magnetic field is such that we approximately get a square lattice, $\beta=0.6$. Furthermore, we choose the Floquet frequency to be ten times larger than the recoil energy, $E_{\text{r}}=0.1$.
First, let us consider the case where $v_{0}=0.25$ corresponding to the most flat adiabatic potential. In this situation the Chern numbers of the first five bands appear to be equal to the unity, as one can see in the left part of Fig. \[fig:bands-chern\]. Thus the Hall current should monotonically increase when filling these bands. This resembles the Quantum Hall effect involving the Landau levels. Second, we check what happens when we leave the regime $v_{0}=0.25$ where the adiabatic potential is flat, and consider lower and higher values of the coupling strength $v_{0}$. Near $v_{0}=0.175$ we find a topological phase transition where the lowest two energy bands touch and their Chern numbers change to $c_{1}=0$ and $c_{2}=2$, while the Chern numbers of the higher bands remain unchanged, illustrated in fig. \[fig:chern\]. In a vicinity of $v_{0}=0.3$ there is another phase transition, where the second and third bands touch, leading to a new distribution of Chern numbers: $c_{1}=1$, $c_{2}=-1$, $c_{3}=3$, $c_{4}=1$. Interestingly the Chern numbers of the second and the third bands jump by two units during such a transition.
![Left: band structure given by the effective Hamiltonian (\[eq:h\_eff\]) for $v_{0}=0.25$, $\beta=0.6$ and $E_{\text{r}}=0.1$. Right: The band gap $\Delta_{12}$ between the first and second bands for $E_{\text{r}}=0.1$ and various values of $v_{0}$ and $\beta$.[]{data-label="fig:bands-chern"}](fig5){width="85.00000%"}
![Dependence of Chern number on the coupling strength $v_{0}$ for $\beta=0.6$ and $E_{\text{r}}=0.1$. Here we present the Chern numbers $c_{1}$, $c_{2}$ and $c_{3}$ of the three lowest bands.[]{data-label="fig:chern"}](fig6){width="50.00000%"}
Finally, we explore the robustness of the topological bands. The right part of Fig. \[fig:bands-chern\] shows the dependence of the band gap $\Delta_{12}$ between the first and second bands on the coupling strength $v_{0}$ and the potential gradient $\beta$. One can see that the band gap is maximum for $v_{0}=0.25$ when the adiabatic potential is the most flat. The gap increases by increasing the gradient $\beta$, simultaneously extending the range of the $v_{0}$ values where the band gap is nonzero. Therefore to observe the topological bands, one needs to take a proper value of the Raman coupling $v_{0}\approx0.25$ and a sufficiently large gradient $\beta$, such as $\beta=0.6$.
We now make some numerical estimates to confirm that this scheme is reasonable. We consider an ensemble of $^{87}{\rm Rb}$ atoms, with $\left|{\uparrow}\right>=\left|f=2,m_{F}=2\right>$ and $\left|\downarrow\right>=\left|{f=1,m_{F}=1}\right>$; the relative magnetic moment of these hyperfine states is $\approx2.1\ {\rm MHz}/{\rm G}$, where $1\
{\rm G}=10^{-4}\ {\rm T}$. For a reasonable magnetic field gradient of $300\
\mathrm{G}/\mathrm{cm}$, this leads to the $\Delta^{\prime}/\hbar\approx2\pi\times600\
\mathrm{MHz}/\mathrm{cm}=2\pi\times60\ {\rm kHz}/\mu{\rm m}$ detuning gradient. For $^{87}{\rm Rb}$ with $\lambda=790\ {\rm nm}$ laser fields the recoil frequency is $\omega_{r}/2\pi=3.5\ \mathrm{kHz}$. Along with the driving frequency $\omega=10\omega_{r}$, this provides the dimensionless energy gradient $\beta=\Delta^{\prime}/(\hbar\omega k_{0})\approx1.3$, allowing easy access to the topological bands displayed in Fig. \[fig:bands-chern\].
Loading into dressed states
---------------------------
Adiabatic loading into this lattice can be achieved by extending the techniques already applied to loading in to Raman dressed states [@Lin2009a]. The loading technique begins with a BEC in the lower energy $\downarrow$ state in a uniform magnetic field $B_{0}$. Subsequently one slowly ramps on a single off resonance RF coupling field and the adiabatically ramp the RF field to resonance (at frequency $\delta\omega$). This RF dressed state can be transformed into a resonant Raman dressed by ramping on the Raman lasers (with only the $\omega_{0}+\delta\omega$ frequency on the $k^{-}$ laser beam) while ramping off the RF field. The loading procedure then continues by slowly ramping on the remaining frequency components on the $k^{-}$ beam, and finally by ramping on the magnetic field gradient (essentially according in the lattice sites from infinity). This procedure leaves the BEC in the $q=0$ crystal momentum state in a single Floquet band.
Conclusions
===========
Initial proposals [@Juzeliunas2006; @Spielman2009; @Gunter2009] and experiments [@Lin2009b] with geometric gauge potentials were limited by the small spatial regions over which these existed. Here we described a proposal that overcomes these limitations using laser coupling reminiscent of a frequency comb: temporally pulsed Raman coupling. Typically, techniques relying on temporal modulation of Hamiltonian parameters to engineer lattice parameters suffer from micro-motion driven heating. Because our method is applied to atoms initially in free space, with no optical lattice present, there are no a priori resonant conditions that would otherwise constrains the modulation frequency to avoid transitions between original Bloch bands [@Weinberg15PRA].
Still, no technique is without its limitations, and this proposal does not resolve the second standing problem of Raman coupling techniques: spontaneous emission process from the Raman lasers. Our new scheme extends the spatial zone where gauge fields are present by adding side-bands to Raman lasers, ultimately leading to a $\propto\sqrt{N}$ increase in the required laser power (where $N$ is the number of frequency tones), and therefore the spontaneous emission rate. As a practical consequence it is likely that this technique would not be able reach the low entropies required for many-body topological matter in alkali systems [@Goldman2014], but straightforward implementations with single-lasers on alkaline-earth clock transitions [@Fallani16PRL; @Kolkowitz2016socSr] are expected to be practical.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Immanuel Bloch, Egidijus Anisimovas and Julius Ruseckas for helpful discussions. This research was supported by the Lithuanian Research Council (Grant No. MIP-086/2015). I. B. S. was partially supported by the ARO’s Atomtronics MURI, by AFOSR’s Quantum Matter MURI, NIST, and the NSF through the PCF at the JQI.
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| ArXiv |
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author:
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[C.G.Bao$^1$, Y.X.Liu$^2$]{}\
[$^1$ Department of Physics,Zhongshan University, Guangzhou,510275,P.R.China.]{}\
[$^2$ Department of Physics, Beijing University, Beijing, 100871, P.R.China]{}
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ABSTRACT: The inherent nodal structures of the wavefunctions of 6-nucleon systems are investigated. A group of six low-lying states (including the ground states) dominated by total orbital angular momentum L=0 components are found, the quantum numbers of each of these states are deduced. In particular, the spatial symmetries of these six states are found to be mainly the {4,2} and {2,2,2}.
PACS: 21.45.+v, 02.20.-a, 27.20.+n
As a few-body system the 6-body system has been scarcely investigated theoretically due to the complexity arising from the 15 spatial degrees of freedom. The existing related literatures concern mainly the ground states and a few resonances \[1-5\]. The study of the character of the excited states is very scarce. On the other hand, the particles of 6-body systems are neither too few nor too many. The study of them is attractive because it may lead to an understanding of the connection between the few-body theory and the modal theories for nuclei. Before solving the 6-body Schrödinger equation precisely, if we can have some qualitative understanding of the spectrum, it would be very helpful. This understanding , together with the results from calculations and experiments, will lead to a complete comprehension of the physics underlying the spectrum. In \[6\] the qualitative feature of 4-nucleon systems has been studied based on symmetry. In this paper we shall generalize the idea of \[6\] to extract qualitative character of the low-lying states of 6-nucleon systems.
There are two noticeable findings in \[6\]. (i) The ground state is dominated by total orbital angular momentum L=0 component, while all the resonances below the 2n+2p threshold are dominated by L=1 components, there is a very large gap lying between them. Experimentally, this gap is about 20 MeV. This fact implies that the collective rotation is difficult to be excited. (ii) The internal wavefunctions (the wavefunction relative to a body-frame) of all the states below the 2n+2p threshold do not contain nodal surfaces. This fact implies that the excitation of internal oscillation takes a very large energy. Therefore, ti would be reasonable to assume that the L=0 nodeless component will be also important in the low-lying spectrum of the 6-nucleon systems.
It was found in \[7,8\] that a specific kind of nodal surfaces may be imposed on the wavefunctions by symmetry. Let $\Psi $ be an eigenstate. Let $A$ denotes a geometric configuration. In some cases $A$ may be invariant to specific combined operations $O_i$ ( i=1 to m). For example, when $A$ is a regular octahedron (OCTA) for a 6-body system, then $A$ is invariant to a rotation about a 4-fold axis of the OCTA by 90$^{\circ }$ together with a cyclic permutation of four particles. In this case we have
$$\stackrel{\wedge }{O_i}\Psi (A)=\Psi (O_iA)=\Psi (A)\hspace{1.0in}(1)$$
Owing to the inherent transformation property of $\Psi $ (the property with respect to rotation, inversion, and permutation), (1) always can be written in a matrix form (as we shall see) and appears as a set of homogeneous linear algebra equations. They impose a very strong constraint on $\Psi $ so that $\Psi $ may be zero at $A$. This is the origin of this specific kind of nodal surfaces, they are called the inherent nodal surfaces (INS).
The INS appear always at geometric configurations with certain geometric symmetry. For a 6-body system the OCTA is the configuration with the strongest geometric symmetry. Let us assume that the six particles form an OCTA. Let k’ be a 4-fold axis of the OCTA, and let the particles 1,2,3, and 4 form a square surrounding k’. Let $R_\delta ^{k^{\prime }}$ denote a rotation about k’ by the angle $\delta $ (in degree), let $p(1432)$ denotes a cyclic permutation. Evidently, the OCTA is invariant to
$$O_1=p(1432)R_{-90}^{k^{\prime }}\hspace{1.0in}(2)$$
Let $p_{ij}$ denotes an interchange of the locations of particles i and j, $%
P $ denotes a space inversion. The OCTA is also invariant to
$$O_2=p_{13}p_{24}p_{56}P\hspace{1.0in}(3)$$
Let i’ be an axis vertical to k’ and parallel to an edge of the above square; say, parallel to $\stackrel{\rightarrow }{r_{12}}$. Then the OCTAis also invariant to
$$O_3=p_{14}p_{23}p_{56}R_{180}^{i^{\prime }}.\hspace{1.0in}(4)$$
Let $OO^{\prime }$ be a 3-fold axis of the OCTA, where $O$ denotes the center of mass. Let particles 2,5, and 3 form a regular triangle surroundung the $OO^{\prime }$; 1,4, and 6 form another triangle. Then the OCTA is also invariant to
$$O_3=p(253)p(146)R_{-120}^{oo^{\prime }}\hspace{1.0in}(5)$$
Besides, the OCTA is also invariant to some other operators, e.g., the $%
p(152)p(364)R_{-120}^{oo"}$ (where $OO"$ is another 3-fold axis). However, since the rotations about two different 3-fold axes are equivalent, one can prove that this additional operator does not introduce new constraints, and the operators $O_1$ to $O_4$ are sufficient to specify the constraints arising from symmetry.
Let an eigenstate of a 6-nucleon system with a given total angular momentum J, parity $\Pi $, and total isospin T be written as $$\Psi =\sum_{L,S}\Psi _{LS}\hspace{1.0in}(6)$$ where S is the total spin, $$\Psi _{LS}=\sum_{\lambda i}F_{LSM}^{\lambda i}\chi _S^{\stackrel{\symbol{126}%
}{\lambda }i}\hspace{1.0in}(7)$$ Where $M$ is the Z-component of L, $F_{LSM}^{\lambda i}$ is a function of the spatial coordinates, which is the i$^{th}$ basis function of the $%
\lambda -$representation of the S$_6$ permutation group. The $\chi _S^{%
\stackrel{\sim }{\lambda }i}$ is a basis function in the spin-isospin space with a given S and T and belonging to the $\stackrel{\sim }{\lambda }-$representation, the conjugate of $\lambda .$ In (7) the allowed $\lambda $ are listed in Table 1, they depend on S and T \[9\].
S T $\lambda $
--- --- ---------------------------------------------------------------------
0 0 {1$^6$}, {2,2,1,1},{3,3},{4,1,1}
1 0 {2,1$^4$}, {3,1$^3$}, {2,2,2}, {3,2,1}, {4,2}
2 0 {2,2,1,1}, {3,2,1}
3 0 {2,2,2}
0 1 {2,1$^4$}, {3,1$^3$}, {2,2,2}, {3,2,1}, {4,2}
1 1 {1$^6$}, {2,1$^4$}, 2{2,2,1,1}, {3,1$^3$}, 2{3,2,1}, {3,3}, {4,1,1}
2 1 {2,1$^4$}, {2,2,1,1}, {3,1$^3$}, {2,2,2}, {3,2,1}
3 1 {2,2,1,1}
Tab.1, The allowed representation $\lambda $ in (7)
From k’ and i’ defined before one can introduce a body frame i’-j’-k’. In the body-frame the $F_{LSM}^{\lambda i}$ can be expanded $$F_{LSM}^{\lambda i}(123456)=\sum_Q D_{QM}^L(-\gamma ,-\beta ,-\alpha
)F_{LSQ}^{\lambda i}(1^{\prime }2^{\prime }3^{\prime }4^{\prime }5^{\prime
}6^{\prime })\hspace{1.0in}(8)$$ Where $\alpha \beta \gamma $ are the Euler angles to specify the collective rotation, $D_{QM}^L$ is the well known Wigner function, Q are the projection of L along the k’-axis. The (123456) and (1’2’3’4’5’6’) specifies that the coordinates are relative to a fixed frame and the body-frame, respectively.
Since the $F_{LSQ}^{\lambda i}$ span a representation of the rotation group, space inversion group, and permutation group, the invariance of the OCTA to the operations $O_1$ to $O_4$ leads to four sets of equations. For example, from $$\hat O_1F_{LSQ}^{\lambda i}(A)=F_{LSQ}^{\lambda i}(O_1A)=F_{LSQ}^{\lambda
i}(A)\hspace{1.0in} (9)$$ where $F_{LSQ}^{\lambda i}(A)$ denotes that the coordinates in $%
F_{LSQ}^{\lambda i}$ are given at an OCTA, for all $Q$ with $|Q|\leq L$ we have $$\sum_{i^{\prime }}[g_{ii^{\prime }}^\lambda (p(1234))e^{-i\frac \pi
2Q}-\delta _{ii^{\prime }}]F_{LSQ}^{\lambda i^{\prime }}(A)=0\hspace{1.0in}%
(10)$$ where $g_{ii^{\prime }}^\lambda $ are the matrix elements belonging to the representation $\lambda $, which are known from the textbooks of group theory (e.g., refer to \[10\]). From $\hat O_2$ and $\hat O_4,$ we have $$\sum_{i^{\prime }}[g_{ii^{\prime }}^\lambda (p_{13}p_{24}p_{56})\Pi -\delta
_{ii^{\prime }}]F_{LSQ}^{\lambda i^{\prime }}(A)=0\hspace{1.0in}(11)$$ and $$\sum_{Q^{\prime }i^{\prime }}[(-1)^Lg_{ii^{\prime }}^\lambda
(p_{14}p_{23}p_{56})\delta _{\stackrel{-}{Q}Q^{\prime }}-\delta _{ii^{\prime
}}\delta _{QQ^{\prime }}]F_{LSQ^{\prime }}^{\lambda i^{\prime }}(A)=0\ %
\hspace{1.0in}(12)$$ where $\stackrel{-}{Q}=-Q.$ It is noted that $$R_{-120}^{oo^{\prime }}=R_\theta ^{j^{\prime }}R_{-120}^{k^{\prime }}R_{%
\stackrel{-}{\theta }}^{j^{\prime }}\hspace{1.0in}(13)$$ where $\theta =\arccos (\sqrt{\frac 13})$. Thus from $\hat O_3$ we have $$\sum_{Q^{\prime }i^{\prime }}[g_{ii^{\prime }}^\lambda
[p(235)p(164)]\sum_{Q^{\prime \prime }}D_{QQ"}^L(0,\theta ,0)e^{-i\frac{2\pi
}3Q"}D_{Q^{\prime }Q"}^L(0,\theta ,0)-\delta _{ii^{\prime }}\delta
_{QQ^{\prime }}]F_{LSQ^{\prime }}^{\lambda i^{\prime }}(A)=0\hspace{1.0in}%
(14)$$ Eq.(10), (11), (12), and (14) are the equations that the $F_{LSQ}^{\lambda
i}(A)$ have to fulfilled. In some cases there is one or more than one nonzero solution(s) (i.e., not all the $F_{LSQ}^{\lambda i}(A)$ are zero) to all these equations . But in some other cases, there are no nonzero solutions. In the latter case, the $\Psi _{LS}$ has to be zero at the OCTA configurations disregarding their size and orientation. Accordingly, an INS emerges and the OCTA is not accessible. Evidently, the above equations depend on and only on L, $\Pi $, and $\lambda .$ Therefore the existence of the INS does not at all depend on dynamics (e.g., not on the interaction, mass, etc.).
Since the search of nonzero solutions of linear equations is trivial, we shall neglect the details but give directly the results of the L=0 components in the second and fourth columns of Tab.2
0$^{+}$ 0$^{+}$ 0$^{-}$ 0$^{-}$
------------ --------- --------- --------- ---------
$\lambda $ OCTA C-PENTA OCTA C-PENTA
{6} 1 1 0 0
{5,1} 0 1 0 0
{4,2} 1 1 0 0
{3,3} 0 1 0 0
{2,2,2} 1 1 1 0
{2,2,1,1} 0 1 0 0
{2,1$^4$} 0 1 0 0
{1$^6$} 0 1 0 0
{3,2,1} 0 2 0 0
{4,1,1} 0 0 0 0
{3,1$^3$} 0 0 1 0
Tab.2, The accessibility of the OCTA (regular octahedron) and the C-PENTA (regular centered-pentagon) to the L$^\Pi =0^{+}$ and $0^{-}$ wavefunctions with different spatial permutation symmetry $\lambda $. Where the figures in the blocks are the numbers of independent nonzero solutions. The figure 0 implies that nonzero solutions do not exist.
The INS existing at the OCTA may even extend beyond the OCTA. For example, when the shape in Fig.1a is prolonged along k’, then the shape is called a prolonged-octahedron. This shape (denoted by $B$ ) is invariant to $O_1,O_2,$ and $O_4$, but not to $O_3$. Hence, the $F_{LSQ^{\prime }}^{\lambda
i^{\prime }}(B)$ should fulfill only (10) to (12), but not (14). When nonzero common solutions of (10), (11), (12), and (14) do not exist, while nonzero solutions of only (10) to (12) also do not exist, the INS extends from the OCTA to the prolonged-octahedrons. An OCTA has many ways to deform;e.g., instead of a square, the particles 1,2,3, and 4 form a rectangle or form a diamond, etc.. Hence, the INS at the OCTA has many possibilities to extend. How it extend is determined by the (L$\Pi \lambda$) of the wavefunction. Thus, in the coordinate space, the OCTA is a source where the INS may emerge and extend to the neighborhood surrounding the OCTA. This fact implies that specific inherent nodal structure exists. The details of the inherent nodal structure will not be concerned in this paper. However, it is emphasized that for a wavefunction, if the OCTA is accessible, all the shapes in the neighborhood of the OCTA are also accessible, therefore this wavefunction is inherent nodeless in this domain.
Another shape with also a stronger geometric symmetry is a regular centered-pentagons(C-PENTA, the particle 6 is assumed to be located at the center of mass O). Let k’ be the 5-fold axis. The C-PENTA is invariant to (i) a rotation about k’ by $\frac{2\pi }5$ together with a cyclic permutation of the five particles of the pentagon , (ii) a rotation about k’ by $\pi $ together with a space inversion, (iii) a rotation about i’ by $\pi
$ together with $p_{14}p_{23}$ (here i’ is the axis vertical to k’ and connecting O and particle 5). These invariances will lead to constraints embodied by sets of homogeneous equations, and therefore the accessibility of the C-PENTA can be identified as also given in Tab.2.
In addition to the OCTA, the C-PENTA is another source where the INS may emerge and extend to its neighborhood; e.g., extend to the pentagon-pyramid as shown in Fig.1b with h$\neq $0. There are also other sources. For example, the one at the regular hexagons. However, among the 15 bonds, 12 can be optimized at an OCTA, 10 at a pentagon-pyramid, but only 6 at a hexagon. Therefore in the neighborhood of the hexagon (and also other regular shapes) the total potential energy is considerably higher. Since the wavefunctions of the low-lying states are mainly distributed in the domain with a relatively lower potential energy, we shall concentrate only in the domains surrounding the OCTA and the C-PENTA.
When (L$\Pi \lambda $) =(0+{6}), (0+{4,2}), or (0+{2,2,2}), the wavefunction can access both the OCTA and the C-PENTA (refer to Table 2). These and only these wavefunctions are inherent-nodeless in the two most important domains, and they should be the dominant components for the low-lying states. All the other L=0 components must contain at least an INS resulting in a great increase in energy. From Tab.1 it is clear that the (0+{6}) component is not allowed, while the (0+{4,2}) component can be contained in \[S,T\]=\[1,0\] and \[0,1\] states, and the (0+{2,2,2}) component can be contained in \[S,T\]=\[1,0\], \[3,0\], \[0,1\], and \[2,1\] states. When \[S,T\]=\[1,0\] , the $\lambda $ can be {4,2} or {2,2,2}, therefore two J$%
^\Pi =1^{+}$ partner-states with their spatial wavefunctions orthogonal to each other exist, each of them is a specific mixture of {4,2} and {2,2,2}. Similarly, two partner-states with \[S,T\]=\[0,1\] and J$^\Pi =0^{+}$ exist also. When \[S,T\]=\[3,0\] or \[2,1\], the $\lambda $ has only one choice, therefore in each case only one state exists. Thus we can predict that there are totally six low-lying states dominated by L=0 components without nodal surfaces as listed in Tab.3, where the L,S, and $\lambda $ are only the quantum numbers of the dominant component.
$\lambda $ E
-- --- --- --- -- ------------------- ------
{4,2} and {2,2,2} 0
{4.2} and {2,2,2} 5.65
{2,2,2} 2.19
0 2 + 4.31
{4,2} and {2,2,2} 3.56
{4,2} and {2,2,2}
{2,2,2} 5.37
Tab.3, Prediction of the quantum numbers of low-lying states (dominated by L=0 components) of the 6-nucleon systems based on symmetry. The last column is the energies (in MeV) of the states of $^6$Li taken from \[11\].
It is expected that these low-lying states should be split by the nuclear force. Owing to the interference of the {4,2} and {2,2,2} components, there would be an larger energy gap lying between the two partner-states of each pair. Ajzenberg-selove has made an analysis on $^6$Li based on experimental data \[11\], the results are listed in Tab.3. Although our analysis is based simply on symmetry, but the results of the two analyses are close. For the T=0 states, there are two J$^\Pi =$ 1$^{+}$ states (\[S,T\]=\[1,0\]) in \[11\] with a split, they are just the expected partners. The split is so large (5.65 MeV) that the lower one becomes the ground state while the higher one becomes the highest state of this group. There is a T=0 state in \[11\] at 2.19 MeV with exactly the predicted quantum numbers J$^\Pi
=3^{+}$. Nonetheless, there is a T=0 state in \[11\] at 4.31 MeV with J$^\Pi $ = 2$^{+}$, which do not appear in our analysis. May be this state is dominated by L=1 component, may be there is another origin to be clarified.
For the T=1 states, one of the expected partners with J$^\Pi =0^{+}$ (\[S,T\]=\[0,1\]) was found in \[11\] at 3.56 MeV . However, the other partner ( it would be considerably higher) has not yet been identified in \[11\], this is an open problem. Nonetheless, if this state exists, the structure of its spatial wavefunction would be similar to the T=0 state at 5.65 MeV . The third expected T=1 state was found in \[11\] at 5.37MeV with exactly the predicted J$^\Pi =2^{+}$.
In summary we have explained the origin of the quantum numbers of the low-lying states of 6-nucleon systems. The explanation is very different from that based on the shell model \[12,13\]. For example, according to our analysis, the J$^\Pi =3^{+}$ state at 2.19 MeV has S=3 and L=0. On the contrary, in the shell model the four nucleons in the 1s orbit must have their total spin zero and total isospin zero; therefore this state should have S $\leq 1$ and L $\geq 2.$ However, it is noted that the 2$_1^{+}$ state (having S=0 and L=2) of the $^{12}C$ lies at 4.44 MeV \[14\]. Since the $%
^6$Li is considerably lighter and smaller than the $^{12}$C, the L=2 state of $^6$Li should be much higher than 4.44 MeV due to having a much smaller moment of inertia. Therefore the 3$^{+}$ state at 2.19 MeV is difficult to be explained as a L $\geq $ 2 state. In particular, it is found that the {2,2,2} component is important; however this component is suppressed by the shell model. Thus, our analysis raises a challenge to the shell model in the case that the number of nucleons is not large enough. Evidently, much work should be done to clarify the physics underlying these systems.
It has been shown that sources of INS may exist in the quantum states. Nonetheless, there are essentially inherent-nodeless components of wavefunctions (each with a specific set of (L$\Pi \lambda $)). They are the most important building blocks to constitute the low-lying states. The identification of these particularly favorable components is a key to understand the low-lying spectrum.
The idea of this paper can be generalized to investigate different kinds of systems, thereby we can understand them in an unified way.
ACKNOWLEDGEMENT: This work is supported by the NNSF of the PRC, and by a fund from the National Educational Committee of the PRC.
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| ArXiv |
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abstract: |
The continuous-time random walk (CTRW) model is useful for alleviating the computational burden of simulating diffusion in actual media. In principle, isotropic CTRW only requires knowledge of the step-size, $P_l$, and waiting-time, $P_t$, distributions of the random walk in the medium and it then generates presumably equivalent walks in free space, which are much faster. Here we test the usefulness of CTRW to modelling diffusion of finite-size particles in porous medium generated by loose granular packs. This is done by first simulating the diffusion process in a model porous medium of mean coordination number, which corresponds to marginal rigidity (the loosest possible structure), computing the resulting distributions $P_l$ and $P_t$ as functions of the particle size, and then using these as input for a free space CTRW. The CTRW walks are then compared to the ones simulated in the actual media.
In particular, we study the normal-to-anomalous transition of the diffusion as a function of increasing particle size. We find that, given the same $P_l$ and $P_t$ for the simulation and the CTRW, the latter predicts incorrectly the size at which the transition occurs. We show that the discrepancy is related to the dependence of the effective connectivity of the porous media on the diffusing particle size, which is not captured simply by these distributions.
We propose a correcting modification to the CTRW model – adding anisotropy – and show that it yields good agreement with the simulated diffusion process. We also present a method to obtain $P_l$ and $P_t$ directly from the porous sample, without having to simulate an actual diffusion process. This extends the use of CTRW, with all its advantages, to modelling diffusion processes of finite-size particles in such confined geometries.
author:
- Shahar Amitai
- Raphael Blumenfeld
date: 'Received: date / Accepted: date'
title: 'Modifying continuous-time random walks to model finite-size particle diffusion in granular porous media'
---
Introduction
============
Diffusion plays a key role in a wide range of natural and technological processes. A textbook modelling of such processes is the consideration of the diffusion of a single memory-free particle in a given medium. The nature of such a random walk is governed by three probability density functions (PDFs): of the step size, $P_l(l_i)$; of the step direction, $P_n(\hat{n}_i)$; and of the waiting time between steps, $P_t(t_i)$. These PDFs are, in principle, position dependent, but it is standard practice to derive (or postulate) them assuming position-independence and that $P_n(\hat{n}_i)$ is uniform. The diffusion is then modelled as a continuous-time random walk (CTRW) in free space. Specifically, the CTRW is constructed by adding vectors of uniformly random orientations, whose lengths are chosen from $P_l$, at time intervals chosen from $P_t$. Averaging over sufficiently many such independent processes, the dependence of the mean square distance (MSD) on time satisfies ${\langle \vec{x}^2 \rangle}= Dt^\alpha$. In normal diffusion $\alpha = 1$ and $D$ is the standard diffusion coefficient. But when $P_l$ and/or $P_t$ are very wide, the diffusion might become anomalous ($\alpha \ne 1$). In particular, when $P_t$ has a slowly decaying algebraic tail and $P_l$ does not, the random walk is sub-diffusive ($\alpha < 1$) [@Scher1975; @Scher1991]. Alternatively, if $P_l$ has a slowly decaying algebraic tail and $P_t$ does not, the random walk is super-diffusive ($\alpha > 1$), resembling a Lévy flight [@Mandelbrot1983]. Diffusion processes that have the same value of $\alpha$ are said to be in the same *universality class* [@Kadanoff1966].
Anomalous diffusion can arise from different sources, which can only be identified by going beyond the MSD. When single particle tracking is possible, the movement can be evaluated by the time-averaged MSD (TAMSD), $\delta^2(t, T)$ [@Metzler2014].While the MSD is the ensemble average of the squared distance, made during a time interval $t$, over different realisations, the TAMSD, $\delta^2(t, T)$, is the average of the same quantity along [*a single trajectory*]{} of length $T$. Within the model of sub-diffusive CTRW, the TAMSD satisfies $\langle \delta^2 \rangle \sim t \cdot T^{\alpha - 1}$, where the angular brackets denote a further ensemble average. In contrast, the MSD is sub-linear in $t$, which makes CTRW non-ergodic – the time-average and ensemble-average differ. In particular, the dependence of the TAMSD on $T$ points to the ageing nature of CTRW [@Metzler2014].
A key feature of sub-diffusive CTRW is the randomness of its TAMSD. Since $P_t$ is scale free, the longest waiting times each individual trajectory encounters vary significantly, as do the amplitudes of the individual TAMSDs. To quantify this, we define the amplitude scatter, $\xi = \delta^2 / \langle \delta^2 \rangle$. For ergodic processes (e.g. $\alpha = 1$) its PDF is $P(\xi) = \delta(\xi - 1)$ for sufficiently long trajectory times. But within CTRW this PDF broadens as $\alpha$ decreases. Defining the ergodicity breaking (EB) parameter, ${{\rm EB}}= \langle \xi^2 \rangle - \langle \xi \rangle^2$, it can be derived analytically for CTRW processes as a monotonically decreasing function of $\alpha$.
Another cause for sub-diffusion is walking in a fractal-like environment [@Gefen1983; @Pandey1984]. Such environment is characterised by a network of narrow passages and dead ends at different length scales, which hinder the walk. Unlike CTRW, this process is stationary and therefore ergodic. The TAMSD, like the MSD, is sub-linear in $t$, independent of $T$ and its ${{\rm EB}}$ parameter vanishes.
Using CTRW to model diffusion in confined geometries, such as porous media formed by either sintered or unconsolidated granular materials, is very attractive [@Berkowitz2006; @Bijeljic2006; @Wong2004; @DeAnna2013] because it alleviates the need to simulate directly the dynamics of particles within the pore space, reducing significantly the computational burden. In addition, it alleviates finite-size errors due to finite samples. This practice is based on the common assumption that $P_l$, $P_t$ and $P_n$ alone control the random walk’s universality class. The common procedure is to find first the forms of these distributions in a specific medium, using either small simulations or analytic derivation under some assumptions, and then use these to carry out a many-step CTRW in free space. It is then presumed that the CTRW yields the same universality class as the diffusion in the confined geometry.
The first aim of this paper is to demonstrate that this does not apply when the size of the diffusing particle is comparable to throat sizes. We do so by analysing trajectories of individual particles diffusing in a porous sample and show statistical deviations from CTRW predictions. We also compare these simulations with an equivalent CTRW model. We show that the effective change in the medium’s connectivity with varying particle size affects directly the nature and universality class of the diffusion process. We conclude that the sub-diffusion is the result of CTRW on a percolation cluster. Indeed, a combination of underlying mechanisms, leading to sub-diffusion, has also been observed in [@Tabei2013; @Weigel2011; @Jeon2011; @Yamamoto2014].
The second aim of the paper is to propose a method to correct for the topological effect, which makes it possible to still use CTRW, with its advantages, to model diffusion of any finite size particle in confined geometries. To maximise the range of validity of our results (see discussion below), we consider very high porosity porous media. These correspond to marginally rigid assemblies of frictional particles, whose mean coordination number is four [@Blumenfeld2015]. The least confined of these are model systems whose each particle has exactly four contacts.
The structure of this paper is the following. In section \[sec:sample\] we describe the simulated porous samples. In section \[sec:diffusion\_in\_sample\] we describe the diffusion process, and discuss the effects of particle size. We perform statistical analysis of the particle trajectories and show disagreements with some predictions of the CTRW model. In section \[sec:diffusion\_in\_free\_space\] we describe the equivalent CTRW simulations and show that they yield different behaviour in spite of having the same step-length and waiting-time distributions. We propose an explanation for this discrepancy. In section \[sec:memory\] we propose a modification to the conventional CTRW model to alleviate this problem, making it more suitable for modelling diffusion of finite size particles in confined geometries. We conclude in section \[sec:conclusions\] with a discussion of the results.
The porous sample {#sec:sample}
=================
To simulate a three-dimensional porous granular assembly of coordination number four, we first generate an open-cell structure, using Surface Evolver as follows [@Brakke1992; @Wang2006]. Initially, $N$ seed points are distributed randomly and uniformly within a cube, and the cube’s space is Voronoï-tessellated to determine the cell associated with each point. A cell around a point consists of the volume of all spatial coordinates closest to it. The resulting cellular structure is then evolved with Surface Evolver to minimise the total surface area of the cell surfaces. This procedure is used commonly to model dry foams and cellular materials whose dynamics are dominated by surface tension. The result is an equilibrated foam-like structure, comprising cells, membranes, edges and vertices. A membrane is a surface shared by two neighbouring cells, an edge is the line where three membranes coincide, and a vertex is the point where four edges coincide.
![Pseudo-grains around the foam vertices.[]{data-label="fig:pseudo_grains"}](pseudo_grain.png){width="40.00000%"}
Next, we construct a tetrahedron around every vertex by connecting the mid-points of the four edges emanating from it [@Frenkel2009]. Neighbouring tetrahedra are in contact in the sense that they share the mid-point of an edge. This construction results in a pseudo-granular structure of volume fraction $\phi = 34\%$, in which every tetrahedron represents a pseudo-grain in contact with exactly four others [@Blumenfeld2006] (see fig. \[fig:pseudo\_grains\]). Since neighbouring pseudo-grains share the mid-point of the edge between them, the tetrahedra structure is topologically homeomorphic to the original structure. The void space surrounded by the pseudo-grains is still cellular, but a cell surface now consists of triangular facets – the faces of the pseudo-grains surrounding it – and throats – the skewed polygons remaining of the original cell membranes. The membranes over the throats are disregarded, resulting in an open-cell porous structure, in which the throats are the openings between neighbouring cells. The pseudo-grains volumes are smaller than those of real convex grains, which curve out into the cells of this structure. This increases the pore volume and forms a limiting case, which establishes the validity of our results for any porous medium, as will be discussed in the concluding section.
Diffusion in the porous sample {#sec:diffusion_in_sample}
==============================
We model the diffuser as a sphere of radius $r$, measured in units of the average effective throat radius, $r_0$. We start by considering particles that are considerably smaller than the smallest throat in the structure. The particle cannot enter the tetrahedral pseudo-grains, but only move from cell $c$ to cell $c'$ through their shared throat. The simulation progresses by moving the particle from one cell, $c$, to a neighbouring cell, $c'$. Each such an event is a step, $\vec{l}_{c,c'}$, namely a vector extending between the centres of these cells. We define a waiting time, $t_c$, which is the number of time steps spent in cell $c$ before a jump occurs. Inside a cell, the particle is assumed to undergo Brownian motion and $t_c$ is proportional to (i) the square of the effective cell radius, $R_c \equiv \left(\frac{3v_c}{4\pi}\right)^{1/3}$, where $v_c$ is the cell volume, and (ii) the inverse of the fraction of the cell’s open surface through which the particle can pass to neighbouring cells, $S_c\equiv \frac{{A^{\rm (t)}_c}}{{A^{\rm (t)}_c}+ {A^{\rm (f)}_c}}$. This is since the particle, on average, has to make $1/S_c$ journeys of length $R_c$ until it goes through a throat rather than hits a facet of a pseudo-grain. The effective area of a throat is the area through which a particle of radius $r$ can pass and ${A^{\rm (t)}_c}$ (${A^{\rm (f)}_c}$) is the total area of throats (facets) that make the surface of the cell. ${A^{\rm (t)}_c}$ is then the sum of the effective throat areas accessible for the particle to go through. Thus, the waiting time within a cell is $$\begin{aligned}
\label{eq:waiting_time}
t_c \equiv \frac{R_c^2}{2 d S_c} = \frac{1}{2d} \left( \frac{3v_c}{4\pi} \right)^{2/3} \left(1 + \frac{{A^{\rm (f)}_c}}{{A^{\rm (t)}_c}} \right) {{\hspace{0.25cm}},}\end{aligned}$$ where $d$ is the local diffusion coefficient, which, using the Stokes-Einstein relation, is inversely proportional to the particle radius, $d = (r_0 / r) d_0$.
The probability to exit cell $c$ into $c'$, $P(c' \mid c)$, is proportional to the area of the throat between them. To reduce finite-size effects, we wrap the sample around with periodic boundary conditions and let the particle travel larger distances by re-entering the sample. This means that the same cell may occur at different locations along the random walk. To avoid distorting the statistics by using the same cell too many times, we stop the process once a cell has occurred at more than five different locations.
We emphasise that we do not simulate the diffusion within the cell – each step in our simulation corresponds to a transition of the particle from one cell to another, and the waiting time associated with the step is calculated from the cell properties. This process constitutes a random walk on a graph, whose nodes are the cell centres. After waiting for a period of time $t_c$ in cell $c$, determined by eq. (\[eq:waiting\_time\]), the particle makes a step to a neighbour cell, according to $P(c' \mid c)$.
Before continuing, it is instructive to put the problem in thermodynamic context. The cell can be regarded as a potential well of height $\Delta E$, and the probability to escape from it is $P(t) = P_0 e^{-t / \tau}$. We can then use Kramer’s escape rate formula, $$\begin{aligned}
\label{eq:kramer}
\tau = \frac{2 \pi k T}{d \sqrt{U"(a)U"(b)}} e^{\Delta E / kT} {{\hspace{0.25cm}},}\end{aligned}$$ where $k$ is the Boltzmann constant, $T$ is the temperature and $U"(a)$ and $U"(b)$ are the second derivative of the potential at the bottom and top of the well, respectively. Interpreting $t_c$ as the half-life of the particle in the cell, $t_c = \tau \ln{2}$, we can combine eq. (\[eq:waiting\_time\]) and (\[eq:kramer\]) to get: $$\begin{aligned}
\frac{2 \ln(2) \pi k T}{\sqrt{U"(a)U"(b)}} e^{\Delta E / kT} &= \frac{1}{2 S_c(r)} \left( \frac{3v_c}{4\pi} \right)^{2/3} .\end{aligned}$$ Assuming that all cells have the same effective potential $U$, we get: $$\begin{aligned}
\frac{\Delta E}{kT} = {\rm Const.} - \ln{T} + \ln{ \frac{v_c^{2 / 3}}{S_c(r)} } {{\hspace{0.25cm}}.}\end{aligned}$$ This equation establishes the height of the effective barrier in terms of the cell volume and the fraction of its surface through which the particle can escape.
Note that the particle’s mean free path within a cell is assumed to be well smaller than the cell size, regardless of the particle size, and therefore that Knudsen diffusion [@Knudsen1909; @Clausing1930] need not be considered. However, even if this assumption is not borne out, this would only modify the coefficient $d$ in eq. (\[eq:waiting\_time\]), which is arbitrary anyway in our simulations. Also note that the above assumption, $P(t) \sim e^{-t/\tau}$, means that typical escape times do not deviate much from the mean or half-life time. This justifies our choice of taking $t_c$ as a representative.
\
For each walk we calculate the particle’s position at time $t$, relative to the origin, $\vec{x}(t) = \sum_{n=1}^{N(t)} \vec{l}_n$, where $N(t)$ is the number of steps made before time $t$. We then calculate the MSD, ${\langle \vec{x}^2 \rangle}$, as a function of $t$, where the angular brackets denote average over 1000 walks. Figure \[fig:small\_particle\_in\_sample\] shows ${\langle \vec{x}^2 \rangle}$ vs. $t$ for a particle of size $r = r_0 / 100$. The linear relation indicates normal diffusion, with a diffusion coefficient of $D = (0.65 \pm 0.03)d$ ($d = 100 d_0$). The inset figure shows a narrowly bounded $P_t$.
#### Large particles
We next consider particles of sizes comparable to $r_0$. Such particles diffuse differently due to two effects. One is delay and trapping inside cells. The larger $r$ is the lower the probability of passing through any particular throat, since the effective area, which the particle can pass through, is smaller. This reduces the overall probability to exit a cell, increasing the waiting times spent inside cells. As a result, while the waiting times are narrowly bounded for a small particle, $P_t(t_i)$ develops a power-law tail for sufficiently large particles, $P_t(t_i > t^{(0)}) \sim t_i^{-\beta}$, with $\beta$ a function of $r$. The second effect is that the topology changes with particle size; as it increases, the probability of passing through some throats vanishes identically, changing the system’s connectivity for this particle.
Fig. \[fig:large\_particles\_waiting\_time\] shows $P_t$ for a few large particles. We see that $\beta$ decreases with $r$, corresponding to longer waiting times. Fig. \[fig:large\_particles\_msd\] shows the MSD vs. time for the same particles. We see that beyond a certain particle size ($r \sim 1.2r_0$) $\alpha$ starts to decrease – the diffusion becomes anomalous. To quantify this relation, we choose a larger set of radii and plot $\alpha(r)$ vs. $\beta(r)$ (blue dots in fig. \[fig:alpha\_vs\_beta\]). We see that for smaller particles $\beta \gg 2$ and $\alpha=1$, corresponding to normal diffusion with narrow waiting-time PDFs. As $r$ increases, $\beta$ decreases and the walks eventually become sub-diffusive with $\alpha < 1$. We measure a transition at $\beta_t^{\rm (sample)} = 2.53 \pm 0.03$.
A short comment on scaling windows is due – $\alpha$ is evaluated along $t \in (10^{3.5}, 10^5)$ (see fig. \[fig:large\_particles\_msd\]). At much longer times, the diffusion is normal for all particle sizes. This is merely a consequence of the finite system size – there are no cells with longer waiting times than $\sim \! 10^5$.
![Two sets of simulations – diffusion in a porous sample (blue dots) and CTRW in unconfined space (green triangles). Each simulation (dot) represents a particular particle size, and is described by the power-law of the waiting-time distribution, $\beta$, and the anomaly parameter, $\alpha$. Small (large) particles appear at the top right (bottom left) corner of the graph – they experience a narrow (wide) waiting-time distribution and undergo normal (sub-) diffusion. The blue and green lines are fits of the form $\alpha = 1 - \frac{1}{2} \exp \{ - (\beta - \beta_t - \frac{1}{2}) / \tau \}$, with ($\beta_t^{\rm (sample)} = 2.53$, $\tau=0.42$) and ($\beta_t^{\rm (CTRW)} = 2.01$, $\tau=0.40$), respectively. The theoretical CTRW prediction of a universality class transition at $\beta=2$ is denoted by the black dashed line. The red lines mark the range of $\beta$’s that corresponds to the percolation threshold of the sample. This range matches the universality-class transition for confined diffusion. Error bars denote $2 \sigma$ ranges, and were established by the variance of 20 independent measurements.[]{data-label="fig:alpha_vs_beta"}](alpha_vs_beta.pdf){width="50.00000%"}
To investigate the diffusion process further, we calculate the TAMSD, $\delta^2$, and its average over 1000 walks, $\langle \delta^2 \rangle$. Fig. \[fig:tamsd\_vs\_delta\] and \[fig:tamsd\_vs\_time\] show $\langle \delta^2(t, T) \rangle$ vs. the time lag, $t$, and vs. the overall trajectory time, $T$, respectively. $\langle \delta^2 \rangle$ is sub-linear in $t$, deviating from the linear $t$-dependence in the CTRW model, and it is nearly independent of $T$. This behaviour is the same as for a random walk on a fractal. In contrast, the amplitude scatter, $\xi$, follows the CTRW prediction: Fig. \[fig:normalised\_tamsd\] shows the PDF of $\xi$ for three different particle sizes of order $r_0$. The dramatic broadening of the PDF as $r$ increases indicates that the process is non-ergodic and that there is ageing [@Metzler2014]. Fig. \[fig:ergodicity\_breaking\] shows that the corresponding EB parameters, i.e. the variance of these PDFs, follow the theoretical expectation from CTRW.
Modelling the diffusion as isotropic CTRW {#sec:diffusion_in_free_space}
=========================================
Next we show that an attempt to simulate this process with straightforward isotropic CTRW fails. This may not come as a surprise, as the statistical analysis showed some deviations from the traditional CTRW, but it is still constructive to describe the CTRW simulation to better understand its adjustments in section \[sec:memory\]. For such a simulation we use the $r$-dependent PDFs, $P_l$ and $P_t$, that the particle experiences while diffusing in the confined structure. Recall that these PDFs refer to the transition between cells, rather than the movement within a cell. One way to obtain these PDFs is to measure them empirically during a diffusion process. However, more efficient is to compute them directly from the structural statistics of the porous medium, as we outline next.
A derivation of $P_l$ and $P_t$ from structural statistics should be made cautiously because a straight-forward histogram of the waiting times of all cells in the sample ignores the inherent correlation between the probability to visit a cell, $P(c)$, and the waiting time [@Edery2013]. Cells that are difficult to get out of (long waiting times) tend to have a lower probability of getting into and are therefore visited less frequently. In particular, some cells are completely inaccessible for particles above a certain size.
To this end we use the observation that, to a very good accuracy in our diffusion process, $P(c)$ is linear in the cell’s total effective throat area, ${A^{\rm (t)}_c}$. This observation, which is independent of the cell volume, can be seen over 3.5 orders of magnitude in fig. \[fig:visits\_vs\_total\_throat\_area\]. Furthermore, this holds for all particle sizes, both well smaller and comparable to $r_0$. This allows us to estimate $P(c)$ for a particular particle size by using the effective ${A^{\rm (t)}_c}$ – a direct structural characteristic of the medium. In principle, one expects the visiting probability to be correlated with the visiting probabilities of neighbouring cells, but fig. \[fig:visits\_vs\_total\_throat\_area\] shows that this effect is negligible.
![Visiting probability, $P(c)$, vs. the cell’s total effective throat area, ${A^{\rm (t)}_c}$, in a diffusion process with $r = r_0$.[]{data-label="fig:visits_vs_total_throat_area"}](visits_vs._throat_area.pdf){width="50.00000%"}
We can now estimate more accurately $P_t$, and, particularly, the power-law $\beta$, for every particle size, by manipulating the waiting-time histogram as follows. For every bin consisting of the waiting times of cells $\{c_1, ... , c_n\}$, we multiply the bin’s height by $\sum_1^n P(c_i)$ and normalise the histogram into a PDF. This suppresses long waiting times in $P_t$.
Collecting the statistics of all possible steps in the sample, we get a PDF described well by the Gaussian $P_l(l_i) \sim \exp\{-(l_i - \bar{l})^2/2\sigma^2\}$, with $\sigma / \bar{l} = 0.15 \pm 0.02$ (see fig. \[fig:step\_length\_dist\]). As $P_l$ is narrow, it does not affect the universality class of the walk. $P_l$ stays narrow, and indeed nearly constant, for all particle sizes, as the cells positions are fixed. Note that it is also possible to derive $P_l$ analytically, given the cell volume distribution and nearest neighbour volume-volume correlations. This, however, is somewhat downstream from the main thrust of this paper.
We are now able to obtain accurate step-length and waiting-time PDFs for every particle size, which could be used as input into unconstrained CTRW. Fig. \[fig:alpha\_vs\_beta\] shows results of CTRW simulations (green triangles) using these PDFs. The fit to this curve (solid green line) differs from the theoretical prediction [@Montroll1965], $\alpha = \beta - 1$, due to finite time and size effects. Our measured transition for CTRW is $\beta_t^{\rm (CTRW)} = 2.01 \pm 0.02$, in agreement with the theoretical prediction.
A key observation is that the CTRW exhibits a normal-to-anomalous transition at a lower values of $\beta$ than in the actual sample. Since both processes have the same step-length and waiting-time distributions, but one is performed on a graph and the other in free space, the discrepancy must stem from the connectivity of the sample, which the CTRW model cannot account for. This is supported by the statistical analysis of the diffusion in the porous sample (section \[sec:diffusion\_in\_sample\]), that show behaviours typical to random walks on a fractal.
As mentioned above, the size of the diffusing particle determines the effective throat sizes, and hence the connectivity of the porous sample. Moreover, above some size there is no path percolating between the sample’s boundaries. Fig. \[fig:cluster\_size\_vs\_particle\_radius\] shows the dependence of the percolating accessible volume on $r$, where a range of radii around the percolation threshold is marked by red lines. The same range is marked in fig. \[fig:alpha\_vs\_beta\]. We see that the universality class transition in the sample occurs within this range. This is more evidence that the connectivity plays an important role in determining the universality class. Specifically, around the percolation threshold the incipient cluster assumes a fractal-like structure, further inducing sub-diffusion [@Gefen1983; @Pandey1984]. We conclude that our simulations of diffusion in porous media are best described as CTRW on a percolation cluster.
![The percentage of cells belonging to the incipient-cluster vs. the particle radius (in units of $r_0$). The percolation range is marked in red. The corresponding $\beta$ range is marked in fig. \[fig:alpha\_vs\_beta\].[]{data-label="fig:cluster_size_vs_particle_radius"}](cluster_size_vs._r.pdf){width="50.00000%"}
Anisotropic CTRW {#sec:memory}
================
To retain the usefulness of the CTRW model, it would be desirable to modify it to capture the effect of connectivity. We next propose such a modification, inspired by the PDF of the angle between successive steps in the porous sample (fig. \[fig:direction\_pdf\]). There is a finite probability to make a backward step, i.e. go back through the throat the particle entered a cell, $P_{\rm back} \equiv P_\theta(\theta = \pi)$. For very small particles ${P_{\rm back}}= 0.087\pm 0.001$. This is in contrast to the conventional CTRW model, where the next step direction is uniformly distributed. Moreover, we see that ${P_{\rm back}}> 1/13.7 \approx 0.073$, which is the inverse of the average number of throats per cell, and is the expected value for ${P_{\rm back}}$ when all steps are equiprobable. This is because the mean size of an entrance throat is larger than the mean size of all the throats. The enhanced backward step probability can be regarded as a correlation between successively visited throats. As $r$ increases, the total available throat area decreases and ${P_{\rm back}}$ increases, as can be seen in the right panel of fig. \[fig:direction\_pdf\]. For very large particles, ${P_{\rm back}}$ dominates the walk. This can be seen as the ‘lowest order’ effect of connectivity, which we next try to capture within CTRW.
![Left: the PDF of $\cos(\theta)$, with $\theta$ the angle between successive steps of a diffusion process in the porous sample for $r = r_0 / 100$. The singular point at $cos(\theta) = -1$ marks the finite probability of a backward step ${P_{\rm back}}= 0.087 \pm 0.001$. In addition, $P(-1 < \cos(\theta) < -0.88) = 0$. Right: the dependence of ${P_{\rm back}}$ on $r$. For large particles, backward steps dominate the walk.[]{data-label="fig:direction_pdf"}](relative_angle_and_p_back.pdf){width="50.00000%"}
We examine two methods to modify the CTRW model, both introducing a backward bias. In the first, we add to $P_l$ and $P_t$ a third distribution, $P_\theta$, of step direction relative to the previous step. In this method, the particle ‘remembers’ the direction of the previous step, and a relative angle is chosen from the non-uniform distribution $P_\theta(\theta)$, e.g. the one in fig. \[fig:direction\_pdf\]. $P_\theta$ can be calculated directly from the porous structure for any particle radius $r$, similarly to $P_l$ and $P_t$ (see section \[sec:diffusion\_in\_free\_space\]). A random step is then made at the angle $\theta$ relative to the previous step direction. The waiting time and step length at each step are chosen as before. This method introduces a correlation between consecutive steps, which is present unavoidably in the diffusion of large particles in the sample. Testing this method by a set of simulations, we find that the universality class transition occurs at $\beta_t^{\rm (anisotropic)} = 2.1 \pm 0.03$ – closer to the one measured in the porous sample.
The second method is more drastic: at every step of the CTRW we construct a new cell, according to the structural statistics of the porous sample. We choose the cell’s volume, the number of throats, and the throats’ areas from the corresponding distributions, derived from the sample. Using these and the particle size, we calculate the waiting time for the new cell. Then we choose randomly an accessible exit throat. The probability to exit through the throat is proportional to its effective area. The particle then makes a step in the direction of the exit throat. Another cell is then constructed around it. The step length is the sum of the radii of these two cells. This process is then iterated as many times as required.
A key feature of this method is that the particle ‘remembers’ the exit direction, the last used cell and the exit throat. The latter is used as one of the throats in constructing the next cell. If this throat is chosen again then the last step is retraced. We refer to this method as DA for the two variables that the particle ‘remembers’ – step direction and throat area.
Within the DA process, the particle may pass back and forth several times through a large throat before it moves on and loses memory of this throat. This process correlates not only successive steps, as the previous method does, but also successive backward steps.
Using the DA model, we obtain a universality-class transition at $\beta_t^{\rm (DA)} = 2.50 \pm 0.03$, in excellent agreement with the original simulation results. Thus, this model captures much better the particle size-driven topological change.
An important feature of the DA model is that ${P_{\rm back}}$ is higher than in the porous sample. To understand this, consider two cells in the porous sample, connected by a large throat, and connected to the rest of the structure by smaller throats. Once the particle enters one of these cells, it is likely to move back and forth several times before it emerges. However, the smaller the throats leading to this pair of cells, the less likely is the particle to enter in the first place. This means that the occurrence frequency of such sub-structures, which increase ${P_{\rm back}}$, is suppressed. In contrast, once a particle passes through a relatively large throat in the DA model, then, other than this throat, an entire new cell is generated for each step. This results in a higher probability that the particle oscillates across such a throat. This feature appears to compensate for other, more complex, missing topological features, making the DA a better model for the diffusion process.
As a further investigation of the proposed methods, we present the step-length correlation function for the different models, all using $r = 1.2r_0$ (fig. \[fig:step\_length\_corr\]). The correlation in the sample is mainly due to the fact that each two consecutive steps enter and exit a certain cell, $c$. If $c$ is small, then the two steps will tend to be short, and vice versa, leading to positive correlation. In addition, a high ${P_{\rm back}}$ means that many consecutive steps are of exactly the same length. This further increases the correlation for larger particles. As expected, the traditional CTRW exhibits no correlations. Our first adjustment to CTRW, introducing the possibility of a backward step, adds correlation. However, since this is only a one-step correlation it decays exponentially. The increase in correlation within the DA process is because of the increased probability for a long sequence of backward steps, as discussed above. As a result, the DA correlation function does not decay exponentially, agreeing better with the diffusion simulation in the sample. The correlation of the DA at a step distance of one is higher than that in the simulated diffusion process because its ${P_{\rm back}}$ is higher, yet the DA correlation decays faster with the number of steps since it lacks the more involved topological correlations.
![Step-length correlation function for the four types of simulations discussed in the paper, all with $r = 1.2r_0$. Solid lines are decaying exponentials. The correlation function of the first adjustment to CTRW decays exponentially, and the second adjustment (DA) – more slowly.[]{data-label="fig:step_length_corr"}](step_length_corr_r=1.2.pdf){width="50.00000%"}
Conclusions {#sec:conclusions}
===========
To conclude, we compared between two numerical models of diffusion of a finite size particle in porous media: a direct simulation of the diffusion process in a computer-generated sample, and what is commonly believed to be an equivalent CTRW. We first presented a method to construct the representative PDFs of the step-length and waiting-time, $P_l$ and $P_t$, given the particle size and statistical information about the structure of the porous media alone. Using the same particle size dependent $P_l$ and $P_t$ in both models, we analysed the transition from regular to anomalous diffusion. We showed that the the two models give different results – while the CTRW simulations follow the theoretical prediction, up to finite-time effects, with a transition to sub-diffusion at $\beta \approx 2$, the diffusion simulations in the confined geometry exhibit a transition at $\beta \approx 2.5$. We established that the difference stems from changes to the effective connectivity available to the particle with increasing size. This particle size-driven change in connectivity is not taken into consideration in the CTRW model. We supported this conclusion by investigation of the time average MSD and by showing that the transition in universality class occurs at the same range of particle sizes that corresponds to the percolation transition. Our findings show unequivocally that the discrepancy between the two models is *not* due to different waiting-time distributions, since using identical distributions do lead to different universality classes.
It is important to comment on the range of validity of our results. Increasing the particle size can be regarded, alternatively, as shrinking the porous structure, while keeping the particle size unchanged. Evidently, the smaller the pore space the more restricted the diffusing particle is and the larger the discrepancy between the simulated diffusion and the CTRW. Thus, by starting from a medium with a very large pore space, we established that our results hold true for a wide range of porous media with lower porosity.
Wong et al. [@Wong2004] studied experimentally a related process of trace particles diffusing in biological networks of entangled F-actin filaments. There, the diffusion of particles, of size comparable to the typical network mesh size, is sub-linear and $P_t$ decays algebraically. The universality class they observe is a function of the particle-to-mesh size ratio, in agreement with our results. However, they do not observe size-driven topological effects and their values of $\alpha$ and $\beta$ are in good agreement with the CTRW model. This is because of the flexibility of the gel-like network, which allows trapped particles to eventually escape by deforming the filament network, a phenomenon also modelled recently in [@Godec2014]. The rigidity of the structure considered here precludes this particle escape mechanism and is the reason for this difference. A potentially related process of large particles, diffusing in rigid porous media, was studied experimentally in [@Skaug2015]. That study focused on hydrodynamic in-pore effects, which might be interesting to eventually include in our model.
The main advantages of the CTRW model are that it overcomes potential finite-size problems and is less demanding computationally. However, as we have demonstrated here, this is achieved at the expense of ignoring topological information about local connectivity. To preserve the advantages of CTRW, these need to be taken into consideration. To this end, we introduced two anisotropic CTRW models. One includes memory of the last step direction and a non-uniform distribution of step direction. The other, the DA model, adds memory of the area of the last throat visited, effectively correlating successive backward steps. The DA model shows a universality-class transition at $\beta_t^{\rm (DA)} = 2.50 \pm 0.03$, in good agreement with the one measured in our simulations of the diffusion process in the confined geometry of a porous medium. We conclude that the DA model is a better alternative to the traditional CTRW for modelling diffusion of finite size particles in such media. It combines the CTRW advantages, overcoming the finiteness of the sample and convenience of application, with a better capturing of topological effects.\
[**Acknowledgement**]{}: SA is grateful for support from the Alan Howard Scholarship.\
[**Conflict of interest**]{}: The authors declare that they have no conflict of interest.
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| ArXiv |
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abstract: 'From Crofton’s formula for Minkowski tensors we derive stereological estimators of translation invariant surface tensors of convex bodies in the $n$-dimensional Euclidean space. The estimators are based on one-dimensional linear sections. In a design based setting we suggest three types of estimators. These are based on isotropic uniform random lines, vertical sections, and non-isotropic random lines, respectively. Further, we derive estimators of the specific surface tensors associated with a stationary process of convex particles in the model based setting.'
author:
- 'Astrid Kousholt[^1]'
- Markus Kiderlen
- Daniel Hug
bibliography:
- 'litteratur.bib'
title: Surface tensor estimation from linear sections
---
**Keywords** Crofton formula, Minkowski tensor, stereology, isotropic random line, anisotropic random line, vertical section estimator, minimal variance estimator, stationary particle process, stereological estimator\
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**MSC2010** 60D05, 52A22, 53C65, 62G05, 60G55
Introduction
============
In recent years, there has been an increasing interest in Minkowski tensors as descriptors of morphology and shape of spatial structures of physical systems. For instance, they have been established as robust and versatile measures of anisotropy in [@Beisbart2002; @SM10; @Schroder-Turk2013]. In addition to the applications in materials science, [@Beisbart] indicates that the Minkowski tensors lead to a putative taxonomy of neuronal cells. From a pure theoretical point of view, Minkowski tensors are, likewise, interesting. This is illustrated by Alesker’s characterization theorem [@Alesker1999], stating that the basic tensor valuations (products of the Minkowski tensors and powers of the metric tensor) span the space of tensor-valued valuations satisfying some natural conditions.
This paper presents estimators of certain Minkowski tensors from measurements in one-dimensional flat sections of the underlying geometric structure. We restrict attention to translation invariant Minkowski tensors of convex bodies, more precisely, to those that are derived from the top order surface area measure; see Section \[prelim\] for a definition. As usual, the estimators are derived from an integral formula, namely the Crofton formula for Minkowski tensors. We adopt the classical setting where the sectioning space is affine and integrated with respect to the motion invariant measure. Rotational Crofton formulae where the sectioning space is a linear subspace and the rotation invariant measure on the corresponding Grassmannian is used, are established in [@A-C2013]. The latter formulae were the basis for local stereological estimators of certain Minkowski tensors in [@Jensen2013] (for $j \in \{1, \dots, n-1\}, s,r \in \{0,1\}$ and $j=n, s=0, r \in {{\mathbb N}}$ in the notation of and , below).
Kanatani [@Kanatani1984; @Kanatani1984a] was apparently the first to use tensorial quantities to detect and analyse structural anisotropy via basic stereological principles. He expresses the expected number $N(m)$ of intersections per unit length of a probe with a test line of given direction $m$ as the cosine transform of the spherical distribution density $f$ of the surface of the given probe in ${{\mathbb R}}^n$ for $n=2,3$. The relation between $N$ and $f$ is studied by expanding $f$ into spherical harmonics and by using the fact that these are eigenfunctions of the cosine transform. In order to express his results independently of a particular coordinate system, Kanatani uses tensors. For a fixed $s$, he considers the vector space $V_s$ of all symmetric tensors spanned by the elementary tensor products $u^{\otimes s}$ of vectors $u$ from the unit sphere $S^{n-1}$. Let $\hat T$ denote the deviator part (or trace-free part) of some symmetric tensor $T$. The tensors $\widehat{(u^{\otimes k})}$, for $k\le s$ and $u\in S^{n-1}$, then span $V_s$ and the components of $\widehat{(u^{\otimes k})}$ with respect to an orthonormal basis of ${{\mathbb R}}^n$ are spherical harmonics of degree $k$, when considered as functions of $u$. Hence, $u\mapsto \widehat{(u^{\otimes k})}$ is an eigenfunction of the cosine transform (Kanatani calls it ‘Buffon transform’), which in fact is the underlying integral transform when considering Crofton integrals with lines, as we shall see below in . In [@Kanatani1984b; @Kanatani1985], he suggests to use these ‘fabric tensors’ to detect surface motions and the anisotropy of the crack distribution in rock.
General Crofton formulas in ${{\mathbb R}}^n$ with arbitrary dimensional flats and for general Minkowski tensors (defined in ) of arbitrary rank are given in [@Hug]. Theorem \[thm\] is a special case of one of these results, for translation invariant surface tensors and one-dimensional sections, that is, sections with lines. In comparison to [@Hug], we get simplified constants in the case considered and obtain this result by an elementary independent proof. In contrast to Kanatani’s approach, our proof does not rely on spherical harmonics. Here we focus on relative Crofton formulas in which the Minkowski tensors of the sections with lines are calculated relative to the section lines and not in the ambient space (Crofton formulas of the second type may be called extrinsic Crofton formulas). A quite general investigation of integral geometric formulas for translation invariant Minkowski tensors, including extrinsic Crofton formulas, is provided in [@BernigHug].
In Theorem \[thm\] we prove that the relative Crofton integral for tensors of arbitrary even rank $s$ of sections with lines is equal to a linear combination of surface tensors of rank at most $s$. From this we deduce by the inversion of a linear system that any translation invariant surface tensor of even rank $s$ can be expressed as a Crofton integral. The involved measurement functions then are linear combinations of relative tensors of rank at most $s$. This implies that the measurement functions only depend on the convex body through the Euler characteristic of the intersection of the convex body and the test line.
Our results do not allow to write surface tensors of odd rank as Crofton integrals based on sections with lines. This drawback is not a result of our method of proof. Indeed, apart from the trivial case of tensors of rank one, there does not exist a translation invariant or a bounded measurement function that expresses a surface tensor of odd rank as a Crofton integral; see Theorem \[s ulige\] for a precise statement of this fact.
In Section \[sec estimation\] the integral formula for surface tensors of even rank is transferred to stereological formulae in a design based setting. Three types of unbiased estimators are discussed. Section \[IUR\] describes an estimator based on isotropic uniform random lines. Due to the structure of the measurement function, it suffices to observe whether the test line hits or misses the convex body in order to estimate the surface tensors. However, the resulting estimators possess some unfortunate statistical properties. In contrast to the surface tensors of full dimensional convex bodies, the estimators are not positive definite. For convex bodies, which are not too eccentric (see ), this problem is solved by using $n$ orthogonal test lines in combination with a measurement of the projection function of order $n-1$ of the convex body.
In applications it might be inconvenient or even impossible to construct the isotropic uniform random lines, which are necessary for the use of the estimator described above. Instead, it might be a possibility to use vertical sections; see Definition \[VUR\]. A combination of Crofton’s formula and a result of Blaschke-Petkantschin type allows us to formulate a vertical section estimator. The estimator, which is discussed in Section \[sec VUR\], is based on two-dimensional vertical flats.
The third type of estimator presented in the design based setting is based on non-isotropic linear sections; see Section \[Sec noniso\]. For a fixed convex body in ${{\mathbb R}}^2$ there exists a density for the distribution of test line directions in an importance-sampling approach that leads to minimal variance of the non-isotropic estimator, when we consider one component of a rank 2 tensor, interpreted as a matrix. In practical applications, this density is not accessible, as it depends on the convex body, which is typically unknown. However, there does exist a density independent of the underlying convex body yielding an estimator with smaller variance than the estimator based on isotropic uniform random lines. If *all* components of the tensor are sought for, the non-isotropic approach requires three test lines, as two of the four components of a rank 2 Minkowski tensor coincide due to symmetry. It should be avoided to use a density suited for estimating one particular component of the tensor to estimate any other component, as this would increase variance of the estimator. In this situation, however, a smaller variance can be obtained by applying an estimator based on *three* isotropic random lines (each of which can be used for the estimation of *all* components of the tensor).
In Section \[SecModel\] we turn to a model-based setting. We discuss estimation of the *specific (translation invariant) surface tensors* associated with a stationary process of convex particles; see for a definition. In [@RSRS06] the problem of estimating the area moment tensor (rank $2$) associated with a stationary process of convex particles via planar sections is discussed. We consider estimators of the specific surface tensors of arbitrary even rank based on one-dimensional linear sections. Using the Crofton formula for surface tensors, we derive a rotational Crofton formula for the specific surface tensors. Further, the specific surface tensor of rank $s$ of a stationary process of convex particles is expressed as a rotational average of a linear combination of specific tensors of rank at most $s$ of the sectioned process.
Preliminaries {#prelim}
=============
We work in the $n$-dimensional Euclidean vector space ${{\mathbb R}^n}$ with inner product $ {\langle {\cdot} , {\cdot} \rangle} $ and induced norm $ {\|}\cdot {\|}$. Let $B^n:=\{x \in {{\mathbb R}^n}\mid {\|}x {\|}\leq 1\}$ be the unit ball and $ S^{n-1}:=\{x \in {{\mathbb R}^n}\mid {\|}x {\|}=1\} $ the unit sphere in ${{\mathbb R}^n}$. By $\kappa_n$ and $\omega_{n}$ we denote the volume and the surface area of $ B^n $, respectively. The Borel $\sigma$-algebra of a topological space $X$ is denoted by ${\mathcal{B}}(X)$. Further, let $\lambda$ denote the $n$-dimensional Lebesgue measure on ${{\mathbb R}}^n$, and for an affine subspace $E$ of ${{\mathbb R}}^n$, let $\lambda_E$ denote the Lebesgue measure defined on $E$. The $k$-dimensional Hausdorff measure is denoted by ${\mathcal{H}}^k$. For $A \subseteq {{\mathbb R}}^n$, let $\dim A$ be the dimension of the affine hull of $A$.
Let ${\mathbb{T}}^p$ be the vector space of symmetric tensors of rank $p$ over ${{\mathbb R}}^n.$ For symmetric tensors $a\in {\mathbb{T}}^{p_1}$ and $b\in {\mathbb{T}}^{p_2}$, let $ab\in {\mathbb{T}}^{p_1+p_2}$ denote the symmetric tensor product of $a$ and $b$. Identifying $x \in {{\mathbb R}}^n$ with the rank 1 tensor $z \mapsto {\langle {z} , {x} \rangle}$, we write $x^p \in {\mathbb{T}}^p$ for the $p$-fold symmetric tensor product of $x$. The metric tensor $Q \in {\mathbb{T}}^2$ is defined by $Q(x,y)={\langle {x} , {y} \rangle}$ for $x,y \in {{\mathbb R}}^n$, and for a linear subspace $L$ of ${{\mathbb R}}^n$, we define $Q(L) \in {\mathbb{T}}^2$ by $Q(L)(x,y)={\langle {p_L(x)} , {p_L(y)} \rangle}$, where $p_L\colon {{\mathbb R}}^n \rightarrow L$ is the orthogonal projection on $L$.
As general references on convex geometry and Minkowski tensors, we use [@Schneider93] and [@Hug]. Let ${{\mathcal K}}^n$ denote the set of convex bodies (that is, compact, convex sets) in ${{\mathbb R}}^n$. In order to define the Minkowski tensors, we introduce the support measures $\Lambda_0(K, \cdot), \dots, \Lambda_{n-1}(K, \cdot)$ of a non-empty, convex body $ K \in{{\mathcal K}}^n$. Let $ p(K,x) $ be the metric projection of $x \in {{\mathbb R}^n}$ on a non-empty convex body $ K $, and define $ u(K,x):=\frac{x-p(K,x)}{{\|}x-p(K,x){\|}} $ for $x \notin K$. For $ \epsilon >0 $ and a Borel set $\nobreak{A \in {\mathcal{B}}({{\mathbb R}}^n \times S^{n-1}})$, the Lebesgue measure of the local parallel set $$M_\epsilon(K,A):=\{x \in (K+\epsilon B^n) \setminus K \mid (p(K,x),u(K,x)) \in A \}$$ of $K$ is a polynomial in $\epsilon$, hence $$\lambda(M_\epsilon(K,A))=\sum_{k=0}^{n-1}\epsilon^{n-k}\kappa_{n-k}\Lambda_k(K,A).$$ This local version of the Steiner formula defines the support measures $ \Lambda_0(K, \cdot), \allowbreak \dots, \Lambda_{n-1}(K,\cdot) $ of a non-empty convex body $ K\in{{\mathcal K}}^n$. If $K=\emptyset$, we define the support measures to be the zero measures. The intrinsic volumes $ V_0(K), \dots, V_{n-1}(K)$ of $ K $ appear as total masses of the support measures, $V_j(K)=\Lambda_j(K,{{\mathbb R}^n}\times S^{n-1})$ for $j=0, \dots, n-1$. Furthermore, the area measures $ S_0(K, \cdot), \dots, S_{n-1}(K,\cdot) $ of $ K $ are rescaled projections of the corresponding support measures on the second component. More explicitly, they are given by $$\binom{n}{j}S_j(K,\omega)=n \kappa_{n-j}\Lambda_j(K,{{\mathbb R}^n}\times \omega)$$ for $ \omega \in {\mathcal{B}}(S^{n-1}) $ and $j=0, \dots, n-1$.
For a non-empty convex body $K\in{{\mathcal K}}^n$, $r,s \in {{\mathbb N}}_0$, and $ j \in \{0,1,\dots, n-1\} $, we define the *Minkowski tensors* as $$\label{Mtensor}
\Phi_{j,r,s}(K) :=\frac{\omega_{n-j}}{r!s! \omega_{n-j+s}} \int_{{{\mathbb R}}^n \times S^{n-1}} x^r u^s\, \Lambda_j (K,d(x,u))$$ and $$\label{volumeTensor}
\Phi_{n,r,0}(K):=\frac{1}{r!}\int_{K}x^r \,\lambda(dx).$$ The definition of the Minkowski tensors is extended by letting $\Phi_{j,r,s}(K)=0$, if $j \notin\{0,1,\dots, n\}$, or if $r$ or $s$ is not in ${{\mathbb N}}_0$, or if $j=n$ and $s \neq 0$. For $j=n-1$, the tensors are called surface tensors. In the present work, we only consider translation invariant surface tensors which are obtained for $r=0$. In [@Hug] the functions $Q^m\Phi_{j,r,s}$ with $m,r,s \in {{\mathbb N}}_0$ and either $ j \in \{0,\dots, n-1\} $ or $ (j,s)=(n,0) $ are called the basic tensor valuations.
For $k \in \{1, \dots,n \}$, let ${\mathcal{L}}^n_k$ be the set of $k$-dimensional linear subspaces of ${{\mathbb R}}^n$, and let ${\mathcal{E}^n}_k$ be the set of $ k $-dimensional affine subspaces of ${{\mathbb R}}^n$. For $L \in {\mathcal{L}}^n_k$, we write $L^\perp$ for the orthogonal complement of $L$. For $E \in {\mathcal{E}^n}_k$, let ${\pi(E)}$ denote the linear subspace in ${\mathcal{L}}^n_k$ which is parallel to $E$, and we define $ E^\perp :={\pi(E)}^\perp$. The sets ${\mathcal{L}}^n_k$ and ${\mathcal{E}}^n_k$ are endowed with their usual topologies and Borel $\sigma$-algebras. Let $\nu^n_k$ denote the unique rotation invariant probability measure on ${\mathcal{L}}^n_k$, and let $\mu_k^n$ denote the unique motion invariant measure on ${\mathcal{E}^n}_k$ normalized so that $\nobreak{\mu^n_k(\{E \in {\mathcal{E}^n}_k \vert E \cap B^n \neq \emptyset\})}=\kappa_{n-k}$ (see, e.g., [@Weil]).
If $K \in {{\mathcal K}}^n$ is non-empty and contained in an affine subspace $E \in {\mathcal{E}^n}_k$, for some $k \in \{1, \dots, n\}$, then the Minkowski tensors can be evaluated in this subspace. For a linear subspace $ L \in {\mathcal{L}}_k^n $, let $ \pi_L \colon S^{n-1} \setminus L^\perp \rightarrow L \cap S^{n-1}$ be given by $$\pi_L(u):=\frac{p_L(u)}{{\|}p_L(u){\|}}.$$ Then we define the $ j $th support measure $ \Lambda_j^{(E)}(K, \cdot) $ of $ K $ relative to $ E $ as the image measure of the restriction of $\Lambda_j(K, \cdot)$ to $ {{\mathbb R}^n}\times (S^{n-1} \setminus E^\perp) $ under the mapping $ {{\mathbb R}^n}\times (S^{n-1} \setminus E^\perp) \rightarrow {{\mathbb R}^n}\times ({\pi(E)}\cap S^{n-1}) $ given by $(x,u) \mapsto (x,\pi_{{\pi(E)}}(u))$.
For a non-empty convex body $K \in {{\mathcal K}}^n$, contained in an affine subspace $ E \in {\mathcal{E}^n}_k $, for some $k \in \{1, \dots, n\}$, we define $$\Phi_{j,r,s}^{(E)}(K) :=\frac{\omega_{k-j}}{r!s! \omega_{k-j+s}} \int_{E \times (S^{n-1} \cap {\pi(E)}) } x^r u^s \, \Lambda^{(E)}_j (K,d(x,u))$$ for $ r,s \in {{\mathbb N}}_0 $ and $ j \in \{0,\dots, k-1\} $, and $$\Phi_{k,r,0}^{(E)}(K):=\frac{1}{r!}\int_{K} x^r \,
\lambda_E(dx).$$ As before, the definition is extended by letting $\Phi^{(E)}_{j,r,s}(K)=0$ for all other choices of $j,r$ and $s$, and for $K=\emptyset$.
In [@Hug], Crofton integrals of the form $$\int_{{\mathcal{E}^n}_k} \Phi_{j,r,s}^{(E)}(K \cap E) \, \mu_k^n (dE),$$ where $K \in {{\mathcal K}}^n$, $r,s \in {{\mathbb N}}_0$ and $0 \leq j \leq k \leq n-1$, are expressed as linear combinations of the basic tensor valuations. When $j=k$ the integral formula becomes $$\label{j=k}
\int_{{\mathcal{E}^n}_k} \Phi_{k,r,s}^{(E)}(K \cap E) \, \mu_k^n (dE)=
\begin{cases}
\Phi_{n,r,0}(K) & \text{if } s=0, \\
0 & \text{otherwise},
\end{cases}$$ see [@Hug Theorem 2.4]. In the case where $j < k$, the formulas become lengthy with coefficients in the linear combinations that are difficult to evaluate, see [@Hug Theorem 2.5 and 2.6]. In the following, we are interested in using the integral formulas for the estimation of the surface tensors, and therefore we need more explicit integral formulas. We only treat the special case where $k=1$, that is, we consider integrals of the form $$\int_{{\mathcal{E}^n}_1} \Phi_{j,r,s}^{(E)}(K \cap E) \, \mu_1^n (dE).$$ Since $\dim(E)=1$, the tensor $\Phi_{j,r,s}^{(E)}(K)$ is by definition the zero function when $j>1$, so the only non-trivial cases are $j=0$ and $j=1$. When $j=1$ formula gives a simple expression for the integral. In the case where $j=0$ and $r=0$, we provide an independent and elementary proof of the integral formula, which also leads to explicit and fairly simple constants.
Linear Crofton formulae for tensors {#CroftonSec}
===================================
We start with the main result of this section, which provides a linear Crofton formula relating an average of tensor valuations defined relative to varying section lines to a linear combination of surface tensors.
\[crofton-like\] Let $K \in {{\mathcal K}}^n$. If $s \in {{\mathbb N}}_0$ is even, then $$\label{thm}
\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,s}(K\cap E)\, \mu_1^n(dE) = \frac{2 \omega_{n+s+1}}{ \pi s!\omega_{s+1}^2\omega_n} \sum_{k=0}^{\frac{s}{2}} c_{k}^{(\frac{s}{2})} Q^{\frac{s}{2}-k} \Phi_{n-1,0,2k}(K),$$ with constants $$c_{k}^{(m)}=(-1)^{k} \binom{m}{k} \frac{(2k)! \, \omega_{2k+1}}{1-2k}$$ for $m \in {{\mathbb N}}_0$ and $k=0, \dots, m$.
For odd $s \in {{\mathbb N}}_0$ the Crofton integral on the left-hand side is zero.
Before we give a proof of Theorem $\ref{crofton-like}$, let us consider the measurement function $\MT[s]$ on the left-hand side of . Let $k \in \{1, \dots, n\}$. Slightly more general than in , we choose $s \in {{\mathbb N}}_0$ and $E \in {\mathcal{E}}_k^n$. Then $$\Phi^{(E)}_{0,0,s}(K\cap E)
=\frac{1}{s!\omega_{k+s}}\int_{S^{n-1}\cap {\pi(E)}}u^s\, {\mathcal{H}}^{k-1}(du)\, V_0(K\cap E),$$ since the surface area measure of order 0 of a non-empty set is up to a constant the invariant measure on the sphere. From calculations equivalent to [@TVCB (24)-(26)] (or from a special case of Lemma 4.3 in [@Hug]) we get that $$\label{metric}
\int_{S^{n-1}\cap {\pi(E)}}u^s\, \mathcal{H}^{k-1}(du)
=\begin{cases}
2\frac{\omega_{s+k}}{\omega_{s+1}}Q({\pi(E)})^{\frac{s}{2}} & \text{if $s$ is even}, \\
0 & \text{if $s$ is odd}.
\end{cases}$$ Hence $$\label{Simpel form for T}
\Phi^{(E)}_{0,0,s}(K\cap E)
=\frac{2}{s! \omega_{s+1}}\cdot Q({\pi(E)})^{\frac{s}{2}}V_0(K\cap E),$$ when $s$ is even, and $\Phi^{(E)}_{0,0,s}(K\cap E)=0$ when $s$ is odd. This implies that the Crofton integral in is zero for odd $s$, and the tensors $\T[s]$ are hereby not accessible in this situation. This is even true for more general measurement functions; see Theorem \[s ulige\]. To show Theorem \[crofton-like\] we can restrict to even $s$ from now on.
In the proof of Theorem \[crofton-like\] we use the following identity for binomial sums.
\[binomial\] Let $m,n \in {{\mathbb N}}_0$. Then $$\sum_{j=0}^m(-1)^j \frac{\binom{2n}{2j}\binom{n-j}{m-j}}{\binom{n-\frac{1}{2}}{j}}
=
\frac{\binom{n}{m}}{1-2m}.$$
Lemma \[binomial\] can be proven by using the identity $$\label{hjaelpebinomial}
\sum_{j=0}^k (-1)^j \frac{\binom{2n}{2j} \binom{n-j}{m-j}}{\binom{n-\frac{1}{2}}{j}}
=
\frac{(-1)^k(2k+1)\binom{2n}{2(k+1)} \binom{n-k-1}{m-k-1}}{(2m-1)\binom{n-\frac{1}{2}}{k+1}}-\frac{\binom{n}{m}}{(2m-1)},$$ where $n, k \in {{\mathbb N}}_0$, and $m \in {{\mathbb N}}$ such that $k<m$. Identity follows by induction on $k$.
Let $K \in {{\mathcal K}}^n$ and let $s \in {{\mathbb N}}_0$ be even. If $n=1$, formula follows from the identity $$\label{binom2}
\sum_{j=0}^m (-1)^j\frac{\binom{m}{j}}{1-2j}=\frac{\sqrt{\pi} \,\Gamma(m+1)}{\Gamma(m+\frac{1}{2})}$$ with $m=\frac{s}{2}$. The left-hand side of is a sum of alternating terms of the same form as the right-hand side of the binomial sum in Lemma \[binomial\]. Using Lemma \[binomial\] and then changing the order of summation yields .
Now assume that $n\geq2$. Using we can rewrite the integral as $$\begin{aligned}
&\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,s}(K\cap E)\, \mu_1^n(dE)\\
&\qquad =\frac{2}{s!\omega_{s+1}}
\int_{{\mathcal{L}}_1^n} Q(L)^{\frac{s}{2}} \int_{{L^{\perp}}} V_0(K \cap (L+x)) \,\lambda_{{L^{\perp}}}(dx)\,\nu^n_1(dL)
\\
&\qquad =\frac{2}{s!\omega_{s+1}\omega_n} \int_{S^{n-1}}u^sV_{n-1}(K\mid u^\perp)\, {\mathcal{H}}^{n-1}(du),\end{aligned}$$ by the convexity of $K$ and an invariance argument for the second equality. Cauchy’s projection formula (see, e.g., [@Gardner (A.43)]) and Fubini’s theorem then imply that $$\begin{aligned}
\label{eq1}
&\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,s}(K\cap E)\, \mu_1^n(dE)\nonumber \\
&\qquad =\frac{1}{s!\omega_{s+1}\omega_n}
\int_{S^{n-1}}\int_{S^{n-1}} u^s |\langle u,v\rangle|\, \mathcal{H}^{n-1}(du)\, S_{n-1}(K,dv).\end{aligned}$$
We now fix $v \in S^{n-1}$ and simplify the inner integral by introducing spherical coordinates (see, e.g, [@SH66]). Then $$\begin{aligned}
&\int_{S^{n-1}}u^s |\langle u,v\rangle|\, \mathcal{H}^{n-1}(du)\\
&\qquad =\int_{-1}^1\int_{S^{n-1}\cap v^\perp}(1-t^2)^{\frac{n-3}{2}}(tv+\sqrt{1-t^2}w)^s|t|\,{\mathcal{H}}^{n-2}(dw) \, dt
\\
&\qquad=\sum_{j=0}^s \binom{s}{j}v^j \int_{-1}^1 (1-t^2)^{\frac{n-3}{2}}t^j\sqrt{1-t^2}^{s-j} |t| \,dt \int_{S^{n-1}\cap v^\perp} w^{s-j}\,{\mathcal{H}}^{n-2}(dw).\end{aligned}$$ The integral with respect to $t$ is zero if $j$ is odd. If $j$ is even, then it is equal to the beta integral $$B\bigg(\frac{j+2}{2},\frac{n+s-j-1}{2}\bigg)
=\frac{2\omega_{n+s+1}}{\omega_{j+2}\,\omega_{n+s-j-1}}.$$ Hence, since $s$ is even, we conclude from that $$\begin{aligned}
&\int_{S^{n-1}}u^s |\langle u,v\rangle|\, {\mathcal{H}}^{n-1}(du)=4 \omega_{n+s+1} \sum_{j=0}^{\frac{s}{2}} \binom{s}{2j}v^{2j} \frac{1}{\omega_{2j+2} \, \omega_{s-2j+1}}Q(v^{\perp})^{\frac{s-2j}{2}}
\\
&\qquad =4 \omega_{n+s+1} \sum_{j=0}^{\frac{s}{2}} \sum_{i=0}^{\frac{s}{2}-j} (-1)^{i} \binom{s}{2j} \binom{\frac{s}{2}-j}{i} \frac{1}{\omega_{2j+2} \, \omega_{s-2j+1}} Q^{\frac{s}{2}-j-i}v^{2(i+j)},\end{aligned}$$ where we have used that $Q(v^{\perp})=Q-v^2$. Substituting this into and by the definition of $\T[2(i+j)]$, we obtain that $$\label{S}
\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,s}(K\cap E)\, \mu_1^n(dE)
=\frac{4 \omega_{n+s+1}}{s!\omega_{s+1}\omega_n} \, S,$$ where $$S= \sum_{j=0}^{\frac{s}{2}} \sum_{i=0}^{\frac{s}{2}-j} (-1)^{i} \binom{s}{2j} \binom{\frac{s}{2}-j}{i} \frac{(2(i+j))!\omega_{2(i+j)+1}}{\omega_{2j+2} \, \omega_{s-2j+1}} Q^{\frac{s}{2}-j-i} \Phi_{n-1,0,2(i+j)}(K).$$ Re-indexing and changing the order of summation, we arrive at $$\begin{aligned}
S&=
\frac{\Gamma(\frac{s}{2}+\frac{1}{2})}{4 \pi^{\frac{s+3}{2}}} \sum_{k=0}^{\frac{s}{2}} (-1)^k (2k)! \omega_{2k+1} Q^{\frac{s}{2}-k}\Phi_{n-1,0,2k}(K)
\\
&\qquad \times \sum_{j=0}^{k}(-1)^{j}\binom{s}{2j}\binom{\frac{s}{2}-j}{k-j} \binom{\frac{s-1}{2}}{j}^{-1}
\\
&=
\frac{1}{2 \pi \omega_{s+1}}\sum_{k=0}^{\frac{s}{2}} (-1)^k \binom{\frac{s}{2}}{k}\frac{(2k)! \, \omega_{2k+1}}{1-2k} Q^{\frac{s}{2}-k}\Phi_{n-1,0,2k}(K),\end{aligned}$$ where we have used Lemma \[binomial\] with $n=\frac{s}{2}$ and $m=k$.
Setting $s=2$ we immediately get the following corollary.
Let $K \in {{\mathcal K}}^n$. Then $$\int_{\mathcal{E}^n_1}\Phi^{(E)}_{0,0,2}(K\cap E)\, \mu_1^n(dE)
=a_n\bigg(\Phi_{n-1,0,2}(K) + \frac{1}{4\pi} Q V_{n-1}(K)\bigg),$$ where $$a_n=\frac{\Gamma(\frac{n}{2})}{2\Gamma(\frac{n+3}{2})\sqrt{\pi}}.$$
The Crofton formula in Theorem \[crofton-like\] expresses the integral of the measurement function $\MT[s]$ as a linear combination of certain surface tensors of $K \in {{\mathcal K}}^n$. This could, in principle, be used to obtain unbiased stereological estimators of the linear combinations. However, it is more natural to ask what measurement one should use in order to obtain $\T[s]$ as a Crofton-type integral. For even $s$ the tensor $\T[s]$ appears in the last term of the sum on the right-hand side of $\eqref{thm}$. But surface tensors of lower rank appear in the remaining terms of the sum. Therefore, we need to express the lower rank tensors $\T$ for $k=0, \dots, \frac{s}{2}-1$ as integrals. This can be done by using Theorem $\ref{crofton-like}$ with $s=2k$ for $k=0, \dots, \frac{s}{2}-1$. This way, we get $\frac{s}{2} + 1$ linear equations, which give rise to the linear system $$\begin{aligned}
&\begin{pmatrix}
C_0\integral[0] \\
C_2\integral[2]\\
\vdots \\
\\
C_s\integral
\end{pmatrix}
= C
\begin{pmatrix}
\T[0] \\
\T[2] \\
\\
\vdots \\
\\
\T[s]
\end{pmatrix}\end{aligned}$$ where $$C=\begin{pmatrix}
c_0^{(0)} & 0 & 0 & \dots & 0 \\
c_0^{(1)}Q & c_1^{(1)} & 0 & & \vdots \\
\vdots &&\ddots & & 0\\
&&&&\\
c_0^{(\frac{s}{2})}Q^{\frac{s}{2}} & c_1^{(\frac{s}{2})}Q^{\frac{s}{2}-1} & \dots &c_{\frac{s}{2}-1}^{(\frac{s}{2})}Q & c_{\frac{s}{2}}^{(\frac{s}{2})}
\end{pmatrix}$$ and $C_j=\frac{\pi j! \omega_{j+1}^2 \omega_n}{2\omega_{n+j+1}}$ for $j=0,2,4, \dots, s$. Our aim is to express $\T[s]$ as an integral, hence we have to invert the system. Notice that the constants $c_i^{(i)}$ are non-zero, which ensures that the system actually is invertible. The system can be inverted by the matrix $$\label{invers matrix}
D=
\begin{pmatrix}
d_{00} & 0 & 0 & \dots && 0 \\
d_{10}Q & d_{11} & 0 & &&\vdots \\
d_{20}Q^2 & d_{21}Q & d_{22} & 0 \\
\vdots & & & \ddots & & 0 \\
d_{\frac{s}{2}0}Q^{\frac{s}{2}} & d_{\frac{s}{2}1}Q^{\frac{s}{2}-1} & \dots & & & d_{\frac{s}{2} \frac{s}{2}}
\end{pmatrix},$$ where $d_{ii}=\frac{1}{c_i^{(i)}}$ for $i=0, \dots, \frac{s}{2}$, and $d_{ij}=-\frac{1} {c_i^{(i)}}\sum_{k=j}^{i-1}c_k^{(i)}d_{kj}$ for $i=1, \dots, \frac{s}{2}$ and $j=0, \dots, i-1$. In particular, we have $$\label{reffinal}
\T[s] = \sum_{j=0}^\frac{s}{2} d_{\frac{s}{2}j} Q^{\frac{s}{2}-j}C_{2j} \integral[2j].$$ Notice that only the dimension of the matrix depends on $s$, hence we get the same integral formulas for the lower rank tensors for different choices of $s$. Formula and the above considerations give the following ‘inverse’ version of the Crofton’s formula.
\[crofton2\] Let $K \in {{\mathcal K}}^n$ and let $s \in {{\mathbb N}}_0$ be even. Then $$\label{thm2}
\int_{{\mathcal{E}^n}_1} G_s({\pi(E)})V_0(K \cap E)\, \mu_1^n (dE) = \T[s],$$ where $$G_{2m}(L):=\sum_{j=0}^m \frac{2 d_{mj} C_{2j} }{(2j)! \, \omega_{2j+1}} Q^{m-j}Q(L)^j$$ for $L \in {\mathcal{L}_1^n}$ and $m \in {{\mathbb N}}_0$.
It should be remarked that the measurement function in is just a linear combination of the relative tensors of even rank at most $s$, but we prefer the present form to indicate the dependence on $K$ more explicitly.
\[s er 4\] For $s=4$ the matrices are $$C=
\begin{pmatrix}
2 & 0 & 0 \\
2Q & 8 \pi & 0 \\
2Q^2 & 16 \pi Q & -\frac{64 \pi^2}{3}
\end{pmatrix}$$ and $$\label{invers matrix ex}
D=
\begin{pmatrix}
\frac{1}{2} & 0 & 0 \\[1ex]
-\frac{1}{8 \pi}Q & \frac{1}{8 \pi} & 0 \\[1ex]
-\frac{3}{64 \pi^2}Q^2 & \frac{3}{32 \pi^2}Q & -\frac{3\pi^2}{64}
\end{pmatrix}.$$ Since $C_0=\frac{2 \pi \omega_n}{\omega_{n+1}}$, $C_2=\frac{16 \pi^3 \omega_{n}}{\omega_{n+3}}$ and $C_4=\frac{256 \pi^5 \omega_n}{3 \omega_{n+5}}$, we have $$G_4(L)= -\frac{\omega_n}{32 \pi \omega_{n+1}}\big(3Q^2 - 6(n+1)QQ(L) + \pi^4(n+1)(n+3)Q(L)^2\big) ,$$ and $$G_2(L)=\frac{\omega_n}{4\omega_{n+1}} \bigg((n+1)Q(L)-Q \bigg)$$ for $L \in {\mathcal{L}_1^n}.$
In Theorem \[crofton2\] we only considered the situation, where $s$ is even. It is natural to ask whether $\T[s]$ can also be written as a linear Crofton integral when $s$ is odd. The case $s=1$ is trivial, as the tensor $\Phi_{n-1,0,1}(K)=0$ for all $K\in{{\mathcal K}}^n$. If $n=1$, then $\T[s]=0$ for all odd $s$, since the area measure of order 0 is the Hausdorff measure on the sphere. Apart from these trivial examples, $\Phi_{n-1,0,s}$ cannot be written as a linear Crofton-type integral, when $s$ is odd and the measurement function satisfies some rather weak assumptions. This is shown in Theorem \[s ulige\].
\[s ulige\] Let $n \geq 2$ and let $s>1$ be odd. Then there exists neither a translation invariant nor a bounded measurable measurement function $\nobreak{\alpha \colon {{\mathcal K}}^n \rightarrow {\mathbb{T}}^s}$ such that $$\label{eq}
\int_{{\mathcal{E}^n}_1} \alpha(K \cap E) \, \mu_1^n(dE)=\T[s]$$ for all $K \in {{\mathcal K}}^n.$
Let $\alpha \colon {{\mathcal K}}^n \rightarrow {\mathbb{T}}^s$ be a measurable and bounded function that satisfies equation . Since $\mu_1^n(\{E \in {\mathcal{E}^n}_1 \mid E \cap K = \emptyset\})=\infty$ for $K \in {{\mathcal K}}^n$, we have $\alpha(\emptyset)=0$. Now define the averaged function $$\alpha_r(M)=\frac{1}{V_n(rB^n)}\int_{rB^n}\alpha(M+x) \, \lambda (dx), \qquad M \in {{\mathcal K}}^n,$$ for $r > 0$. Since $\alpha$ is measurable and bounded, the average function $\alpha_r$ is well-defined. Clearly $\alpha_r(\emptyset)=0$. Using Fubini’s theorem, the invariance of $\mu_1^n$ and the fact that $\Phi_{n-1,0,s}$ is translation invariant, we get that $$\int_{{\mathcal{E}^n}_1} \alpha_r(K \cap E) \, \mu_1^n (dE)
=\frac{1}{V_n(rB^n)} \int_{rB^n} \Phi_{n-1,0,s}(K + x) \, \lambda(dx)
=\T[s].$$ Let $K \in {{\mathcal K}}^n$ be such that $K \subseteq B^n$. Since $K \cap E$ is either the empty set or a a line segment in $B^n$ when $E \in {\mathcal{E}^n}_1$, there exists a vector $z_E \in {{\mathbb R}}^n$ with ${\|}z_E {\|}\leq 2$ such that $-(K \cap E)=(K \cap E) + z_E$. Let ${\mathcal{A}}=\{E \in {\mathcal{E}^n}_1 \mid B^n \cap E \neq \emptyset\}$, let $B_1\Delta B_2$ denote the symmetric difference of two sets $B_1,B_2$, and assume that $|\alpha|\le M$ for some constant $M$. Then $$\begin{aligned}
& \big|\T[s] - \Phi_{n-1,0,s}(-K)\big|
=\bigg| \int_{{\mathcal{A}}} \alpha_r(K \cap E) - \alpha_r(-(K \cap E)) \, \mu_1^n (dE) \bigg|
\\
&\qquad \leq \frac{1}{V_n(rB^n)} \int_{\mathcal{A}}\bigg| \int_{rB^n} \alpha((K \cap E) + x)\, \lambda(dx) \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad - \int_{rB^n+z_E} \alpha((K \cap E) + x)\, \lambda (dx) \bigg| \,\mu_1^n (dE)\\
&\qquad \leq \frac{1}{V_n(rB^n)} \int_{{\mathcal{A}}}\int_{(rB^n+z_E)\Delta (rB^n)} \left|\alpha((K\cap E)+x)\right|\, \lambda(dx)\, \mu_1^n (dE)\\
&\qquad\leq \frac{2M}{V_n(rB^n)} \int_{{\mathcal{A}}} V_n((rB^n+z_E)\setminus (rB^n))\, \mu_1^n (dE)\\
&\qquad\leq 2M\frac{ (r+2)^n-r^n}{r^n}\kappa_{n-1}\longrightarrow 0\qquad\text{as $r \rightarrow \infty$. }\end{aligned}$$ Here we used that $(rB^n+z_E)\setminus (rB^n)\subseteq (r+2)B^n\setminus (rB^n)$ and $\mu^n_1({\mathcal{A}})=\kappa_{n-1}$.
Hence, we get $\T[s]=\Phi_{n-1,0,s}(-K)$. Since $s$ is odd, we also have $\T[s]=-\Phi_{n-1,0,s}(-K)$. Therefore $\T[s]=0$, which is not the case for all $K \subseteq B^n$, since $s>1$. Then, by contradiction, cannot be satisfied by a bounded measurement function, when $s>1$ is odd.
Now assume that $\alpha$ is translation invariant and satisfies equation . As $-(K \cap E)$ is a translation of $K \cap E$, we have $$\int_{{\mathcal{E}^n}_1} \alpha(-K \cap E) \, \mu_1^n (dE)
= \int_{{\mathcal{E}^n}_1} \alpha(-(K \cap E)) \, \mu_1^n (dE)
=\int_{{\mathcal{E}^n}_1} \alpha(K \cap E) \, \mu_1^n (dE),$$ implying $\Phi_{n-1,0,s}(-K)=\T[s]=-\Phi_{n-1,0,s}(-K)$, and hereby we obtain that $\T[s]=0$ for all $K \in {{\mathcal K}}^n$. This is a contradiction as before.
Design based estimation {#sec estimation}
=======================
In this section we use the integral formula in Theorem \[crofton2\] to derive unbiased estimators of the surface tensors $\T[s]$ of $K \in {{\mathcal K}}^n$, when $s$ is even. We assume throughout this chapter that $n \geq 2$. Three different types of estimators based on 1-dimensional linear sections are presented. First, we establish estimators based on isotropic uniform random lines, then estimators based on random lines in vertical sections and finally estimators based on non-isotropic uniform random lines.
Estimation based on isotropic uniform random lines {#IUR}
--------------------------------------------------
In this section we construct estimators of $\T[s]$ based on isotropic uniform random lines. Let $K \in {{\mathcal K}}^n.$ We assume that (the unknown set) $K$ is contained in a compact reference set $A \subseteq {{\mathbb R}}^n$, the latter being known. Now let $E$ be an *isotropic uniform random (IUR) line in ${{\mathbb R}}^n$ hitting $A$*, i.e., the distribution of $E$ is given by $$\label{IURdistribution}
{\mathbb{P}}(E \in {\mathcal{A}})= c_1(A) \int_{{\mathcal{A}}} {\textbf{1}}(E' \cap A \neq \emptyset)\,\mu_1^n(dE')$$ for ${\mathcal{A}}\in {\mathcal{B}}({\mathcal{E}^n}_1),$ where $c_1(A)$ is the normalizing constant $$c_1(A)=\bigg(\int_{{\mathcal{E}^n}_1} {\textbf{1}}(E' \cap A \neq \emptyset)\,\mu_1^n(dE')\bigg)^{-1}.$$ By with $ s=0 $ the normalizing constant becomes $c_1(A)=\frac{\omega_n }{2 \kappa_{n-1}}V_{n-1}(A)^{-1}$, when $A$ is a convex body. Then Theorem \[crofton2\] implies that $$\label{Estimator}
c_1(A)^{-1} G_{s}({\pi(E)})V_0(K \cap E)$$ is an unbiased estimator of $\T[s]$, when $s$ is even.
\[example\] Using the expressions of $G_2$ and $G_4$ in Example $\ref{s er 4}$ we get that $$\begin{aligned}
-\frac{V_{n-1}(A)}{32\pi^2}\big(3Q^2 - 6(n+1)QQ(L) + \pi^4(n+1)(n+3)Q(L)^2\big) V_0(K \cap E)\end{aligned}$$ is an unbiased estimator of $\T[4],$ and $$\begin{aligned}
\label{s er 2}
\frac{V_{n-1}(A)}{4\pi}\bigg((n+1)Q({\pi(E)})-Q \bigg) V_0(K \cap E)\end{aligned}$$ is an unbiased estimator of $\T[2]$, when $A$ is a convex body. For $n=3$, these estimators read $$-\frac{V_2(A)}{32 \pi^2}\bigg(3 Q^2-24QQ({\pi(E)})+24\pi^4Q({\pi(E)})^2\bigg)V_0(K \cap E)$$ and $$\label{IURn3}
\frac{V_2(A)}{\pi} \bigg(Q({\pi(E)})-\frac{1}{4}Q\bigg)V_0(K \cap E).$$
An investigation of the estimators in Example \[example\] shows that they possess some unfavourable statistical properties. If $K \cap E = \emptyset$ the estimators are simply zero. Furthermore, if $K \cap E \neq \emptyset,$ the matrix representation of the estimator of $\T[2]$ is, in contrast to $ \T[2] $, not positive semi-definite. In fact, the eigenvalues of the matrix representation of $(n+1)Q({\pi(E)})-Q$ are $n$ (with multiplicity $ 1 $) and $-1$ (with multiplicity $n-1$). It is not surprising that estimators based on the measurement of one single line, are not sufficient, when we are estimating tensors with many unknown parameters. To improve the estimators, they can be extended in a natural way to use information from $N$ *IUR* lines for some $N \in {{\mathbb N}}$. In addition, the integral formula can be rewritten in the form $$\begin{aligned}
\label{Alternativ}
\T[s]&=\int_{{\mathcal{L}_1^n}} \int_{L^\perp} G_s(L)V_0(K \cap (L+x))\, \lambda_{{L^{\perp}}}(dx) \, \nu_1^n (dL) \nonumber \\
&=\int_{{\mathcal{L}_1^n}} G_s(L) V_{n-1}(K \vert L^\perp) \, \nu_1^n (dL),\end{aligned}$$ which implies that $$\label{projestimator}
\frac{1}{N}\sum_{i=1}^NG_s(L_i)V_{n-1}(K \vert L_i^\perp)$$ is an unbiased estimator of $\T[s]$, when $L_1, \dots L_N \in {\mathcal{L}_1^n}$ are $ N $ isotropic lines (through the origin) for an $ {N \in {{\mathbb N}}} $. When $K$ is full-dimensional this estimator never vanishes. In the case where $s=2$ the estimator becomes $$\label{EstimatorN}
\frac{1}{N}\frac{\omega_n}{4\omega_{n+1}}\sum_{i=1}^N\big((n+1)Q(L_i)-Q \big) V_{n-1}(K \vert L_i^\perp).$$ In stereology it is common practice to use orthogonal test lines. If we set $ N=n $ and let $ L_1, \dots, L_n $ be isotropic, pairwise orthogonal lines, then the estimator becomes positive definite exactly when $$\label{posdefcondition}
(n+1)V_{n-1}(K \vert L_i^\perp) > \sum_{j=1}^n V_{n-1}(K \mid L_j^\perp)$$ for all $i=1, \dots, n.$ This is a condition on $ K $ requiring that $ K $ is not too eccentric. A sufficient condition for to hold makes use of the radius $ R(K) $ of the smallest ball containing $ K $ and the radius $ r(K) $ of the largest ball contained in $ K $. If $$\frac{r(K)}{R(K)}> \bigg(1-\frac{1}{n}\bigg)^{\frac{1}{n-1}},$$ then is satisfied, and hence the estimator with $ n $ orthogonal, isotropic lines is positive definite. In $ {{\mathbb R}}^2 $ this means that $ 2r(K) > R(K) $ is sufficient for a positive definite estimator , and in particular for all ellipses for which the length of the longer main axis does not exceed twice the length of the smaller main axis, yields positive definite estimators. For ellipses, this criterion is also necessary as the following example shows.
Consider the situation where $ n=2 $ and $ K $ is an ellipse, $ K=\{x \in {{\mathbb R}}^2 \mid x^\top B x \leq 1\} $, given by the matrix $$B=\begin{pmatrix}
\alpha^{-2} & 0 \\ 0 & (k\alpha)^{-2}
\end{pmatrix},$$ where $ \alpha > 0 $ and $ k \in (0,1] $. The parameter $ k $ determines the eccentricity of $ K $. If $ k \in (\frac{1}{2},1] $, and $ L_1 $ and $ L_2 $ are orthogonal, isotropic random lines in $ {{\mathbb R}}^2 $, the estimator becomes positive definite by the above considerations. Now let $ k \in [0,1/2] $. Since $ n=2 $, each pair of orthogonal lines is determined by a constant $ \phi \in [0, \frac{\pi}{2}) $ by letting $ L_1 = u_{\phi}^\perp $ and $ L_2=u_{\phi+ \frac{\pi}{2}}^\perp $, where $ u_{\phi}=(\cos(\phi),\sin(\phi))^\top $. Then $$V_{n-1}(K \mid L_1^\perp)=2h(K,u_{\phi})=2 \alpha \sqrt{\cos^2(\phi)+k^2 \sin^2(\phi)}$$ and $$V_{n-1}(K \mid L_2^\perp ) = 2 \alpha \sqrt{\sin^2(\phi)+k^2 \cos^2(\phi)}.$$ Condition is satisfied if and only if $$\phi \in [\sin^{-1}\bigg(\sqrt{\frac{1-4k^2}{5(1-k^2)}}\bigg), \cos^{-1}\bigg(\sqrt{\frac{1-4k^2}{5(1-k^2)}}\bigg) ],$$ and the probability that the estimator is positive definite, when $ L_1 $ and $ L_2 $ are orthogonal, isotropic lines (corresponding to $ \phi $ being uniformly distributed on $ [0,\frac{\pi}{2}] $) is $$\frac{2}{\pi}\bigg(\cos^{-1}\bigg(\sqrt{\frac{1-4k^2}{5(1-k^2)}}\bigg)-\sin^{-1}\bigg(\sqrt{\frac{1-4k^2}{5(1-k^2)}}\bigg)\bigg),$$ which converges to $ \frac{2}{\pi} \big(\cos^{-1}(\sqrt{\frac{1}{5}})- \sin^{-1}(\sqrt{\frac{1}{5}}) \big) \approx 0.41 $ as $ k $ converges to 0.
In $ {{\mathbb R}}^2 $ the estimator can alternatively be combined with a systematic sampling approach with $ N $ isotropic random lines. Let $ N \in {{\mathbb N}}$, and let $ \phi_0 $ be uniformly distributed on $ [0,\frac{\pi}{N}] $. Moreover, let $ \phi_i=\phi_0 + i \frac{\pi}{N} $ for $ i=1, \dots, N-1 $. Then $ u_{\phi_0}, \dots, u_{\phi_{N-1}} $ are $ N $ systematic isotropic uniform random directions in the upper half of $ S^1 $, where $ u_{\phi}=(\cos(\phi),\sin(\phi))^\top $. As the estimator is a tensor of rank 2, it can be identified with the symmetric $2 \times 2$ matrix, where the $(i,j)$’th entry is the estimator evaluated at $(e_i,e_j)$, where $(e_1,e_2)$ is the standard basis of ${{\mathbb R}}^2$. The estimator becomes $$\label{syst}
S_N(K, \phi_0)=\frac{1}{N}\sum_{i=0}^{N-1} \begin{pmatrix}
3\cos^2(\phi_i)-1 & 3\cos(\phi_i) \sin(\phi_i) \\
3\cos(\phi_i) \sin(\phi_i) & 3\sin^2(\phi_i)-1
\end{pmatrix} V_1(K \mid u_{{\phi}_i}^\perp).$$
\[exsyst\] To investigate how the estimator $ S_N(K, \phi_0) $ performs we estimate the probability that the estimator is positive definite for three different origin-symmetric convex bodies in $ {{\mathbb R}}^2 $; a parallelogram, a rectangle, and an ellipse. Thus let $$\begin{aligned}
K_1&=\mathrm{conv}\{ (1,\epsilon),(-1,\epsilon),(-1,-\epsilon),(1,-\epsilon)\}, \\
K_2&=\mathrm{conv}\{ (1,0),(0,\epsilon),(-1,0),(0,-\epsilon)\} \\
\intertext{and}
K_3&= \{x \in {{\mathbb R}}^2 \mid x^\top \begin{pmatrix}
1 & 0 \\ 0 & \frac{1}{\sqrt{\epsilon}}
\end{pmatrix} x \leq 1\} \end{aligned}$$ with $\epsilon=0.1$. The support functions, and hence the intrinsic volumes $V_1(K_i \vert u_\phi^\perp)$, of $ K_1, K_2$ and $ K_3 $ have simple analytic expressions, and the estimator $ S_N(K_i, \phi_0) $ can be calculated for $ \phi_0 \in [0,\frac{\pi}{N}] $ and $ i=1,2,3 $. The eigenvalues of the estimators can be calculated numerically, and the probability that the estimators $ S_N(K_i,\phi_0) $ are positive definite, when $ \phi_0 $ is uniformly distributed on $ [0, \frac{\pi}{N}] $, can hereby be estimated. For each choice of $ N $, the estimate of the probability is based on $ 500 $ equally spread values of $ \phi_0 $ in $ [0,\frac{\pi}{N}] $. The estimate of the probability that $ S_N(K_i,\phi_0) $ is positive definite is plotted against the number of equidistant lines $ N $ for $i=1,2,3$ in Figure \[Plot of prob for posdef\]. The plots in Figure \[Plot of prob for posdef\] show that even though we consider rather eccentric shapes, the number $N$ of lines needed to get a positive definite estimator with probability $1$ is in all cases less than $7$.
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](R)
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](Rectangle)
\
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](Par)
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](Parallelogram)
\
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](E)
![The probability that $ S_N(K_i,\phi_0) $ is positive definite for $ i=1,2,3 $, when $ \phi_0 $ is uniformly distributed on $ [0,\frac{\pi}{N}] $ plotted against the number of equidistant lines $ N $.[]{data-label="Plot of prob for posdef"}](Ellipse)
To apply the estimator it is only required to observe whether the test line hits or misses the convex body $K$. The estimator requires more sophisticated information in terms of the projection function. In the following example the coefficient of variation of versions of the estimators and are estimated and compared in a three-dimensional set-up.
\[3dimex\] Let $K_{l}'$ be the prolate spheroid in ${{\mathbb R}}^3$ with main axis parallel to the standard basis vectors $e_1,e_2$ and $e_3$, and corresponding lengths of semi-axes $\lambda_1=\lambda_2=1$ and $\lambda_3=l$. For $l=1, \dots, 5$, let $K_l$ denote the ellipsoid obtained by rotating $K_l'$ first around $e_1$ with an angle $\frac{3\pi}{16}$, and then around $e_2$ with an angle $\frac{5\pi}{16}$. Note, that the eccentricity of $K_l$ increases with $l$. In this example, based on simulations, we estimate and compare the coefficient of variation *(CV)* of the developed estimators of $\Phi_{2,0,2}(K_l)$ for $l=1, \dots, 5$.
Formula provides an unbiased estimator of the tensor $\Phi_{2,0,2}(K_l)$ for $l=1, \dots, 5$. The estimator is based on one *IUR* line hitting a reference set $A$, and can in a natural way be extended to an estimator based on three orthogonal *IUR* lines hitting $A$. We estimate the variance of both estimators. Let, for $l=1, \dots,5$, the reference set $A_l$ be a ball of radius $R_l>0$. The choice of the reference set influences the variance of the estimator. In order to minimize this effect in the comparison of the CV’s, the radii of the reference sets are chosen such that the probability that a test line hits $K_l$ is constant for $l=1, \dots, 5$. By formula the probability that an *IUR* line hitting $A_l$ hits $K_l$ is $\frac{V_{2}(K_l)}{V_2(A_l)}$. The radius is chosen, such that this probability is $\frac{1}{7}$. We further estimate the variance of the projection estimator based on one isotropic line and on three orthogonal isotropic lines.
As $\Phi_{2,0,2}(K_l)$ is a tensor of rank 2, it can be identified with the symmetric $3 \times 3$ matrix $\{\Phi_{2,0,2}(K_l)(e_i,e_j)\}_{i,j=1}^3$. Thus, in order to estimate $\Phi_{2,0,2}(K_l)$, the matrix $\{\hat{\Phi}_{2,0,2}(K_l)(e_i,e_j)\}_{i,j=1}^3$ is calculated. Here, $\hat{\Phi}_{2,0,2}(K_l)$ refers to any of the four estimators described above. Due to symmetry, there are six different components of the matrices.
The estimates of the variances are based on 1500-10000 estimates of the tensor, depending on the choice of the estimator and the eccentricity of $K_l$. Using the estimates of the variances, we estimate the absolute value of the CV’s by $$\widehat{CV}_{ij}=\frac{\sqrt{\widehat{\text{Var}}(\hat{\Phi}_{2,0,2}(K_l)(e_i,e_j))}}{|\Phi_{2,0,2}(K_l)(e_i,e_j)|},$$ for $i,j=1,2,3$ and $l=1, \dots, 5$. As $K_l$ is an ellipsoid, the tensor $\Phi_{2,0,2}(K_l)$ can be calculated numerically. The CV’s of the four estimators are plotted in Figure \[4variances\] for each of the six different components of the associated matrix. As $K_1$ is a ball, the off-diagonal elements of the matrix associated with $\Phi_{2,0,2}(K_1)$ are zero. Thus, the CV is in this case calculated only for the estimators of the diagonal-elements.
The projection estimators give, as expected, smaller CV’s, than the estimators based on the Euler characteristic of the intersection between the test lines and the ellipsoid. For the estimators based on one test line the CV of the projection estimator is typically around $38\%$ of the corresponding estimator . For the estimators based on three orthogonal test lines, the CV of the projection estimator is typically $9\%$ of the estimator , when $l=2,\dots,5$. Due to the fact that $K_1$ is a ball, the variance of the projection estimator based on three orthogonal lines is 0, when $l=1$.
It is interesting to compare the increase of efficiency when using the estimator based on three orthogonal test lines instead of three i.i.d. test lines. The CV of an estimator based on three i.i.d. test lines is $\frac{1}{\sqrt{3}}$ of the CV of the estimator , (the “$+$” signs in Figure \[4variances\]). The CV, when using three orthogonal test lines, is typically around $92\%$ of that CV. For $l=2,\dots, 5$, the CV’s of the projection estimator based on three orthogonal lines, are typically $20\%$ of the CV, when using three i.i.d lines, indicating that spatial random systematic sampling increases precision without extra workload.
The CV’s of the estimators of the diagonal-elements $\Phi_{2,0,2}(K_l)(e_i,e_i)$ are almost constant in $l$. Hence the eccentricity of $K_l$ does not affect the CV’s for these choices of $l$. There is a decreasing tendency of the CV’s of the estimators of the off-diagonal elements. This might be explained by the fact that the true value of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ is close to zero, when $i \neq j$ and $l$ is small.
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV11ny)
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV22ny)
\
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV33ny)
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV12ny)
\
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV13ny)
![The estimated coefficients of variation $\widehat{CV}_{ij}$ of the estimators of $\Phi_{2,0,2}(K_l)(e_i,e_j)$ plotted against $l$ for $\nobreak{i,j \in \{1,2,3\}}$. The CV of the estimator based on *one* line is designated by “$+$”, while the CV of the corresponding estimator based on *three* lines is designated by “$\bullet$”. The CV of the projection estimator is designated by “$\circ$” and “$\square$” for one and three lines, respectively.[]{data-label="4variances"}](CV23ny)
The above example shows that only the projection estimator based on three orthogonal test lines has a satisfactory precision. For $l=2$ the CV’s are approximately $\frac{1}{3}$ for the diagonal-elements and $1$ for the off-diagonal elements. Further variance reduction of the projection estimator can be obtained by using a larger number of systematic random test directions. For $n=2$ this can be effectuated by choosing equidistant points on the upper half circle; see . For $n=3$ the directions must be chosen evenly spread; see [@Leopardi2006] for details.
If the projections are not available or too costly to obtain, systematic sampling in the position of the test lines with given orientations can be applied. In ${{\mathbb R}}^2$ this corresponds to a Steinhaus-type estimation procedure (see e.g. [@Jensen2005]). In ${{\mathbb R}}^3$ the fakir method described in [@L.1998] can be applied.
Estimation based on vertical sections {#sec VUR}
-------------------------------------
In the previous section we constructed an estimator of $\T[s]$ based on isotropic uniform random lines. As described in [@Markus], it is sometimes inconvenient or impossible to use the *IUR* design in applications. For instance, in biology when analysing skin tissue, it might be necessary to use sample sections, which are normal to the surface of the skin, so that the different layers become clearly distinguishable in the sample. Instead of using *IUR* lines it is then a possibility to use vertical sections introduced by Baddeley in [@Baddeley1983]. The idea is to fix a direction (the normal of the skin surface), and only consider flats parallel to this direction. After randomly selecting a flat among these flats, we want to pick a line in the flat in such a way that this line is an isotropic uniform random line in ${{\mathbb R}}^n$. Like in the classical formulae for vertical sections, we select this line in a non-uniform way according to a Blaschke-Petkantschin formula (see ). This idea is used to deduce estimators of $\T[s]$ from the Crofton formula .
When introducing the concept of vertical sections we use the following notation. For $ 0 \leq k \leq n $ and $ L \in {\mathcal{L}}^n_k $, let $$\cLL=
\begin{cases}
\{M \in {\mathcal{L}}_r^n \mid M \subseteq L \} & \text{if } 0 \leq r \leq k \\
\{M \in {\mathcal{L}}_r^n \mid L \subseteq M \} & \text{if } k < r \leq n,
\end{cases}$$ and, similarly, let ${\mathcal{E}}^E_r= \{F \in {\mathcal{E}^n}_r \mid F \subseteq E \} $ for $ E \in {\mathcal{E}}^n_k $ and $ 0 \leq r \leq k$. Let $ \nu^L_r$ denote the unique rotation invariant probability measure on $ {\mathcal{L}}^L_r $, and let $ \mu^E_r $ denote the motion invariant measure on $ {\mathcal{E}}^E_r $ normalized as in [@Weil].
Let $L_{0} \in {\mathcal{L}_1^n}$ be fixed. This is the *vertical axis* (the normal of the skin surface in the example above). Let the reference set $A \subseteq {{\mathbb R}}^n$ be a compact set.
\[VUR\] Let $1 < k < n$. A random $k$-flat $H$ in ${{\mathbb R}}^n$ is called **a vertical uniform random (VUR) $k$-flat hitting $A$** if the distribution of $H$ is given by $$P(H \in {\mathcal{A}}) = c_2(A) \int_{\cLLn[k]} \int_{A \mid L^\perp} {\textbf{1}}(L+x \in {\mathcal{A}})\, \lambda_{{L^{\perp}}}(dx) \, \nu_k^{L_0}(dL)$$ for ${\mathcal{A}}\in {\mathcal{B}}({\mathcal{E}^n}_k)$, where $c_2(A)>0$ is a normalizing constant.
The distribution of $H$ is concentrated on the set $$\{E \in {\mathcal{E}^n}_{k} \mid E \cap A \neq \emptyset, L_0 \subseteq {\pi(E)}\}.$$ When the reference set $A$ is a convex body, the normalizing constant becomes $$c_2(A)=\binom{n-1}{k-1}\frac{\kappa_{n-1}}{\kappa_{k-1}\kappa_{n-k}}\frac{1}{V_{n-k}(A\vert L_0^\perp)}.$$ (Note that we do not indicate the dependence of $c_2(A)$ on $k$ by our notation.) This can be shown, e.g., by using the definition of $\nu_k^{L_0}$ together with [@Weil (13.13)], Crofton’s formula in the space $L_0^{\perp}$, and the equality $$\label{v0 lighed}
{\textbf{1}}_{A \vert L^\perp}(x)=V_0((A\vert L_0^\perp) \cap (x+L))$$ for $A \in {{\mathcal K}}^n$, $L \in \cLLn[k]$ and $x \in L^\perp$. For later use note that when $k=2$ the normalizing constant becomes $$\label{normalizing constant}
c_2(A)=\frac{\omega_{n-1}}{2 \kappa_{n-2}V_{n-2}(A \vert L_0^\perp)}.$$
To construct an estimator, which is based on a vertical uniform random flat, we cannot use Theorem \[crofton2\] immediately as in the *IUR*-case. It is necessary to use a Blaschke-Petkantschin formula first; see [@Markus (2.8)]. It states that for a fixed $L_0 \in {\mathcal{L}_1^n}$ and an integrable function $f \colon {\mathcal{E}}^n_1 \rightarrow {{\mathbb R}}$, we have $$\begin{aligned}
\label{B-P}
\int_{{\mathcal{E}^n}_1} f(E) \, \mu_1^n(dE)& = \frac{\pi \omega_{n-1}}{\omega_n} \int _{{\mathcal{L}}^{L_0}_2} \int_{M^\perp} \int _{{\mathcal{E}}_1^{M+x}}
f(E) \sin (\angle(E,L_0))^{n-2} \nonumber
\\
&\qquad\qquad\times \mu_1^{M+x}(dE) \, \lambda_{M^{\perp}}(dx) \, \nu_2^{L_0} (dM),\end{aligned}$$ where $\angle(E_1,E_2)$ is the (smaller) angle between $\pi(E_1)$ and $ \pi(E_2)$ for two lines $E_1,E_2 \in {\mathcal{E}}^n_1$. For $K \in {{\mathcal K}}^n$ and even $s \in {{\mathbb N}}_0$, equation can be applied coordinate-wise to the mapping $E \mapsto \MT[s]$ and combined with the Crofton formula in Theorem \[crofton-like\]. The result is an integral formula for two-dimensional vertical sections.
\[VUR crofton\] Let $L_0 \in {\mathcal{L}_1^n}$ be fixed. If $K \in {{\mathcal K}}^n$ and $s \in {{\mathbb N}}_0$ is even, then $$\begin{aligned}
\label{VUR crofton formel}
&\int _{{\mathcal{L}}^{L_0}_2} \int_{M^\perp} \int _{{\mathcal{E}}_1^{M+x}} \Phi^{(E)}_{0,0,s}(K \cap E) \sin(\angle(E,L_0))^{n-2} \, \mu_{1}^{M+x} (dE) \, \lambda_{M^{\perp}} (dx) \, \nu_{2}^{L_0} (dM)
\nonumber \\
&\qquad\qquad = \frac{2 \omega_{n+s+1}}{ s! \pi^2 \omega_{n-1} \omega_{s+1}^2 } \sum_{k=0}^{\frac{s}{2}} c_k^{(\frac{s}{2})} Q^{\frac{s}{2}-k} \Phi_{n-1,0,2k}(K),\end{aligned}$$ where the constants $c_k^{(m)}$ are given in Theorem $\ref{crofton-like}$. For odd $s \in {{\mathbb N}}_0$ the integral on the left-hand side is zero.
If Theorem \[crofton-like\] is replaced by Theorem \[crofton2\] in the above line of arguments, we obtain an explicit measurement function for vertical sections leading to one single tensor.
\[VURcrofton2\] Let $L_0 \in {\mathcal{L}_1^n}$ be fixed. If $K \in {{\mathcal K}}^n$ and $s\in {{\mathbb N}}_0$ is even, then $$\begin{aligned}
\frac{\omega_n}{\pi \omega_{n-1}}\T[s]&=\int _{{\mathcal{L}}^{L_0}_2} \int_{M^\perp} \int _{{\mathcal{E}}_1^{M+x}} G_s({\pi(E)}) V_0(K \cap E)
\\
& \qquad \times \sin(\angle(E,L_0))^{n-2} \, \mu_1^{M+x}(dE) \, \lambda_{M^{\perp}}(dx) \, \nu_2^{L_0}(dM),\end{aligned}$$ where $G_s$ is given in Theorem $\ref{crofton2}$.
Let $s \in {{\mathbb N}}_0$ be even and assume that $K \in {{\mathcal K}}^n$ is contained in a reference set $A \in {{\mathcal K}}^n$. Using Theorem \[VURcrofton2\] we are able to construct unbiased estimators of the tensors $\T[s]$ of $K$ based on a vertical uniform random 2-flat. If $H$ is an *VUR* 2-flat hitting $A$ with vertical direction $L_0 \in {\mathcal{L}_1^n}$, then it follows from Theorem \[VURcrofton2\] and that $$V_{n-2}(A \vert L_0^\perp)
\int_{{\mathcal{E}}_1^{H}} G_s({\pi(E)})V_0(K \cap E)\sin(\angle(E,L_0))^{n-2}\, \mu_1^H(dE)$$ is an unbiased estimator of $\T[s]$. Hence the surface tensors can be estimated by a two-step procedure. First, let $H$ be a *VUR* 2-flat hitting the convex body $A$ with vertical direction $L_0$. Given $H$, the integral $$\label{integral}
\int_{{\mathcal{E}}_1^{H}} G_s({\pi(E)})V_0(K \cap E)\sin(\angle(E,L_0))^{n-2} \,\mu_1^H(dE)$$ is estimated in the following way. Let $E \in {\mathcal{E}}^H_1$ be an *IUR* line in $H$ hitting $A$, i.e. the distribution of $E$ is given by $$P(E \in {\mathcal{A}})= c_3(A) \int_{{\mathcal{A}}} {\textbf{1}}(A \cap E \neq \emptyset) \, \mu_1^H (dE), \qquad {\mathcal{A}}\in {\mathcal{B}}({\mathcal{E}}_1^H),$$ where $$c_3(A)=\frac{\pi}{2}V_{1}(A \cap H)^{-1}$$ is the normalizing constant. The integral is then estimated unbiasedly by $$\label{Integral estimator}
c_3(A)^{-1}G_s({\pi(E)})V_0(K \cap E) \sin(\angle(E,L_0))^{n-2}.$$
Consider the case $s=2$. Let $H$ be a *VUR* 2-flat hitting $A \in {{\mathcal K}}^n$ with vertical direction $L_0$. Given $H$, let $E$ be an *IUR* line in $H$ hitting $A$. Then $$\frac{ \kappa_{n-2}V_{n-2}(A \vert L_0^\perp)V_1(A \cap H)}{ \omega_{n+1} }\bigg((n+1)Q({\pi(E)})-Q \bigg)V_0(K \cap E) \sin(\angle(E,L_0))^{n-2}$$ is an unbiased estimator of $\T[2]$.
Using [@Weil (13.13)] and an invariance argument, the integral can alternatively be expressed by means of the support function of $K$ in the following way $$\begin{aligned}
&\quad ~ \int_{{\mathcal{E}}_1^{H}} G_s({\pi(E)})V_0(K \cap E)\sin(\angle(E,L_0))^{n-2} \,\mu_1^H (dE)
\\
&=\frac{1}{\omega_2}\int_{S^{n-1} \cap {\pi (H)}} G_s(u^\perp \cap {\pi (H)})\sin(\angle(u^\perp \cap {\pi (H)},L_0))^{n-2}
\\ & \qquad \qquad \quad \times \int_{[u]} V_0(K \cap H \cap (u^\perp + x)) \, \lambda_{[u]}(dx) \, {\mathcal{H}}^{1}(du) \nonumber
\\
&=\frac{1}{\omega_2}\int_{S^{n-1} \cap {\pi (H)}} G_s(u^\perp \cap {\pi (H)}) \cos(\angle(u,L_0))^{n-2} w(K \cap H, u) \,{\mathcal{H}}^{1}(du),\end{aligned}$$ where $ [u] $ denotes the linear hull of a unit vector $ u $, and $$w(M,u)=h(M,u)+h(M,-u)$$ is the width of $M \in {{\mathcal K}}^n$ in direction $u$. Hence, given $H$, $$\label{Alternative integral estimator}
G_s(U^\perp \cap {\pi (H)})\cos(\angle(U,L_0))^{n-2} w(K \cap H, U)$$ is an unbiased estimator of the integral if $U$ is uniform on $S^{n-1} \cap {\pi (H)}$. As in the *IUR* set-up in Section \[IUR\] we have two estimators: an estimator , where it is only necessary to observe whether the random line $E$ hits or misses $K$, and the alternative estimator $\eqref{Alternative integral estimator}$, which requires more information. The latter estimator has a better precision at least when the reference set $A$ is large. Variance reduction can be obtained by combining the estimators with a systematic sampling approach.
Estimation based on non-isotropic random lines {#Sec noniso}
----------------------------------------------
In this section we consider estimators based on non-isotropic random lines. It is well-known from the theory of importance sampling, that variance reduction of estimators can be obtained by modifying the sampling distribution in a suitable way (see, e.g., [@Asmussen]). The estimators in this section are developed with inspiration from this theory. Let again $K \in {{\mathcal K}}^n$, and let $f \colon {\mathcal{L}_1^n}\rightarrow [0,\infty)$ be a density with respect to the invariant measure $\nu_1^n$ on ${\mathcal{L}_1^n}$ such that $f$ is positive $\nu^n_1$-almost surely. Then by Theorem \[crofton2\] we have trivially $$\label{density integral}
\int_{{\mathcal{E}^n}_1} \frac{G_s({\pi(E)})V_0(K \cap E)}{f({\pi(E)})} \, f({\pi(E)})\,\mu_1^n(dE) = \T[s].$$ Let $A \subseteq {{\mathbb R}}^n$ be a compact reference set containing $K$, and let $E$ be an $f$-weighted random line in ${{\mathbb R}}^n$ hitting A, that is, the distribution of $E$ is given by $$\begin{aligned}
P(E \in {\mathcal{A}}) = c_4(A) \int_{{\mathcal{A}}} {\textbf{1}}(E \cap A \neq \emptyset) f({\pi(E)}) \, \mu_1^n (dE)\end{aligned}$$ for ${\mathcal{A}}\in {\mathcal{B}}({\mathcal{E}^n}_1)$, where $$c_4(A)=\bigg(\int_{{\mathcal{E}^n}_1} {\textbf{1}}(E \cap A \neq \emptyset) f({\pi(E)})\, \mu_1^n (dE)\bigg)^{-1}$$ is a normalizing constant. Then $$\frac{c_4(A)^{-1} G_{s}({\pi(E)})V_0(K \cap E)}{f({\pi(E)})}$$ is an unbiased estimator of $\T[s]$. Notice that if we let the density $ f $ be constant, then this procedure coincides with the *IUR* design in Section \[IUR\].
Our aim is to decide, which density $f$ should be used in order to decrease the variance of the estimator of $\T[s]$. Furthermore, we want to compare this variance with the variance of the estimator based on an *IUR* line. From now on, we restrict the investigation to the situation where $ n=2 $ and $ s=2 $. Furthermore, we assume that the reference set $ A $ is a ball in $ {{\mathbb R}}^2 $ of radius $ R $ for some $ R > 0 $. Then $ c_4(A)=(2R)^{-1} $ independently of $f$.
Since $ \Phi_{1,0,2}(K) $ can be identified with a symmetric $ 2 \times 2 $ matrix, we have to estimate three unknown components. We consider the variances of the three estimators separately. The components of the associated matrix of $G_2(L)$ for $L \in {\mathcal{L}}^n_1$ is defined by $$g_{ij}(L)=G_2(L)(e_i,e_j),$$ for $i,j=1,2$, where $(e_1,e_2)$ is the standard basis of ${{\mathbb R}}^2$. More explicitly, by Example \[s er 4\], the associated matrix of $G_2(L)$ of the line $L=[u]$, for $u \in S^1$, is $$\{g_{ij}([u])\}_{ij}=
\frac{3}{8}
\begin{pmatrix}
u_1^2 - \frac{1}{3} & u_1 u_2 \\
u_1u_2 & u_2^2-\frac{1}{3}
\end{pmatrix}.$$ Now let $${\hat{\varphi}_{ij}(K \cap E)}:= 2R \, g_{ij}({\pi(E)})V_0(K \cap E).$$ Then $$\label{estnon}
\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f({\pi(E)})}$$ is an unbiased estimator of $ \Phi_{1,0,2}(K)(e_i,e_j) $, when $ E $ is an $ f $-weighted random line in $ {{\mathbb R}}^2 $ hitting $ A $.
For a given $K \in {{\mathcal K}}^2$ the weight function $f$ minimizing the variance of the estimators of the form can be determined.
\[lemmaopt\] For a fixed $K \in {{\mathcal K}}^2$ with $\dim K \geq 1$ and $i,j \in \{1,2\}$, the estimator has minimal variance if and only if $f=f_K^*$ holds $\nu^2_1-a.s.$, where $$f_K^*(L)\propto \sqrt{2RV_1(K\vert L^{\perp})}\, \vert g_{ij}(L) \vert$$ is a density with respect to $\nu^2_1$ that depends on $i,j$ and $K$.
As $K$ is compact, $f^*_K$ is a well-defined probability density, and since $\dim K \geq 1$, the density $f^*_K$ is non-vanishing $\nu^2_1$-almost surely. The second moment of the estimator is $$\label{2momentberegning}
{\mathbb{E}}_f \bigg(\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f({\pi(E)})}\bigg)^2
= 2R \int_{{\mathcal{L}}^2_1}V_1(K \vert L^\perp) \frac{g_{ij}(L)^2}{f(L)}\, \nu_1^2(dL),$$ where ${\mathbb{E}}_f$ denotes expectation with respect to the distribution of an $f$-weighted random line in ${{\mathbb R}}^2$ hitting $A$. The right-hand side of is the second moment of the random variable $$\frac{\sqrt{2RV_1(K \vert {L^{\perp}})}\,g_{ij}(L)}{f(L)},$$ where the distribution of the random line $L$ has density $f$ with respect to $\nu^2_1$. By [@Asmussen Chapter 5, Theorem 1.2] the second moment of this variable is minimized, when $f$ is proportional to $\sqrt{2RV_1(K \vert {L^{\perp}})}\, |g_{ij}(L)|$. Since the proof of [@Asmussen Chapter 5, Theorem 1.2] follows simply by an application of Jensen’s inequality to the function $t \mapsto t^2$, equality can be characterized due to the strict convexity of this function, (see, e.g., [@Gardner (B.4)]). Equality holds if and only if $\sqrt{2RV_1(K \vert {L^{\perp}})}\,|g_{ij}(L)|$ is a constant multiple of $f(L)$ (or equivalently $f = f_K^*$) almost surely.
The proof of Lemma \[lemmaopt\] generalizes directly to arbitrary dimension $n$. As a consequence of Lemma \[lemmaopt\], we obtain that for any convex body $K \in {{\mathcal K}}^2$, optimal non-isotropic sampling provides a strictly smaller variance of the estimator than isotropic sampling. Indeed, noting that with a constant function $f$ reduces to the usual estimator (with $n=2$, $A=RB^2$) based on *IUR* lines, this follows from the fact that $f^*_K$ cannot be constant. If $f^*_K$ was constant almost surely, then $V_1(K \vert u^\perp) \propto |g_{ij}([u])|^{-2} $ for almost all $u \in S^1$. The left-hand side is essentially bounded, whereas the right-hand side is not. This is a contradiction.
A further consequence of Lemma \[lemmaopt\] is that there does not exist an estimator of the form independent of $K$ that has uniformly minimal variance for all $K \in {{\mathcal K}}^2$ with $\dim K \geq 1$. Unfortunately, $ f_K^* $ is not accessible, as it depends on $ K $, which is typically unknown. Even though estimators of the form cannot have uniformly minimal variance for all $K \in {{\mathcal K}}^2$ with $\dim K \geq 1$, we now show that there is a non-isotropic sampling design which always yields smaller variance than the isotropic sampling design. Let $$f^*(L) \propto |g_{ij}(L)|$$ be a density with respect to $\nu^2_1$. As $ |g_{ij}(L)| $ is bounded and non-vanishing for $ \nu^2_1$-almost all $ L $, $ f^* $ is well-defined and non-zero $ \nu^2_1$-almost everywhere. For convex bodies of constant width, the density $f^*$ coincides with the optimal density $f^*_K$.
\[onecomponent\] Let $ K \in {{\mathcal K}}^2 $, and let $ A = RB^2 $ for some $ R>0 $ be such that $ K \subseteq A $. Then $$\label{varineq}
\text{Var}_{f^*}\bigg(\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f^*({\pi(E)})}\bigg)
<
\text{Var}_{IUR}\big({\hat{\varphi}_{ij}(K \cap E)}\big).$$
Using the fact that both estimators are unbiased, it is sufficient to show that there is a $0 < \lambda < 1$ with $$\label{Eineq}
{\mathbb{E}}_{f^*}\bigg(\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f^*({\pi(E)})}\bigg)^2 \leq
\lambda \, {\mathbb{E}}_{IUR}\big({\hat{\varphi}_{ij}(K \cap E)}\big)^2,$$ for all $K \in {{\mathcal K}}^2$. Using , the left-hand side of this inequality is $$2R \int_{{\mathcal{L}}^2_1}|g_{ij}(L)| \, \nu_1^2 (dL) \int_{{\mathcal{L}}^2_1} |g_{ij}(L)|V_{1}(K \vert L^\perp) \, \nu_1^2 (dL)$$ and the right-hand side is $$2R \int_{{\mathcal{L}}^2_1} g_{ij}(L)^2 \, V_{1}(K \vert L^\perp) \, \nu_1^2 (dL).$$
Since $u \mapsto V_1(K \vert u^{\perp})$ is the support function of an origin-symmetric zonoid, the inequality holds if $$\begin{aligned}
\label{support ineq}
& \int_0^{2 \pi}|g_{ij}([u_{\phi}])| \, \frac{d\phi}{2\pi} \, \int_0^{2\pi} |g_{ij}([u_{\phi}])|\, h(Z,u_{\phi}) \, \frac{d\phi}{2\pi} \nonumber
\\
& \quad \leq \lambda \, \int_0^{2 \pi} g_{ij}([u_{\phi}])^2 h(Z,u_{\phi}) \, \frac{d\phi}{2\pi}\end{aligned}$$ for any origin-symmetric zonoid $ Z $. Here $ u_{\phi} = (\cos(\phi),\sin(\phi))^\top $ for $ \phi \in [0,2\pi] $. As support functions of zonoids can be uniformly approximated by support functions of zonotopes (see, e.g., [@Schneider93 Theorem 1.8.14]) and the integrals in depend linearly on these support functions, it is sufficient to show for all origin-symmetric line segments $Z$ of length two. Hence, we may assume that $Z$ is an origin-symmetric line segment with endpoints $\pm(\cos(\gamma), \sin(\gamma))^\top$, where $\gamma \in [0,\pi)$. We now substitute the support function $$h(Z,u_{\phi})=|\cos(\phi-\gamma)|$$ for $\phi \in [0,2\pi),$ into .
First, we consider the estimation of the first diagonal element of $\Phi_{1,0,2}(K)$, that is, $i,j=1$ and $g_{ij}([u_\phi])= \frac{3}{8}(\cos^2(\phi)-\frac{1}{3}) $ for $ \phi \in [0,2\pi] $. The integrals in then become $$P_{f^*}(\gamma):= \frac{3}{8}\int_0^{2\pi}|\cos^2(\phi)-\frac{1}{3}| \, \frac{d\phi}{2 \pi} \, \frac{3}{8} \int_{0}^{2\pi}|\cos^2(\phi)-\frac{1}{3}| |\cos(\phi-\gamma)| \, \frac{d\phi}{2\pi}$$ and $$P_{IUR}(\gamma):=\frac{9}{64}\int_0^{2\pi}\bigg(\cos^2(\phi)-\frac{1}{3}\bigg)^2 |\cos(\phi- \gamma)| \, \frac{d\phi}{2 \pi}.$$ Let $ \kappa = \arccos(\frac{1}{\sqrt{3}}) $. Then $$M:= \frac{3}{8}\int_0^{2\pi}|\cos^2(\phi)-\frac{1}{3}| \, \frac{d\phi}{2 \pi}
=\frac{\sqrt{2}+\kappa}{4\pi}-\frac{1}{16},$$ and elementary, but tedious calculations show that $$\begin{aligned}
P_{f^*}(\gamma)&=\frac{M}{ \pi} \bigg(\frac{2\sqrt{2}}{3\sqrt{3}}\cos(\gamma) - \frac{1}{4}\cos^2(\gamma) \bigg){\textbf{1}}_{[0,\frac{\pi}{2}-\kappa]}(\gamma)
\\
&\qquad +
\frac{M}{\pi}\bigg(\frac{1}{4}\cos^2(\gamma) + \frac{1}{3\sqrt{3}}\sin(\gamma) \bigg){\textbf{1}}_{(\frac{\pi}{2}-\kappa,\frac{\pi}{2}]}(\gamma)\end{aligned}$$ for $ \gamma \in [0,\frac{\pi}{2}] $. Further, $P_{f^*}(\gamma)=P_{f^*}(\pi-\gamma) $ for $\gamma \in [\frac{\pi}{2},\pi] $. For the *IUR* estimator we get that $$P_{IUR}(\gamma)=\frac{1}{20\pi}\bigg(-\frac{3}{8}\cos^4(\gamma) + \cos^2(\gamma) + \frac{1}{2} \bigg)$$ for $ \gamma \in [0,\frac{\pi}{2}] $, and $ P_{IUR}(\gamma)=P_{IUR}(\pi -\gamma) $ for $ \gamma \in [\frac{\pi}{2},\pi] $. The functions $ P_{f^*} $ and $ P_{IUR} $ are plotted in Figure \[Secondmoments\]. Basic calculus for the comparison of these two functions shows that $P_{f^*} < P_{IUR}$. This implies that $P_{f^*} \leq \lambda P_{IUR}$, where $\nobreak{\lambda=\max_{\gamma \in [0,\pi]} \frac{P_{f^*(\gamma)}}{P_{IUR}(\gamma)}}$ is smaller than one as $P_{f^*}$ and $P_{IUR}$ are continuous on the compact interval $[0,\pi]$. Hereby is satisfied for $i=j=1$. Interchanging the roles of the coordinate axes in yields the same result for $i=j=2$.
We now consider estimation of the off-diagonal element, that is, $i=1$, $j=2$. Then the left-hand and the right-hand side of become $$\label{Qfs}
Q_{f^*}(\gamma)=\frac{3}{8} \int_0^{2\pi}|\cos(\phi)\sin(\phi)| \, \frac{d\phi}{2 \pi}
\,\frac{3}{8} \int_{0}^{2\pi} |\cos(\phi)\sin(\phi)| |\cos(\phi-\gamma)| \, \frac{d\phi}{2 \pi}$$ and $$\label{M}
Q_{IUR}(\gamma)=\frac{9}{64}\int_0^{2\pi}\cos^2(\phi)\sin^2(\phi) |\cos(\phi- \gamma)| \,\frac{d\phi}{2 \pi}$$ for $ \gamma \in [0,\pi] $. We have $$\frac{3}{8} \int_0^{2\pi}|\cos(\phi)\sin(\phi)| \, \frac{d\phi}{2 \pi} = \frac{3}{8\pi},$$ and then $$Q_{f^*}(\gamma) = \frac{3}{32\pi^2}\bigg(\sin(\gamma)+\cos(\gamma)-\sin(\gamma)\cos(\gamma) \bigg)$$ for $ \gamma \in [0,\frac{\pi}{2}] $, and $ Q_{f^*}(\gamma)=Q_{f^*}(\gamma-\frac{\pi}{2}) $ for $ \gamma \in [\frac{\pi}{2},\pi] $. For $ \gamma \in [0,\pi]$ we further find that $$Q_{IUR}(\gamma) = \frac{3}{320 \pi}\bigg( 4-\frac{1}{2}\sin^2(2\gamma) \bigg).
$$ The functions $ Q_{IUR} $ and $ Q_{f^*} $ are plotted in Figure \[Secondmoments2\]. Basic calculus shows that $$\label{minmaxf}
\min_{0 \leq \gamma \leq \pi} Q_{f^*} = \frac{3}{32 \pi^2}\left(\sqrt{2}-\frac{1}{2}\right),
\qquad
\max_{0 \leq \gamma \leq \pi}Q_{f^*}=\frac{3}{32 \pi^2},$$ and $$\label{maxminIUR}
\min_{0 \leq \gamma \leq \pi} Q_{IUR} = \frac{21}{640 \pi},
\qquad
\max_{0 \leq \gamma \leq \pi} Q_{IUR} = \frac{3}{80 \pi}.$$ Hence $$Q_{f^*}(\gamma) \leq \frac{3}{32 \pi^2} \le \lambda \frac{21}{640 \pi} \leq \lambda \, Q_{IUR}(\gamma)$$ for $\gamma \in [0,\pi]$ with $\lambda=\frac{3}{\pi}< 1$. Hereby holds for all zonotopes $Z$ and $i=1,j=2$, and the claim is shown.
![The straight line is $ P_{IUR} $, the dashed line is $ P_{f^*} $, and the dash-dotted line is $ P_{opt} $.[]{data-label="Secondmoments"}](P.pdf)
![The straight line is $ Q_{IUR} $, the dashed line is $ Q_{f^*} $, and the dash-dotted line is $ Q_{opt} $.[]{data-label="Secondmoments2"}](Q.pdf)
If $ E $ is an $ f^* $-weighted random line suited for estimating one particular component of $ \Phi_{1,0,2}(K) $, then $ E $ should not be used to estimate any of the other components, as this would increase the variance of these estimators considerably. Hence, if we estimate all of the components of the tensor using the estimator based on $ f^* $-weighted lines, we need three lines; one for each component. If we want to compare this approach with an estimation procedure based on *IUR* lines, requiring the same workload, we will use *three* *IUR* lines. Note however, that all three *IUR* lines can be used to estimate *all* three components of the tensor. This implies that we should actually compare the variance of the estimator based on *one* $ f^* $-weighted random line with the variance of an estimator based on *three* *IUR* lines. It turns out that the estimator based on *three* independent *IUR* lines has always smaller variance, than the estimator based on *one* $ f$-weighted line, no matter how the density $f$ is chosen.
Let $ K \in {{\mathcal K}}^2 $, and let $ A =RB^2 $ with some $ R > 0 $ be such that $ K \subseteq A $. Let $ f $ be a density with respect to $ \nu^2_1 $, which is non-zero $ \nu^2_1 $-almost everywhere. Let $ E_1,E_2$ and $ E_3 $ be independent *IUR* lines in $ {{\mathbb R}}^2 $ hitting $ A $. Then $$Var\bigg(\frac{1}{3}\sum_{k=1}^3 \hat{\varphi}_{ij}(K \cap E_k) \bigg)
<
Var_{f}\bigg(\frac{{\hat{\varphi}_{ij}(K \cap E)}}{f(\pi(E))}\bigg)$$ for $i,j \in \{1,2\}$.
By Theorem \[onecomponent\], the variance of the estimator is bounded from below by the variance of the same estimator with $f=f_K^*$. Hence, it is sufficient to compare the second moments of $$\frac{1}{3}\sum_{k=1}^3 \hat{\varphi}_{ij}(K \cap E_k)$$ and with $f=f_K^*$. The latter is $$2R \bigg(\int_{{\mathcal{L}}^2_1} |g_{ij}(L)| \sqrt{V_1(K \vert L^\perp)} \, \nu^2_1 (dL) \bigg)^2,$$ so let $$P_{opt}(\gamma):= \bigg(\frac{3}{8} \int_0^{2\pi} |\cos^2(\phi)-\frac{1}{3}| \sqrt{|\cos(\phi - \gamma)|} \, \frac{d\phi}{2 \pi} \bigg)^2$$ and $$Q_{opt}(\gamma):= \bigg( \frac{3}{8} \int_0^{2\pi}|\cos(\phi)\sin(\phi)| \sqrt{|\cos(\phi - \gamma)|} \, \frac{d\phi}{2 \pi} \bigg)^2$$ for $ \gamma \in [0,\pi] $. Using the notation of the previous proofs, by , , and we have $$Q_{opt}(\gamma) \geq \bigg(\frac{8 \pi Q_{f^*}(\gamma)}{3}\bigg)^2 \geq \frac{9-4\sqrt{2}}{64 \pi^2} > \frac{1}{80\pi} \geq \frac{1}{3}Q_{IUR}(\gamma)$$ for $\gamma \in [0,\pi]$. Likewise, $ P_{opt}(\gamma) \geq \bigg(\frac{P_{f^*}(\gamma)}{M}\bigg)^2$. Elementary analysis shows that $$\min_{0\leq \gamma \leq \frac{\pi}{2}-\kappa}\bigg(\frac{P_{f^*}(\gamma)}{M}\bigg)^2=\frac{25}{324 \pi^2}
> \frac{3}{160\pi} = \max_{0\leq \gamma \leq \frac{\pi}{2} - \kappa} \frac{1}{3} P_{IUR}(\gamma),$$ and that $$\bigg(\frac{P_{f^*}(\gamma)}{M}\bigg)^2-\frac{1}{3}P_{IUR}(\gamma)
\geq
\bigg(\frac{P_{f^*}(\frac{\pi}{2})}{M}\bigg)^2-\frac{1}{3}P_{IUR}(\frac{\pi}{2}) > 0$$ on $[\frac{\pi}{2}-\kappa,\frac{\pi}{2}]$. Hence $P_{opt}>\frac{1}{3}P_{IUR}$ on $[0,\pi]$, and the assertion is proved.
This leads to the following conclusion: If one single component of the tensor $\T[2]$ is to be estimated for unknown $K$, the estimator with $f=f^*$ is recommended, as its variance is strictly smaller than the one from isotropic sampling (where $f$ is a constant). If all components are sought for, the estimator based on three *IUR* lines should be preferred.
Model based estimation {#SecModel}
======================
In this section we derive estimators of the specific surface tensors associated with a stationary process of convex particles based on linear sections. In [@RSRS06], Schneider and Schuster treat the similar problem of estimating the area moment tensor ($ s=2 $) associated with a stationary process of convex particles using planar sections.
Let $X$ be a stationary process of convex particles in ${{\mathbb R}}^n$ with locally finite (and non-zero) intensity measure, intensity $\gamma >0$ and grain distribution ${\mathbb{Q}}$ on ${{\mathcal K}}_0:=\{K \in {{\mathcal K}}^n \mid c(K)=0\}$; see, e.g., [@Weil] for further information on this basic model of stochastic geometry. Here $c \colon {{\mathcal K}}^n \setminus \{\emptyset\} \rightarrow {{\mathbb R}}^n$ is the center of the circumball of $K$. Since $ X $ is a stationary process of convex particles, the intrinsic volumes $V_0, \dots, V_n$ are $ {\mathbb{Q}}$-integrable by [@Weil Theorem 4.1.2]. For $j \in \{0, \dots, n-1\}$ and $s \in {{\mathbb N}}_0$ the tensor valuation $\Phi_{j,0,s}$ is measurable and translation invariant on ${{\mathcal K}}^n$, and since, by , $$|\jT[s](e_{i_1}, \dots, e_{i_s})| \leq \frac{\omega_{n-j}}{s!\omega_{n-j+s}}V_{j}(K),$$ it is coordinate-wise ${\mathbb{Q}}$-integrable. The *$j$th specific (translation invariant) tensor of rank s* can then be defined as $$\label{defmeansurface}
\joT[s]:=\gamma \int_{{{\mathcal K}}_0}\jT[s] \,{\mathbb{Q}}(dK)$$ for $j \in \{0, \dots, n-1\}$ and $s \in {{\mathbb N}}_0$. For $j=n-1$, the specific tensors are called the specific surface tensors. Notice that $\oT[2]=\frac{1}{8 \pi} \overline{T}(X)$, where $\overline{T}(X)$ is the mean area moment tensor described in [@RSRS06]. By [@Weil Theorem 4.1.3] the specific tensors of $X$ can be represented as $$\label{intuitive1}
\joT=\frac{1}{\lambda(B)} \; {\mathbb{E}}\, \sum_{\mathclap{\substack{K \in X \\ c(K)\in B}}} \, \jT[s],$$ where $B \in {\mathcal{B}}({{\mathbb R}}^n)$ with $0 < \lambda(B) < \infty$.
In the following we restrict to $j=n-1$ and discuss the estimation of $\oT[s]$ from linear sections of $X$. We assume from now on that $n\geq 2$. For $L \in {\mathcal{L}_1^n}$ we let $X \cap L :=\{K \cap L \mid K \in X, K \cap L \neq \emptyset\}$ be the stationary process of convex particles in $L$ induced by $X$. Let $\gamma_L$ and ${\mathbb{Q}}_L$ denote the intensity and the grain distribution of $ X \cap L $, respectively. The tensor valuation $\Phi_{0,0,s}^{(L)}$ is measurable and ${\mathbb{Q}}_L$-integrable on $ K_0^{(L)} :=\{K \in {{\mathcal K}}_0 \mid K \subseteq L\}$. We can thus define $$\oMT:=\gamma_L \int_{{{\mathcal K}}_0^{(L)}} \Phi_{0,0,s}^{(L)}(K) \, {\mathbb{Q}}_L (dK).$$ This deviates in the special case $\overline{T}^{(L)}(X \cap L) = 8 \pi \, \overline{\Phi}_{0,0,2}^{(L)}(X \cap L)$ from the definition in [@RSRS06] due to a misprint there. An application of yields, $$\label{simpel form, model}
\oMT= \frac{2}{s! \omega_{s+1}}Q(L)^{\frac{s}{2}} \gamma_L$$ for even $s$, and $\oMT=0$ for odd $s$.
\[modelthm\] Let $X$ be a stationary process of convex particles in ${{\mathbb R}}^n$ with positive intensity. If $s \in {{\mathbb N}}_0$ is even, then $$\label{model integral}
\int_{{\mathcal{L}_1^n}} \oMT \, \nu_1^n (dL)= \frac{2 \omega_{n+s+1}}{\pi s! \omega_{s+1}^2 \omega_n} \sum_{k=0}^{\frac{s}{2}}c_k^{(\frac{s}{2})}Q^{\frac{s}{2}-k}\,\oT[2k],$$ where the constants $c_k^{(\frac{s}{2})}$ for $k=0, \dots, \frac{s}{2}$ are given in Theorem \[crofton-like\].
Let $L \in {\mathcal{L}_1^n}$, and let $\gamma_L$ be the intensity of the stationary process $X \cap L$. If $B \subseteq L$ is a Borel set with $\lambda_L(B)=1$, then an application of Campbell’s theorem and Fubini’s theorem yields $$\begin{aligned}
\gamma_L&=
{\mathbb{E}}\; \sum_{\mathclap{\substack{K \in X \\ K \cap L \neq \emptyset}}} \; {\textbf{1}}(c(K \cap L) \in B)
\\
&=\gamma \int_{{{\mathcal K}}_0} \int_{{L^{\perp}}} V_0(K \cap (L + x)) \, \lambda_{{L^{\perp}}}(dx) \, {\mathbb{Q}}(dK),\end{aligned}$$ where $\gamma$ and ${\mathbb{Q}}$ are the intensity and the grain distribution of $X$. Then, implies that $$\oMT=\gamma \int_{{{\mathcal K}}_0} \int_{{L^{\perp}}} \Phi_{0,0,s}^{(L+z)}(K \cap (L+z)) \, \lambda_{{L^{\perp}}}(dz) \,{\mathbb{Q}}(dK),$$ and by Fubini’s theorem we get $$\begin{aligned}
\int_{{\mathcal{L}_1^n}} \oMT \, \nu^n_1(dL)
=\gamma \int_{{{\mathcal K}}_0} \int_{{\mathcal{E}^n}_1} \Phi^{(E)}_{0,0,s}(K \cap E) \,\mu_1^n(dE)\, {\mathbb{Q}}(dK). \label{Eq}\end{aligned}$$ Now Theorem \[crofton-like\] yields the stated integral formula .
A combination of equation and equation immediately gives the following Theorem \[model2\], which suggests an estimation procedure of the specific surface tensor $\oT$ of the stationary particle process $X$.
\[model2\] Let $X$ be a stationary process of convex particles in ${{\mathbb R}}^n$ with positive intensity. If $s \in {{\mathbb N}}_0$ is even, then $$\label{inversmodel}
\int_{{\mathcal{L}_1^n}} \sum_{j=0}^{\frac{s}{2}} d_{\frac{s}{2}\,j}C_{2j} Q^{\frac{s}{2}-j} \oMT[2j] \, \nu^n_1(dL)= \oT,$$ where $d_{\frac{s}{2}\,j}$ and $ C_{2j} $ for $ j=0, \dots, \frac{s}{2} $ are given before Theorem \[crofton2\].
Using , we can reformulate the integral formula in the form $$\int_{{\mathcal{L}_1^n}} G_s(L) \gamma_L \, \nu^n_1(dL)= \oT,$$ where $G_s$ is given in Theorem \[crofton2\].
In the case where $ s=2 $ formula becomes $$\int_{{\mathcal{L}}_1^n} \frac{2 \pi^2\omega_n}{\omega_{n+3}}\oMT[2]-\frac{\omega_n}{4 \omega_{n+1}}Q \oMT[0] \, \nu_1^n(dL)=\oT[2].$$ Up to a normalizing factor $ 2 \pi $ in the constant in front of $ \overline{\Phi}_{0,0,2}^{(L)} $, this formula coincides with formula (7) in [@RSRS06], when $ n=2 $. Apparently the normalizing factor got lost, when Schneider and Schuster used [@TVCB (36)], which is based on the spherical Lebesgue measure. In [@RSRS06], Schneider and Schuster use the *normalized* spherical Lebesgue measure.
**Acknowledgements** The authors acknowledge support by the German research foundation (DFG) through the research group “Geometry and Physics of Spatial Random Systems” under grants HU1874/2-1, HU1874/2-2 and by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from The Villum Foundation.
[^1]: E-mail: [email protected]
| ArXiv |
---
abstract: 'We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding $I\colon \mathbb{R}^n\to{\mathop{\mathrm{Ham}}\nolimits}(M)$ with respect to the fragmentation norm on the group ${\mathop{\mathrm{Ham}}\nolimits}(M)$ of Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega)$. As an application, we provide an answer to Brandenbursky’s question on fragmentation norms on ${\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g)$, where $\Sigma_g$ is a closed Riemannian surface of genus $g\geq 2$.'
address:
- 'Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan'
- 'Department of Mathematical Sciences, Tokyo Metropolitan University, Tokyo 192-0397, Japan'
author:
- Morimichi Kawasaki
- Ryuma Orita
title: Disjoint superheavy subsets and fragmentation norms
---
Introduction {#section:intro}
============
Background and definition
-------------------------
Let $(M,\omega)$ be a symplectic manifold. Let ${\mathop{\mathrm{Ham}}\nolimits}(M)$ denote the group of compactly supported Hamiltonian diffeomorphisms of $M$. In his well-known work [@Ba], Banyaga proved the simplicity of ${\mathop{\mathrm{Ham}}\nolimits}(M)$ when $M$ is a closed symplectic manifold. The key ingredient was the proof of the fragmentation lemma for this group, which, in turn, allows us to define fragmentation norms on ${\mathop{\mathrm{Ham}}\nolimits}(M)$ as follows. Let $\mathcal{U}=\{U_{\lambda}\}_{\lambda}$ be an open covering of $M$. The fragmentation lemma implies that for every $\phi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ there exists a positive integer $n$ such that $\phi$ can be represented as a product of $n$ diffeomorphisms $\theta_i\in{\mathop{\mathrm{Ham}}\nolimits}(U_{\lambda_i})$, where $\lambda_i\in\lambda$ and $1\leq i\leq n$. For $\phi\neq\mathrm{id}_M$, its *fragmentation norm* $\|\phi\|_{\mathcal{U}}$ with respect to the covering $\mathcal{U}$ is defined to be the minimal number of factors in such a product. We set $\|\phi\|_{\mathcal{U}}=0$ when $\phi=\mathrm{id}_M$. Accordingly, the fragmentation norm with respect to an open subset $U$ of $M$ is defined as follows. We consider an open covering $\mathcal{U}_U$ consisting of all open subsets $V$ such that $\psi(V)\subset U$ for some $\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$. The fragmentation norm $\|\phi\|_U$ of $\phi$ is defined to be $\|\phi\|_{\mathcal{U}_U}$.
Entov and Polterovich [@EP03] provided a lower bound for the quantitative fragmentation norm [@EP03], using primarily the Oh–Schwarz spectral invariant constructed using Hamiltonian Floer homology. Subsequently, Burago, Ivanov, and Polterovich [@BIP] provided a lower bound for the fragmentation norm itself, also using the Oh–Schwarz spectral invariant, but their argument had a different basis; see also [@E Section 4.4]. In addition, Lanzat [@L] and Monzner, Vichery, and Zapolsky [@MVZ] provided lower bounds for the fragmentation norms in the case in which $M$ is an open symplectic manifold, basing their strategies on arguments from [@EP03]. In addition, Brandenbursky and Brandenbursky–Kȩdra [@Br; @BK] provided a lower bound for the fragmentation norm using a Polterovich quasi-morphism whose construction does not involve Floer theory.
Recently, fragmentation norms have been receiving considerable attention, because they are known to be related to the open problem of the simplicity of the group of compactly supported measure-preserving homeomorphisms of an open disk in the Euclidean plane [@LR; @EPP].
In the present paper, we provide a lower bound for the fragmentation norm and construct a bi-Lipschitz embedding $I\colon \mathbb{R}^n\to{\mathop{\mathrm{Ham}}\nolimits}(M)$ with respect to the fragmentation norm on ${\mathop{\mathrm{Ham}}\nolimits}(M)$. Our strategy of the proof is based on the work of Entov, Polterovich, and Py [@EPP]. As an application, we provide an answer to Brandenbursky’s question [@Br Remark 1.5]. The solution involves both Hamiltonian and Lagrangian Floer theory.
Principal results
-----------------
Let $(M,\omega)$ be a symplectic manifold. Let $\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$ denote the universal cover of ${\mathop{\mathrm{Ham}}\nolimits}(M)$. Here we define subadditive invariants on $\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$ as a generalization of the Oh–Schwarz spectral invariant and the Lagrangian spectral invariant.
A function $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ is called a *subadditive invariant* if it satisfies the subadditivity condition, i.e., $c(\tilde{\phi}\tilde{\psi})\leq c(\tilde{\phi})+c(\tilde{\psi})$ for any $\tilde{\phi},\tilde{\psi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$.
Polterovich and Rosen introduced a function similar to our subadditive invariant [@PR Section 3.4]. However, in addition to subadditivity, they assumed conjugation invariance. In this paper, we do not make that assumption, because in Section \[section:lagspecinv\], we deal with Lagrangian spectral invariants, which are not conjugation invariant.
Let $N$ be a positive integer. The *oscillation norm* ${\mathop{\mathrm{osc}}\nolimits}$ on $\mathbb{R}^N$ is defined to be ${\mathop{\mathrm{osc}}\nolimits}(r_1,\ldots,r_N)=\max_{i,j}\lvert r_i-r_j\rvert$ for $(r_1,\ldots,r_N)\in\mathbb{R}^N$. We refer to Section \[section:preliminaries\] for the definitions of the notions appearing in the following principal theorems.
\[theorem:main\] Let $c_0,c_1,\ldots,c_N\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be subadditive invariants descending asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$. Let $U$ be an open subset of $M$ satisfying the normally bounded spectrum condition with respect to $c_i$ for all $i=0,1,\ldots,N$. Let $X_0,X_1,\ldots,X_N$ be mutually disjoint closed subsets of $M$ such that each $X_i$ is $c_i$-superheavy. Then there exists a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{R}^N,{\mathop{\mathrm{osc}}\nolimits})\to({\mathop{\mathrm{Ham}}\nolimits}(M),\|\cdot\|_U).$$
When $c_0,c_1,\ldots,c_N$ are quasi-morphisms, we obtain a stronger result than Theorem \[theorem:main\].
\[theorem:main2\] Let $c_0,c_1,\ldots,c_N\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be subadditive invariants descending asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$ that are quasi-morphisms. Let $U$ be an open subset of $M$ satisfying the asymptotically vanishing spectrum condition with respect to $c_i$ for all $i=0,1,\ldots,N$. Let $X_0,X_1,\ldots,X_N$ be mutually disjoint closed subsets of $M$ such that each $X_i$ is $c_i$-superheavy. Then, there exists a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{R}^N,{\mathop{\mathrm{osc}}\nolimits})\to({\mathop{\mathrm{Ham}}\nolimits}(M),\|\cdot\|_U).$$
Concerning the fragmentation norm $\|\cdot\|_{\mathcal{U}}$ with respect to an open covering $\mathcal{U}$, we have the following theorem.
\[theorem:main3\] Let $c_0,c_1,\ldots,c_N\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be subadditive invariants descending asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$. Let $\mathcal{U}=\{U_{\lambda}\}_{\lambda}$ be an open covering of $M$ such that each $U_{\lambda}$ satisfies the bounded spectrum condition with respect to $c_i$ for all $i=0,1,\ldots,N$. Let $X_0,X_1,\ldots,X_N$ be mutually disjoint closed subsets of $M$ such that each $X_i$ is $c_i$-superheavy. Then, there exists a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{R}^N,{\mathop{\mathrm{osc}}\nolimits})\to({\mathop{\mathrm{Ham}}\nolimits}(M),\|\cdot\|_{\mathcal{U}}).$$
We prove Theorems \[theorem:main\], \[theorem:main2\], and \[theorem:main3\] in Section \[section:pf of thm\].
Applications {#section:app}
============
In this section, we provide applications of our principal theorems.
Let $(M,\omega)$ be a symplectic manifold and $X$ a subset of $M$. An open subset $U\subset M$ is called *displaceable from $X$* if there exists a Hamiltonian $H\colon S^1\times M\to\mathbb{R}$ such that $\varphi_H(U)\cap\overline{X}=\emptyset$, where $\varphi_H$ is the Hamiltonian diffeomorphism generated by $H$ and $\overline{X}$ is the topological closure of $X$. $U\subset M$ is called *$($abstractly$)$ displaceable* if $U$ is displaceable from $U$ itself.
$B^{2n}$
--------
We consider the $2n$-dimensional ball $$B^{2n}=\left\{\,(p,q)\in\mathbb{R}^n\times\mathbb{R}^n{\mathrel{}\middle|\mathrel{}}\lvert p\rvert^2+\lvert q\rvert^2<1\,\right\}\subset\mathbb{R}^{2n}$$ equipped with the symplectic form $dp_1\wedge dq_1+\cdots+dp_n\wedge dq_n$, where $p=(p_1,\ldots,p_n)$ and $q=(q_1,\ldots,q_n)$. We have the following corollary of Theorem \[theorem:main2\].
\[b2n\] For any open ball $B(r)\subset\mathbb{R}^{2n}$ of radius $r<1$ centered at $0\in\mathbb{R}^{2n}$ and any positive integer $N$, there exists a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{R}^N,{\mathop{\mathrm{osc}}\nolimits})\to({\mathop{\mathrm{Ham}}\nolimits}(B^{2n}),\|\cdot\|_{B(r)}).$$
We prove Corollary \[b2n\] in Section \[section:pf of cor\].
Entov, Polterovich, and Py implicitly proved a similar statement when $r$ is sufficiently small [@EPP].
$S^2\times S^2$
---------------
We consider the product $S^2\times S^2$ with the symplectic form $\mathrm{pr}_1^{\ast}\omega_1+\mathrm{pr}_2^{\ast}\omega_1$, where $\omega_1$ is a symplectic form on $S^2$ with $\int_{S^2}\omega_1=1$ and $\mathrm{pr}_1, \mathrm{pr}_2\colon S^2\times S^2\to S^2$ are the first and second projections, respectively. Let $E$ denote the equator of $S^2$.
We have the following corollary of Theorem \[theorem:main2\]
\[s2s2\] Let $U$ be an open subset of $S^2\times S^2$ that is either abstractly displaceable or displaceable from $E\times E$. Then, for any positive integer $N$, there exists a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{R}^N,{\mathop{\mathrm{osc}}\nolimits})\to({\mathop{\mathrm{Ham}}\nolimits}(S^2\times S^2),\|\cdot\|_U).$$
We prove Corollary \[s2s2\] in Section \[section:pf of cor\].
$\mathbb{C}P^2$
---------------
Let $(\mathbb{C}P^2,\omega_{\mathrm{FS}})$ be two-dimensional complex projective space equipped with the Fubini–Study form. Then, the real projective space $\mathbb{R}P^2$ is naturally embedded in $(\mathbb{C}P^2,\omega_{\mathrm{FS}})$ as a Lagrangian submanifold. The *Clifford torus* $L_C$ is the Lagrangian submanifold $$L_C=\left\{\,[z_0:z_1:z_2]\in\mathbb{C}P^2{\mathrel{}\middle|\mathrel{}}\lvert z_0\rvert=\lvert z_1\rvert=\lvert z_2\rvert\,\right\}.$$ By [@BEP], $L_C$ is a *stem* in the sense of [@EP06 Definition 2.3]. There is another Lagrangian submanifold $L_W$ constructed by Wu [@W] that is disjoint from $\mathbb{R}P^2$. We call $L_W$ the *Chekanov torus*. Although there are some other Lagraingian submanifolds of $\mathbb{C}P^2$ called the Chekanov torus [@CS; @Gad; @BC], Oakley and Usher proved that they are all Hamiltonian isotopic [@OU].
We have the following corollary of Theorem \[theorem:main2\].
\[cp2\] Let $U$ be an open subset of $\mathbb{C}P^2$ satisfying one of the following conditions:
1. $U$ is abstractly displaceable.
2. $U$ is displaceable from $\mathbb{R}P^2$ and $L_W$.
3. $U$ is displaceable from $L_C$.
Then, there exists a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{R},\lvert\cdot\rvert)\to({\mathop{\mathrm{Ham}}\nolimits}(\mathbb{C}P^2),\|\cdot\|_U).$$
Here $\lvert\cdot\rvert$ is the absolute value.
Surfaces
--------
Let $(\Sigma_g,\omega)$ be a closed Riemannian surface $\Sigma_g$ of genus $g$ with an area form $\omega$. Brandenbursky studied fragmentation norms on ${\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g)$ under some assumptions.
\[Brandenbursky\] Let $g$ be a positive integer with $g\geq 2$ and $U$ be a contractible open subset of $\Sigma_g$. Then, there exists a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{Z}^{2g-2},\|\cdot\|_{\mathrm{word}})\to({\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g),\|\cdot\|_U).$$
Here $\|\cdot\|_{\mathrm{word}}$ is the word metric on $\mathbb{Z}^{2g-2}$. We point out that Burago, Ivanov, and Polterovich [@BIP] already proved the existence of a bi-Lipschitz injective homomorphism $I\colon(\mathbb{Z},\lvert\cdot\rvert)\to({\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g),\|\cdot\|_U)$, where $g$ is positive and $U$ is displaceable. Relating to Theorem \[Brandenbursky\], Brandenbursky asked whether one can construct a bi-Lipschitz injective homomorphism $\mathbb{Z}^N\to{\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g)$ for any $N$ and any $g\geq 2$ [@Br Remark 1.5]. As a corollary of Theorem \[theorem:main\], we solve his problem and generalize Theorem \[Brandenbursky\].
\[surface main theorem\] Let $g$ be a positive integer. Let $U$ be a contractible open subset of $\Sigma_g$ and $N$ a positive integer. Then, there exists a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{R}^N,{\mathop{\mathrm{osc}}\nolimits})\to({\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g),\|\cdot\|_U).$$
Since all norms on a finite-dimensional vector space are equivalent, the restriction of ${\mathop{\mathrm{osc}}\nolimits}$ to $\mathbb{Z}^{2g-2}\subset\mathbb{R}^{2g-2}$ is equivalent to the word metric on $\mathbb{Z}^{2g-2}$.
Moreover, as a corollary of Theorem \[theorem:main3\], we prove the following result.
\[annulus frag\] Let $g$ be a positive integer and $C$ be a non-contractible simple closed curve in $\Sigma_g$. Let $\mathcal{U}=\{U_\lambda\}_{\lambda}$ be an open covering such that each $U_\lambda$ is displaceable from $C$. Then, for any positive integer $N$, there exists a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{R}^N,{\mathop{\mathrm{osc}}\nolimits})\to({\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g),\|\cdot\|_{\mathcal{U}}).$$
We prove Corollaries \[surface main theorem\] and \[annulus frag\] in Section \[section:pf of cor\].
Let $\mathbb{N}$ denote the set of positive integers. For $\vec{g}=(g_1,\ldots,g_n)\in\mathbb{N}^n$, let $(\Sigma_{\vec{g}},\omega)$ denote the product manifold $\Sigma_{\vec{g}}=\Sigma_{g_1}\times\cdots\times\Sigma_{g_n}$ with a symplectic form $\omega$. Entov and Polterovich constructed a partial Calabi quasi-morphism (see Section \[section:calabi\] for the definition) on ${\mathop{\mathrm{Ham}}\nolimits}(\Sigma_{\vec{g}})$ for any $\vec{g}\in\mathbb{N}^n$ by using the Oh–Schwarz spectral invariant [@EP06]. They asked whether one can construct a Calabi quasi-morphism on ${\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g)$ for positive $g$. Py gave a positive answer to their question. Moreover, he constructed an infinite family of linearly independent Calabi quasi-morphisms on ${\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g)$ for positive $g$ [@Py6; @Py6-2]. Brandenbursky provided another construction of such an infinite family for $g\geq 2$ [@Br]. Brandenbursky, Kedra, and Shelukhin [@BKS] also provided a construction of Calabi quasi-morphisms in case $g=1$. In this paper, we prove the following theorem.
\[calabi on surface\] For any $\vec{g}\in\mathbb{N}^n$, the dimension of the space of partial Calabi quasi-morphisms on ${\mathop{\mathrm{Ham}}\nolimits}(\Sigma_{\vec{g}})$ is infinite.
We prove Theorem \[calabi on surface\] in Section \[section:calabi\].
Preliminaries {#section:preliminaries}
=============
In this section, we provide the defnitions appearing in Sections \[section:intro\] and \[section:app\], and review their properties. Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold.
Conventions and notation
------------------------
For a Hamiltonian $H\colon S^1\times M\to\mathbb{R}$ with compact support, we set $H_t=H(t,\cdot)$ for $t\in S^1$. The *mean value* of $H$ is defined to be $$\langle H\rangle={\mathop{\mathrm{Vol}}\nolimits}(M)^{-1}\int_0^1\int_M H_t\omega^n\,dt,$$ where ${\mathop{\mathrm{Vol}}\nolimits}(M)=\int_M\omega^n$ is the volume of $(M,\omega)$. A Hamiltonian $H$ is called *normalized* if $\langle H\rangle=0$. The *Hamiltonian vector field* $X_{H_t}$ associated with $H_t$ is a time-dependent vector field defined by the formula $$\omega(X_{H_t},\cdot)=-dH_t.$$ The *Hamiltonian isotopy* $\{\varphi_H^t\}_{t\in\mathbb{R}}$ associated with $H$ is defined by $$\begin{cases}
\varphi_H^0=\mathrm{id}_M,\\
\frac{d}{dt}\varphi_H^t=X_{H_t}\circ\varphi_H^t\quad \text{for all}\ t\in\mathbb{R},
\end{cases}$$ and its time-one map $\varphi_H=\varphi_H^1$ is referred to as the *Hamiltonian diffeomorphism $($with compact support$)$* generated by $H$. Let ${\mathop{\mathrm{Ham}}\nolimits}(M)$ and $\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$ denote the group of Hamiltonian diffeomorphisms of $M$ with compact support and its universal cover, respectively. An element of $\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$ is represented by a path in ${\mathop{\mathrm{Ham}}\nolimits}(M)$ starting from the identity. Hence, for every Hamiltonian $H\colon S^1\times M\to\mathbb{R}$ with compact support, its Hamiltonian isotopy $\{\varphi_H^t\}_{t\in\mathbb{R}}$ defines an element $\tilde{\varphi}_H\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$. Let $\mathbbm{1}$ denote the identity of $\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$, i.e., the homotopy class of the constant path $t\mapsto\mathrm{id}_M$ in ${\mathop{\mathrm{Ham}}\nolimits}(M)$.
For an open subset $U$ of $M$, let $\mathcal{H}(U)$ be the subset of $C^{\infty}(S^1\times M)$ consisting of all Hamiltonians supported in $S^1\times U$.
Subadditive invariants and superheaviness
-----------------------------------------
Let $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be a subadditive invariant. We define a map $\sigma_c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ as $$\sigma_c(\tilde{\phi})=\lim_{k\to\infty}\frac{c(\tilde{\phi}^k)}k.$$ The limit exists by subadditivity property.
\[definition:descend\] Let $\pi\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to{\mathop{\mathrm{Ham}}\nolimits}(M)$ denote the natural projection.
1. We say that a subadditive invariant $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ *descends to ${\mathop{\mathrm{Ham}}\nolimits}(M)$* if $c$ induces a map $\bar{c}\colon{\mathop{\mathrm{Ham}}\nolimits}(M)\to\mathbb{R}$ such that $c=\bar{c}\circ\pi$.
2. We say that a subadditive invariant $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ *descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$* if the map $\sigma_c$ induces a map $\bar{\sigma}_c\colon{\mathop{\mathrm{Ham}}\nolimits}(M)\to\mathbb{R}$ such that $\sigma_c=\bar{\sigma}_c\circ\pi$.
By definition, every subadditive invariant descending to ${\mathop{\mathrm{Ham}}\nolimits}(M)$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$.
Given two subadditive invariants, we can prove the following proposition.
\[comparing descending\] Let $c,c'\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be subadditive invariants. Assume that $c'$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$ and $c(\tilde{\varphi}_H)\leq c'(\tilde{\varphi}_H)$ holds for any Hamiltonian $H\colon S^1\times M\to\mathbb{R}$. Then, $c$ also descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$.
To prove Proposition \[comparing descending\], we first prove the following lemma.
\[equiv of descending\] Let $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be a subadditive invariant. Then, $c$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$ if and only if $\sigma_c|_{\pi_1({\mathop{\mathrm{Ham}}\nolimits}(M))}=0$.
The “only if” part follows immediately from the definition of descending asymptotically. Accordingly, we prove the “if” part and assume that $\sigma_c|_{\pi_1({\mathop{\mathrm{Ham}}\nolimits}(M))}=0$. Take $\tilde{\phi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$ and $\tilde{\psi}\in\pi_1\bigl({\mathop{\mathrm{Ham}}\nolimits}(M)\bigr)$. By subadditivity, for any positive integer $k$, $$\label{eq:beforedivide}
c(\tilde{\phi}^k)-c(\tilde{\psi}^{-k})\leq c(\tilde{\phi}^k\tilde{\psi}^k)\leq c(\tilde{\phi}^k)+c(\tilde{\psi}^k).$$ Since $\pi_1\bigl({\mathop{\mathrm{Ham}}\nolimits}(M)\bigr)$ is a connected topological group with respect to the $C^{\infty}$-topology, $\pi_1\bigl({\mathop{\mathrm{Ham}}\nolimits}(M)\bigr)$ lies in the center of $\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$. Here note that the fundamental group $\pi_1(G)$ of a connected topological group $G$ lies in the center of its universal cover $\widetilde{G}$ (see, for example, [@P Theorem 15]). Hence, $c(\tilde{\phi}^k\tilde{\psi}^k)=c\bigl((\tilde{\phi}\tilde{\psi})^k\bigr)$. Dividing by $k$ and passing to the limit as $k\to\infty$ yields $$\sigma_c(\tilde{\phi})-\sigma_c(\tilde{\psi}^{-1})\leq \sigma_c(\tilde{\phi}\tilde{\psi})\leq \sigma_c(\tilde{\phi})+\sigma_c(\tilde{\psi}).$$ Since $\tilde{\psi}\in\pi_1\bigl({\mathop{\mathrm{Ham}}\nolimits}(M)\bigr)$ and $\sigma_c|_{\pi_1({\mathop{\mathrm{Ham}}\nolimits}(M))}=0$, we conclude that $\sigma_c(\tilde{\phi}\tilde{\psi})=\sigma_c(\tilde{\phi})$. Since $\sigma_c(\tilde{\phi}\tilde{\psi})=\sigma_c(\tilde{\phi})$ for any $\tilde{\phi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$ and any $\tilde{\psi}\in\pi_1\bigl({\mathop{\mathrm{Ham}}\nolimits}(M)\bigr)$, $c$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$.
By Lemma \[equiv of descending\], it is sufficient to show that $\sigma_c|_{\pi_1({\mathop{\mathrm{Ham}}\nolimits}(M))}=0$. Let $H\colon S^1\times M\to\mathbb{R}$ be a Hamiltonian generating $\mathrm{id}_M\in{\mathop{\mathrm{Ham}}\nolimits}(M)$. Then, $\tilde{\varphi}_H\tilde{\varphi}_H^{-1}=\mathbbm{1}$. By subadditivity, $$c(\mathbbm{1})\leq c(\tilde{\varphi}_H^k)+c(\tilde{\varphi}_H^{-k}).$$ Dividing by $k$ and passing to the limit as $k\to\infty$ yields $$0\leq \sigma_c(\tilde{\varphi}_H)+\sigma_c(\tilde{\varphi}_H^{-1}).$$ Since $c'$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$, $$\sigma_c(\tilde{\varphi}_H)\leq\sigma_{c'}(\tilde{\varphi}_H)=\bar{\sigma}_{c'}(\mathrm{id}_M)=0.$$ Similarly, $$\sigma_c(\tilde{\varphi}_H^{-1})\leq\sigma_{c'}(\tilde{\varphi}_H^{-1})=\bar{\sigma}_{c'}(\mathrm{id}_M)=0.$$ Thus, $$\sigma_c(\tilde{\varphi}_H)=\sigma_c(\tilde{\varphi}_H^{-1})=0.\qedhere$$
A closed subset $X$ of $M$ is called *$c$-superheavy* if $$\inf_{S^1\times X}H\leq\sigma_c(\tilde{\varphi}_H)\leq\sup_{S^1\times X}H$$ for any normalized Hamiltonian $H\colon S^1\times M\to\mathbb{R}$.
By definition, we have the following result.
\[proposition:shv\] Let $X$ be a $c$-superheavy subset of $M$. Then, for any $\alpha\in\mathbb{R}$ and any normalized Hamiltonian $H\colon S^1\times M\to\mathbb{R}$ with $H|_{S^1\times X}\equiv \alpha$, $$\sigma_c(\tilde{\varphi}_H)=\alpha.$$
Spectrum conditions
-------------------
We define three kinds of spectrum conditions. We recall that the mean value $\langle H\rangle$ of a Hamiltonian $H\colon S^1\times M\to\mathbb{R}$ is given by $$\langle H\rangle={\mathop{\mathrm{Vol}}\nolimits}(M)^{-1}\int_0^1\int_M H_t\omega^n\,dt.$$
### Normally bounded spectrum condition
Let $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be a subadditive invariant.
\[definition:NBSC\] An open subset $U$ of $M$ satisfies the *normally bounded spectrum condition with respect to $c$* if there exists a positive number $K>0$ such that for any $F\in\mathcal{H}(U)$ and any $\tilde{\psi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$, $$\label{eq:NBSC}
c(\tilde{\psi}^{-1}\tilde{\varphi}_F\tilde{\psi})+\langle F\rangle\leq K.$$
If we put $c(H)=c(\tilde{\varphi}_H)+\langle H\rangle$ for a Hamiltonian $H\colon S^1\times M\to\mathbb{R}$, then the inequality can be written as $c(F\circ\psi)\leq K$. However, in this paper, we avoid this notation for simplicity.
Definition \[definition:NBSC\] is equivalent to the existence of a positive number $K>0$ such that for any $\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ and any $F\in\mathcal{H}\bigl(\psi(U)\bigr)$, $$\label{eq:NBSC2}
c(\tilde{\varphi}_F)+\langle F\rangle\leq K$$ since the Hamiltonian diffeomorphism generated by $F\circ\psi$ is $\psi^{-1}\varphi_F\psi$.
\[MVZ\] When $c$ is an Oh–Schwarz spectral invariant, the normally bounded spectrum condition is equivalent to the bounded spectrum condition (see Definition \[definition:BSC\]) since Oh–Schwarz spectral invariants are conjugation invariant. The normally bounded spectrum condition was introduced by Monzner, Vichery, and Zapolsky [@MVZ].
\[proposition:VPNBSC\] Let $U$ be an open subset of $M$ satisfying the normally bounded spectrum condition with respect to $c$. For any $\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ and any $F\in\mathcal{H}\bigl(\psi(U)\bigr)$, $$\sigma_c(\tilde{\varphi}_F)=-\langle F\rangle.$$
Let $\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ and $F\in\mathcal{H}\bigl(\psi(U)\bigr)$. Note that the Hamiltonian $-F\circ\varphi_F$ generates $\tilde{\varphi}_F^{-1}$ and satisfies $\langle -F\circ\varphi_F\rangle=-\langle F\rangle$. Since $U$ satisfies the normally bounded spectrum condition with respect to $c$, we can choose a positive number $K>0$ such that for any $k\in\mathbb{Z}$, $$c(\tilde{\varphi}_F^k)+\langle kF\rangle\leq K \quad\text{and}\quad c(\tilde{\varphi}_F^{-k})+\langle -kF\rangle\leq K.$$ By subadditivity, $$c(\mathbbm{1})\leq c(\tilde{\varphi}_F^k)+c(\tilde{\varphi}_F^{-k}).$$ Therefore, $$-K+c(\mathbbm{1})\leq -c(\tilde{\varphi}_F^{-k})-\langle -kF\rangle+c(\mathbbm{1})\leq c(\tilde{\varphi}_F^k)+\langle kF\rangle\leq K.$$ Dividing by $k$ and passing to the limit as $k\to\infty$ yields $$\sigma_c(\tilde{\varphi}_F)+\langle F\rangle=\lim_{k\to\infty}\frac{c(\tilde{\varphi}_F^k)+k\langle F\rangle}k=0.\qedhere$$
The following proposition is useful in the proof of Theorem \[theorem:main\].
\[proposition:defect1\] Let $U$ be an open subset of $M$ satisfying the normally bounded spectrum condition with respect to $c$. Then, there exists a positive number $K>0$ such that for any $\tilde{\phi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$, any $\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ and any $F\in\mathcal{H}\bigl(\psi(U)\bigr)$, $$\left\lvert\sigma_c(\tilde{\varphi}_F\tilde{\phi})-\sigma_c(\tilde{\varphi}_F)-\sigma_c(\tilde{\phi})\right\rvert\leq K.$$
To prove Proposition \[proposition:defect1\], we first prove the following lemma.
\[lemma:defect\] There exists a positive number $K>0$ such that for any $\tilde{\phi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$, any $\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ and any $F\in\mathcal{H}\bigl(\psi(U)\bigr)$, $$\left\lvert c(\tilde{\varphi}_F\tilde{\phi})-c(\tilde{\phi})+\langle F\rangle\right\rvert\leq K.$$
Let $\tilde{\phi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$, $\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ and $F\in\mathcal{H}\bigl(\psi(U)\bigr)$. Since $U$ satisfies the normally bounded spectrum condition with respect to $c$, we can choose a positive number $K>0$ such that $$c(\tilde{\varphi}_F)+\langle F\rangle\leq K\quad\text{and}\quad c(\tilde{\varphi}_F^{-1})-\langle F\rangle\leq K.$$ By subadditivity, $$c(\tilde{\phi})\leq c(\tilde{\varphi}_F^{-1})+c(\tilde{\varphi}_F\tilde{\phi})\quad\text{and}\quad c(\tilde{\varphi}_F\tilde{\phi})\leq c(\tilde{\varphi}_F)+c(\tilde{\phi}).$$ Therefore, $$-K\leq -c(\tilde{\varphi}_F^{-1})+\langle F\rangle\leq c(\tilde{\varphi}_F\tilde{\phi})-c(\tilde{\phi})+\langle F\rangle\leq c(\tilde{\varphi}_F)+\langle F\rangle\leq K.\qedhere$$
Let $\tilde{\phi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$, $\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ and $F\in\mathcal{H}\bigl(\psi(U)\bigr)$. For an integer $k$, decompose $(\tilde{\varphi}_F\tilde{\phi})^k$ into $$(\tilde{\varphi}_F\tilde{\phi})^k =\tilde{\varphi}_F(\tilde{\phi}\tilde{\varphi}_F\tilde{\phi}^{-1})\cdots(\tilde{\phi}^{k-1}\tilde{\varphi}_F\tilde{\phi}^{-k+1})\tilde{\phi}^k.$$ Since $\tilde{\phi}^i\tilde{\varphi}_F\tilde{\phi}^{-i}=\tilde{\varphi}_{F\circ\phi^{-i}}$ for all $i=0,1,\ldots,k-1$, Lemma \[lemma:defect\] implies that there exists a positive number $K>0$ such that $$\left\lvert c\bigl((\tilde{\varphi}_F\tilde{\phi})^k\bigr)-c\bigl((\tilde{\phi}\tilde{\varphi}_F\tilde{\phi}^{-1})\cdots(\tilde{\phi}^{k-1}\tilde{\varphi}_F\tilde{\phi}^{-k+1})\tilde{\phi}^k\bigr)+\langle F\rangle\right\rvert\leq K,$$ $$\left\lvert c\bigl((\tilde{\phi}\tilde{\varphi}_F\tilde{\phi}^{-1})\cdots(\tilde{\phi}^{k-1}\tilde{\varphi}_F\tilde{\phi}^{-k+1})\tilde{\phi}^k\bigr)-c\bigl((\tilde{\phi}^2\tilde{\varphi}_F\tilde{\phi}^{-2})\cdots(\tilde{\phi}^{k-1}\tilde{\varphi}_F\tilde{\phi}^{-k+1})\tilde{\phi}^k\bigr)+\langle F\circ\phi^{-1}\rangle\right\rvert\leq K,$$ $$\cdots$$ $$\left\lvert c\bigl((\tilde{\phi}^{k-1}\tilde{\varphi}_F\tilde{\phi}^{-k+1})\tilde{\phi}^k\bigr)-c\bigl(\tilde{\phi}^k\bigr)+\langle F\circ\phi^{-k+1}\rangle\right\rvert\leq K.$$ Therefore, since $\langle F\circ\phi^{-i}\rangle=\langle F\rangle$ for all $i=0,1,\ldots,k-1$, by the triangle inequality, $$\begin{aligned}
&\left\lvert c\bigl((\tilde{\varphi}_F\tilde{\phi})^k\bigr)-c(\tilde{\phi}^k)+k\langle F\rangle\right\rvert\\
&=\left\lvert c\bigl((\tilde{\varphi}_F\tilde{\phi})^k\bigr)-c(\tilde{\phi}^k)+\langle F\rangle+\langle F\circ\phi^{-1}\rangle+\cdots+\langle F\circ\phi^{-k+1}\rangle\right\rvert\\
&\leq\left\lvert c\bigl((\tilde{\varphi}_F\tilde{\phi})^k\bigr)-c\bigl((\tilde{\phi}\tilde{\varphi}_F\tilde{\phi}^{-1})\cdots(\tilde{\phi}^{k-1}\tilde{\varphi}_F\tilde{\phi}^{-k+1})\tilde{\phi}^k\bigr)+\langle F\rangle\right\rvert\\
&+\left\lvert c\bigl((\tilde{\phi}\tilde{\varphi}_F\tilde{\phi}^{-1})\cdots(\tilde{\phi}^{k-1}\tilde{\varphi}_F\tilde{\phi}^{-k+1})\tilde{\phi}^k\bigr)-c\bigl((\tilde{\phi}^2\tilde{\varphi}_F\tilde{\phi}^{-2})\cdots(\tilde{\phi}^{k-1}\tilde{\varphi}_F\tilde{\phi}^{-k+1})\tilde{\phi}^k\bigr)+\langle F\circ\phi^{-1}\rangle\right\rvert\\
&+\cdots\\
&+\left\lvert c\bigl((\tilde{\phi}^{k-1}\tilde{\varphi}_F\tilde{\phi}^{-k+1})\tilde{\phi}^k\bigr)-c\bigl(\tilde{\phi}^k\bigr)+\langle F\circ\phi^{-k+1}\rangle\right\rvert\\
&\leq kK.\end{aligned}$$ Dividing by $k$ and passing to the limit as $k\to\infty$ yields $$\left\lvert\sigma_c(\tilde{\varphi}_F\tilde{\phi})-\sigma_c(\tilde{\phi})+\langle F\rangle\right\rvert =\lim_{k\to\infty}\frac{\left\lvert c\bigl((\tilde{\varphi}_F\tilde{\phi})^k\bigr)-c(\tilde{\phi}^k)+k\langle F\rangle\right\rvert}k\leq K.$$ Then, Proposition \[proposition:VPNBSC\] completes the proof of Proposition \[proposition:defect1\].
### Bounded spectrum condition
Let $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be a subadditive invariant.
\[definition:BSC\] An open subset $U$ of $M$ satisfies the *bounded spectrum condition with respect to $c$* if there exists a positive number $K>0$ such that for any $F\in\mathcal{H}(U)$, $$\label{eq:BSC}
c(\tilde{\varphi}_F)+\langle F\rangle\leq K.$$
Note that the normally bounded spectrum condition implies the bounded spectrum condition.
### Asymptotically vanishing spectrum condition
Let $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be a subadditive invariant.
An open subset $U$ of $M$ satisfies the *asymptotically vanishing spectrum condition with respect to $c$* if for any $F\in\mathcal{H}(U)$, $$\sigma_c(\tilde{\varphi}_F)+\langle F\rangle=0.$$
\[remark:AVSC\] An argument similar to the proof of Proposition \[proposition:VPNBSC\] shows that the asymptotically vanishing spectrum condition is weaker than the bounded spectrum condition.
\[disp and vanish\] Every open subset of $M$ displaceable from a $c$-superheavy subset satisfies the asymptotically vanishing spectrum condition with respect to $c$.
To prove Proposition \[disp and vanish\], we first prove
\[sigma is ci\] For any $\tilde{\phi},\tilde{\psi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$, $\sigma_c(\tilde{\phi}^{-1}\tilde{\psi}\tilde{\phi})=\sigma_c(\tilde{\psi})$.
Let $k$ be an integer. By subadditivity, $$-c(\tilde{\phi}^{-1})-c(\tilde{\phi})\leq c(\tilde{\phi}^{-1}\tilde{\psi}^k\tilde{\phi})-c(\tilde{\psi}^k)\leq c(\tilde{\phi}^{-1})+c(\tilde{\phi}).$$ Since $(\tilde{\phi}^{-1}\tilde{\psi}\tilde{\phi})^k=\tilde{\phi}^{-1}\tilde{\psi}^k\tilde{\phi}$, dividing by $k$ and passing to the limit as $k\to\infty$ yields $$\sigma_c(\tilde{\phi}^{-1}\tilde{\psi}\tilde{\phi})=\sigma_c(\tilde{\psi}).\qedhere$$
Let $X$ be a $c$-superheavy subset of $M$. Let $U$ be an open subset displaceable from $X$. By assumption, we can take $\phi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ such that $\phi(U)\cap X=\emptyset$. Since $X$ is $c$-superheavy, for any $F\in\mathcal{H}\bigl(\phi(U)\bigr)$, $$0=\inf_{S^1\times X}F\leq\sigma_c(\tilde{\varphi}_F)+\langle F\rangle\leq\sup_{S^1\times X}F=0.$$ Hence, $\phi(U)$ satisfies the asymptotically vanishing spectrum condition with respect to $c$. Lemma \[sigma is ci\] implies that $U$ also satisfies the asymptotically vanishing spectrum condition with respect to $c$.
Delicate Banyaga fragmentation lemma
====================================
To prove the principal theorems, we use the following folklore lemma which is a slightly delicate version of Banyaga’s fragmentation lemma (see also [@Ka3 Lemma 2.1]).
\[lemma:uniform\] Let $(M,\omega)$ be a symplectic manifold, $K$ a compact subset of $M$ and $\mathcal{U}$ an open cover of $M$. Then, there exists a positive number $N_{K,\mathcal{U}}$ such that $\|\varphi_H\|_\mathcal{U}\leq N_{K,\mathcal{U}}$ for any $C^1$-small Hamiltonian $H\colon [0,1]\times M\to\mathbb{R}$ with ${\mathop{\mathrm{supp}}\nolimits}(H)\subset [0,1]\times K$.
Since $K$ is compact, we can take finite open coverings $\mathcal{V}=\{V_i\}_{i=1,\ldots,\ell}$ and $\mathcal{V}'=\{V'_i\}_{i=1,\ldots,\ell}$ of $K$ such that
- for any $i$, $\overline{V_i}\subset V'_i$,
- for any $i$ there exists $U_i\in\mathcal{U}$ such that $\overline{V'_i}\subset U_i$.
Take a partition of unity $\{\rho_i\colon K\to[0,1]\}_{i=1,\ldots,\ell}$ subordinated to $\mathcal{V}$ (i.e., ${\mathop{\mathrm{supp}}\nolimits}(\rho_i)\subset V_i$ for any $i$). We then define functions $\chi_j\colon K\to[0,1]$ ($j=0,1,\ldots,\ell$) as $$\chi_j=
\begin{cases}
0 & \text{if\: $j=0$},\\
\sum_{i=1}^j\rho_i & \text{if\: $j=1,\ldots,\ell$}.
\end{cases}$$ For a Hamiltonian $H\colon [0,1]\times M\to\mathbb{R}$ with ${\mathop{\mathrm{supp}}\nolimits}(H)\subset [0,1]\times K$, we define Hamiltonians $H^j$ ($j=0,1,\ldots,\ell$) and $L^j$ ($j=1,\ldots,\ell$) as $$H^j(t,x)=\chi_j(x)H(t,x)$$ and $$L^j(t,x)=-H^{j-1}\bigl(t,\varphi_{H^{j-1}}^t(x)\bigr)+H^j\bigl(t,\varphi_{H^{j-1}}^t(x)\bigr)$$ for $(t,x)\in [0,1]\times K$, respectively. Since ${\mathop{\mathrm{supp}}\nolimits}(H)\subset [0,1]\times K$, we can regard $H^j$ and $L^j$ as smooth functions on $[0,1]\times M$. Fix $j=1,\ldots,\ell$. Note that $L^j$ generates the Hamiltonian diffeomorphism $\varphi_{H^{j-1}}^{-1}\varphi_{H^j}$ and thus $\varphi_{H^j}=\varphi_{H^{j-1}}\varphi_{L^j}$. Since $H^\ell=H$ and $H^0=0$, $$\varphi_H=\varphi_{H^\ell}=\varphi_{H^{\ell-1}}\varphi_{L^{\ell}}=\cdots=\varphi_{H^0}\varphi_{L^1}\cdots\varphi_{L^\ell}=\varphi_{L^1}\cdots\varphi_{L^\ell}.$$
Now, we claim that ${\mathop{\mathrm{supp}}\nolimits}(L^{j})\subset [0,1]\times\overline{V'_j}$ if $H$ is $C^1$-small. Since $H^{j-1}$ is also $C^1$-small, $(\varphi_{H^{j-1}}^t)^{-1}(V_{j})\subset V'_j$. Suppose that $x\notin\overline{V'_j}$. Then, $(\varphi_{H^{j-1}}^t)(x)\notin\overline{V_j}$ and in particular, $(\varphi_{H^{j-1}}^t)(x)\notin{\mathop{\mathrm{supp}}\nolimits}(\rho_j)$. Since $\chi_j=\sum_{i=1}^{j}\rho_i$, $\chi_{j-1}\bigl(\varphi_{H^{j-1}}^t(x)\bigr)=\chi_{j}\bigl(\varphi_{H^{j-1}}^t(x)\bigr)$ for any $t$. Hence, $L^j(t,x)=0$ for any $x\notin \overline{V'_j}$ and any $t\in[0,1]$. This completes the proof of the claim.
By ${\mathop{\mathrm{supp}}\nolimits}(L^{j})\subset [0,1]\times\overline{V'_j}$ and the second condition on $\mathcal{V}'$, ${\mathop{\mathrm{supp}}\nolimits}(L^{j})\subset [0,1]\times U_j$. Therefore, since $\varphi_H=\varphi_{L^1}\cdots\varphi_{L^\ell}$, $$\|\varphi_H\|_\mathcal{U}\leq \ell.$$ Thus, we can take $\ell$ as $N_{K,\mathcal{U}}$ in Lemma \[lemma:uniform\].
Proof of the principal theorems {#section:pf of thm}
===============================
In this section, we prove Theorems \[theorem:main\], \[theorem:main2\], and \[theorem:main3\].
Proof of Theorem \[theorem:main\]
---------------------------------
For $i=1,\ldots,N$, we choose a normalized time-independent Hamiltonian $H_i\colon M\to\mathbb{R}$ such that $H_i|_{X_i}\equiv 1$ and $X_j\cap{\mathop{\mathrm{supp}}\nolimits}(H_i)=\emptyset$ whenever $j\in\{0,1,\ldots,N\}\setminus\{i\}$. We define an injective homomorphism $I\colon\mathbb{R}^N\to{\mathop{\mathrm{Ham}}\nolimits}(M)$ to be $$I(r_1,\ldots,r_N)=\varphi_{r_1H_1+\cdots+r_N H_N}.$$ Hence, it is enough to show that $I$ is bi-Lipschitz.
We fix $i=0,1,\ldots,N$ and set $r_0=0$. Since $X_i$ is $c_i$-superheavy and $(r_1H_1+\cdots+r_NH_N)|_{X_i}\equiv r_i$, Proposition \[proposition:shv\] implies that $$\label{eq:shv}
\sigma_{c_i}(\tilde{\varphi}_{r_1H_1+\cdots+r_N H_N})=r_i.$$
We set $\alpha=\|\varphi_{r_1H_1+\cdots+r_N H_N}\|_U$. Let us designate that $$\varphi_{r_1H_1+\cdots+r_N H_N}=\varphi_{F_1}\cdots\varphi_{F_{\alpha}},$$ where $F_{\ell}\in\mathcal{H}\bigl(\phi_{\ell}(U)\bigr)$ for some $\phi_{\ell}\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ for $\ell=1,\ldots,\alpha$. Since $U$ satisfies the normally bounded spectrum condition with respect to $c_i$, Proposition \[proposition:defect1\] implies that there exists a positive number $K_i>0$ such that $$\lvert\sigma_{c_i}(\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})-\sigma_{c_i}(\tilde{\varphi}_{F_1})-\sigma_{c_i}(\tilde{\varphi}_{F_2}\cdots\tilde{\varphi}_{F_{\alpha}})\rvert\leq K_i,$$ $$\lvert\sigma_{c_i}(\tilde{\varphi}_{F_2}\cdots\tilde{\varphi}_{F_{\alpha}})-\sigma_{c_i}(\tilde{\varphi}_{F_2})-\sigma_{c_i}(\tilde{\varphi}_{F_3}\cdots\tilde{\varphi}_{F_{\alpha}})\rvert\leq K_i,$$ $$\cdots$$ $$\lvert\sigma_{c_i}(\tilde{\varphi}_{F_{\alpha-1}}\tilde{\varphi}_{F_{\alpha}})-\sigma_{c_i}(\tilde{\varphi}_{F_{\alpha-1}})-\sigma_{c_i}(\tilde{\varphi}_{F_{\alpha}})\rvert\leq K_i.$$ Therefore, by Proposition \[proposition:VPNBSC\] and the triangle inequality, $$\label{eq:mdefect}
\left\lvert\sigma_{c_i}(\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}}) +\sum_{\ell=1}^{\alpha}\langle F_{\ell}\rangle\right\rvert\leq(\alpha-1)K_i<\alpha K_i.$$
Now, we define a map $\sigma'_i\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ to be $\sigma'_i=\sigma_{c_i}-\sigma_{c_0}$. Then, by and , we obtain $$\sigma'_i(\tilde{\varphi}_{r_1H_1+\cdots+r_N H_N})=r_i-r_0=r_i,$$ and $$\lvert\sigma'_i(\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})\rvert <\alpha (K_i+K_0).$$
Since the natural projection $\pi\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to{\mathop{\mathrm{Ham}}\nolimits}(M)$ is a group homomorphism, $$\pi(\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}}) =\varphi_{F_1}\cdots\varphi_{F_{\alpha}} =\varphi_{r_1H_1+\cdots+r_N H_N}.$$ By assumption, $c_i$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$. Hence, the map $\sigma'_i=\sigma_{c_i}-\sigma_{c_0}$ induces a map $\bar{\sigma}'_i\colon{\mathop{\mathrm{Ham}}\nolimits}(M)\to\mathbb{R}$ such that $\sigma'_i=\bar{\sigma}'_i\circ\pi$. Thus, $$\begin{aligned}
\lvert r_i\rvert &=\left\lvert\sigma'_i(\tilde{\varphi}_{r_1H_1+\cdots+r_N H_N})\right\rvert =\lvert\bar{\sigma}'_i(\varphi_{F_1}\cdots\varphi_{F_{\alpha}})\rvert\\
&=\lvert\sigma'_i(\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})\rvert <\alpha(K_i+K_0).\end{aligned}$$ Hence, $$\|\varphi_{r_1H_1+\cdots+r_N H_N}\|_U=\alpha >(K_i+K_0)^{-1}\lvert r_i\rvert.$$ Therefore, $$\|\varphi_{r_1H_1+\cdots+r_N H_N}\|_U >N^{-1}\left(\max_{1\leq i\leq N}K_i+K_0\right)^{-1}(\lvert r_1\rvert+\cdots+\lvert r_N\rvert).$$
On the other hand, since ${\mathop{\mathrm{supp}}\nolimits}(H_i)$ is compact for any $i$, by Lemma \[lemma:uniform\], there exist positive numbers $N_{i,U}$ and $\varepsilon$ such that for any $0\leq t\leq\varepsilon$, $$\|\varphi_{tH_i}\|_U=\|\varphi_{tH_i}\|_{\mathcal{U}_U}\leq N_{i,U}.$$ Set $N_U=\max_i{N_{i,U}}$. For each $i=1,\ldots,N$, choose a non-negative integer $a_i$ and a non-negative number $b_i$ with $b_i<\varepsilon$ such that $r_i=a_i\varepsilon+b_i$. Then, $$\begin{aligned}
\|\varphi_{r_1H_1+\cdots+r_N H_N}\|_U &\leq\|\varphi_{r_1H_1}\|_U+\cdots+\|\varphi_{r_NH_N}\|_U\\
&\leq\|\varphi_{\varepsilon H_1}^{a_1}\varphi_{b_1H_1}\|_U+\cdots+\|\varphi_{\varepsilon H_N}^{a_N}\varphi_{b_NH_N}\|_U\\
&\leq (a_1+\cdots+a_N+N)N_U.\end{aligned}$$ This completes the proof of Theorem \[theorem:main\].
Proof of Theorem \[theorem:main2\]
----------------------------------
Since the proof is almost same as that of Theorem \[theorem:main\], we provide only the necessary changes.
Let $H_1,\ldots,H_N$ and $I\colon\mathbb{R}^N\to{\mathop{\mathrm{Ham}}\nolimits}(M)$ be chosen as in the proof of Theorem \[theorem:main\]. We fix $i=0,1,\ldots,N$ and set $r_0=0$. We set $\alpha=\|\varphi_{r_1H_1+\cdots+r_N H_N}\|_U$ and $$\varphi_{r_1H_1+\cdots+r_N H_N}=(\phi_1^{-1}\varphi_{F_1}\phi_1)\cdots(\phi_{\alpha}^{-1}\varphi_{F_{\alpha}}\phi_{\alpha}),$$ where $F_{\ell}\in\mathcal{H}(U)$ and $\phi_{\ell}\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ for $\ell=1,\ldots,\alpha$. Since $U$ satisfies the asymptotically vanishing spectrum condition with respect to $c_i$, Lemma \[sigma is ci\] implies that $$\sigma_{c_i}(\tilde{\phi}_{\ell}^{-1}\tilde{\varphi}_{F_{\ell}}\tilde{\phi}_{\ell}) =\sigma_{c_i}(\tilde{\varphi}_{F_{\ell}}) =-\langle F_{\ell}\rangle$$ for all $\ell$. We define a map $\sigma'_i\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ to be $\sigma'_i=\sigma_{c_i}-\sigma_{c_0}$. Then, $$\label{eq:AVP}
\sigma'_i(\tilde{\phi}_{\ell}^{-1}\tilde{\varphi}_{F_{\ell}}\tilde{\phi}_{\ell})=-\langle F_{\ell}\rangle+\langle F_{\ell}\rangle=0$$ for all $\ell$.
On the other hand, since the homogenization of a quasi-morphism is also a quasi-morphism (see, for example, [@Ca]), there exists a positive number $K_i>0$ such that $$\left\lvert\sigma_{c_i}(\tilde{\phi}\tilde{\psi})-\sigma_{c_i}(\tilde{\phi})-\sigma_{c_i}(\tilde{\psi})\right\rvert<K_i$$ for any $\tilde{\phi},\tilde{\psi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$. Hence, we obtain $$\label{eq:mdefect2}
\left\lvert\sigma'_i(\tilde{\phi}\tilde{\psi})-\sigma'_i(\tilde{\phi})-\sigma'_i(\tilde{\psi})\right\rvert<K_i+K_0$$ for any $\tilde{\phi},\tilde{\psi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$. Using and several times yields $$\left\lvert\sigma'_i\left((\tilde{\phi}_1^{-1}\tilde{\varphi}_{F_1}\tilde{\phi}_1)\cdots(\tilde{\phi}_{\alpha}^{-1}\tilde{\varphi}_{F_{\alpha}}\tilde{\phi}_{\alpha})\right)\right\rvert <(\alpha-1)(K_i+K_0)<\alpha (K_i+K_0).$$ Then, the remainder of the proof follows the same path as in Theorem \[theorem:main\].
Proof of Theorem \[theorem:main3\]
----------------------------------
Let $H_1,\ldots,H_N$ and $I\colon\mathbb{R}^N\to{\mathop{\mathrm{Ham}}\nolimits}(M)$ be chosen as in the proof of Theorem \[theorem:main\]. We fix $i=1,\ldots,N$ and set $r_0=0$. We define a map $\sigma'_i\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ to be $\sigma'_i=\sigma_{c_i}-\sigma_{c_0}$. Then, by , $$\sigma'_i(\widetilde{\varphi}_{r_1H_1+\cdots+r_N H_N})=r_i-r_0=r_i.$$
We set $\alpha=\|\varphi_{r_1H_1+\cdots+r_N H_N}\|_{\mathcal{U}}$. Let us denote $$\varphi_{r_1H_1+\cdots+r_N H_N}=\varphi_{F_1}\cdots\varphi_{F_{\alpha}},$$ where $F_{\ell}\in\mathcal{H}(U_{\lambda_{\ell}})$ for some $U_{\lambda_{\ell}}\in\mathcal{U}$.
Since $U_{\lambda_{\ell}}$ satisfies the bounded spectrum condition with respect to $ c_i$ and $ c_0$ for all $\ell$, there exist positive numbers $K_i,K_0>0$ such that $$\label{eq:bdd}
c_i(\tilde{\varphi}_{F_{\ell}})+\langle F_{\ell}\rangle<K_i \quad\text{and}\quad c_0(\tilde{\varphi}_{F_{\ell}}^{-1})-\langle F_{\ell}\rangle<K_0$$ for all $\ell$. We claim that $$\left\lvert\sigma'_i(\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})\right\rvert <\alpha(K_i+K_0).$$ Indeed, by subadditivity, $$c_i\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right)\leq k c_i(\tilde{\varphi}_{F_1})+\cdots+k c_i(\tilde{\varphi}_{F_{\alpha}}),$$ $$c_0\left((\tilde{\varphi}_{F_{\alpha}}^{-1}\cdots\tilde{\varphi}_{F_1}^{-1})^k\right)\leq k c_0(\tilde{\varphi}_{F_1}^{-1})+\cdots+k c_0(\tilde{\varphi}_{F_{\alpha}}^{-1}),$$ and $$- c_0\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right)\leq c_0\left((\tilde{\varphi}_{F_{\alpha}}^{-1}\cdots\tilde{\varphi}_{F_1}^{-1})^k\right)-c(\mathbbm{1}).$$ By combining with , we obtain $$\begin{aligned}
c_i\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right)- c_0\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right) &\leq k\sum_{\ell=1}^{\alpha}\left( c_i(\tilde{\varphi}_{F_{\ell}})+ c_0(\tilde{\varphi}_{F_{\ell}}^{-1})\right)-c(\mathbbm{1})\\
&<k\alpha(K_i+K_0)-c(\mathbbm{1}).\end{aligned}$$ Similarly, $$c_0\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right)- c_i\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right)
<k\alpha(K_i+K_0)-c(\mathbbm{1}).$$ Therefore,$$\left\lvert c_i\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right)- c_0\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right)\right\rvert <\lvert k\alpha(K_i+K_0)-c(\mathbbm{1})\rvert.$$ Thus, dividing by $k$ and passing to the limit as $k\to\infty$ yields $$\lvert\sigma'_i(\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})\rvert =\lim_{k\to\infty}\frac{\left\lvert c_i\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right)- c_0\left((\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})^k\right)\right\rvert}k <\alpha(K_i+K_0).$$
Since the natural projection $\pi\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to{\mathop{\mathrm{Ham}}\nolimits}(M)$ is a group homomorphism, $$\pi(\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})=\varphi_{F_1}\cdots\varphi_{F_{\alpha}}=\varphi_{r_1H_1+\cdots+r_N H_N}.$$ By assumption, $c_i$ and $c_0$ descend asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$. Hence, the map $\sigma'_i=\sigma_{c_i}-\sigma_{c_0}$ induces a map $\bar{\sigma}'_i\colon{\mathop{\mathrm{Ham}}\nolimits}(M)\to\mathbb{R}$ such that $\sigma'_i=\bar{\sigma}'_i\circ\pi$. Thus, $$\lvert r_i\rvert=\lvert\sigma'_i(\tilde{\varphi}_{r_1H_1+\cdots+r_N H_N})\rvert =\lvert\bar{\sigma}'_i(\varphi_{F_1}\cdots\varphi_{F_{\alpha}})\rvert =\lvert\sigma'_i(\tilde{\varphi}_{F_1}\cdots\tilde{\varphi}_{F_{\alpha}})\rvert <\alpha(K_i+K_0).$$ Then, the remainder of the proof follows the same path as in Theorem \[theorem:main\].
Lagrangian spectral invariants {#section:lagspecinv}
==============================
Lagrangian spectral invariants for monotone Lagrangian submanifolds were defined by Leclercq and Zapolsky [@LZ]. In this section, we review their properties and prove the corollaries given in Section \[section:app\].
Let $(M,\omega)$ be a closed symplectic manifold. Let $L$ be a monotone Lagrangian submanifold of $(M,\omega)$ with minimal Maslov number $N_L\geq 2$ (For the definitions of the monotonicity and the minimal Maslov number of a Lagrangian submanifold, see [@Oh96; @BC; @LZ] for example). We fix a commutative ring $R$. Assuming that $L$ is relatively $\mathrm{Pin}^{\pm}$ (see [@Za Section 7.1] for the definition), Zapolsky defined the *Lagrangian quantum homology* [^1] ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L;R)$ of $L$ [@Za Sections 4 and 7.3]. Moreover, he defined the *Lagrangian Floer homology* ${\mathop{\mathrm{HF}}\nolimits}_{\ast}(L;R)$ of $L$ and proved that there exists an isomorphism ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L;R)\cong{\mathop{\mathrm{HF}}\nolimits}_{\ast}(L;R)$ called the *Piunikhin–Salamon–Schwarz isomorphism* [@Za Sections 5 and 7.3]. His work generalizes that of Oh and that of Biran and Cornea [@Oh96; @BC].
We assume that ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L;R)\neq0$. Following [@LZ Section 3], one can define the Lagrangian spectral invariant $c^{(L;R)}\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ associated with the fundamental class $[L]\in{\mathop{\mathrm{QH}}\nolimits}_{\ast}(L;R)$. Moreover, Leclercq and Zapolsky proved that $c^{(L;R)}$ is a subadditive invariant [@LZ Theorem 41].
To prove Corollaries \[cp2\], \[surface main theorem\], \[annulus frag\] and Theorem \[calabi on surface\], we first prove the following result.
\[Lagrangian descending\] If $(M,\omega)$ is either $(\mathbb{C}P^n,\omega_{\mathrm{FS}})$ or $(\Sigma_{\vec{g}},\omega)$, then the Lagrangian spectral invariant $c^{(L;R)}\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$.
One can also define the quantum homology [^2] ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(M;R)$ of the ambient manifold $(M,\omega)$ [@Za Sections 4.5 and 7.2]. Let $c^{(M;R)}\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be the Oh–Schwarz spectral invariant associated with the fundamental class $[M]\in {\mathop{\mathrm{QH}}\nolimits}_{\ast}(M;R)$. $c^{(M;R)}$ is also a subadditive invariant (see, for example, [@Oh05 Theorem I]).
Now, we have the *quantum module action* $$\bullet\colon{\mathop{\mathrm{QH}}\nolimits}_{\ast}(M;R)\otimes{\mathop{\mathrm{QH}}\nolimits}_{\ast}(L;R)\to{\mathop{\mathrm{QH}}\nolimits}_{\ast}(L;R)$$ (see [@Za Section 7.4], [@LZ Section 2.5.3]). [@LZ Proposition 5] then yields the following inequality as a corollary.
\[ham and lagr\] For any Hamiltonian $H\colon S^1\times M\to\mathbb{R}$, $$c^{(L;R)}(\tilde{\varphi}_H)\leq c^{(M;R)}(\tilde{\varphi}_H).$$
As a consequence of Schwarz [@Sch], $c^{(\Sigma_{\vec{g}};R)}$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(\Sigma_{\vec{g}})$. As a consequence of Entov and Polterovich [@EP03], $c^{(\mathbb{C}P^n;R)}$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(\mathbb{C}P^n)$. Thus, Theorem \[Lagrangian descending\] follows from Propositions \[comparing descending\] and \[ham and lagr\].
When $c^{(M;R)}\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ is a quasi-morphism, Proposition \[ham and lagr\] enables us to prove the following proposition.
\[quasiquasi\] If $c^{(M;R)}$ is a quasi-morphism, then $c^{(L;R)}$ is as well.
For the sake of brevity, we write $c^L=c^{(L;R)}$ and $c^M=c^{(M;R)}$. By subadditivity, $$c^L(\tilde{\phi}\tilde{\psi})-c^L(\tilde{\phi})-c^L(\tilde{\psi})\leq 0$$ for any $\tilde{\phi},\tilde{\psi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$. Hence, it is sufficient to show that there exists a positive number $K>0$ such that $$c^L(\tilde{\phi}\tilde{\psi})-c^L(\tilde{\phi})-c^L(\tilde{\psi})>-K$$ for any $\tilde{\phi},\tilde{\psi}\in\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)$.
Since $c^M$ is a quasi-morphism, there exists a positive number $C>0$ such that $$c^M(\mathbbm{1})-c^M(\tilde{\psi})-c^M(\tilde{\psi}^{-1})>-C.$$ Then, subadditivity and Proposition \[ham and lagr\] imply that $$\begin{aligned}
c^L(\tilde{\phi}\tilde{\psi})-c^L(\tilde{\phi})-c^L(\tilde{\psi}) &\geq -c^L(\tilde{\psi})-c^L(\tilde{\psi}^{-1})\\
&\geq -c^M(\tilde{\psi})-c^M(\tilde{\psi}^{-1}) >-c^M(\mathbbm{1})-C.\qedhere\end{aligned}$$
To prove Corollaries \[cp2\], \[surface main theorem\], and \[annulus frag\] and Theorem \[calabi on surface\], we use the following propositions.
\[L is shv\] $L$ is $c^{(L;R)}$-superheavy.
\[relative disp\] Any open subset $U\subset M$ displaceable from $L$ satisfies the bounded spectrum condition with respect to $c^{(L;R)}$.
\[displaceable SC\] Any abstractly displaceable open subset $U\subset M$ satisfies the normally bounded spectrum condition with respect to $c^{(M;R)}$ and $c^{(L;R)}$.
Proof of corollaries {#section:pf of cor}
====================
In this section, we prove Corollaries \[s2s2\], \[cp2\], \[surface main theorem\], and \[annulus frag\]. For the sake of brevity, let $\mathbb{Z}_2$ denote the field $\mathbb{Z}/2\mathbb{Z}$ below.
Proof of Corollary \[b2n\]
--------------------------
We think of the ball $B^{2n}$ as embedded in $\mathbb{C}^n$ and consider the mutually disjoint tori $$T_{\delta}=\left\{\,(w_1,\ldots,w_n)\in\mathbb{C}^n{\mathrel{}\middle|\mathrel{}}\lvert w_i\rvert^2=\frac{1}{\delta(n+1)}\ \text{for any}\ i=1,\ldots,n\,\right\},$$ $0<\delta\leq 1$, where $w_1,\ldots,w_n$ are the standard complex coordinates on $\mathbb{C}^n$. Let $(\mathbb{C}P^n,\omega_{\mathrm{FS}})$ be $n$-dimensional complex projective space and $L_C=\{\lvert z_0\rvert=\cdots=\lvert z_n\rvert\}\subset\mathbb{C}P^n$ the Clifford torus. For a positive number $\delta$ with $\delta\in(\frac{n}{n+1},1]$, Biran, Entov, and Polterovich [@BEP Section 4] constructed a conformally symplectic embedding $\vartheta_{\delta}\colon B^{2n}\to\mathbb{C}P^n$ satisfying $\vartheta_{\delta}(T_{\delta})=L_C$. The embeddings $\vartheta_{\delta}\colon B^{2n}\to\mathbb{C}P^n$, $\delta\in(\frac{n}{n+1},1]$, induce homomorphisms $(\vartheta_{\delta})_{\ast}\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(B^{2n})\to\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(\mathbb{C}P^n)$.
Let $c^{\mathbb{C}P^n}=c^{(\mathbb{C}P^n;\mathbb{Z}_2)}\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(\mathbb{C}P^n)\to\mathbb{R}$ be the Oh–Schwarz spectral invariant associated with $[\mathbb{C}P^n]\in{\mathop{\mathrm{QH}}\nolimits}_{\ast}(\mathbb{C}P^n;\mathbb{Z}_2)$. According to [@EP03 Theorem 3.1], $c^{\mathbb{C}P^n}$ is a quasi-morphism. Therefore, the functions $c'_{\delta}=c^{\mathbb{C}P^n}\circ(\vartheta_{\delta})_{\ast}\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(B^{2n})\to\mathbb{R}$, $\delta\in(\frac{n}{n+1},1]$, are subadditive invariants and quasi-morphisms.
Biran, Entov, and Polterovich proved that there exists a constant $c_n$ such that $T_{\delta}$ is $c_{\delta}$-superheavy, where $c_{\delta}=c_n\cdot c'_{\delta}$ for any $\delta\in(\frac{n}{n+1},1]$. Since $T_\delta\cap B(r)=\emptyset$ holds for any $\delta$ with $\delta\in(\frac{n}{n+1},(\frac{n}{n+1})\cdot r^{-2})$, by Proposition \[disp and vanish\], the open ball $B(r)$ of radius $r<1$ satisfies the asymptotically vanishing condition with respect to $c_{\delta}$ for any $\delta\in(\frac{n}{n+1},(\frac{n}{n+1})\cdot r^{-2})$. Thus, Theorem \[theorem:main2\] completes the proof of Corollary \[b2n\].
Proof of Corollary \[s2s2\]
---------------------------
First we recall the definition of stems. Let $(M,\omega)$ be a closed symplectic manifold. Let $\mathbb{A}$ be a finite-dimensional Poisson-commutative subspace of $C^{\infty}(M)$. Let $\Phi\colon M\to\mathbb{A}^{\ast}$ be the moment map given by $F(x)=\langle\Phi(x),F\rangle$ for $x\in M$ and $F\in\mathbb{A}$.
A closed subset $X$ of $M$ is called a *stem* if there exists a finite-dimensional Poisson-commutative subspace $\mathbb{A}$ of $C^{\infty}(M)$ such that $X$ is a fiber of $\Phi$ and each non-trivial fiber of $\Phi$, other than $X$, is displaceable.
The proof of the following theorem is quite similar to that of [@EP09 Theorem 1.8].
Every stem is $c$-superheavy, where $c$ is a Lagrangian spectral invariant defined in [@LZ] or a spectral invariant defined in [@FOOO].
Fukaya, Oh, Ohta, and Ono [@FOOO] defined a family of bulk-deformed Oh–Schwarz spectral invariants $\{c_\rho\}_{\rho\in[0,1/2)}$ on $\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(S^2\times S^2)$ and proved that any $c_\rho$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(S^2\times S^2)$. They also constructed a family of mutually disjoint Lagrangian submanifolds $T(\rho)$ ($\rho\in[0,1/2)$) and proved that each $T(\rho)$ is $c_{\rho}$-superheavy [@FOOO Theorem 23.4]. It is known that when $U$ is abstractly displaceable, $U$ satisfies the bounded spectrum condition with respect to $c_{\rho}$ for any $\rho$ (see also Proposition \[displaceable SC\]).
On the other hand, $E\times E\subset S^2\times S^2$ is a stem. In particular, $E\times E$ is $c_{\rho}$-superheavy for any $\rho$. Hence, Proposition \[disp and vanish\] implies that if $U$ is displaceable from $E\times E$, then $U$ satisfies the asymptotically vanishing spectrum condition with respect to $c_\rho$ for any $\rho$.
Therefore, in any case, $U$ satisfies the asymptotically vanishing spectrum condition with respect to $c_\rho$ for any $\rho$ (see also Remark \[remark:AVSC\]). Since $c_\rho$ is known to be a quasi-morphism for any $\rho$, Theorem \[theorem:main2\] completes the proof of Corollary \[s2s2\].
Proof of Corollary \[cp2\]
--------------------------
By Biran and Cornea’s work [@BC Corollary 1.2.11 (ii)], ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(\mathbb{R}P^2;\mathbb{Z}_2)\cong{\mathop{\mathrm{HF}}\nolimits}_{\ast}(\mathbb{R}P^2;\mathbb{Z}_2)\cong\mathbb{Z}_2$ (see also [@LZ Section 2.6.1]) [^3]. Moreover, Leclercq and Zapolsky [@LZ Section 2.6.3] showed that ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L_W;\mathbb{Z})\neq 0$. Let $c^{\mathbb{R}P^2}=c^{(\mathbb{R}P^2;\mathbb{Z}_2)}$ and $c^{L_W}=c^{(L_W;\mathbb{Z})}$.
By Theorem \[Lagrangian descending\], $c^{\mathbb{R}P^2}$ and $c^{L_W}$ descend asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(\mathbb{C}P^2)$. According to [@EP03 Theorem 3.1], the Oh–Schwarz spectral invariant $c^{(\mathbb{C}P^2;R)}$ is a quasi-morphism for $R=\mathbb{Z}_2$ and $\mathbb{Z}$. Hence, by Proposition \[quasiquasi\], $c^{\mathbb{R}P^2}$ and $c^{L_W}$ are also quasi-morphisms. Moreover, by Proposition \[L is shv\], $\mathbb{R}P^2$ and $L_W$ are superheavy with respect to $c^{\mathbb{R}P^2}$ and $c^{L_W}$, respectively.
When $U$ is abstractly displaceable (case (i)), Proposition \[displaceable SC\] ensures that $U$ satisfies the normally bounded spectrum condition with respect to $c^{\mathbb{R}P^2}$ and $c^{L_W}$.
When $U$ is displaceable from $\mathbb{R}P^2$ and $L_W$ (case (ii)), Proposition \[relative disp\] ensures that $U$ satisfies the bounded spectrum condition with respect to $c^{\mathbb{R}P^2}$ and $c^{L_W}$.
In addition, the Clifford torus $L_C$ is a stem [@BEP]. In particular, $L_C$ is superheavy with respect to $c^{\mathbb{R}P^2}$ and $c^{L_W}$. Hence, Proposition \[disp and vanish\] implies that if $U$ is displaceable from $L_C$ (case (iii)), then $U$ satisfies the asymptotically vanishing spectrum condition with respect to $c^{\mathbb{R}P^2}$ and $c^{L_W}$.
Therefore, in any case, $U$ satisfies the asymptotically vanishing spectrum condition with respect to $c^{\mathbb{R}P^2}$ and $c^{L_W}$. Since $\mathbb{R}P^2\cap L_W=\emptyset$, we conclude that $c^{\mathbb{R}P^2}$, $c^{L_W}$, $\mathbb{R}P^2$, $L_W$ and $U$ satisfy the assumption of Theorem \[theorem:main2\] for $N=1$. This completes the proof of Corollary \[cp2\].
We do not need Theorem \[Lagrangian descending\] to prove Corollary \[cp2\] if we use the well-known fact that $\pi_1\bigl({\mathop{\mathrm{Ham}}\nolimits}(\mathbb{C}P^2)\bigr)=0$ (see [@G]). We provide a more general argument here for future works.
Proof of Corollary \[surface main theorem\]
-------------------------------------------
We use the following result to prove Corollary \[surface main theorem\].
\[master thesis\] For any positive integer $g$, there exists a positive number $K$ such that $$c^{(\Sigma_g;\mathbb{Z}_2)}(\tilde{\varphi}_F)+\langle F\rangle\leq K$$ for any contractible open subset $U$ of $\Sigma_g$ and any $F\in\mathcal{H}(U)$.
Let $C$ be a non-contractible simple closed curve in the surface $\Sigma_g$. We choose symplectomorphisms $f_1,\ldots,f_N$ of $(\Sigma_g,\omega)$ to ensure that the subsets $C$, $f_1(C),\ldots,f_N(C)$ are mutually disjoint. We fix $i=0,1,\ldots,N$. We set $L_i=f_i(C)$, where $f_0=\mathrm{id}_{\Sigma_g}$. Then, ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L_i;\mathbb{Z}_2)\cong{\mathop{\mathrm{HF}}\nolimits}_{\ast}(L_i;\mathbb{Z}_2)$ does not vanish. Let $c^{L_i}=c^{(L_i;\mathbb{Z}_2)}$ denote the associated Lagrangian spectral invariant. Remark \[MVZ\], Propositions \[master thesis\] and \[ham and lagr\] imply that $U$ satisfies the normally bounded spectrum condition with respect to $c^{L_i}$ for any $i$.
Then, Theorem \[Lagrangian descending\] and Proposition \[L is shv\] ensure that $c^{L_0},\ldots,c^{L_N}$, $L_0,\ldots,L_N$ and $U$ satisfy the assumption of Theorem \[theorem:main\]. This completes the proof of Corollary \[surface main theorem\].
We do not need Theorem \[Lagrangian descending\] to prove Corollary \[surface main theorem\] if we use the well-known fact that $\pi_1\bigl({\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g)\bigr)=0$ for positive $g$ (see [@Po01 Section 7.2.B]). We provide a more general argument here for future works.
Proof of Corollary \[annulus frag\]
-----------------------------------
In the proof of Corollary \[surface main theorem\], we constructed mutually disjoint Lagrangian submanifolds $L_0,\ldots,L_N\subset(\Sigma_g,\omega)$ such that ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L_i;\mathbb{Z}_2)$ does not vanish. By the construction of $L_0,\ldots,L_N$ and the assumption on the covering $\mathcal{U}$, each $U_{\lambda}$ is displaceable from $L_i$.
Then, Theorem \[Lagrangian descending\] and Propositions \[L is shv\] and \[relative disp\] ensure that $c^{L_0},\ldots,c^{L_N}$, $L_0,\ldots,L_N$ and $\mathcal{U}=\{U_\lambda\}_{\lambda}$ satisfy the assumption of Theorem \[theorem:main3\]. This completes the proof of Corollary \[annulus frag\].
Partial Calabi quasi-morphisms {#section:calabi}
==============================
Let $(M,\omega)$ be a closed symplectic manifold. Given an open subset $U\subset M$ such that $\omega|_U$ is exact, we recall that the *Calabi homomorphism* is a homomorphism $\mathrm{Cal}_U\colon{\mathop{\mathrm{Ham}}\nolimits}(U)\to\mathbb{R}$ defined by $$\mathrm{Cal}_U(\varphi_F)=\int_0^1\int_U F_t\omega^n\,dt.$$
\[definition of Calabi\] A *partial Calabi quasi-morphism* is a function $\mu\colon{\mathop{\mathrm{Ham}}\nolimits}(M)\to\mathbb{R}$ satisfying the following conditions.
Stability
: For any Hamiltonians $H,K\colon S^1\times M\to\mathbb{R}$, $$\int_0^1\min_M(H_t-K_t)\,dt\leq\frac{\mu(\varphi_H)-\mu(\varphi_K)}{{\mathop{\mathrm{Vol}}\nolimits}(M)}\leq\int_0^1\max_M(H_t-K_t)\,dt.$$
Partial homogeneity
: $\mu(\phi^k)=k\mu(\phi)$ for any $\phi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ and $k\in\mathbb{Z}_{\geq 0}$.
Partial quasi-additivity
: Given a displaceable open subset $U\subset M$, there exists a positive number $K>0$ such that $$\lvert\mu(\phi\psi)-\mu(\phi)-\mu(\psi)\rvert\leq K\min\{\|\phi\|_U,\|\psi\|_U\}$$ for any $\phi,\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$.
Calabi property
: For any displaceable open subset $U\subset M$ such that $\omega|_U$ is exact, the restriction of $\mu$ to ${\mathop{\mathrm{Ham}}\nolimits}(U)$ coincides with the Calabi homomorphism $\mathrm{Cal}_U$.
Let $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be a subadditive invariant descending asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(M)$ (see Definition \[definition:descend\]). Let $U$ be an open subset of $M$ satisfying the normally bounded spectrum condition with respect to $c$ (see Definition \[definition:NBSC\]). We can generalize Proposition \[proposition:defect1\] as follows.
\[proposition:defect\] There exists a positive number $K>0$ such that $$\lvert\bar{\sigma}_c(\phi\psi)-\bar{\sigma}_c(\phi)-\bar{\sigma}_c(\psi)\rvert\leq K\min\{\|\phi\|_U,\|\psi\|_U\}$$ for any $\phi,\psi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$.
We assume, without loss of generality, that $\|\phi\|_U\leq\|\psi\|_U$. We represent $\phi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ as $\phi=\phi_1\cdots\phi_{\alpha}$ with $\|\phi_i\|_U=1$ for all $i$. We claim that $$\lvert\bar{\sigma}_c(\phi\psi)-\bar{\sigma}_c(\phi)-\bar{\sigma}_c(\psi)\rvert\leq C(2\alpha-1)$$ for some $C>0$. Then, the proposition follows if we set $K=2C$. We prove the claim by induction on $\alpha=\|\phi\|_U$.
When $\alpha=1$, we can choose a Hamiltonian $F$ such that $\varphi_F=\phi$ and $F\in\mathcal{H}\bigl(\theta(U)\bigr)$ for some $\theta\in{\mathop{\mathrm{Ham}}\nolimits}(M)$. Then, Proposition \[proposition:defect1\] implies that $$\lvert\bar{\sigma}_c(\phi\psi)-\bar{\sigma}_c(\phi)-\bar{\sigma}_c(\psi)\rvert =\lvert\bar{\sigma}_c(\varphi_F\psi)-\bar{\sigma}_c(\varphi_F)-\bar{\sigma}_c(\psi)\rvert \leq C$$ for some $C>0$. This proves the claim.
We assume that the claim holds for $\|\phi\|_U=\alpha$. For $\phi\in{\mathop{\mathrm{Ham}}\nolimits}(M)$ with $\|\phi\|_U=\alpha+1$, we decompose it into $\phi=\phi_{\alpha}\phi_1$ where $\|\phi_{\alpha}\|_U=\alpha$ and $\|\phi_1\|_U=1$. By the induction hypothesis, $$\lvert\bar{\sigma}_c(\phi\psi)-\bar{\sigma}_c(\phi_{\alpha})-\bar{\sigma}_c(\phi_1\psi)\rvert\leq C(2\alpha-1).$$ Moreover, since $\|\phi_1\|_U=1$, $$\lvert\bar{\sigma}_c(\phi_1\psi)-\bar{\sigma}_c(\phi_1)-\bar{\sigma}_c(\psi)\rvert\leq C
\quad\text{and}\quad
\lvert\bar{\sigma}_c(\phi_1)+\bar{\sigma}_c(\phi_{\alpha})-\bar{\sigma}_c(\phi)\rvert\leq C.$$ Hence, $$\lvert\bar{\sigma}_c(\phi\psi)-\bar{\sigma}_c(\phi)-\bar{\sigma}_c(\psi)\rvert\leq C(2\alpha+1).$$ This completes the proof of Proposition \[proposition:defect\].
Since the fragmentation norm $\|\cdot\|_{\mathcal{U}}$ with respect to a covering $\mathcal{U}$ is not conjugation invariant in general, we cannot prove a proposition corresponding to Proposition \[proposition:defect\] in the same manner (see also the proof of Proposition \[proposition:defect1\]).
Proof of Theorem \[calabi on surface\]
--------------------------------------
For $\vec{g}=(g_1,\ldots,g_n)\in\mathbb{N}^n$, we recall that $(\Sigma_{\vec{g}},\omega)$ is the product manifold $\Sigma_{\vec{g}}=\Sigma_{g_1}\times\cdots\times\Sigma_{g_n}$ equipped with a symplectic form $\omega$.
Let $C_i$ be a non-contractible simple closed curve in the surface $\Sigma_{g_i}$, and let $C$ denote the Lagrangian submanifold $C_1\times\cdots\times C_n$ of $(\Sigma_{\vec{g}},\omega_{\vec{A}})$.
For all positive integers $N$, we choose symplectomorphisms $f_1,\ldots,f_N$ of $(\Sigma_{\vec{g}},\omega_{\vec{A}})$ to ensure that the subsets $C$, $f_1(C),\ldots,f_N(C)$ are mutually disjoint. We fix $i=0,1,\ldots,N$. We set $L_i=f_i(C)$, where $f_0=\mathrm{id}_{\Sigma_{\vec{g}}}$. Then, ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L_i;\mathbb{Z}_2)\cong{\mathop{\mathrm{HF}}\nolimits}_{\ast}(L_i;\mathbb{Z}_2)$ does not vanish. Let $c_i=c^{(L_i;\mathbb{Z}_2)}\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(\Sigma_{\vec{g}})\to\mathbb{R}$ denote the Lagrangian spectral invariant associated with $[L_i]\in{\mathop{\mathrm{QH}}\nolimits}_{\ast}(L_i;\mathbb{Z}_2)$. By Theorem \[Lagrangian descending\], $c_i$ descends asymptotically to ${\mathop{\mathrm{Ham}}\nolimits}(\Sigma_{\vec{g}})$.
Now, we define a function $\mu_i\colon{\mathop{\mathrm{Ham}}\nolimits}(\Sigma_{\vec{g}})\to\mathbb{R}$ by $\mu_i=-{\mathop{\mathrm{Vol}}\nolimits}(\Sigma_{\vec{g}})\cdot\bar{\sigma}_{c_i}$. By definition, $\mu_i$ satisfies partial homogeneity. By Proposition \[displaceable SC\], any displaceable open subset of $\Sigma_{\vec{g}}$ satisfies the normally bounded spectrum condition with respect to $c_i$. Hence, Proposition \[proposition:defect\] implies partial quasi-additivity. Moreover, the Calabi property follows from Proposition \[proposition:VPNBSC\]. In fact, for any displaceable open subset $U$ such that $\omega|_U$ is exact, and any Hamiltonian $F\in\mathcal{H}(U)$, $$\mu_i(\varphi_F)=-{\mathop{\mathrm{Vol}}\nolimits}(\Sigma_{\vec{g}})\cdot\bar{\sigma}_{c_i}(\varphi_F) ={\mathop{\mathrm{Vol}}\nolimits}(\Sigma_{\vec{g}})\cdot\langle F\rangle=\mathrm{Cal}_U(\varphi_F).$$ Finally, [@LZ Theorem 41] ensures the stability of $\mu_i$. Hence, $\mu_i$ is a partial Calabi quasi-morphism.
By construction, $\mu_0,\mu_1,\ldots,\mu_N$ are linearly independent. This completes the proof of Theorem \[calabi on surface\].
Problems
========
The authors are yet to find the answers to the following problems.
Let $(M,\omega)$ be a closed symplectic manifold. Let $c\colon\widetilde{{\mathop{\mathrm{Ham}}\nolimits}}(M)\to\mathbb{R}$ be either the Oh–Schwarz spectral invariant or the Lagrangian spectral invariant defined in [@LZ]. Does there exist an open subset $U$ of $M$ satisfying the asymptotically vanishing spectrum condition with respect to $c$ but not the normally bounded spectrum condition?
Related to Corollary \[annulus frag\], we pose the following problem.
Let $(\Sigma_g,\omega)$ be a closed Riemann surface of positive genus $g$ with a symplectic form $\omega$. Let $C$ be a non-contractible simple closed curve in $\Sigma_g$ and $U$ an open subset of $\Sigma_g$ displaceable from $C$. Does there exist a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{Z},\lvert\cdot\rvert)\to({\mathop{\mathrm{Ham}}\nolimits}(\Sigma_g),\|\cdot\|_U)\text{?}$$
The following problem is also related to Corollary \[annulus frag\].
Let $(\mathbb{T}^2=\mathbb{R}/\mathbb{Z}\times\mathbb{R}/\mathbb{Z},\omega)$ be a 2-torus with a symplectic form $\omega$. Let $U$ be an open neighborhood of $(\{0\}\times\mathbb{R}/\mathbb{Z})\cup(\mathbb{R}/\mathbb{Z}\times\{0\})$ and $V$ a contractible open subset of $\mathbb{T}^2$ with $\mathbb{T}^2=U\cup V$. We consider the open covering $\mathcal{U}=\{U,V\}$ of $\mathbb{T}^2$. Does there exist a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{Z},\lvert\cdot\rvert)\to({\mathop{\mathrm{Ham}}\nolimits}(\mathbb{T}^2),\|\cdot\|_{\mathcal{U}})\text{?}$$
Related to Corollary \[cp2\], we pose the following problem.
Let $(\mathbb{C}P^n,\omega_{\mathrm{FS}})$ be $n$-dimensional complex projective space with the Fubini–Study form $\omega_{\mathrm{FS}}$. Let $L_C$ be the Clifford torus in $\mathbb{C}P^n$ and $U$ an open subset of $\mathbb{C}P^n$ displaceable from $L_C$. Let $(\Sigma_g,\omega)$ be a closed Riemann surface of positive genus $g$ with a symplectic form $\omega$ and $C$ a non-contractible simple closed curve in $\Sigma_g$. We consider the product manifold $(\mathbb{C}P^n\times\Sigma_g,\omega_{\mathrm{FS}}\oplus\omega)$, the Lagrangian submanifold $L_C\times C$, and the open subset $\widehat{U}=U\times\Sigma_g$ of $\mathbb{C}P^n\times\Sigma_g$. Does there exist a bi-Lipschitz injective homomorphism $$I\colon(\mathbb{Z},\lvert\cdot\rvert)\to({\mathop{\mathrm{Ham}}\nolimits}(\mathbb{C}P^n\times\Sigma_g),\|\cdot\|_{\widehat{U}})\text{?}$$
By Proposition \[relative disp\], $\widehat{U}$ satisfies the bounded spectrum condition with respect to $c^{(L_C\times C;R)}$ for any ring $R$. However, by an argument similar to [@EP09], we see that $c^{(L_C\times C;R)}$ is not a quasi-morphism and that we therefore cannot apply Theorem \[theorem:main2\].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Professor Yong-Geun Oh and Takahiro Matsushita for some advice. Especially, Takahiro Matsushita read our draft seriously and gave a lot of comments on writing. A part of this work was carried out while the first named author was visiting NCTS (Taipei, Taiwan) and the second named author was visiting IBS-CGP (Pohang, Korea). They would like to thank the institutes for their warm hospitality and support.
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[^1]: In Leclercq and Zapolsky’s terminology, our Lagrangian quantum homology ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L;R)$ (resp., Lagrangian Floer homology ${\mathop{\mathrm{HF}}\nolimits}_{\ast}(L;R)$) is the *quotient* Lagrangian quantum homology ${\mathop{\mathrm{QH}}\nolimits}_{\ast}^{\pi_2^0(M,L)}(L;R)$ (resp., *quotient* Lagrangian Floer homology ${\mathop{\mathrm{HF}}\nolimits}_{\ast}^{\pi_2^0(M,L)}(L;R)$), where $\pi_2^0(M,L)$ is the kernel of $[\omega]\colon \pi_2(M,L)\to\mathbb{R}$.
[^2]: Similar to the above, our quantum homology ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(M;R)$ is Zapolsky’s *quotient* quantum homology ${\mathop{\mathrm{QH}}\nolimits}_{\ast}^{\pi_2^0(M)}(M;R)$, where $\pi_2^0(M)$ is the kernel of $[\omega]\colon \pi_2(M)\to\mathbb{R}$.
[^3]: Given a Lagrangian submanifold $L$ of a symplectic manifold $(M,\omega)$, our Lagrangian quantum homology ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L;\mathbb{Z}_2)$ is actually Biran and Cornea’s ${\mathop{\mathrm{QH}}\nolimits}_{\ast}(L;\Lambda)$ where $\Lambda=\mathbb{Z}_2[t,t^{-1}]$.
| ArXiv |
---
abstract: 'In the $\phi $-mapping theory, the topological current constructed by the order parameters can possess different inner structure. The difference in topology must correspond to the difference in physical structure. The transition between different structures happens at the bifurcation point of the topological current. In a self-interaction two-level system, the change of topological particles corresponds to change of energy levels.'
address: |
$^1$ Institute of Applied Physics and Computational Mathematics,\
P.O. Box 8009(28), Beijing 100088, P.R. China\
$^2$ Institute of Theoretical Physics, Department of Physics,\
Lanzhou University, 730000, P. R. China
author:
- 'Li-Bin Fu$^1$, Jie Liu$^1$, Shi-Gang Chen$^1$, and Yi-Shi Duan$^2$'
title: 'The configuration of a topological current and physical structure: an application and paradigmatic evidence'
---
0.5cm In recent years, topology has established itself as an important part of the physicist’s mathematical arsenal [@zh1]. The concepts of the topological particle and its current have been widely used in particle physics [@duan1; @hha] and topological defect theory [@zh4]. Here, the topological particles are regarded as abstract particles, such as monopoles and the points defects.
In this paper, we give a new understanding in topology and physics. Many physics system can be described by employing the order parameters. By making use of the $\phi $-mapping theory, we find that the topological current constructed by the order parameters can possess different inner structure. The topological properties are basic properties for a physics system, so the difference in configuration of the topological current must correspond to the difference in physical structure.
Considering a $(n+1)$-dimensional system with $n$-component vector order parameter field $\vec \phi ({\bf x}),$ where ${\bf x}=(x^0,x^1,x^2,\cdots
x^n)$ correspond to local coordinates. The direction unit field of $\vec \phi
$ is defined by $$n^a=\frac{\phi ^a}{||\phi ||},\quad a=1,2,\cdots n \label{c1unit}$$ where $$||\phi ||=(\phi ^a\phi ^a)^{1/2}.$$ The topological current of this system is defined by $$j^\mu (x)=\frac{\in ^{\mu \mu _1\cdots \mu _n}}{A(S^{n-1})(d-1)!}\in
_{a_1\cdots a_n}\partial _{\mu _1}n^{a_1}\cdots \partial _{\mu _n}n^{a_n}
\label{firstcurr}$$ where $A(S^{n-1})$ is the surface area of $(n-1)$-dimensional unit sphere $%
S^{n-1}$. Obviously, the current is identically conserved, $$\partial _\mu j^\mu =0.$$ If we define a Jacobians by $$\in ^{a_1\cdots a_n}D^\mu (\frac \phi x)=\in ^{\mu \mu _1\cdots \mu
_n}\partial _{\mu _1}\phi ^{a_1}\cdots \partial _{\mu _n}\phi ^{a_n},
\label{firstjac}$$ as has been proved before [@topc], this current takes the form as $$j^\mu =\delta (\vec \phi )D^\mu (\frac \phi x). \label{deltfirstcurr}$$ Then, we can obtain $$j^\mu =\sum_{i=1}^l\beta _i\eta _i\delta (\vec x-\vec z_i(x^0))\frac{%
dz_i^\mu }{dx^0},$$ in which $z_i(x^0)$ are the zero lines where $\vec \phi ({\bf x)}=0,$ the positive integer $\beta _i$ and $\eta _i=sgnD(\frac \phi {\vec x})$ are the Hopf index and Brouwer degree of $\phi $-mapping [@zh17] respectively, and $l$ is the total number of the zero lines$.$ This current is similar to a current of point particles and the $i$-th one with the charge $\beta
_i\eta _i,$ and the zero lines $z_i(x)$ are just the trajectories of the particles, for convenience we define these point particles as topological particles. Then the total topological charge of the system is $$Q=\int_Mj^0d^nx=\sum_{i=1}^l\beta _i\eta _i,$$ here $M$ is a $n$-dimensional spatial space for a given $x^0.$ This is a topological invariant and corresponds to some basic conditions of this physical system. However, it is important that the inner structure of the topological invariant can be constructed in different configurations, i.e., the number of the topological particles and their charge can be changed. This change in configuration of the topological current must correspond to some change in physical structure.
All of the above discussion is based on the condition that $$D\left( \frac \phi x\right) =\left. D^0\left( \frac \phi x\right) \right|
_{z_i}\neq 0.$$ When $\left. D\left( \frac \phi x\right) \right| _{z_i}=0$ at some points $%
p_i^{*}=z^{*}(x_c^0)$ at $x^0=x_c^0$ along the zero line $z_i(x^0)$, it is shown that there exist several crucial cases of branch process, which correspond to the topological particle generating or annihilating at limit points and splitting, encountering or merging at the bifurcation points. A vast amount of literature has been devoted to discussing these features of the evolution of the topological particles [@bifur]. Here, we will not spend more attention on describing these evolution, but put our attention on the physical substance of these processes.
As we have known before, all of these branch processes keep the total topological charge conserved, but it is very important that these branch processes change the number and the charge of the topological particles. i.e. change the inner structure of the topological current. In our point of view, the different configuration of topological current corresponds to the different physical structure.
We consider $x^0$ as a parameter $\lambda $ of a physics system. Let us define $$\left. f_i(\lambda )=D^0\left( \frac \phi x\right) \right| _{z_i}$$ As $\lambda $ changing, the value of $f_i(\lambda )$ changes along the zero lines $z_i(\lambda ).$ At a critical point $\lambda =\lambda _c,$ when $%
f_i(\lambda _c)=0,$ we know that the inner structure of the topological current will be changed in some way, at the same time the physical structure will also be changed, i.e., the physical structure when $\lambda <\lambda _c$ will be different from the one when $\lambda >\lambda _c.$ The transition between these structures occurs at the bifurcation points where $f_i(\lambda
)=0.$
As an application and example, let us consider a self-interacting two-level model introduced in Ref. [@wuniu]. The nonlinear two-level system is described by the dimensionless Schrödinger equation $$i\frac \partial {\partial t}\left(
\begin{array}{c}
a \\
b
\end{array}
\right) =H(\gamma )\left(
\begin{array}{c}
a \\
b
\end{array}
\right) \label{a}$$ with the Hamiltonian given by $$H(\gamma )=\left(
\begin{array}{cc}
\frac \gamma 2+\frac C2(|b|^2-|a|^2) & \frac V2 \\
\frac V2 & -\frac \gamma 2-\frac C2(|b|^2-|a|^2)
\end{array}
\right) , \label{b}$$ in which $\gamma $ is the level separation, $V$ is the coupling constant between the two levels, and $C$ is the nonlinear parameter describing the interaction. The total probability $|a|^2+|b|^2$ is conserved and is set to be $1$.
We assume $a=|a|e^{i\varphi _1(t)}$, $b=|b|e^{i\varphi _2(t)},$ the fractional population imbalance and relative phase can be defined by $$z(t)=|b|^2-|a|^2,\;\;\;\;\;\varphi (t)=\varphi _2(t)-\varphi _1(t).
\label{bal}$$ From Eqs. (\[a\]) and (\[b\]), we obtain $$\frac d{dt}z(t)=-V\sqrt{1-z^2(t)}\sin [\varphi (t)] \label{ceq1}$$ $$\frac d{dt}\varphi (t)=\gamma +Cz(t)+\frac{Vz(t)}{\sqrt{1-z^2(t)}}\cos
[\varphi (t)]. \label{ceq2}$$
If we chose $x=2|a||b|\cos (\varphi ),$ $y=2|a||b|\sin (\varphi )$, it is easy to see that $x^2+y^2+z^2=1$ by considering $|a|^2+|b|^2=1,$ which describes a unit sphere $S^2\ $with $z$ and $\varphi $ a pair of co-ordinates$.$ We define a vector field on this unit sphere: $$\phi ^1=-V\sqrt{1-z^2}\sin (\varphi ), \label{f1}$$ $$\phi ^2=\gamma +Cz+\frac{Vz}{\sqrt{1-z^2}}\cos (\varphi ) \label{f2}$$ Apparently, there are singularities at the two pole points $z=\pm
1,$ which make the vector $\vec \phi $ is discontinuous at these points. However, the direction unit vector $\vec n$ is continuous. In the $\phi $-mapping theory, we only need the unit vector $\vec
n$ is continuous and differentiable on the whole sphere $S^2$ (at the zero points of $\vec \phi $, the differential of $\vec n$ is a general function)$,$ and the vector $\vec \phi $ is continuous and differentiable at the neighborhoods of its zero points. Then from $\phi $-mapping theory, we can obtain a topological current as $$\vec j=\sum_{i=1}^l\beta _i\eta _i\delta (\varphi -\varphi _i(\gamma
))\delta (z-z_i(\gamma ))\frac{dz_i}{d\gamma }\left| _{p_i}\right. ,$$ in which $p_i=p_i(z_i,\varphi _i)$ is the trajectory of the $i$-th topological particle $P_i$ and $$\eta _i=sgn\left( D(\gamma )\right) =sgn\left( \det \left. \left(
\begin{array}{cc}
\partial \phi ^1/\partial \varphi & \partial \phi ^2/\partial \varphi \\
\partial \phi ^1/\partial z & \partial \phi ^2/\partial z
\end{array}
\right) \right| _{p_i}\right) .$$ From Eqs. (\[f1\]) and (\[f2\]), it is easy to see that $\vec \phi $ is single-valued on $S^2,$ which states that Hopf index $\beta _i=1$ $%
(i=1,2,\cdots ,l)$ here. It can be proved that the total charge of this system is just the Euler number of $S^2$ which is $2$ [@lishen]. This is a topological invariant of $S^2$ which corresponds to the basic condition of this system: $|a|^2+|b|^2=1.$ The following discussion can show that this topological invariant can possess different configuration when $\gamma $ changes. This difference in topology corresponds to the change in adiabatic levels of this nonlinear system.
We can prove that every topological particle corresponds to an eigenstate of the nonlinear two-level system. By solving $\vec \phi =0$ form (\[f1\]) and (\[f2\]), we find there are two different cases for discussing.
Case 1. For $|C/V|\leq 1,$ there only two topological particles $P_1$ and $%
P_2$, which locate on the line $\varphi =0$ and $\varphi =\pi $ respectively as shown in the upper panel of Fig.1. All of them with topological charge $%
+1,$ and $D(\gamma )|_{P_{1,2}}>0$ for any $\gamma .$ Correspondingly, in this case, there are only two adiabatic energy levels in this nonlinear two-level mode for various $\gamma $ [@wuniu], as shown in the lower panel of Fig.1, $P_1$ corresponds to the upper level and $P_2$ corresponds to the lower level.
Case 2, For $C/V>1,$ two more topological particles can appear when $\gamma $ lies in a window $-\gamma _c<\gamma <\gamma _c$. The boundary of the window can be obtained by assuming $D(\gamma )|_{P_i}=0,$ yielding $$\gamma _c=(C^{2/3}-V^{2/3})^{3/2}. \label{gc}$$ The striking feature happens at $\gamma =-\gamma _c,$ there exists a critical point $T_1^{*}(z_c,\pi )$ with $D(\gamma _c)|_{T_1^{*}}=0,$ as we have shown in Ref. [@bifur], we can prove that this point is a limit point, and a pair of topological particles $P_3$ and $P_4$ generating with opposite charge $-1$ and $+1$ respectively$,$ both of the new topological particles lie on the line $\varphi =\pi $. One of the original topological particle, $P_2$ with charge $+1$ on the line $\varphi =\pi ,$ moves smoothly up to $\gamma =\gamma _c,$ where it collides with $P_3$ and annihilates with it at another limit point $T_2^{*}(-z_c,\pi ).$ The other, $P_1,$ which lies on the line $\varphi =0$, still moves safely with $\gamma .$
As pointed out by B. Wu and Q. Niu [@wuniu], when the interaction is strong enough $(C/V>1),$ a loop appears at the tip of the lower adiabatic level when $C/V>1$ while $-\gamma _c\leq \gamma \leq \gamma _c$. We show the interesting structure in Fig. 2 in which $C/V=2$. For $\gamma <-\gamma _c,$ there are two adiabatic levels, the upper level corresponds to the topological particle $P_1$, the lower one corresponds to the topological particle $P_2;$ for $\gamma >\gamma _c$, there are also only two adiabatical levels, but at this time the lower one corresponds to $P_4$ while the upper one still corresponds to $P_1.$ The arc part of the loop on the tip of lower level when $-\gamma _c<\gamma <\gamma _c$ just corresponds to $P_3,$ which merges with the level corresponding to $P_4$ at the point $M$ on the left and with the one corresponding to $P_1$ at the point $T$ on the right$.$
From the above discussion, one finds that when the structure of the topological current is changed by generating or annihilating a pair of topological particles ( the upper panel of Fig. 2), at the same time, the physical structure is changed by adding two energy levels or subtracting two energy levels respectively (the lower panel of Fig.2). The critical behaviors happen at the limit points where $D(\gamma _c)|_{T_{1,2}^{*}}=0$ .
In fact, this nonlinear two-level model is of comprehensive interest for it associates with a wide range of concrete physical systems, e.g., BEC in an optical lattice[@becol1] or in a double-well potential[@dwpp1], and the motion of small polarons[@polar]. So, the relation between topological particles and physical inner structure can be observed in experimental methods. Here we propose a perspective system for observing the striking phenomenon: a Bose-Einstein condensate in a double-well potential[@dwpp1; @RS99]. The amplitudes of general occupations $N_{1,2}(t)$ and phases $\varphi _{1,2}$ obey the nonlinear two-mode Schrödinger equations, approximately[@RS99], $$\begin{aligned}
i\hbar \frac{\partial \phi _1}{\partial t} &=&(E_1^0+U_1N_1)\phi _1-K\phi _2
\nonumber \\
i\hbar \frac{\partial \phi _2}{\partial t} &=&(E_2^0+U_2N_2)\phi _2-K\phi _1\end{aligned}$$ with $\phi _{1,2}=\sqrt{N_{1,2}}exp(i\varphi _{1,2})$, and total number of atoms $N_1+N_2=N_T$, is conserved. Here $E_{1,2}^0$ are zero-point energy in each well, $U_{1,2}N_{1,2}$ are proportional to the atomic self-interaction energy, and $K$ describes the amplitude of the tunnelling between the condensates. After introducing the new variables $z(t)=(N_2(t)-N_1(t))/N_T$ and $\varphi =\varphi _2-\varphi _1$, one also obtain an equations having the same form as Eqs. (\[a\]) and (\[b\]) except for the parameters replaced by $$\gamma =-[(E_1^0-E_2^0)-(U_1-U_2)N_T/2]/\hbar ,$$ $$V=2K/\hbar ,\;C=(U_1+U_2)N_T/2\hbar .$$ With these explicit expressions, our theory and results can be directly applied to this system without intrinsic difficulty. In this system the topological particles can be located by the stable occupation and relative phase $\varphi =0$ or $\varphi =\pi $ for a give parameter $\gamma .$ And, one can draw the zero line of each topological particle by giving different $%
\gamma .$ We hope our discussions will stimulate the experimental works in the direction.
We note that for a system the global property (topology) is given, the interesting feature is that under the same topology the topological configuration can be different, this difference must correspond the different physical structure. This relation between topological configuration and physical structure gives an important property to classify some physical system which contains many different structures.
We thank Prof. B.Y. Ou for useful discussions. This project was supported by Fundamental Research Project of China.
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Figure caption
==============
Fig. 1. (a) The projecting of trajectory of topological particles on $%
(z-\gamma )$ plane for $C/V=0.$ $P_i$ denotes the $i$-th topological particle. (b). The energy levels for $C/V=0.$ Each level is labelled by the topological particle which corresponds to it.
Fig. 2. (a) The projecting of trajectory of topological particles on $%
(z-\gamma )$ plane for $C/V=2.$ $P_i$ denotes the $i$-th topological particle. (b). The energy levels for $C/V=2.$ Each level is labelled by the topological particle which corresponds to it.
| ArXiv |
---
abstract: |
We report spectroscopic observations of the 2.63 day, detached, F-type main-sequence eclipsing binary . We use our observations together with existing $uvby$ photometric measurements to derive accurate absolute masses and radii for the stars good to better than 1.5%. We obtain masses of $M_1 = 1.269 \pm 0.017~M_{\sun}$ and $M_2 =
0.7542 \pm 0.0059~M_{\sun}$, radii of $R_1 = 1.477 \pm 0.012~R_{\sun}$ and $R_2 = 0.7232 \pm 0.0091~R_{\sun}$, and effective temperatures of $6770 \pm 150$ K and $5020 \pm 150$ K for the primary and secondary stars, respectively. Both components appear to have their rotations synchronized with the motion in the circular orbit. A comparison of the properties of the primary with current stellar evolution models gives good agreement for a metallicity of ${\rm [Fe/H]} = -0.17$, which is consistent with photometric estimates, and an age of about 2.2 Gyr. On the other hand, the K2 secondary is larger than predicted for its mass by about 4%. Similar discrepancies are known to exist for other cool stars, and are generally ascribed to stellar activity. The system is in fact an X-ray source, and we argue that the main site of the activity is the secondary star. Indirect estimates give a strength of about 1 kG for the surface magnetic field on that star. A previously known close visual companion to is shown to be physically bound, making the system a hierarchical triple.
author:
- 'Jane C. Bright and Guillermo Torres,'
title: 'Absolute dimensions of the F-type eclipsing binary V2154 Cygni'
---
Introduction {#sec:introduction}
============
(also known as HD 203839, HIP 105584, BD+47 3386, and TYC 3594-1060-1; $V = 7.77$) is a 2.63 day eclipsing binary discovered by the [*Hipparcos*]{} team [@Perryman:1997], and found independently in 1996 by [@Martin:2003] in the course of a search for variable stars in the open cluster M39. Light curves in the $uvby$ Strömgren system were published by [@Rodriguez:2001], but the physical properties of the components were not derived by them because spectroscopy was lacking. The only spectroscopic work we are aware of are brief reports by [@Kurpinska:2000] listing preliminary values for the velocity amplitudes, and by [@Oblak:2004] giving preliminary masses and radii, though details of those analyses are unavailable. The very unequal depths of the eclipses ($\sim$0.3 mag for the primary and $\sim$0.05 mag for the secondary) suggest stars of rather different masses, making it an interesting object for followup because of the increased leverage for the comparison with stellar evolution models. This motivated us to carry out our own high-resolution spectroscopic observations of this star, which we report here. is known from [*Tycho-2*]{} observations to have a close, 047 visual companion about two magnitudes fainter than the binary [$\Delta B_T = 2.18$ mag, $\Delta V_T = 2.15$ mag; @Fabricius:2000]. We show below that it is physically associated, making a hierarchical triple system.
While the primary of the eclipsing pair is an early F star, the secondary is a much smaller K star in the range where previous observations have shown discrepancies with models [see, e.g., @Torres:2013]. The measured radii of such stars are sometimes larger than predicted, and their temperatures cooler than expected, both presumably due to the effects of magnetic activity and/or spots [e.g., @Chabrier:2007; @Morales:2010]. therefore presents an opportunity to determine accurate physical properties of the stars in a system with a mass ratio significantly different from unity, and to investigate any discrepancies with theory in connection with measures of stellar activity.
The layout of our paper is as follows. Our new spectroscopic observations are reported in Section \[sec:spectroscopy\], followed by a brief description in Section \[sec:photometry\] of the [@Rodriguez:2001] photometric measurements we incorporate into our analysis. The light curve fits are presented in Section \[sec:analysis\], along with consistency checks to support the accuracy of the results. With the spectroscopic and photometric parameters we then derive the physical properties of the system, given in Section \[sec:dimensions\], and compare them with current models of stellar structure and stellar evolution (Section \[sec:models\]). We discuss the results in the context of available activity measurements in Section \[sec:discussion\], and conclude with some final thoughts in Section \[sec:conclusions\].
Spectroscopic observations and analysis {#sec:spectroscopy}
=======================================
was placed on our spectroscopic program in October of 2001, and observed through June of 2007 with two nearly identical echelle instruments [Digital Speedometer; @Latham:1992] on the 1.5m telescope at the Oak Ridge Observatory in the town of Harvard (MA), and on the 1.5m Tillinghast reflector at the Fred L. Whipple Observatory on Mount Hopkins (AZ). Both instruments (now decommissioned) used intensified photon-counting Reticon detectors providing spectral coverage in a single echelle order 45 Å wide centered on the b triplet at 5187 Å. The resolving power delivered by these spectrographs was $R \approx 35,\!000$, and the signal-to-noise ratios achieved for the 80 usable observations of range from about 20 to 67 per resolution element of 8.5 . Wavelength solutions were carried out by means of exposures of a thorium-argon lamp taken before and after each science exposure, and reductions were performed with a custom pipeline. Observations of the evening and morning twilight sky were used to place the observations from the two instruments on the same velocity system and to monitor instrumental drifts [@Latham:1992].
Visual inspection of one-dimensional cross-correlation functions for each of our spectra indicated the presence of a star much fainter than the primary that we initially assumed was the secondary in . However, subsequent analysis with the two-dimensional cross-correlation algorithm TODCOR [@Zucker:1994] showed those faint lines to be stationary, while a third set of even weaker lines was noticed that moved in phase with the orbital period. This is therefore the secondary in the eclipsing pair, and the stationary lines correspond to the visual companion mentioned in the Introduction, as we show later, which falls within the 1 slit of the spectrograph. Consequently, for the final velocity measurements we used an extension of TODCOR to three dimensions [referred to here as TRICOR; @Zucker:1995] that uses three different templates, one for each star. In the following we refer to the binary components as stars 1 and 2, and to the tertiary as star 3. The templates were selected from a large library of synthetic spectra based on model atmospheres by R. L. Kurucz [see @Nordstrom:1994; @Latham:2002], computed for a range of temperatures ($T_{\rm eff}$), surface gravities ($\log g$), rotational broadenings ($v \sin i$, when seen in projection), and metallicities (\[m/H\]).
We selected the optimum parameters for the templates as follows, adopting solar metallicity throughout. For the primary star we ran a grid of one-dimensional cross-correlations against synthetic spectra over a wide range of temperatures and $v \sin i$ values [see @Torres:2002], for a fixed $\log g$ of 4.0 that is sufficiently close to our final estimate presented later. The best match, as measured by the cross-correlation coefficient averaged over all exposures, was obtained for interpolated values of $T_{\rm eff} =
6770 \pm 150$ K and $v \sin i = 26 \pm 2$ . The secondary and tertiary stars are faint enough (by factors of 25 and 9, respectively; see below) that they do not affect these results significantly. For the secondary the optimal $v \sin i$ from grids of TRICOR correlations was $12 \pm 2$ . However, due to its faintness we were unable to establish its temperature from the spectra themselves, so we relied on results from the light curve analysis described later in Section \[sec:analysis\]. The central surface brightness ratio $J$ provides an accurate measure of the temperature ratio between stars 1 and 2. Using the primary temperature from above, the $J$ value for the $y$ band, and the visual flux calibration by [@Popper:1980], we obtained $T_{\rm eff} = 5020 \pm 150$ K. The surface gravity was adopted as $\log g = 4.5$, appropriate for a main-sequence star of this temperature. For the tertiary we again adopted $\log g = 4.5$, and grids of correlations with TRICOR for a range of temperatures indicated a preference for a value of 5500 K, to which we assign a conservative uncertainty of 200 K. Similar correlation grids varying $v \sin i$ indicated no measurable line broadening for the tertiary, so we adopted $v \sin i = 0$ kms, with an estimated upper limit of 2 .
Radial velocities were then measured with TRICOR using values for the template parameters ($T_{\rm eff}$, $v \sin i$) in our library nearest to those given above: 6750 K and 25 for the primary, 5000 K and 12 for the secondary, and 5500 K and 0 for the tertiary. The light ratios we determined from our spectra are $L_2/L_1 = 0.036 \pm 0.004$ and $L_3/L_1 = 0.108 \pm
0.012$, corresponding to the mean wavelength of our observations (5187 Å).
[lccccccc]{} 51874.5314 & 36.96 & 1.11 & $-$20.74 & 9.28 & 17.32 & 1.85 & 0.4586\
52109.6581 & $-$41.58 & 0.67 & 128.03 & 5.55 & 19.35 & 1.11 & 0.8387\
52123.6422 & 79.58 & 0.65 & $-$85.40 & 5.42 & 19.62 & 1.08 & 0.1546\
52130.5621 & $-$52.47 & 0.75 & 135.05 & 6.22 & 18.83 & 1.24 & 0.7851\
52151.5379 & $-$53.69 & 0.69 & 143.48 & 5.73 & 19.73 & 1.14 & 0.7588
Because our spectra are only 45 Å wide, systematic errors in the velocities can result from lines shifting in and out of this window as a function of orbital phase [see @Latham:1996]. To estimate this effect we followed a procedure similar to that of [@Torres:1997] and created artificial triple-lined spectra based on our adopted templates, which we then processed with TRICOR in the same way as the real spectra. A comparison of the input and output velocities showed a phase-dependent pattern with maximum shifts of about 0.2 for the primary, 6 for the secondary, and 1.2 for the tertiary. We applied these shifts as corrections to the individual raw velocities, and the final measurements including all corrections are listed in Table \[tab:rvs\], along with their uncertainties. The velocities of the third star appear constant within their uncertainties, and have a mean of $+19.31 \pm
0.13$ (weighted average). A similar correction for systematic errors was applied to the light ratios, and is already included in the values reported above.
A weighted least-squares orbital fit to the primary and secondary velocities gives the elements and derived quantities presented in Table \[tab:specorbit\], where a circular orbit has been assumed. Tests allowing for eccentricity gave results consistent with zero, in agreement with similar experiments below based on the light curves. Initial solutions in which we included a possible systematic offset between the primary and secondary velocities, as may arise, e.g., from template mismatch, also gave a value consistent with zero. The observations and orbital fit are shown in Figure \[fig:rvs\]. The tertiary velocities, represented with triangles, are seen to be very close to the center-of-mass velocity, supporting the physical association.
[lc]{}\
$P$ (days) & 2.6306359 $\pm$ 0.0000039\
$T_{\rm max}$ (HJD$-$2,400,000) & 52973.58847 $\pm$ 0.00091\
$\gamma$ () & +19.408 $\pm$ 0.076\
$K_1$ () & 72.699 $\pm$ 0.092\
$K_2$ () & 122.298 $\pm$ 0.723\
$e$ & 0.0 (fixed)\
\
$M_1\sin^3 i$ ($M_{\sun}$) & 1.268 $\pm$ 0.017\
$M_2\sin^3 i$ ($M_{\sun}$) & 0.7535 $\pm$ 0.0059\
$q\equiv M_2/M_1$ & 0.5944 $\pm$ 0.0036\
$a_1\sin i$ (10$^6$ km) & 2.6298 $\pm$ 0.0033\
$a_2\sin i$ (10$^6$ km) & 4.424 $\pm$ 0.026\
$a \sin i$ ($R_{\sun}$) & 10.140 $\pm$ 0.038\
\
$N_{\rm obs}$ & 80\
Time span (days) & 2381.4\
Time span (cycles) & 905.3\
$\sigma_1$ () & 0.69\
$\sigma_2$ () & 5.74\
$\sigma_3$ () & 1.21
Photometric observations {#sec:photometry}
========================
The light curves used for our analysis are those published by [@Rodriguez:2001][^1], and were obtained between July and November of 1998 with the 0.9m telescope at the Sierra Nevada Observatory (Spain). The 852 observations were made on 28 nights using $uvby$ filters, with HD204626 () as the comparison star and HD204977 () as the check star. The standard deviations of the difference in magnitude between the comparison and check stars, which may be taken as an indication of the precision of the observations, were 0.0085, 0.0035, 0.0032, and 0.0043 mag for $u$, $v$, $b$, and $y$, respectively.
Light curve analysis {#sec:analysis}
====================
For the analysis of the light curves of this well-detached system we have adopted the Nelson-Davis-Etzel model [@Popper:1981; @Etzel:1981], as implemented in the JKTEBOP code[^2] [@Southworth:2013]. The free parameters of the fit are the period $P$ and reference epoch of primary minimum $T_{\rm min}$, the central surface brightness ratio $J \equiv J_2/J_1$, the sum of the relative radii $r_1+r_2$ normalized to the semimajor axis, the radius ratio $k
\equiv r_2/r_1$, the inclination angle $i$, and a magnitude zero point $m_0$. Because of the presence of the third star in the aperture we also included the third light parameter $L_3$ (fractional brightness of star 3 divided by the total light, at phase 0.25 from primary eclipse). The mass ratio was held fixed at the spectroscopic value ($q
= 0.5944$). Linear limb-darkening coefficients ($u_1$, $u_2$) were interpolated from the tables by [@Claret:2000] using the JKTLD code[^3] [@Southworth:2008], and gravity-darkening coefficients ($y_1$, $y_2$) were taken from the tabulations by [@Claret:2011] for the properties of the primary and secondary given earlier. Experiments with quadratic limb-darkening gave no improvement, so the linear law was used throughout. Initial fits that included the eccentricity as an additional free parameter indicated a value that was not significantly different from zero, consistent with the spectroscopic evidence, so the orbit was assumed to be circular.
[lcccc]{} $P$ (days) & 2.630607 (+14/$-$19) & 2.6306290 (+61/$-$75) & 2.6306305 (+56/$-$59) & 2.6306316 (+74/$-$81)\
$T_{\rm min}$ (HJD$-$2,400,000) & 51048.61797 (+28/$-$22) & 51048.61815 (+12/$-$14) & 51048.61814 (+11/$-$14) & 51048.61808 (+15/$-$12)\
$r_1+r_2$ & 0.2169 (+26/$-$17) & 0.2172 (+12/$-$14) & 0.2167 (+13/$-$15) & 0.2163 (+13/$-$14)\
$k \equiv r_2/r_1$ & 0.492 (+17/$-$14) & 0.486 (+18/$-$9) & 0.492 (+15/$-$14) & 0.473 (+24/$-$4)\
$i$ (deg) & 88.76 (+32/$-$79) & 88.39 (+61/$-$33) & 88.57 (+43/$-$48) & 87.79 (+82/$-$6)\
$J$ & 0.120 (+11/$-$9) & 0.1267 (+84/$-$58) & 0.193 (+11/$-$9) & 0.246 (+16/$-$11)\
$L_3$ & 0.087 (+35/$-$59) & 0.054 (+51/$-$29) & 0.093 (+34/$-$47) & 0.028 (+87/$-$1)\
$m_0$ (mag) & 0.33069 (+45/$-$45) & 0.69957 (+40/$-$32) & 0.44492 (+37/$-$31) & 0.20558 (+44/$-$29)\
\
$r_1$ & 0.1454 (+24/$-$22) & 0.1462 (+13/$-$23) & 0.1453 (+20/$-$23) & 0.1469 (+11/$-$31)\
$r_2$ & 0.0715 (+16/$-$12) & 0.0710 (+15/$-$8) & 0.0714 (+12/$-$13) & 0.0695 (+21/$-$2)\
$L_2/L_1$ & 0.0264 (+21/$-$23) & 0.0280 (+22/$-$10) & 0.0435 (+22/$-$24) & 0.0513 (+60/$-$5)\
$\sigma$ (mmag) & 8.56 & 3.58 & 3.28 & 3.79\
\
$u_1$ & 0.722 & 0.748 & 0.696 & 0.615\
$u_2$ & 0.929 & 0.892 & 0.854 & 0.768\
$y_1$ & 0.393 & 0.354 & 0.305 & 0.260\
$y_2$ & 1.157 & 0.892 & 0.672 & 0.581
Separate solutions for each of the $uvby$ bands are presented in Table \[tab:LCfits1\]. As the errors provided by JKTEBOP are strictly internal and do not capture systematic components that may result, e.g., from red noise, the uncertainties given in the table were computed with the residual permutation (“prayer bead”) method, as follows. We shifted the residuals from the original fits by an arbitrary number of time indices (with wraparound), and added them back into the computed curves to create artificial data sets that preserve any time-correlated noise that might be present in the original data. We generated 500 such data sets for each of the passbands and fitted them with JKTEBOP. In each solution we simultaneously perturbed all of the quantities that were initially held fixed. We did this by adding Gaussian noise to the mass ratio corresponding to its measured error ($\sigma_q = 0.0036$), and Gaussian noise with $\sigma = 0.1$ to the limb-darkening and gravity-darkening coefficients. The standard deviations of the resulting distributions for each parameter were adopted as the uncertainties for the light curve elements.
[lc]{} $r_1+r_2$ & 0.21696 $\pm$ 0.00087\
$k \equiv r_2/r_1$ & 0.4895 $\pm$ 0.0083\
$i$ (deg) & 88.55 $\pm$ 0.28\
$r_1$ & 0.1457 $\pm$ 0.0010\
$r_2$ & 0.07129 $\pm$ 0.00060\
$P$ (days) & 2.6306303 $\pm$ 0.0000038\
$T_{\rm min}$ (HJD$-$2,400,000) & 51048.618122 $\pm$ 0.000075
[cccc]{} $u$ & 0.119 (+12/$-$10) & 0.075 (+22/$-$13) & 0.0259 (+19/$-$16)\
$v$ & 0.1272 (+86/$-$82) & 0.069 (+22/$-$22) & 0.02851 (+75/$-$67)\
$b$ & 0.193 (+11/$-$11) & 0.086 (+22/$-$21) & 0.04311 (+81/$-$75)\
$y$ & 0.250 (+12/$-$14) & 0.101 (+21/$-$17) & 0.0560 (+13/$-$10)
The results from the four passbands are fairly consistent within their uncertainties, with a few exceptions: (1) The ephemeris ($P$, $T_{\rm
min}$) seems rather different for the $u$ band, which is the fit with the largest scatter. The fact that the $uvby$ measurements are simultaneous indicates this is almost certainly due to systematic errors affecting $u$ that are not uncommon. (2) The geometric parameters (most notably $k$ and $i$, and to a lesser extent $r_1+r_2$) seem systematically different for the $y$ band. Several features of that fit make us suspicious of these quantities, and of $L_3$ as well. In particular, $L_3$ is significantly lower than in the other bands, which runs counter to expectations given that the third star is cooler (redder) than the primary, and so its flux contribution ought to be larger in $y$, not smaller. Third light is always strongly (and positively) correlated with the inclination angle and with $k$ in this case, and indeed we see that both $i$ and $k$ are also low. Grids of JKTEBOP solutions over a range of fixed values of $k$ show that for all $k$ values the radius sum in the $y$ band is always considerably smaller than in the other three bands, which agree well among each other. Finally, we note that the $y$-band error bars for $k$, $i$, and $L_3$ are all highly asymmetric (always much larger in the direction toward the average of the $uvb$ results), which is not the case in the other bands. These features are symptomatic of strong degeneracies in $y$ that make the results highly prone to biases. We have therefore chosen not to rely on the geometric parameters from the $y$ band.
Weighted averages of the photometric period and epoch (excluding the $u$ band) and of the geometric parameters (excluding the $y$ band) are given in Table \[tab:LCfits2\]. The photometric period agrees well with the spectroscopic one, within the errors. The final solutions for the wavelength-dependent quantities were carried out by holding the ephemeris and geometry fixed to these values, and the results are collected in Table \[tab:LCfits3\]. We illustrate these final fits in Figure \[fig:LC\], where the secondary eclipse is seen to be total.
Consistency checks {#sec:consistency}
------------------
The spectroscopic light ratios reported in Section \[sec:spectroscopy\] ($L_2/L_1$ and $L_3/L_1$), which are independent of the light curve analysis above, offer an opportunity to test the accuracy of the light curve solutions. For the necessary flux transformation between the 5187 Å spectral window and the slightly redder Strömgren $y$ band (5470 Å) we used synthetic spectra from the PHOENIX library by [@Husser:2013], along with our adopted effective temperatures and surface gravities from Section \[sec:spectroscopy\], integrating the model fluxes over both passbands. An additional quantity that is needed to properly scale the spectral energy distributions is the radius ratio.
As a sanity check we first used these spectra coupled with our measured radius ratio of $k = 0.4895$ to calculate the $y$-band light ratio between the primary and secondary, and obtained $L_2/L_1 =
0.055$, in good agreement with our light curve value. The flux ratio we then infer at 5187 Å based on the same parameters is 0.039, which is consistent with the spectroscopic measurement of $0.036 \pm
0.004$ (see Figure \[fig:bands\]).
The scaling of the energy distributions of the tertiary and primary components requires knowledge of the radius ratio between those two stars, which our observations do not provide. We estimated it as follows. With our $y$-band light curve results from Table \[tab:LCfits3\] ($L_2/L_1$ and $L_3$) we calculated $L_3/L_1 =
L_3 (1 + L_2/L_1)/(1 - L_3) = 0.119$. We then used the PHOENIX synthetic spectra and varied the radius ratio until we reproduced this value of $L_3/L_1$, which occurred for $R_3/R_1 = 0.56$. With the scaling set in this way, the predicted flux ratio at 5187 Å between the tertiary and primary is 0.100, which again agrees with the spectroscopically measured ratio of $0.108 \pm 0.012$, as illustrated in Figure \[fig:bands\].
These consistency checks between the spectroscopy and the photometry are an indication that the light curve fits are largely free from biases, and support the accuracy of the geometric elements used in the next section to derive the physical properties of the stars.
Absolute dimensions {#sec:dimensions}
===================
The absolute masses and radii of are listed in Table \[tab:dimensions\]. The relative uncertainties are smaller than 1.5% for both components. The combined out-of-eclipse magnitudes of the system from [@Rodriguez:2001] and our fitted light ratios and third-light values enable us to deconvolve the light of the components. For the primary star we obtained the Strömgren indices $b-y = 0.243 \pm 0.035$, $m_1 = 0.139 \pm 0.063$, and $c_1 = 0.528 \pm
0.063$, along with $\beta = 2.691$. With these and the calibrations of [@Crawford:1975] we infer negligible reddening for the system (consistent with its small distance; see below), and an estimated photometric metallicity of ${\rm [Fe/H]} = -0.12$. Photometric estimates of the temperatures may be obtained from the $b-y$ index of the primary and the corresponding value for the secondary of $0.527
\pm 0.046$. The color/temperature calibration of [@Casagrande:2010] leads to values of $6840 \pm 200$ K and $5050
\pm 260$ K that are in good agreement with the spectroscopic values adopted in Section \[sec:spectroscopy\]. The deconvolved color of the third star ($b-y = 0.45 \pm 0.38$) is too uncertain to be useful, though the inferred temperature of $5500 \pm 870$ K again matches the value from Section \[sec:spectroscopy\]. The spectral types corresponding to the adopted temperatures are F2, K2, and G8 for the primary, secondary, and tertiary, respectively.
Additional quantities listed in Table \[tab:dimensions\] include the luminosities, absolute magnitudes, and the distance ($90 \pm 9$ pc), which makes use of the bolometric corrections by [@Flower:1996]. The corresponding parallax, $11.2 \pm 1.1$ mas, is not far from the trigonometric values listed in the [*Hipparcos*]{} catalog ($\pi_{\rm HIP} = 11.77 \pm 0.59$ mas) and in the first data release of [*Gaia*]{} [$\pi_{\rm Gaia} = 13.35 \pm
0.82$ mas; @Brown:2016]. Our measured projected rotational velocities are also quite close to the expected synchronous values ($v_{\rm sync} \sin i$).
As noted earlier, the third star was angularly resolved by the [ *Tycho-2*]{} experiment at a separation of 047 and a measured position angle of 59, at the mean epoch 1991.25. Subsequent astrometric measurements by a number of authors indicate a gradual decrease in the angular separation to 025 in 2010 [@Horch:2010], with no significant change in the position angle. This is inconsistent with being the result of a chance alignment with a background star, as the binary’s fairly large proper motion of 113 mas yr$^{-1}$ measured by [*Gaia*]{} would have carried the companion 2 away in that interval. The direction of motion would suggest a high inclined orbit, or possibly even an edge-on orientation. At our measured 90 pc distance the 047 separation implies a semimajor axis of roughly 42 au and an orbital period of $\sim$160 yr.
[lcc]{} Mass ($M_{\sun}$)& $1.269 \pm 0.017$ & $0.7542 \pm 0.0059$\
Radius ($R_{\sun}$)& $1.477 \pm 0.012$ & $0.7232 \pm 0.0091$\
$\log g$ (cgs)& $4.2028 \pm 0.0089$ & $4.597 \pm 0.012$\
Temperature (K)& $6770 \pm 150$& $5020 \pm 150$\
$\log L/L_{\sun}$& $0.616 \pm 0.039$ & $-0.523 \pm 0.039$\
$BC_{\rm V}$ (mag)& $-0.02 \pm 0.10$& $-0.30 \pm 0.11$\
$M_{\rm bol}$ (mag)& $3.192 \pm 0.098$ & $6.041 \pm 0.097$\
$M_V$ (mag)& $3.17 \pm 0.14$ & $6.34 \pm 0.17$\
$m-M$ (mag)&\
Distance (pc)&\
Parallax (mas)&\
$v_{\rm sync} \sin i$ ()& $28.4 \pm 0.2$& $13.9 \pm 0.2$\
$v \sin i$ ()& $26 \pm 2$& $12 \pm 2$
Comparison with stellar evolution models {#sec:models}
========================================
The accurate properties for are compared with predictions from current stellar evolution theory in Figure \[fig:mistlogg\]. The evolutionary tracks for the measured masses of the components were taken from the grid of MESA Isochrones and Stellar Tracks [MIST; @Choi:2016], which is based on the Modules for Experiments in Stellar Astrophysics package [MESA; @Paxton:2011; @Paxton:2013; @Paxton:2015]. The metallicity in the models was adjusted to ${\rm [Fe/H]} = -0.17$ to provide the best fit to the temperatures of the stars. This composition is not far from the photometric estimate reported earlier. The shaded areas in the figure indicate the uncertainty in the location of the tracks that comes from the errors in the measured masses. Solar-metallicity tracks are shown with dotted lines for reference. The age that best matches the radius of the primary is 2.2 Gyr (see below). An isochrone for this age is shown with a dashed line. The primary star is seen to be almost halfway through its main-sequence phase.
At this relatively old age it is not surprising that we found the components’ rotation to be synchronized with the motion in a circular orbit, as the theoretically expected timescales for synchronization and orbit circularization are $\sim$1 Myr and $\sim$200 Myr, respectively [e.g., @Hilditch:2001].
The radii and temperatures are shown separately as a function of mass in Figure \[fig:mistmassradius\], in which the solid line represents the 2.2 Gyr isochrone for ${\rm [Fe/H]} = -0.17$ that reproduces the measured radius of the primary star at its measured mass. A solar metallicity isochrone for the same age is shown with the dashed line. The secondary star is seen to be larger than predicted for its mass by almost 4%, corresponding to a nearly $\sim$3$\sigma$ discrepancy. Similar deviations from theory are known to be present in other stars with convective envelopes, and are usually attributed to the effects of stellar activity [see, e.g. @Popper:1997; @Torres:2013]. The bottom panel of the figure shows that the temperatures of the two components are consistent with the theoretical values for their mass within the errors. This is somewhat unexpected for the secondary, as stellar activity typically causes both “radius inflation” and “temperature suppression”, though the latter effect is smaller and not as easy to detect.
Aside from the brightness measurements and our spectroscopic estimates of $T_{\rm eff}$ and $v \sin i$ from Section \[sec:spectroscopy\], we have no direct information on the other fundamental physical properties of the tertiary. Based on our spectroscopically measured flux ratio of $L_3/L_1 = 0.108 \pm 0.012$ at 5187 Å and the above best-fit MIST isochrone, we infer $M_3 \approx 0.87~M_{\sun}$, $R_3
\approx 0.80~R_{\sun}$, and $T_{\rm eff} \approx 5490$ K. The temperature is consistent with that estimated directly from our spectra, and the radius ratio $R_3/R_1 \approx 0.54$ is not far from the value we found in a different way at the end of Section \[sec:analysis\].
Discussion {#sec:discussion}
==========
The $\sim$4% discrepancy between the measured and predicted radius for the K2 secondary in is in line with similar anomalies displayed by other late-type stars having significant levels of activity. While we do not detect any temperature suppression that often accompanies radius inflation, the fractional effect in $T_{\rm
eff}$ seen in other cases is typically half that of radius inflation, or only about 100 K in this case, which is smaller than our formal uncertainty.
is an X-ray source listed in the ROSAT All-Sky Survey [@Voges:1999], and is also reported to have shown at least one X-ray flare during those observations [@Fuhrmeister:2003]. This is a clear indication of magnetic activity in the system, though in principle the source could be any of the three stars, or even all three. From the ROSAT count rate of $0.082 \pm 0.012$ counts s$^{-1}$ and the measured hardness ratio (${\rm HR1} = -0.22 \pm 0.14$) we infer an X-ray flux of $F_{\rm X} = 5.9 \times
10^{-13}$ erg cm$^{-2}$ s$^{-1}$, adopting the energy conversion factor recommended by [@Fleming:1995]. Using our distance estimate of 90 pc we then derive an X-ray luminosity of $L_{\rm X} =
5.7 \times 10^{29}$ erg s$^{-1}$.
While fairly common in late-type objects (particularly if rotating rapidly), X-ray emission in stars much earlier than mid-F is generally not easy to explain because they lack sufficiently deep surface convective zones that are typically associated with magnetic activity generated by the dynamo effect. For this reason, X-rays in these stars are most often attributed to an unseen late-type companion [e.g., @Schroder:2007 and references therein], which can easily be hidden in the glare of the primary. Other mechanisms intrinsic to earlier-type stars are possible, such as shocks and instabilities in the radiatively driven winds, although these are not thought to be able to explain variability such as the X-ray flaring mentioned above [see, e.g. @Schmitt:2004; @Balona:2012]. We cannot completely rule out a priori that the primary in is the main source of the X-rays, but its much thinner convective envelope makes this seem far less likely than an origin in a later-type star such as the secondary or tertiary. Indeed, the MIST models indicate that the mass of the convective envelope of the secondary is about 7.2% of its total mass, and that of the tertiary is 4.5% (the value for the Sun is 1.6%), whereas the fractional mass of the primary’s envelope is only $3 \times 10^{-4}$.
The tertiary component in is a possible source for the X-rays, if it were a rapidly rotating star. However, our spectroscopy suggests it is not a fast rotator: we measure $v \sin i <
2$ (Section \[sec:spectroscopy\]), although the projection factor is unknown so it is concievable the equatorial rotation is much faster. To estimate the true rotation period we used the age of the system (2.2 Gyr) along with the gyrochronology relations of [@Epstein:2014] and the estimated $B-V$ color of the star from the MIST isochrones, and inferred $P_{\rm rot} \approx 18$ days. If attributed entirely to the tertiary, the measured X-ray luminosity of would be far in excess (by about an order of magnitude) of what is expected for a star of this mass and rotation period, according to studies of the relationship between stellar activity and rotation [e.g., @Pizzolato:2003]. This argues the X-rays are unlikely to originate mainly in the tertiary, although it is possible it has some small contribution.
We are thus left with the secondary as the most probable site of the bulk of the X-ray emission in . With the bolometric luminosity given in Table \[tab:dimensions\] we compute $\log L_{\rm X}/L_{\rm
bol} = -3.31$, a value that is close to the saturation level seen in very active stars. The study of [@Pizzolato:2003] indicates that this is in fact a typical value for a star of this mass with a rotation period of 2.63 days, supporting our conclusion that the secondary is the active star in the system. If that is the case, this provides a natural explanation for its inflated radius.
Recent stellar evolution models that incorporate the effects of magnetic fields have had some success in explaining radius inflation in stars like the secondary [see, e.g., @Feiden:2012; @Feiden:2013]. To achieve this, those models introduce a tunable parameter that is the average strength of the surface magnetic field, $\langle Bf\rangle$, where $B$ is the photospheric magnetic field strength and $f$ the filling factor. Measurements of magnetic field strengths are very difficult to make in binary systems, let alone in triple-lined systems such as , but they are essential in order to validate the fits that these models provide.
A rough estimate of $\langle Bf\rangle$ for the secondary may be obtained by taking advantage of a power-law relationship shown by [@Saar:2001] to exist between $\langle Bf\rangle$ and the Rossby number, $Ro \equiv P_{\rm rot}/\tau_c$, where $\tau_c$ is the convective turnover time. For consistency with the work of [@Saar:2001], we take $\tau_c$ from the theoretical calculations by [@Gilliland:1986], which give $\tau_c \approx 29$ days for a star with a temperature of 5020 K. The resulting Rossby number, $Ro
\approx 0.091$, together with the relation by [@Saar:2001] then yields $\langle Bf\rangle \approx 1.1$ kG.[^4] An independent way of estimating the magnetic field strength makes use of the X-ray luminosity and the empirical relationship between that quantity and the total unsigned surface magnetic flux, $\Phi = 4\pi
R^2 \langle Bf\rangle$. [@Pevtsov:2003] have shown in a study of magnetic field observations of the Sun and active stars that the relation holds over many orders of magnitude. With an updated version of that relation by [@Feiden:2013], and the measured radius of the secondary, we obtain $\langle Bf\rangle \approx 1.0$ kG, which is similar to our previous result. A magnetic field strength of this order is quite consistent with values measured in many other cool, active single stars [see, e.g., @Cranmer:2011; @Reiners:2012].
Our estimate of $\langle Bf\rangle \approx 1.0$ kG can serve as an input to stellar evolution calculations that model the effects of magnetic fields, and test their ability to match the measured size of the secondary.
is attended by a distant third star that is physically bound: we have shown that it has a similar radial velocity as the eclipsing pair, a brightness perfectly consistent with that expected for a star of its temperature at the same distance as the binary, and a motion on the plane of the sky that is incompatible with a background object but consistent with orbital motion in a highly inclined orbit around the binary (possibly even coplanar with it). The system is thus a hierarchical triple, which is not surprising given that [@Tokovinin:2006] have shown that up to 96% of all solar-type binaries with periods shorter than 3 days have third components.
Conclusions {#sec:conclusions}
===========
Our spectroscopic observations together with existing $uvby$ photometry have enabled us to derive accurate absolute masses and radii for the eclipsing components good to better than 1.5%, despite the faintness of the secondary (only 3.6% of the brightness of the primary). thus joins the ranks of binary systems with the best determined properties [see @Torresetal:2010]. The highly unequal masses provide increased leverage for the comparison with stellar evolution models, and we find that the K2 secondary is about 4% larger than predicted for its mass, though its temperature appears normal. Thus, the star appears overluminous. The detection of the system as an X-ray source is evidence of activity, and we have argued that the source is the secondary component. This would provide at least a qualitative explanation for the radius anomaly, which is also seen in many other active stars with convective envelopes. We would expect the secondary to have significant spot coverage, but the star is much too faint compared to the primary for this to produce a visible effect on the light curves.
is a good test case for recent stellar evolution models that attempt to explain radius inflation in a more quantitative way by including the effects of magnetic fields. To this end, we have provided an estimate of the strength of the surface magnetic field on the secondary ($\sim$1 kG).
Finally, we note that the study of this system would benefit from a detailed chemical analysis of the primary star based on high-resolution spectroscopy with broader wavelength coverage than the 45 Å afforded by the material at our disposal. This would remove the metallicity as a free parameter in the comparison with stellar evolution models, strengthening the results.
[**Note added in proof:**]{} A high-resolution ($R \approx
44,000$) echelle spectrum of with a signal-to-noise ratio of 220 in the b region was obtained recently at the Tillinghast reflector during the second quadrature (HJD 2,458,029.6, phase 0.73). It shows no sign of activity (e.g., H and K or H$\alpha$ emission) in the brighter primary, supporting our contention that this star is not particularly active.
We are grateful to P. Berlind, M. Calkins, D. W. Latham, R. P. Stefanik, and J. Zajac for help in obtaining the spectroscopic observations of , and to R. J. Davis and J. Mink for maintaining the CfA echelle database over the years. We also thank J. Choi for assistance in calculating the extent of stellar envelopes, and the anonymous referee for helpful comments. We acknowledge support from the SAO Research Experience for Undergraduates (REU) program, which is funded by the National Science Foundation (NSF) REU and Department of Defense ASSURE programs under NSF grant AST-1659473, and by the Smithsonian Institution. G.T. acknowledges partial support for this work from NSF grant AST-1509375. This research has made use of the SIMBAD and VizieR databases, operated at CDS, Strasbourg, France, and of NASA’s Astrophysics Data System Abstract Service.
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[^1]: We note, incidentally, that the heliocentric Julian dates in the online electronic files should be corrected by subtracting exactly 790 days.
[^2]: [ http://www.astro.keele.ac.uk/jkt/codes/jktebop.html]{}
[^3]: <http://www.astro.keele.ac.uk/jkt/codes/jktld.html>
[^4]: The same calculation applied to the tertiary star gives $\langle Bf\rangle
\approx 70$ G, which is small compared to the secondary and supports the notion that it is not a very active star. The parameters for the primary star are outside of the range of validity of the [@Saar:2001] relation, but point to a magnetic field strength of only a few Gauss, again suggesting a very low activity level if the sustaining mechanism is the same as in late-type stars.
| ArXiv |
---
abstract:
- |
This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of $k+1$ reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension $k$ which generalizes geodesic subspaces.
Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).
- 'This supplementary material details the notions of Riemannian geometry that are underlying the paper [*Barycentric Subspace Analysis on Manifolds*]{}. In particular, it investigates the Hessian of the Riemannian square distance whose definiteness controls the local regularity of the barycentric subspaces. This is exemplified on the sphere and the hyperbolic space.'
- 'This supplementary material details in length the proof that the flag of linear subspaces found by PCA optimizes the Accumulated Unexplained Variances (AUV) criterion in a Euclidean space.'
address:
- |
Asclepios team, Inria Sophia-Antipolis Méditerrannée\
2004 Route des Lucioles, BP93\
F-06902 Sophia-Antipolis Cedex, France\
- |
Asclepios team, Inria Sophia Antipolis\
2004 Route des Lucioles, BP93\
F-06902 Sophia-Antipolis Cedex, France
- |
Asclepios team, Inria Sophia-Antipolis Méditerranée\
2004 Route des Lucioles, BP93\
F-06902 Sophia-Antipolis Cedex, France\
author:
-
-
-
title:
- Barycentric Subspace Analysis on Manifolds
- |
Supplementary Materials A:\
Hessian of the Riemannian Squared Distance
- 'Supplementary Materials B: Euclidean PCA as an optimization in the flag space'
---
Introduction
============
In a Euclidean space, the principal $k$-dimensional affine subspace of the Principal Component Analysis (PCA) procedure is equivalently defined by minimizing the variance of the residuals (the projection of the data point to the subspace) or by maximizing the explained variance within that affine subspace. This double interpretation is available through Pythagoras’ theorem, which does not hold in more general manifolds. A second important observation is that principal components of different orders are nested, enabling the forward or backward construction of nested principal components.
Generalizing PCA to manifolds first requires the definition of the equivalent of affine subspaces in manifolds. For the zero-dimensional subspace, an intrinsic generalization of the mean on manifolds naturally comes into mind: the Fréchet mean is the set of global minima of the variance, as defined by [@frechet48] in general metric spaces. For simply connected Riemannian manifolds of non-positive curvature, the minimum is unique and is called the Riemannian center of mass. This fact was already known by Cartan in the 1920’s, but was not used for statistical purposes. [@karcher77; @buser_gromovs_1981] first established conditions on the support of the distribution to ensure the uniqueness of a local minimum in general Riemannian manifolds. This is now generally called Karcher mean, although there is a dispute on the naming [@karcher_riemannian_2014]. From a statistical point of view, [@Bhattacharya:2003; @Bhattacharya:2005] have studied in depth the asymptotic properties of the empirical Fréchet / Karcher means.
The one-dimensional component can naturally be a geodesic passing through the mean point. Higher-order components are more difficult to define. The simplest generalization is tangent PCA (tPCA), which amounts unfolding the whole distribution in the tangent space at the mean, and computing the principal components of the covariance matrix in the tangent space. The method is thus based on the maximization of the explained variance, which is consistent with the entropy maximization definition of a Gaussian on a manifold proposed by [@pennec:inria-00614994]. tPCA is actually implicitly used in most statistical works on shape spaces and Riemannian manifolds because of its simplicity and efficiency. However, if tPCA is good for analyzing data which are sufficiently centered around a central value (unimodal or Gaussian-like data), it is often not sufficient for distributions which are multimodal or supported on large compact subspaces (e.g. circles or spheres).
Instead of an analysis of the covariance matrix, [@fletcher_principal_2004] proposed the minimization of squared distances to subspaces which are totally geodesic at a point, a procedure coined Principal Geodesic Analysis (PGA). These Geodesic Subspaces (GS) are spanned by the geodesics going through a point with tangent vector restricted to a linear subspace of the tangent space. However, the least-squares procedure is computationally expensive, so that the authors approximated it in practice with tPCA, which led to confusions between tPCA and PGA. A real implementation of the original PGA procedure was only recently provided by [@sommer_optimization_2013]. PGA is allowing to build a flag (sequences of embedded subspaces) of principal geodesic subspaces consistent with a forward component analysis approach. Components are built iteratively from the mean point by selecting the tangent direction that optimally reduces the square distance of data points to the geodesic subspace. In this procedure, the mean always belongs to geodesic subspaces even when it is outside of the distribution support.
To alleviate this problem, [@huckemann_principal_2006], and later [@huckemann_intrinsic_2010], proposed to start at the first order component directly with the geodesic best fitting the data, which is not necessarily going through the mean. The second principal geodesic is chosen orthogonally to the first one, and higher order components are added orthogonally at the crossing point of the first two components. The method was named Geodesic PCA (GPCA). Further relaxing the assumption that second and higher order components should cross at a single point, [@sommer_horizontal_2013] proposed a parallel transport of the second direction along the first principal geodesic to define the second coordinates, and iteratively define higher order coordinates through horizontal development along the previous modes.
These are all intrinsically forward methods that build successively larger approximation spaces for the data. A notable exception is the concept of Principal Nested Spheres (PNS), proposed by [@jung_analysis_2012] in the context of planar landmarks shape spaces. A backward analysis approach determines a decreasing family of nested subspheres by slicing a higher dimensional sphere with affine hyperplanes. In this process, the nested subspheres are not of radius one, unless the hyperplanes passe through the origin. [@damon_backwards_2013] have recently generalized this approach to manifolds with the help of a “nested sequence of relations”. However, up to now, such a sequence was only known for spheres or Euclidean spaces.
We first propose in this paper new types of family of subspaces in manifolds: barycentric subspaces generalize geodesic subspaces and can naturally be nested, allowing the construction of inductive forward or backward nested subspaces. We then rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). To that end, we propose an extension of the unexplained variance criterion that generalizes nicely to flags of barycentric subspaces in Riemannian manifolds. This leads to a particularly appealing generalization of PCA on manifolds: Barycentric Subspaces Analysis (BSA).
Paper Organization {#paper-organization .unnumbered}
------------------
We recall in Section \[Sec:Geom\] the notions and notations needed to define statistics on Riemannian manifolds, and we introduce the two running example manifolds of this paper: $n$-dimensional spheres and hyperbolic spaces. Exponential Barycentric Subspaces (EBS) are then defined in Section \[Sec:Bary\] as the locus of weighted exponential barycenters of $k+1$ affinely independent reference points. The closure of the EBS in the original manifold is called affine span (this differs from the preliminary definition of [@pennec:hal-01164463]). Equations of the EBS and affine span are exemplified on our running examples: the affine span of $k+1$ affinely independent reference points is the great subsphere (resp. sub-hyperbola) that contains the reference points. In fact, other tuple of points of that subspace generates the same affine span, which is also a geodesic subspace. This coincidence is due to the very high symmetry of the constant curvature spaces.
Section \[Sec:KBS\] defines the Karcher (resp. Fréchet) barycentric subspaces (KBS, resp. FBS) as the local (resp. global) minima of the weighted squared distance to the reference points. As the definitions relies on distances between points and not on tangent vectors, they are also valid in more general non-Riemannian geodesic spaces. For instance, in stratified spaces, barycentric subspaces may naturally span several strata. For Riemannian manifolds, we show that our three definitions are subsets of each other (except possibly at the cut locus of the reference points): the largest one, the EBS, is composed of the critical points of the weighted variance. It forms a cell complex according to the index of the critical points. Cells of positive index gather local minima to form the KBS. We explicitly compute the Hessian on our running spherical and hyperbolic examples. Numerical tests show that the index can be arbitrary, thus subdividing the EBS into several regions for both positively and negatively curved spaces. Thus, the KBS consistently covers only a small portion of the affine span in general and is a less interesting definition for subspace analysis purposes.
For affinely independent points, we show in Section \[Sec:Prop\] that the regular part of a barycentric subspace is a stratified space which is locally a submanifold of dimension $k$. At the limit, points may coalesce along certain directions, defining non local jets[^1] instead of a regular $k+1$-tuple. Restricted geodesic subspaces, which are defined by $k$ vectors tangent at a point, correspond to the limit of the affine span when the $k$-tuple converges towards that jet.
Finally, we discuss in Section \[Sec:BSA\] the use of these barycentric subspaces to generalize PCA on manifolds. Barycentric subspaces can be naturally nested by defining an ordering of the reference points. Like for PGA, this enables the construction of a forward nested sequence of subspaces which contains the Fréchet mean. In addition, BSA also provides backward nested sequences which may not contain the mean. However, the criterion on which these constructions are based can be optimized for each subspace independently but not consistently for the whole sequence of subspaces. In order to obtain a global criterion, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchies of properly embedded linear subspaces of increasing dimension). To that end, we propose an extension of the unexplained variance criterion (the Accumulated Unexplained Variance (AUV) criterion) that generalizes nicely to flags of affine spans in Riemannian manifolds. This results into a particularly appealing generalization of PCA on manifolds, that we call Barycentric Subspaces Analysis (BSA).
Riemannian geometry {#Sec:Geom}
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In Statistics, directional data occupy a place of choice [@dryden2005; @huckemann_principal_2006]. Hyperbolic spaces are also the simplest models of negatively curved spaces which model the space of isotropic Gaussian parameters with the Fisher-Rao metric in information geometry [@costa_fisher_2015]. As non-flat constant curvature spaces, both spherical and hyperbolic spaces are now considered in manifold learning for embedding data [@wilson_spherical_2014]. Thus, they are ideal examples to illustrate the theory throughout this paper.
Tools for computing on Riemannian manifolds
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We consider a differential manifold ${\ensuremath{{\cal M}}}$ endowed with a smooth scalar product ${\ensuremath{ \left<
\:.\:\left|\:.\right.\right> }}_{x}$ called the Riemannian metric on each tangent space $T_{x}{\ensuremath{{\cal M}}}$ at point $x$ of ${\ensuremath{{\cal M}}}$. In a chart, the metric is specified by the dot product of the tangent vector to the coordinate curves: $g_{ij}(x) = {\ensuremath{ \left<
\:\partial_i\:\left|\:\partial_j\right.\right> }}_x$. The Riemannian distance between two points is the infimum of the length of the curves joining these points. Geodesics, which are critical points of the energy functional, are parametrized by arc-length in addition to optimizing the length. We assume here that the manifold is geodesically complete, i.e. that the definition domain of all geodesics can be extended to ${\ensuremath{\mathbb{R}}}$. This means that the manifold has no boundary nor any singular point that we can reach in a finite time. As an important consequence, the Hopf-Rinow-De Rham theorem states that there always exists at least one minimizing geodesic between any two points of the manifold (i.e. whose length is the distance between the two points).
#### Normal coordinate system
From the theory of second order differential equations, we know that there exists one and only one geodesic $\gamma_{(x,v)}(t)$ starting from the point $x$ with the tangent vector $v \in T_{x}{\ensuremath{{\cal M}}}$. The exponential map at point $x$ maps each tangent vector $v \in T_{x}{\ensuremath{{\cal M}}}$ to the point of the manifold that is reached after a unit time by the geodesic: $ \exp_{x}(v) = \gamma_{(x,v)}(1)$. The exponential map is locally one-to-one around $0$: we denote by ${\ensuremath{\overrightarrow{xy}}}=\log_{x}(y)$ its inverse. The injectivity domain is the maximal domain $D(x) \subset T_{x}{\ensuremath{{\cal M}}}$ containing $0$ where the exponential map is a diffeomorphism. This is a connected star-shape domain limited by the tangential cut locus $\partial D(x) = C(x) \subset T_{x}{\ensuremath{{\cal M}}}$ (the set of vectors $t v$ where the geodesic $\gamma_{(x, v)}(t)$ ceases to be length minimizing). The cut locus ${\ensuremath{{\cal C}}}(x) = \exp_{x}(C(x)) \subset {\ensuremath{{\cal M}}}$ is the closure of the set of points where several minimizing geodesics starting from $x$ meet. The image of the domain $D(x)$ by the exponential map covers all the manifold except the cut locus, which has null measure. Provided with an orthonormal basis, exp and log maps realize a normal coordinate system at each point $x$. Such an atlas is the basis of programming on Riemannian manifolds as exemplified in [@pennec:inria-00614990].
#### Hessian of the squared Riemannian distance
On ${\ensuremath{{\cal M}}}\setminus C(y)$, the Riemannian gradient $\nabla^a = g^{ab} \partial_b$ of the squared distance $d^2_y(x)={\ensuremath{\:\mbox{\rm dist}}}^2(x, y)$ with respect to the fixed point $y$ is $\nabla d^2_y(x) = -2 \log_x(y)$. The Hessian operator (or double covariant derivative) $\nabla^2$ is the covariant derivative of the gradient. In a normal coordinate at the point $x$, the Christoffel symbols vanish at $x$ so that the Hessian of the square distance can be expressed with the standard differential $D_x$ with respect to the footpoint $x$: $\nabla^2 d^2_y(x) = -2 (D_x \log_x(y))$. It can also be written in terms of the differentials of the exponential map as $ \nabla^2 d^2_y(x) = ( \left. D\exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}})^{-1} \left. D_x \exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}}$ to explicitly make the link with Jacobi fields. Following [@brewin_riemann_2009], we computed in \[suppA\] the Taylor expansion of this matrix in a normal coordinate system at $x$: $$\label{eq:Diff_log}
\left[ D_x \log_x(y) \right]^a_b = -\delta^a_b + \frac{1}{3} R^a_{cbd} {\ensuremath{\overrightarrow{xy}}}^c {\ensuremath{\overrightarrow{xy}}}^d + \frac{1}{12} \nabla_c R^a_{dbe} {\ensuremath{\overrightarrow{xy}}}^c {\ensuremath{\overrightarrow{xy}}}^d {\ensuremath{\overrightarrow{xy}}}^e + O(\varepsilon^3).$$ Here, $R^a_{cbd}(x)$ are the coefficients of the curvature tensor at $x$ and Einstein summation convention implicitly sums upon each index that appear up and down in the formula. Since we are in a normal coordinate system, the zeroth order term is the identity matrix, like in Euclidean spaces, and the first order term vanishes. The Riemannian curvature tensor appears in the second order term and its covariant derivative in the third order term. Curvature is the leading term that makes this matrix departing from the identity (the Euclidean case) and may lead to the non invertibility of the differential.
#### Moments of point distributions
Let $\{x_0,\ldots x_k\}$ be a set of $k+1$ points on a Manifold provided with weights $(\lambda_0, \ldots \lambda_k)$ that do not sum to zero. We may see these weighted points as the weighted sum of Diracs $\mu(x) = \sum_i \lambda_i \delta_{x_i}(x)$. As this distribution is not normalized and weights can be negative, it is generally not a probability. It is also singular with respect to the Riemannian measure. Thus, we have to take care in defining its moments as the Riemannian log and distance functions are not smooth at the cut-locus.
\[$(k+1)$-pointed / punctured Riemannian manifold\] $ $\
Let $\{x_0, \ldots x_k\} \in {\ensuremath{{\cal M}}}^{k+1}$ be a set of $k+1$ reference points in the $n$-dimensional Riemannian manifold ${\ensuremath{{\cal M}}}$ and $C(x_0, \ldots x_k) = \cup_{i=0}^k C(x_i)$ be the union of the cut loci of these points. We call the object consisting of the smooth manifold ${\ensuremath{{\cal M}}}$ and the $k+1$ reference points a $(k+1)$-pointed manifold. Likewise, we call the submanifold $ {\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}= {\ensuremath{{\cal M}}}\setminus C(x_0, \ldots x_k)$ of the non-cut points of the $k+1$ reference points a $(k+1)$-punctured manifold.
On $ {\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$, the distance to the points $\{x_0, \ldots x_k\}$ is smooth. The Riemannian log function ${\ensuremath{\overrightarrow{x x_i}}} = \log_x(x_i)$ is also well defined for all the points of ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$. Since the cut locus of each point is closed and has null measure, the punctured manifold ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$ is open and dense in ${\ensuremath{{\cal M}}}$, which means that it is a submanifold of ${\ensuremath{{\cal M}}}$. However, this submanifold is not necessarily connected. For instance in the flat torus $(S_1)^n$, the cut-locus of $k+1 \leq n$ points divides the torus into $k^n$ disconnected cells.
\[Weighted moments of a $(k+1)$-pointed manifold\] $ $\
Let $(\lambda_0, \ldots \lambda_k) \in {\ensuremath{\mathbb{R}}}^{k+1}$ such that $\sum_i \lambda_i \not = 0$. We call ${{\underaccent{\bar}{\lambda}}}_i = \lambda_i / (\sum_{j=0}^k \lambda_j)$ the normalized weights. The weighted $p$-th order moment of a $(k+1)$-pointed Riemannian manifold is the $p$-contravariant tensor: $${\mathfrak M}_{p}(x,\lambda) = \sum_i \lambda_i \underbrace{{\ensuremath{\overrightarrow{xx_i}}} \otimes {\ensuremath{\overrightarrow{xx_i}}} \ldots \otimes {\ensuremath{\overrightarrow{xx_i}}}}_{\text{$p$ times}}.$$ The normalized $p$-th order moment is: $\underline{\mathfrak M}_p(x,\lambda) = {\mathfrak M}_p(x,{{\underaccent{\bar}{\lambda}}}) = {\mathfrak M}_p(x,\lambda)/ {\mathfrak M}_0(\lambda).$ Both tensors are smoothly defined on the punctured manifold ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$.
The 0-th order moment ${\mathfrak M}_0(\lambda) = \sum_i \lambda_i = \mathds{1}{^{\text{\tiny T}}}\lambda$ is the mass. The $p$-th order moment is homogeneous of degree 1 in $\lambda$ while the normalized $p$-th order moment is naturally invariant by a change of scale of the weights. For a fixed weight $\lambda$, the first order moment ${\mathfrak M}_1(x,\lambda) = \sum_i \lambda_i {\ensuremath{\overrightarrow{xx_i}}}$ is a smooth vector field on the manifold ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$ whose zeros will be the subject of our interest. The second and higher order moments are smooth $(p,0)$ tensor fields that will be used in contraction with the Riemannian curvature tensor.
#### Affinely independent points on a manifold
In a Euclidean space, $k+1$ points are affinely independent if their affine combination generates a $k$ dimensional subspace, or equivalently if none of the point belong to the affine span of the $k$ others. They define in that case a $k$-simplex. Extending these different definitions to manifolds lead to different notions. We chose a definition which rules out the singularities of constant curvature spaces and which guaranties the existence of barycentric subspaces around reference point. In the sequel, we assume by default that the $k+1$ reference points of pointed manifolds are affinely independent (thus $k \leq n$). Except for a few examples, the study of singular configurations is left for a future work.
A set of $k+1$ points $\{x_0,\ldots x_k\}$ is affinely independent if no point is in the cut-locus of another and if all the sets of $k$ vectors $\{ \log_{x_i}(x_j) \}_{0 \leq j \not = i \leq k} \in T_{x_i}{\ensuremath{{\cal M}}}^k$ are linearly independent. \[def:AffineIndependence\]
Example on the sphere ${\cal S}_n$
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We consider the unit sphere in dimension $n \geq 1$ embedded in ${\ensuremath{\mathbb{R}}}^{n+1}$. The tangent space at $x$ is the space of vectors orthogonal to $x$: $T_x{\cal S}_n = \{ v \in {\ensuremath{\mathbb{R}}}^{n+1}, v{^{\text{\tiny T}}}x =0\}$ and the Riemannian metric is inherited from the Euclidean metric of the embedding space. With these conventions, the Riemannian distance is the arc-length $d(x,y) = \arccos( x{^{\text{\tiny T}}}y)= \theta \in [0,\pi]$. Using the smooth function $f(\theta) = { \theta}/{\sin\theta}$ from $]-\pi;\pi[$ to ${\ensuremath{\mathbb{R}}}$ which is always greater than one, the spherical exp and log maps are: $$\begin{aligned}
\exp_x(v) & = & \cos(\| v\|) x + \sin(\| v\|) v / \| v\| \\
\log_x(y) & = & f(\theta) \left( y - \cos\theta\: x \right)
\quad \text{with} \quad \theta = \arccos(x{^{\text{\tiny T}}}y).\end{aligned}$$
#### Hessian
The orthogonal projection $v=({\ensuremath{\:\mathrm{Id}}}-x x{^{\text{\tiny T}}})w$ of a vector $w \in {\ensuremath{\mathbb{R}}}^{n+1}$ onto the tangent space $T_x{\cal S}_n$ provides a chart around a point $x\in {\cal S}_n$ where we can compute the gradient and Hessian of the squared Riemannian distance (detailed in \[suppA\]). Let $u={({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})y }/ {\sin \theta} = { \log_x(y) }/{\theta}$ be the unit tangent vector pointing from $x$ to $y$, we obtain: $$\begin{aligned}
H_x(y) = \nabla^2 d^2_y(x) & = &2 u u{^{\text{\tiny T}}}+ 2 f( \theta )\cos\theta ({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}}- u u{^{\text{\tiny T}}}).
\label{eq:HessDistSphere2}\end{aligned}$$ By construction, $x$ is an eigenvector with eigenvalue $0$. Then the vector $u$ (or equivalently $\log_x(y) = \theta u$) is an eigenvector with eigenvalue $1$. To finish, every vector which is orthogonal to these two vectors (i.e. orthogonal to the plane spanned by 0, $x$ and $y$) has eigenvalue $ f(\theta)\cos\theta = \theta \cot \theta$. This last eigenvalue is positive for $\theta \in [0,\pi/2[$, vanishes for $\theta = \pi/2$ and becomes negative for $\theta \in ]\pi/2 \pi[$. We retrieve here the results of [@buss_spherical_2001 lemma 2] expressed in a more general coordinate system.
#### Moments of a $k+1$-pointed sphere
We denote a set of $k+1$ point on the sphere and the matrix of their coordinates by $X=[x_0,\ldots x_k]$. The cut locus of $x_i$ is its antipodal point $-x_i$ so that the $(k+1)$-punctured manifold is ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}= {\cal S}_n \setminus -X$. Using the invertible diagonal matrix $F(X,x) = \mbox{Diag}( f( \arccos(x_i {^{\text{\tiny T}}}x) ) )$, the first weighted moment is: $${\mathfrak M}_1(x, \lambda) = \textstyle \sum_i \lambda_i {\ensuremath{\overrightarrow{x x_i}}} = ( {\ensuremath{\:\mathrm{Id}}}- x x{^{\text{\tiny T}}}) X F(X,x) \lambda.
\label{eq:MomemtSphere}$$
#### Affine independence of the reference points
Because no point is antipodal nor identical to another, the plane generated by 0, $x_i$ and $x_j$ in the embedding space is also generated by 0, $x_i$ and the tangent vector $\log_{x_i}(x_j)$. This can be be seen using a stereographic projection of pole $-x_i$ from ${\cal S}_n$ to $T_{x_i} {\cal S}_n$. Thus, 0, $x_i$ and the $k$ independent vectors $\log_{x_i}(x_j)$ ($j \not = i$) generate the same linear subspace of dimension $k+1$ in the embedding space than the points $\{0, x_0,\ldots x_k\}$. We conclude that $k+1$ points on the sphere are affinely independent if and only if the matrix $X=[x_0,\ldots x_k]$ has rank $k+1$.
Example on the hyperbolic space ${\ensuremath{\mathbb{H}}}^n$ {#sec:hyperbolic}
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We now consider the hyperboloid of equation $-x_0^2 + x_1^2 \ldots x_n^2 = -1$ ($x_0 > 0$) embedded in ${\ensuremath{\mathbb{R}}}^{n+1}$ ($n \geq 2$). Using the notation $x=(x_0,\hat x)$ and the indefinite non-degenerate symmetric bilinear form ${\ensuremath{ \left<
\:x\:\left|\:y\right.\right> }}_* = x{^{\text{\tiny T}}}J y= \hat x{^{\text{\tiny T}}}\hat y -x_0 y_0$ with $ J = \mbox{diag}(-1, {\ensuremath{\:\mathrm{Id}}}_n)$, the hyperbolic space ${\ensuremath{\mathbb{H}}}^n$ can be seen as the pseudo-sphere $\|x\|^2_* = \|\hat x\|^2 -x_0^2 = -1$ of radius -1 in the Minkowski space ${\ensuremath{\mathbb{R}}}^{1,n}$. A point can be parametrized by $x=(\sqrt{1+\|\hat x\|^2}, \hat x)$ for $\hat x \in {\ensuremath{\mathbb{R}}}^n$ (Weierstrass coordinates). The restriction of the Minkowski pseudo-metric of the embedding space ${\ensuremath{\mathbb{R}}}^{1,n}$ to the tangent space of $T_x{\ensuremath{\mathbb{H}}}^n$ is positive definite. It defines the natural Riemannian metric on the hyperbolic space. With these conventions, geodesics are the trace of 2-planes passing through the origin and the Riemannian distance is the arc-length $d(x,y) = \operatorname{arccosh}( - {\ensuremath{ \left<
\:x\:\left|\:y\right.\right> }}_* )$. Using the smooth positive function $f_*(\theta) = { \theta}/{\sinh(\theta)}$ from ${\ensuremath{\mathbb{R}}}$ to $]0,1]$, the hyperbolic exp and log maps are: $$\begin{aligned}
\exp_x(v) & = & \cosh(\| v\|_* ) x + {\sinh(\| v\|_* )} v / {\| v\|_* } \\
\log_x(y) & = & f_*(\theta) \left( y - \cosh(\theta) x \right)
\quad \text{with} \quad \theta = \operatorname{arccosh}( -{\ensuremath{ \left<
\:x\:\left|\:y\right.\right> }}_* ).\end{aligned}$$
#### Hessian
The orthogonal projection $v=w + {\ensuremath{ \left<
\:w\:\left|\:x\right.\right> }}_* x = ({\ensuremath{\:\mathrm{Id}}}+ x x{^{\text{\tiny T}}}J) w$ of a vector $w\in {\ensuremath{\mathbb{R}}}^{1,n}$ onto the tangent space at $T_x {\ensuremath{\mathbb{H}}}^n$ provides a chart around the point $x\in {\ensuremath{\mathbb{H}}}^n$ where we can compute the gradient and Hessian of the hyperbolic squared distance (detailed in \[suppA\]). Let $u= { \log_x(y) }/{\theta}$ be the unit tangent vector pointing from $x$ to $y$, the Hessian is: $$H_x(y) = \nabla^2 d^2_y(x) = 2 J \left( u u{^{\text{\tiny T}}}+ \theta \coth \theta (J + x x{^{\text{\tiny T}}}-u u{^{\text{\tiny T}}}) \right) J
\label{eq:GradDistHyper}$$ By construction, $x$ is an eigenvector with eigenvalue $0$. The vector $u$ (or equivalently $\log_x(y) = \theta u$) is an eigenvector with eigenvalue $1$. Every vector orthogonal to these two vectors (i.e. to the plane spanned by 0, $x$ and $y$) has eigenvalue $ \theta \coth \theta \geq 1$ (with equality only for $\theta=0$). Thus, the Hessian of the squared distance is always positive definite. As a consequence, the squared distance is a convex function and has a unique minimum. This was of course expected for a negatively curved space [@bishop_manifolds_1969].
#### Moments of a $k+1$-pointed hyperboloid
We now pick $k+1$ points on the hyperboloid whose matrix of coordinates is denoted by $X=[x_0,\ldots x_k]$. Since there is no cut-locus, the $(k+1)$-punctured manifold is the manifold itself: ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}= {\ensuremath{{\cal M}}}= {{\ensuremath{\mathbb{H}}}}^n$. Using the invertible diagonal matrix $F_*(X,x) = \mbox{Diag}( f_*( \operatorname{arccosh}( -{\ensuremath{ \left<
\:x_i\:\left|\:x\right.\right> }}_* ) ) )$, the first weighted moment is $${\mathfrak M}_1(x, \lambda) = \textstyle \sum_i \lambda_i \log_x(x_i) = ({\ensuremath{\:\mathrm{Id}}}+ x x{^{\text{\tiny T}}}J) X F_*(X,x) \lambda.
\label{eq:MomemtHyperboloid}$$
#### Affine independence
As for the sphere, the origin, the point $x_i$ and the $k$ independent vectors $\log_{x_i}(x_j) \in T_{x_i}{{\ensuremath{\mathbb{H}}}}^n$ ($j \not = i$) generate the same $k+1$ dimensional linear subspace of the embedding Minkowski space ${\ensuremath{\mathbb{R}}}^{1,n}$ than the points $\{x_0, \ldots x_k\}$. Thus, $k+1$ points on the hyperboloid are affinely independent if and only if the matrix $X$ has rank $k+1$.
Exponential Barycentric Subspaces (EBS) and Affine Spans {#Sec:Bary}
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Affine subspaces in a Euclidean space
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In Euclidean PCA, a zero dimensional space is a point, a one-dimensional space is a line, and an affine subspace of dimension $k$ is generated by a point and $k \leq n$ linearly independent vectors. We can also generate such a subspace by taking the affine hull of $k+1$ affinely independent points: $\operatorname{Aff}(x_0,\ldots x_k) =\left\{ x = \sum_i \lambda_i x_i, \text{with} \sum_{i=0}^k \lambda_i = 1\right\}.$ These two definitions are equivalent in a Euclidean space, but turn out to have different generalizations in manifolds.
When there exists a vector of coefficients $\lambda = (\lambda_0, \lambda_1, \ldots, \lambda_k) \in {\ensuremath{\mathbb{R}}}^{k+1}$ (which do not sum to zero) such that $\sum_{i=0}^k \lambda_i (x_i-x) =0,$ then $\lambda$ is called the barycentric coordinates of the point $x$ with respect to the $k$-simplex $\{x_0, \ldots x_k\}$. When points are dependent, some extra care has to be taken to show that the affine span is still well defined but with a lower dimensionality. Barycentric coordinates are homogeneous of degree one:
\[def:Pk\] Barycentric coordinates of $k+1$ points live in the real projective space ${\ensuremath{\mathbb{R}}}P^n = ({\ensuremath{\mathbb{R}}}^{k+1} \setminus \{0\})/{\ensuremath{\mathbb{R}}}^*$ from which we remove the codimension 1 subspace $\mathds{1}^{\perp}$ orthogonal to the point $\mathds{1} = (1:1: \ldots 1)$: $$\textstyle
{\ensuremath{{\cal P}^*_k}}= \left\{ \lambda=(\lambda_0 : \lambda_1 : \ldots : \lambda_k) \in {\ensuremath{\mathbb{R}}}P^n \text{ s.t. } \mathds{1}^{\top}\lambda \not = 0 \right\}.$$
[r]{}[0.5]{} {width="50.00000%"}
Projective points are represented by lines through 0 in Fig.\[fig:P2\]. Standard representations are given by the intersection of the lines with the “upper” unit sphere $S_k$ of ${\ensuremath{\mathbb{R}}}^{k+1}$ with north pole $\mathds{1}/\sqrt{k+1}$ or by the affine $k$-plane of ${\ensuremath{\mathbb{R}}}^{k+1}$ passing through the point $\mathds{1}/(k+1)$ and orthogonal to this vector. This last representation give the normalized weight $ \underline{\lambda}_i= \lambda_i / (\sum_{j=0}^k \lambda_j)$: the vertices of the simplex have homogeneous coordinates $(1 : 0 : ... : 0) \ldots (0 : 0 : ... : 1)$. To prevent weights to sum up to zero, we have to remove the codimension 1 subspace $\mathds{1}^{\perp}$ orthogonal to the projective point $\mathds{1} = (1:1: \ldots 1)$ (blue line in Fig.\[fig:P2\]). This excluded subspace corresponds to the equator of the pole $\mathds{1}/\sqrt{k+1}$ for the sphere representation (points $C$ and $-C$ identified in Fig.\[fig:P2\]), and to the projective completion (points at infinity) of the affine $k$-plane of normalized weights.
EBS and Affine Span in Riemannian manifolds
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\[Barycentric coordinates in a $(k+1)$-pointed manifold\] A point $x \in {\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$ has barycentric coordinates $\lambda \in {\ensuremath{{\cal P}^*_k}}$ with respect to $k+1$ reference affinely independent points if $$\label{eq:Bary}
{\mathfrak M}_1(x,\lambda) = \textstyle \sum_{i=0}^k \lambda_i {\ensuremath{\overrightarrow{x x_i}}} = 0 .$$
Since the Riemannian log function ${\ensuremath{\overrightarrow{x x_i}}} = \log_x(x_i)$ is multiply defined on the cut locus of $x_i$, this definition cannot be extended to the the union of all cut loci $C(x_0, \ldots x_k)$, which is why we restrict the definition to ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$.
The EBS of the affinely independent points $(x_0,\ldots x_k) \in {\ensuremath{{\cal M}}}^{k+1}$ is the locus of weighted exponential barycenters of the reference points in ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$: $$\mbox{EBS}(x_0, \ldots x_k) = \{ x\in {\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}| \exists \lambda \in {\ensuremath{{\cal P}^*_k}}: {\mathfrak M}_1(x,\lambda) =0 \}.$$
The reference points could be seen as landmarks in the manifold. This definition is fully symmetric wit respect to all of them, while one point is privileged in geodesic subspaces. We could draw a link with archetypal analysis [@Cutler:1994:AA] which searches for extreme data values such that all of the data can be well represented as convex mixtures of the archetypes. However, extremality is not mandatory in our framework.
The subspace of barycentric coordinates $\Lambda(x) = \{ \lambda \in {\ensuremath{{\cal P}^*_k}}| {\mathfrak M}_1(x,\lambda) =0 \}$ at point $x \in {\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$ is either void, a point, or a linear subspace of ${\ensuremath{{\cal P}^*_k}}$.
We see that a point belongs to $\operatorname{EBS}(x_0, \ldots x_k)$ if and only if $\Lambda(x) \not = \emptyset$. Moreover, any linear combination of weights that satisfy the equation is also a valid weight so that $\Lambda(x)$ can only be a unique point (dimension 0) or a linear subspace of ${\ensuremath{{\cal P}^*_k}}$. The dimension of the dual space $\Lambda(x)$ is actually controlling the local dimension of the barycentric space, as we will see below.
The discontinuity of the Riemannian log on the cut locus of the reference points may hide the continuity or discontinuities of the exponential barycentric subspace. In order to ensure the completeness and potentially reconnect different components, we consider the closure of this set.
The affine span is the closure of the EBS in ${\ensuremath{{\cal M}}}$: $ \operatorname{Aff}(x_0, \ldots x_k) = \overline{\mbox{EBS}}(x_0, \ldots x_k).$ Because we assumed that ${\ensuremath{{\cal M}}}$ is geodesically complete, this is equivalent to the metric completion of the EBS.
Characterizations of the EBS
----------------------------
Let $Z(x)= [{\ensuremath{\overrightarrow{x x_0}}},\ldots {\ensuremath{\overrightarrow{x x_k}}}]$ be the smooth field of $n\times (k+1)$ matrices of vectors pointing from any point $x \in {\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$ to the reference points. We can rewrite the constraint $\sum_i \lambda_i {\ensuremath{\overrightarrow{x x_i}}} =0$ in matrix form: ${\mathfrak M}_1(x,\lambda) = Z(x)\lambda =0,$ where $\lambda$ is the $k+1$ vector of homogeneous coordinates $\lambda_i$.
\[THM1\] Let $Z(x)=U(x)\: S(x)\: V(x){^{\text{\tiny T}}}$ be a singular decomposition of the $n\times (k+1)$ matrix fields $Z(x)= [{\ensuremath{\overrightarrow{x x_0}}},\ldots {\ensuremath{\overrightarrow{x x_k}}}]$ on ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$ with singular values $\{s_i(x)\}_{0\leq i \leq k}$ sorted in decreasing order. $\mbox{EBS}(x_0, \ldots x_k)$ is the zero level-set of the smallest singular value $s_{k+1}(x)$ and the dual subspace of valid barycentric weights is spanned by the right singular vectors corresponding to the $l$ vanishing singular values: $\Lambda(x) = \operatorname{Span}(v_{k-l}, \ldots v_{k})$ (it is void if $l=0$).
Since $U$ and $V$ are orthogonal matrices, $Z(x)\lambda=0$ if and only if at least one singular value (necessarily the smallest one $s_{k}$) is null, and $\lambda$ has to live in the corresponding right-singular space: $\Lambda(x) = Ker(Z(x))$. If we have only one zero singular value ($s_{k+1}=0$ and $s_k>0$), then $\lambda$ is proportional to $v_{k+1}$. If $l$ singular values vanish, then we have a higher dimensional linear subspace of solutions for $\lambda$.
\[THM5\] Let $G(x)$ be the matrix expression of the Riemannian metric in a local coordinate system and $\Omega(x) = Z(x){^{\text{\tiny T}}}G(x) Z(x)$ be the smooth $(k+1)\times (k+1)$ matrix field on ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$ with components $\Omega_{ij}(x) = {\ensuremath{ \left<
\: {\ensuremath{\overrightarrow{x x_i}}} \:\left|\: {\ensuremath{\overrightarrow{x x_j}}} \right.\right> }}_x$ and $\Sigma(x) = {\mathfrak M}_2(x,\mathds{1} ) = \sum_{i=0}^k {\ensuremath{\overrightarrow{x x_i}}} \: {\ensuremath{\overrightarrow{x x_i}}}{^{\text{\tiny T}}}= Z(x) Z(x){^{\text{\tiny T}}}$be the (scaled) $n \times n$ covariance matrix field of the reference points. $\operatorname{EBS}(x_0, \ldots x_k)$ is the zero level-set of: $\det(\Omega(x))$, the minimal eigenvalue $\sigma_{k+1}^2$ of $\Omega(x)$, the $k+1$ eigenvalue (in decreasing order) of the covariance $\Sigma(x)$.
The constraint ${\mathfrak M}_1(x,\lambda)=0$ is satisfied if and only if: $$\| {\mathfrak M}_1(x,\lambda) \|^2_x = \left\| { \textstyle \sum_i \lambda_i {\ensuremath{\overrightarrow{x x_i}}}} \right\|^2_{x} = {\lambda{^{\text{\tiny T}}}\Omega(x) \lambda} =0.$$ As the function is homogeneous in $\lambda$, we can restrict to unit vectors. Adding this constrains with a Lagrange multiplier to the cost function, we end-up with the Lagrangian ${\cal L}(x, \lambda, \alpha) = \lambda{^{\text{\tiny T}}}\Omega(x) \lambda +\alpha (\lambda{^{\text{\tiny T}}}\lambda -1)$. The minimum with respect to $\lambda$ is obtained for the eigenvector $\mu_{k+1}(x)$ associated to the smallest eigenvalue $\sigma_{k+1}(x)$ of $\Omega(x)$ (assuming that eigenvalues are sorted in decreasing order) and we have $\|{\mathfrak M}_1(x, \mu_{k+1}(x))\|^2_2 = \sigma_{k+1}(x)$, which is null if and only if the minimal eigenvalue is zero. Thus, the barycentric subspace of $k+1$ points is the locus of rank deficient matrices $\Omega(x)$: $$\operatorname{EBS}(x_0, \ldots x_k) = \phi{^{\text{\tiny (-1)}}}(0) \quad \mbox{where} \quad \phi(x) = \det(\Omega(x)).$$
One may want to relate the singular values of $Z(x)$ to the eigenvalues of $\Omega(x)$. The later are the square of the singular values of $G(x)^{1/2}Z(x)$. However, the left multiplication by the square root of the metric (a non singular but non orthogonal matrix) obviously changes the singular values in general except for vanishing ones: the (right) kernels of $G(x)^{1/2}Z(x)$ and $Z(x)$ are indeed the same. This shows that the EBS is an affine notion rather than a metric one, contrarily to the Fréchet / Karcher barycentric subspace.
To draw the link with the $n\times n$ covariance matrix of the reference points, let us notice first that the definition does not assumes that the coordinate system is orthonormal. Thus, the eigenvalues of the covariance matrix depend on the chosen coordinate system, unless they vanish. In fact, only the joint eigenvalues of $\Sigma(x)$ and $G(x)$ really make sense, which is why this decomposition is called the proper orthogonal decomposition (POD). Now, the singular values of $Z(x)=U(x) S(x) V(x){^{\text{\tiny T}}}$ are also the square root of the first $k+1$ eigenvalues of $\Sigma(x) = U(x) S^2(x) U(x){^{\text{\tiny T}}}$, the remaining $n-k-1$ eigenvalues being null. Similarly, the singular values of $G(x)^{1/2}Z(x)$ are the square root of the first $k+1$ joint eigenvalues of $\Sigma(x)$ and $G(x)$. Thus, our barycentric subspace may also be characterized as the zero level-set of the $k+1$ eigenvalue (sorted in decreasing order) of $\Sigma$, and this characterization is once again independent of the basis chosen.
Spherical EBS and affine span {#Sec:SphericalEBS}
-----------------------------
From Eq.(\[eq:MomemtSphere\]) we identify the matrix: $Z(x) = ( {\ensuremath{\:\mathrm{Id}}}- x x{^{\text{\tiny T}}}) X F(X,x).$ Finding points $x$ and weights $\lambda$ such that $Z(x)\lambda=0$ is a classical matrix equation, except for the scaling matrix $F(X,x)$ acting on homogeneous projective weights, which is non-stationary and non-linear in both $X$ and $x$. However, since $F(X,x) = \mbox{Diag}( \theta_i /\sin \theta_i )$ is an invertible diagonal matrix, we can introduce [*renormalized weights*]{} $\tilde{\lambda} = F(X,x) \lambda, $ which leaves us with the equation $ ( {\ensuremath{\:\mathrm{Id}}}- x x{^{\text{\tiny T}}}) X \tilde \lambda=0$. The solutions under the constraint $\|x\|=1$ are given by $(x{^{\text{\tiny T}}}X \tilde{\lambda} ) x = X \tilde{\lambda}$ or more explicitly $x = \pm X \tilde{\lambda} / \| X \tilde{\lambda}\|$ whenever $X \tilde{\lambda} \not = 0$. This condition is ensured if $Ker(X)=\{0\}$. Thus, when the reference points are linearly independent, the point $x \in {\cal M}^*(X) $ has to belong to the Euclidean span of the reference vectors. Notice that for each barycentric coordinate we have two two antipodal solution points. Conversely, any unit vector $x = X\tilde \lambda$ of the Euclidean span of $X$ satisfies the equation $( {\ensuremath{\:\mathrm{Id}}}- x x{^{\text{\tiny T}}}) X \tilde \lambda = (1-\|x\|^2) X \tilde \lambda =0$, and is thus a point of the EBS provided that it is not at the cut-locus of one of the reference points. This shows that $$\operatorname{EBS}(X) = \operatorname{Span}\{x_0, \ldots x_k\} \cap {\cal S}_n \setminus X.$$
Using the renormalization principle, we can orthogonalize the reference points: let $X=U S V{^{\text{\tiny T}}}$ be a singular value decomposition of the matrix of reference vectors. All the singular values $s_i$ are positive since the reference vectors $x_i$ are assumed to be linearly independent. Thus, $\mu = S V{^{\text{\tiny T}}}\tilde{\lambda} = S V{^{\text{\tiny T}}}F(X,x) \lambda$ is an invertible change of coordinate, and we are left with solving $ ( {\ensuremath{\:\mathrm{Id}}}- x x{^{\text{\tiny T}}}) U\mu =0$. By definition of the singular value decomposition, the Euclidean spans of $X$ and $U$ are the same, so that $\operatorname{EBS}(U) = \operatorname{Span}\{x_0, \ldots x_k\} \cap {\cal S}_n \setminus -U$. This shows that the exponential barycentric subspace generated by the original points $X=[x_0, \ldots x_k]$ and the orthogonalized points $U=[u_0, \ldots u_k]$ are the same, except at the cut locus of all these points, but with different barycentric coordinates.
To obtain the affine span, we take the closure of the EBS, which incorporates the cut locus of the reference points: $\operatorname{Aff}(X) = \operatorname{Span}\{x_0, \ldots x_k\} \cap {\cal S}_n$. Thus, for spherical data as for Euclidean data, the affine span only depend on the reference points through the point of the Grassmanian they define.
The affine span $\operatorname{Aff}(X)$ of $k+1$ linearly independent reference unit points $X=[x_0, \ldots x_k]$ on the $n$-dimensional sphere ${\cal S}_n$ endowed with the canonical metric is the great subsphere of dimension k that contains the reference points. \[THM7\]
When the reference points are affinely dependent on the sphere, the matrix $X$ has one or more (say $l$) vanishing singular values. Any weight $\tilde{\lambda} \in \mbox{Ker}(X)$ is a barycentric coordinate vector for any point $x$ of the pointed sphere since the equation $( {\ensuremath{\:\mathrm{Id}}}- x x{^{\text{\tiny T}}}) X \tilde \lambda =0$ is verified. Thus, the EBS is ${\cal S}_n \setminus -X$ and the affine span is the full sphere. If we exclude the abnormal subspace of weights valid for all points, we find that $x$ should be in the span of the non-zero left singular vectors of $X$, i.e. in the subsphere of dimension of dimension $rank(X)-1$ generated the Euclidean span of the reference vectors. This can also be achieved by focusing of the locus of points where $Z(x)$ has two vanishing singular values. This more reasonable result suggests adapting the EBS and affine span definitions for singular point configurations.
Two points on a 2-sphere is an interesting example that can be explicitly worked out. When the points are not antipodal, the rank of $X=[x_0,x_1]$ is 2, and the generated affine span is the one-dimensional geodesic joining the two points. When the reference points are antipodal, say north and south poles, X becomes rank one and one easily sees that all points of the 2-sphere are on one geodesic joining the poles with opposite log directions to the poles. This solution of the EBS definition correspond to the renormalized weight $\tilde \lambda = (1/2 : 1/2) \in Ker(X)$ of the kernel of $X$. However, looking at the locus of points with two vanishing singular values of $Z(x)$ leads to restrict to the north and south poles only, which is a more natural and expected result.
Hyperbolic EBS and affine span {#Sec:HyperbolicEBS}
------------------------------
The hyperbolic case closely follows the spherical one. From Eq., we get the expression of the matrix $ Z(x) = ( {\ensuremath{\:\mathrm{Id}}}+ x x{^{\text{\tiny T}}}J) X F_*(X,x)$. Solving for $Z(x)\lambda=0$ can be done as previously by solving $ ( {\ensuremath{\:\mathrm{Id}}}+ x x{^{\text{\tiny T}}}J) X \tilde \lambda =0$ with the renormalized weights $\tilde{\lambda} = F_*(X,x) \lambda$. This equation rewrites $<x| X \tilde{\lambda}>_* x = - X \tilde{\lambda}$, so that the solution has to be of the form $X \tilde \lambda =0$ or $x = \alpha X \tilde{\lambda}$. When the points are affinely independent, the first form is excluded since $Ker(X)=0$. In order to satisfy the constraint $\|x\|^2_*=-1$ in the second form, we need to have $\alpha^2 = - \|X \tilde{\lambda}\|_*^{-2} >0$ and the first coordinate $[X \tilde{\lambda}]_0$ of $X \tilde{\lambda}$ has to be positive. This defines a cone in the space of renormalized weights from which each line parametrizes a point $x = \text{sgn}( [X \tilde{\lambda}]_0) X \tilde \lambda / \sqrt {\tiny -\|X \tilde{\lambda}\|_*^2}$ of the Hyperbolic EBS. Thus, $\operatorname{Aff}(X)$ is the $k$-dimensional hyperboloid generated by the intersection of the Euclidean span of the reference vectors with the hyperboloid ${\ensuremath{\mathbb{H}}}^n$. Since it is complete, the completion does not add anything to the affine span: $$\operatorname{Aff}(X) = \operatorname{EBS}(X) = \operatorname{Span}\{x_0, \ldots x_k\} \cap {\ensuremath{\mathbb{H}}}^n.$$ As for spheres, we see that the hyperbolic affine span only depend on the reference points through the point of the Grassmanian they define.
The affine span $\operatorname{Aff}(X) = \operatorname{EBS}(X)$ of $k+1$ affinely independent reference points $X=[x_0, \ldots x_k]$ on the $n$-dimensional hyperboloid ${\ensuremath{\mathbb{H}}}^n$ endowed with the canonical Minkowski pseudo-metric of the embedding space ${\ensuremath{\mathbb{R}}}^{1,n}$ is the hyperboloid of dimension $k$ generated by the intersection of the hyperboloid with the hyperplane containing the reference points. \[thm:HyperbolicSpan\]
When the matrix $X$ has one or more vanishing singular values (affine dependance), all the points of the hyperboloid are solutions corresponding to weights from $Ker(X)$. Excluding these abnormal solutions and looking at the locus of points where $Z(x)$ has two vanishing singular values, we find that $x$ should be in the span of the non-zero left singular vectors of $X$, i.e. in the subsphere of dimension of dimension $rank(X)-1$ generated the Euclidean span of the reference vectors.
Fréchet / Karcher Barycentric subspaces {#Sec:KBS}
=======================================
The reformulation of the affine span as the weighted mean of $k+1$ points also suggests a definition using the Fréchet or the Karcher mean, valid in general metric spaces.
Let $({\ensuremath{{\cal M}}}, {\ensuremath{\:\mbox{\rm dist}}})$ be a metric space of dimension $n$ and $(x_0,\ldots x_k) \in {\ensuremath{{\cal M}}}^{k+1}$ be $k+1\leq n+1$ distinct reference points. The (normalized) weighted variance at point $x$ with weight $\lambda \in {\ensuremath{{\cal P}^*_k}}$ is: $\sigma^2(x,\lambda) = \frac{1}{2}\sum_{i=0}^k {{\underaccent{\bar}{\lambda}}}_i {\ensuremath{\:\mbox{\rm dist}}}^2(x, x_i) = \frac{1}{2}\sum_{i=0}^k \lambda_i {\ensuremath{\:\mbox{\rm dist}}}^2(x, x_i) / (\sum_{j=0}^k \lambda_j).$ The Fréchet barycentric subspace of these points is the locus of weighted Fréchet means of these points, i.e. the set of absolute minima of the weighted variance: $$\operatorname{FBS}(x_0, \ldots x_k) = \left\{ \arg\min_{x\in {\ensuremath{{\cal M}}}} \sigma^2(x, \lambda), \: \lambda \in {\ensuremath{{\cal P}^*_k}}\right\}$$ The Karcher barycentric subspaces $\operatorname{KBS}(x_0, \ldots x_k)$ are defined similarly with local minima instead of global ones.
In stratified metric spaces, for instance, the barycentric subspace spanned by points belonging to different strata naturally maps over several strata. This is a significant improvement over geodesic subspaces used in PGA which can only be defined within a regular strata. In the sequel, we only deal with the KBS/FBS of affinely independent points in a Riemannian manifold.
Link between the different barycentric subspaces
------------------------------------------------
In order to analyze the relationship between the Fréchet, Karcher and Exponential barycentric subspaces, we follow the seminal work of [@karcher77]. First, the locus of local minima (i.e. Karcher mean) is a superset of the global minima (Fréchet mean). On the punctured manifold ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$, the squared distance $d^2_{x_i}(x) = {\ensuremath{\:\mbox{\rm dist}}}^2(x, x_i)$ is smooth and its gradient is $\nabla d^2_{x_i}(x) = -2 \log_x(x_i)$. Thus, one recognizes that the EBS equation $\sum_i {{\underaccent{\bar}{\lambda}}}_i \log_x(x_i) =0$ (Eq.(\[eq:Bary\])) defines nothing else than the critical points of the weighted variance: $$FBS\cap {\cal M}^* \subset KBS \cap {\cal M}^* \subset Aff \cap {\cal M}^* = EBS.$$ Among the critical points with a non-degenerate Hessian, local minima are characterized by a positive definite Hessian. When the Hessian is degenerate, we cannot conclude on the local minimality without going to higher order differentials. The goal of this section is to subdivide the EBS into a cell complex according to the index of the Hessian operator of the variance: $$\textstyle
H(x,\lambda) = \nabla^2 \sigma^2(x,\lambda) = - \sum_{i=0}^k {{\underaccent{\bar}{\lambda}}}_i D_x \log_x(x_i).
\label{eq:Hessian}$$
Plugging the value of the Taylor expansion of the differential of the log of Eq.(\[eq:Diff\_log\]), we obtain the Taylor expansion: $$\label{eq:TaylorH}
\left[ H(x, \lambda) \right]^a_b
= \delta^a_b - \frac{1}{3} R^a_{cbd}(x) {\mathfrak M}^{cd}_2(x,{{\underaccent{\bar}{\lambda}}})
- \frac{1}{12} \nabla_c R^a_{dbe}(x) {\mathfrak M}_3^{cde}(x,{{\underaccent{\bar}{\lambda}}}) + O(\varepsilon^4).$$ The key factor in this expression is the contraction of the Riemannian curvature with the weighted covariance tensor of the reference points. This contraction is an extension of the Ricci curvature tensor. Exactly as the Ricci curvature tensor encodes how the volume of an isotropic geodesic ball in the manifold deviates from the volume of the standard ball in a Euclidean space (through its metric trace, the scalar curvature), the extended Ricci curvature encodes how the volume of the geodesic ellipsoid ${\ensuremath{\overrightarrow{xy}}}{^{\text{\tiny T}}}{\mathfrak M}_2(x,{{\underaccent{\bar}{\lambda}}}){^{\text{\tiny (-1)}}}{\ensuremath{\overrightarrow{xy}}} \leq \varepsilon $ deviates from the volume of the standard Euclidean ellipsoid.
In locally symmetric affine spaces, the covariant derivative of the curvature is identically zero, which simplifies the formula. In the limit of null curvature, (e.g. for a locally Euclidean space like the torus), the Hessian matrix $H(x, \lambda)$ converges to the unit matrix and never vanishes. In general Riemannian manifolds, Eq.(\[eq:TaylorH\]) only gives a qualitative behavior but does not provide guaranties as it is a series involving higher order moments of the reference points. In order to obtain hard bounds on the spectrum of $H(x, \lambda)$, one has to investigate bounds on Jacobi fields using Riemannian comparison theorems, as for the proof of uniqueness of the Karcher and Fréchet means (see [@karcher77; @kendall90; @Le:2004; @Afsari:2010; @Yang:2011]).
\[def:NonDegenerate\] An exponential barycenter $x \in \operatorname{EBS}(x_0,\ldots x_k)$ is degenerate (resp. non-degenerate or positive) if the Hessian matrix $H(x,\lambda)$ is singular (resp. definite or positive definite) for all $\lambda$ in the the dual space of barycentric coordinates $\Lambda(x)$. The set of degenerate exponential barycenters is denoted by $EBS^0(x_0,\ldots,x_k)$ (resp. non-degenerate by $EBS^*(x_0,\ldots,x_k)$ and positive by $EBS^+(x_0,\ldots x_k)$).
The definition of non-degenerate and positive points could be generalized to non-critical points (outside the affine span) by considering for instance the right singular space of the smallest singular value of $Z(x)$. However, this would depend on the metric on the space of weights and a renormalization of the weights (such as for spheres) can change the smallest non-zero singular value. Positive points are obviously non-degenerate. In Euclidean spaces, all the points of an affine span are positive and non-degenerate. In positively curved manifolds, we may have degenerate points and non-positive points, as we will see with the sphere example. For negatively curved spaces, the intuition that points of the EBS should all be positive like in Euclidean spaces is also wrong, as we sill see with the example of hyperbolic spaces.
$ $\
\[THM2\] $EBS^+(x_0, \ldots x_k)$ is the set of non-degenerate points of the Karcher barycentric subspace $\operatorname{KBS}(x_0, \ldots x_k)$ on ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$. In other words, the KBS is the positive EBS plus potentially some degenerate points of the affine span and some points of the cut locus of the reference points.
Spherical KBS {#Sec:SphericalKBS}
-------------
In order to find the positive points of the EBS on the sphere, we compute the Hessian of the normalized variance. Using Eq.(\[eq:HessDistSphere2\]) and $u_i= { \log_x(x_i) }/{\theta_i}$, we obtain the Hessian of $\sigma^2(x,\lambda) = \frac{1}{2}\sum_{i=0}^k {{\underaccent{\bar}{\lambda}}}_i {\ensuremath{\:\mbox{\rm dist}}}^2(x, x_i)$: $$\textstyle
H(x, \lambda) = \big(\sum_i {{\underaccent{\bar}{\lambda}}}_i \theta_i \cot\theta_i \big)({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})
+ \sum_i {{\underaccent{\bar}{\lambda}}}_i ( 1- \theta_i \cot\theta_i)u_i u_i{^{\text{\tiny T}}}.$$ As expected, $x$ is an eigenvector with eigenvalue $0$ due to the projection on the tangent space at $x$. Any vector $w$ of the tangent space at $x$ (thus orthogonal to $x$) which is orthogonal to the affine span (and thus to the vectors $u_i$) is an eigenvector with eigenvalue $ \sum_i {{\underaccent{\bar}{\lambda}}}_i \theta_i \cot \theta_i $. Since the Euclidean affine span $\operatorname{Aff}_{{\ensuremath{\mathbb{R}}}^{n+1}}(X)$ has $rank(X) \leq k+1$ dimensions, this eigenvalue has multiplicity $n+1-rank(X) \geq n - k$ when $x\in \operatorname{Aff}(X)$. The last $Rank(X)-1$ eigenvalues have associated eigenvectors within $\operatorname{Aff}_{{\ensuremath{\mathbb{R}}}^{n+1}}(X)$.
[@buss_spherical_2001] have have shown that this Hessian matrix is positive definite for [*positive weights*]{} when the points are within one hemisphere with at least one non-zero weight point which is not on the equator. In contrast, we are interested here in the positivity and definiteness of the Hessian $H(x,\lambda)$ for the positive and negative weights which live in dual space of barycentric coordinates $\Lambda(x)$. This is actually a non trivial algebraic geometry problem. Simulation tests with random reference points $X$ show that the eigenvalues of $H(x, {{\underaccent{\bar}{\lambda}}}(x))$ can be positive or negative at different points of the EBS. The number of positive eigenvalues (the index) of the Hessian is illustrated on Fig. (\[fig:signature\]) for a few configuration of 3 affinely independent reference points on the 2-sphere. This illustrates the subdivision of the EBS on spheres in a cell complex based on the index of the critical point: the positive points of the KBS do not in general cover the full subsphere containing the reference points. It may even be disconnected, contrarily to the affine span which consistently covers the whole subsphere. For subspace definition purposes, this suggests that the affine span might thus be the most interesting definition. For affinely dependent points, the KBS/FBS behave similarly to the EBS. For instance, the weighted variance of $X=[e_1,-e_1]$ on a 2-sphere is a function of the latitude only. The points of a parallel at any specific latitude are global minima of the weighted variance for a choice of $\lambda =(\alpha : 1-\alpha), \: \alpha \in [0,1]$. Thus, all points of the sphere belong to the KBS, which is also the FBS and the affine span. However, the Hessian matrix has one positive eigenvalue along meridians and one zero eigenvalue along the parallels. This is a very non-generic case.
![Signature of the weighted Hessian matrix for different configurations of 3 reference points (in black, antipodal point in red) on the 2-sphere: the locus of local minima (KBS) in brown does not cover the whole sphere and can even be disconnected (first example).[]{data-label="fig:signature"}](Figures/SphereHessian_colorbar "fig:"){height="3.1cm"} ![Signature of the weighted Hessian matrix for different configurations of 3 reference points (in black, antipodal point in red) on the 2-sphere: the locus of local minima (KBS) in brown does not cover the whole sphere and can even be disconnected (first example).[]{data-label="fig:signature"}](Figures/SphereHessian_5_sc "fig:"){height="3.1cm"} ![Signature of the weighted Hessian matrix for different configurations of 3 reference points (in black, antipodal point in red) on the 2-sphere: the locus of local minima (KBS) in brown does not cover the whole sphere and can even be disconnected (first example).[]{data-label="fig:signature"}](Figures/SphereHessian_6_sc "fig:"){height="3.1cm"} ![Signature of the weighted Hessian matrix for different configurations of 3 reference points (in black, antipodal point in red) on the 2-sphere: the locus of local minima (KBS) in brown does not cover the whole sphere and can even be disconnected (first example).[]{data-label="fig:signature"}](Figures/SphereHessian_9_sc "fig:"){height="3.1cm"} ![Signature of the weighted Hessian matrix for different configurations of 3 reference points (in black, antipodal point in red) on the 2-sphere: the locus of local minima (KBS) in brown does not cover the whole sphere and can even be disconnected (first example).[]{data-label="fig:signature"}](Figures/SphereHessian_2_sc "fig:"){height="3.26cm"}
Hyperbolic KBS / FBS {#Sec:HyperbolicKBS}
--------------------
Let $x = X \tilde \lambda$ be a point of the hyperbolic affine span of $X=[x_0,\ldots x_k]$. The renormalized weights $\tilde \lambda$ are related to the original weights through $\lambda = F_*(X,x)^{-1} \tilde \lambda$ and satisfy $\|X \tilde{\lambda}\|_*^2 = -1$ and $\text{sgn}( [X \tilde{\lambda}]_0) >0$. The point $x$ is a critical point of the (normalized) weighted variance. In order to know if this is a local minimum (i.e. a point of the KBS), we compute the Hessian of this weighted variance. Denoting $ u_i = \log_x(x_i) / \theta_i$ with $\cosh \theta_i = -{\ensuremath{ \left<
\:x\:\left|\:x_i\right.\right> }}_*$, and using the Hessian of the square distance derived in Eq., we obtain the following formula: $$\textstyle
H(x,\lambda) = \sum_i {{\underaccent{\bar}{\lambda}}}_i \theta_i \coth \theta_i (J + J x x{^{\text{\tiny T}}}J) +
\sum_i {{\underaccent{\bar}{\lambda}}}_i {(1 - \theta_i \coth \theta_i)} J u_i u_i{^{\text{\tiny T}}}J.$$
As expected, $x$ is an eigenvector with eigenvalue 0 due to the projection on the tangent space at $x$. Any vector $w$ of the tangent space at $x$ which is orthogonal to the affine span (and thus to the vectors $u_i$) is an eigenvector with eigenvalue $\sum_i {{\underaccent{\bar}{\lambda}}}_i \theta_i \coth \theta_i = 1/({\ensuremath{\mathds{1}}}{^{\text{\tiny T}}}\tilde \lambda)$ with multiplicity $n+1-rank(X)$. The last $Rank(X)-1$ eigenvalues have associated eigenvectors within $\operatorname{Aff}_{{\ensuremath{\mathbb{R}}}^{n+1}}(X)$. Simulation tests with random reference points $X$ show these eigenvalues can be positive or negative at different points of $Aff(X)$. The index of the Hessian is illustrated on Fig. (\[fig:signatureHyp\]) for a few configuration of 3 affinely independent reference points on the 2-hyperbolic space. Contrarily to the sphere, we observe only one or two positive eigenvalues corresponding respectively to saddle points and local minima. This subdivision of the hyperbolic affine span in a cell complex shows that the hyperbolic KBS is in general a strict subset of the hyperbolic affine span. We conjecture that there is an exception for reference points at infinity, for which the barycentric subspaces could be generalized using Busemann functions [@busemann_geometry_1955]: it is likely that the FBS, KBS and the affine span are all equal in this case and cover the whole lower dimensional hyperbola.
![Signature of the weighted Hessian matrix for different configurations of 3 reference points on the 2-hyperboloid: the locus of local minima (KBS) in brown does not cover the whole hyperboloid and can be disconnected (last two example).[]{data-label="fig:signatureHyp"}](Figures/HyperboloidHessian_Colorbar "fig:"){height="3cm"} ![Signature of the weighted Hessian matrix for different configurations of 3 reference points on the 2-hyperboloid: the locus of local minima (KBS) in brown does not cover the whole hyperboloid and can be disconnected (last two example).[]{data-label="fig:signatureHyp"}](Figures/HyperboloidHessian_2_sc "fig:"){height="3cm"} ![Signature of the weighted Hessian matrix for different configurations of 3 reference points on the 2-hyperboloid: the locus of local minima (KBS) in brown does not cover the whole hyperboloid and can be disconnected (last two example).[]{data-label="fig:signatureHyp"}](Figures/HyperboloidHessian_4_sc "fig:"){height="3cm"} ![Signature of the weighted Hessian matrix for different configurations of 3 reference points on the 2-hyperboloid: the locus of local minima (KBS) in brown does not cover the whole hyperboloid and can be disconnected (last two example).[]{data-label="fig:signatureHyp"}](Figures/HyperboloidHessian_7_sc "fig:"){height="3cm"} ![Signature of the weighted Hessian matrix for different configurations of 3 reference points on the 2-hyperboloid: the locus of local minima (KBS) in brown does not cover the whole hyperboloid and can be disconnected (last two example).[]{data-label="fig:signatureHyp"}](Figures/HyperboloidHessian_5_sc "fig:"){height="3cm"} ![Signature of the weighted Hessian matrix for different configurations of 3 reference points on the 2-hyperboloid: the locus of local minima (KBS) in brown does not cover the whole hyperboloid and can be disconnected (last two example).[]{data-label="fig:signatureHyp"}](Figures/HyperboloidHessian_8_sc "fig:"){height="3cm"}
Properties of the barycentric subspaces {#Sec:Prop}
=======================================
The EBS exists at each reference point $x_i$ with weight 1 for this point and zero for the others. Moreover, when the points are affinely independent, the matrix $Z(x_i)$ has exactly one zero singular value since column $i$ is $\log_{x_i}(x_i)=0$ and all the other column vectors are affinely independent. Finally, the weighted Hessian matrix boils down to $H(x_i,\lambda) = - \left. D_x \log_{x}(x_i)\right|_{x=x_i} = {\ensuremath{\:\mathrm{Id}}}$ (See e.g. Eq.(\[eq:Diff\_log\])). Thus the reference points are actually local minima of the weighted variance and the KBS exists by continuity in their neighborhood.
Barycentric simplex in a regular geodesic ball
----------------------------------------------
We call the subset of the FBS that has non-negative weights a barycentric simplex. It contains all the reference points, the geodesics segments between the reference points, and of course the Fréchet mean of the reference points. This is the generalization of a geodesic segment for 2 points, a triangle for 3 points, etc. The $(k-l)$-faces of a $k$-simplex are the simplices defined by the barycentric subspace of $k-l+1$ points among the $k+1$. They are obtained by imposing the $l$ remaining barycentric coordinates to be zero. In parallel to this paper, [@weyenberg_statistics_2015] has investigated barycentric simplexes as extensions of principal subspaces in the negatively curved metric spaces of trees under the name Locus of Fréchet mean (LFM), with very interesting results.
\[THM3\] Let $\kappa$ be an upper bound of sectional curvatures of ${\ensuremath{{\cal M}}}$ and $\text{inj}({\ensuremath{{\cal M}}})$ be the radius of injection (which can be infinite) of the Riemannian manifold. Let $X= \{ x_0,\ldots x_k\} \in {\ensuremath{{\cal M}}}^{(k+1)}$ be a set of $k+1\leq n$ affinely independent points included in a regular geodesic ball $B(x,\rho)$ with $\rho < \frac{1}{2}\min\{ \text{inj}({\ensuremath{{\cal M}}}), \frac{1}{2}\pi/\sqrt{\kappa} \} $ ($\pi/\sqrt{\kappa}$ being infinite if $\kappa < 0$). The barycentric simplex is the graph of a $k$-dimensional differentiable function from the non-negative quadrant of homogeneous coordinates $({\ensuremath{{\cal P}^*_k}})^+$ to $B(x,\rho)$ and is thus at most $k$-dimensional. The $(k-l)$-faces of the simplex are the simplices defined by the barycentric subspace of $k-l+1$ points among the $k+1$ and include the reference points themselves as vertices and the geodesics joining them as edges.
The proof closely follows the one of [@karcher77] for the uniqueness of the Riemannian barycenter. The main argument is that $\mu_{(X, \lambda)}(x) = \sum {{\underaccent{\bar}{\lambda}}}_i \delta_{x_i}(x)$ is a probability distribution whose support is included in the strongly convex geodesic ball $B(x,\rho)$. The variance $\sigma^2(x, \lambda) = \frac{1}{2}\sum_i {{\underaccent{\bar}{\lambda}}}_i d^2(x, x_i)$ is strictly convex on that ball and has a unique minimum $x_{\lambda} \in B(x,\rho)$, necessarily the weighted Fréchet mean. This proof of the uniqueness of the weighted Fréchet mean with non-negative weights was actually already present in [@buser_gromovs_1981]. We supplement the proof here by noting that since the Hessian $H(x_{\lambda}, \lambda) = \sum_i {{\underaccent{\bar}{\lambda}}}_i H_i(x_{\lambda})$ is the convex combination of positive matrices, it is positive definite for all $\lambda \in ({\ensuremath{{\cal P}^*_k}})^+$ in the positive quadrant. Thus the function $x_{\lambda}$ is differentiable thanks to the implicit function theorem: $ D_{\lambda} x_{\lambda} = H( x_{\lambda}, \lambda){^{\text{\tiny (-1)}}}Z(x_{\lambda}).$ The rank of this derivative is at most $k$ since $Z(x_{\lambda})=0$, which proves that the graph of the function $x_{\lambda}$ describes at most a $k$ dimensional subset in ${\ensuremath{{\cal M}}}$.
Barycentric simplexes and convex hulls
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In a vector space, a point lies in the convex hull of a simplex if and only if its barycentric coordinates are all non-negative (thus between 0 and 1 with the unit sum constraint). Consequently, barycentric coordinates are often thought to be related to convex hulls. However, in a general Riemannian manifold, the situation is quite different. When there are closed geodesics, the convex hull can reveal several disconnected components, unless one restrict to convex subsets of the manifolds as shown by [@Groisser:2003]. In metric spaces with negative curvature (CAT spaces), [@weyenberg_statistics_2015] displays explicit examples of convex hulls of 3 points which are 3-dimensional rather than 2-dimensional as expected. In fact, the relationship between barycentric simplexes and convex hulls cannot hold in general Riemannian manifolds if the barycentric simplex is not totally geodesic at each point, which happens for constant curvature spaces but not for general Riemannian manifolds.
Local dimension of the barycentric subspaces
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Let $x$ be a point of the $EBS$ with affinely independent reference points. The EBS equation $Z(x)\lambda = 0$ for $\lambda \in \Lambda(x)$ is smooth in $x$ and $\lambda$ so that we can take a Taylor expansion: at the first order, a variation of barycentric coordinates $\delta \lambda$ induces a variation of position $\delta x$ which are linked through $H(x,\lambda) \delta x - Z(x) \delta \lambda =0.$ Thus, at regular points: $$\delta x = H(x,\lambda){^{\text{\tiny (-1)}}}Z(x) \delta \lambda.$$
Let $Z(x)=U(x)S(x)V(x){^{\text{\tiny T}}}$ be a singular value decomposition with singular values sorted in decreasing order. Since $x$ belongs to the EBS, there is at least one (say $m \geq 1$) singular value that vanish and the dual space of barycentric coordinates is $\Lambda(x) = \operatorname{Span}(v_{k-m}, \ldots v_k)$. For a variation of weights $\delta \lambda$ in this subspace, there is no change of coordinates, while any variation of weights in $\operatorname{Span}(v_0, \ldots v_{k-m-1})$ induces a non-zero position variation. Thus, the tangent space of the EBS restricts to the $(k-m)$-dimensional linear space generated by $\{ \delta x_i' = H(x,\lambda){^{\text{\tiny (-1)}}}u_i\}_{0\leq i\leq k-m}$. Here, we see that the Hessian matrix $H(x, \lambda)$ encodes the distortion of the orthonormal frame fields $ u_1(x), \ldots u_k(x)$ to match the tangent space. Since the lower dimensional subspaces are included one the larger ones, we have a stratification of our $k$-dimensional submanifold into $k-1$, $k-2, \ldots 0$-dimensional subsets.
\[THM4\] The non-degenerate exponential barycentric subspace $EBS^*(x_0,\ldots,x_k)$ of $k+1$ affinely independent points is a stratified space of dimension $k$ on ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$. On the $m$-dimensional strata, $Z(x)$ has exactly $k-m+1$ vanishing singular values.
At degenerate points, $H(x, \lambda)$ is not invertible and vectors living in its kernel are also authorized, which potentially raises the dimensionality of the tangent space, even if they do not change the barycentric coordinates. These pathologies do not appear in practice for the constant curvature spaces as we have seen with spherical and hyperbolic spaces, and we conjecture that this is also not the case for symmetric spaces.
Stability of the affine span with respect to the metric power
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The Fréchet (resp. Karcher) mean can be further generalized by taking a power $p$ of the metric to define the $p$-variance $\sigma^{p}(x) = \frac{1}{p} \sum_{i=0}^k {\ensuremath{\:\mbox{\rm dist}}}^{p}(x, x_i)$. The global (resp. local) minima of this $p$-variance defines the median for $p =1$. This suggest to further generalize barycentric subspaces by taking the locus of the minima of the weighted $p$-variance $\sigma^{p}(x,\lambda) = \frac{1}{p} \sum_{i=0}^k {{\underaccent{\bar}{\lambda}}}_i {\ensuremath{\:\mbox{\rm dist}}}^{p}(x, x_i)$. In fact, it turns out that all these “$p$-subspaces” are necessarily included in the affine span, which shows this notion is really central. To see that, we compute the gradient of the $p$-variance at non-reference point of ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$: $$\textstyle
\nabla_x \sigma^{p}(x,\lambda) = - \sum_{i=0}^k {{\underaccent{\bar}{\lambda}}}_i {\ensuremath{\:\mbox{\rm dist}}}^{p -2}(x, x_i) \log_{x}(x_i).$$ Critical points of the $p$-variance satisfy the equation $\sum_{i=0}^k \lambda'_i \log_{x}(x_i) =0$ for the new weights $\lambda'_i = \lambda_i {\ensuremath{\:\mbox{\rm dist}}}^{p -2}(x, x_i) $. Thus, they are still elements of the EBS and changing the power of the metric just amounts to a reparametrization of the barycentric weights.
Restricted geodesic submanifolds are limit of affine spans
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We investigate in this section what is happening when all the points $\{x_i = \exp_{x_0}(\varepsilon w_i)\}_{1\leq i\leq k}$ are converging to $x_0$ at first order along $k$ independent vectors $\{ w_i\}_{1\leq i\leq k}$. Here, we fix $w_0 =0$ to simplify the derivations, but the proof can be easily extended with a suitable change of coordinates provided that $\sum_{i=0}^k w_i =0$. In Euclidean spaces, a point of the affine span $y = \sum_{i=0}^k {{\underaccent{\bar}{\lambda}}}_i x_i$ may be written as the point $y = x + \varepsilon \sum_{i=1}^k {{\underaccent{\bar}{\lambda}}}_i w_i$ of the “geodesic subspace” generated by the family of vectors $\{ w_i\}_{1\leq i\leq k}$. By analogy, we expect the exponential barycentric subspace $\operatorname{EBS}(x_0, \exp_{x_0}(\varepsilon w_1) \ldots \exp_{x_0}(\varepsilon w_k))$ to converge towards the totally geodesic subspace at $x$ generated by the $k$ independent vectors $w_1, \ldots w_k$ of $T_x{\ensuremath{{\cal M}}}$: $$\textstyle
GS(x, w_1, \ldots w_k) = \left\{ \textstyle \exp_{x}\left( \sum_{i=1}^k \alpha_i w_i \right) \in {\ensuremath{{\cal M}}}\text{ for } \alpha \in {\ensuremath{\mathbb{R}}}^k \right\}.$$
In fact, the above definition of the geodesic subspaces (which is the one implicitly used in most of the works using PGA) is too large and may not define a $k$-dimensional submanifold when there is a cut-locus. For instance, it is well known that geodesics of a flat torus are either periodic or everywhere dense in a flat torus submanifold depending on whether the components of the initial velocity field have rational or irrational ratios. This means that the geodesic space generated by a single vector for which all ratio of coordinates are irrational (e.g. $w=(\pi, \pi^2,\ldots \pi^k)$) is filling the full $k$-dimensional flat torus. Thus all the 1-dimensional geodesic subspaces that have irrational ratio of all coordinates minimize the distance to any set of data points in a flat torus of any dimension. In order to have a more meaningful definition and to guaranty the dimensionality of the geodesic subspace, we need to restrict the definition to the points of the geodesics that are distance minimizing.
\[def:RGS\] Let $x \in {\ensuremath{{\cal M}}}$ be a point of a Riemannian manifold and let $W_x = \{ \sum_{i=1}^k \alpha_i w_i, \alpha \in {\ensuremath{\mathbb{R}}}^k\}$ be the $k$-dimensional linear subspace of $T_x{\ensuremath{{\cal M}}}$ generated a $k$-tuple $\{ w_i\}_{1\leq i\leq k} \in (T_x{\ensuremath{{\cal M}}})^k$ of independent tangent vectors at $x$. We consider the geodesics starting at $x$ with tangent vectors in $W_x$, but up to the first cut-point of $x$ only. This generates a submanifold of ${\ensuremath{{\cal M}}}$ called the restricted geodesic submanifold $GS^*(W_x)$: $$\textstyle
GS^*(W_x) = GS^*(x, w_1, \ldots w_k) = \{ \exp_{x}\left( w \right), w\in W_x \cap D(x) \},$$ where $D(x) \subset T_x{\ensuremath{{\cal M}}}$ is the injectivity domain.
It may not be immediately clear that the subspace we define that way is a submanifold of ${\ensuremath{{\cal M}}}$: since $\exp_x$ is a diffeomorphism from $D(x) \subset T_x{\ensuremath{{\cal M}}}$ to ${\ensuremath{{\cal M}}}\setminus {\ensuremath{{\cal C}}}(x)$ whose differential has full rank, its restriction to the open star-shape subset $ W_x \cap D(x)$ of dimension $k$ is a diffeomorphism from that subset to the restricted geodesic subspace $GS^*(W_x)$ which is thus an open submanifolds of dimension $k$ of ${\ensuremath{{\cal M}}}$. This submanifold is generally not geodesically complete.
\[THM6\] The restricted geodesic submanifold $GS^*(W_{x_0}) = \{ \exp_{x_0}\left( w \right), w\in W_{x_0} \cap D(x_0) \}$ is the limit of the $EBS(x_0, x_1(\varepsilon), \ldots x_k(\varepsilon))$ when the points $x_i(\varepsilon) = \exp_{x_0}(\varepsilon w_i)$ are converging to $x_0$ at first order in $\varepsilon$ along the tangent vectors $w_i$ defining the $k$-dimensional subspace $W_{x_0} \subset T_{x_0}{\ensuremath{{\cal M}}}$. These limit points are parametrized by barycentric coordinates at infinity in the codimension 1 subspace $\mathds{1}^{\perp}$, the projective completion of ${\ensuremath{{\cal P}^*_k}}$ in ${\ensuremath{\mathbb{R}}}P^k$, see Definition \[def:Pk\].
The proof is deferred to Appendix A because of its technicality. We conjecture that the construction can be generalized using techniques from sub-Riemannian geometry to higher order derivatives when the first order derivative do not span a $k$-dimensional subspace. This would mean that we could also see some non-geodesic decomposition schemes as limit cases of barycentric subspaces, such as splines on manifolds [@crouch_dynamic_1995; @machado_higher-order_2010; @Gay-Balmaz:2012:10.1007/s00220-011-1313-y].
#### Example on spheres and hyperbolic spaces
In spheres (resp. hyperbolic spaces), the restricted geodesic subspace $GS^*(W_{x})$ describes a great subsphere (resp. a great hyperbola), except for the cut-locus of the base-point $x$ in spheres. Thus, points of $GS^*(W_{x})$ are also points of the affine span generated by $k+1$ affinely independent reference points of this subspace. When all the reference points $x_i = \exp_{x}(\varepsilon w_i)$ coalesce to a single point $x$ along the tangent vectors $W = [w_0,\ldots w_k]$ (with $W {\ensuremath{\mathds{1}}}=0$), we find that solutions of the EBS equation are of the form $y = x + W ( \varepsilon \tilde \lambda / {\ensuremath{\mathds{1}}}{^{\text{\tiny T}}}\tilde \lambda) + O(\varepsilon^2)$, which describes the affine hyperplane generated by $x$ and $W$ in the embedding Euclidean (resp. Minkowski) space. The weights $\mu = \varepsilon \tilde \lambda / {\ensuremath{\mathds{1}}}{^{\text{\tiny T}}}\tilde \lambda$ converge to points at infinity (${\ensuremath{\mathds{1}}}{^{\text{\tiny T}}}\mu =0$) of the affine k-plane of normalized weights.
When reference points coalesce with an additional second order acceleration orthogonally to the subspace $W_x$, we conjecture that the affine span is not any more a great subspheres but a smaller one. This would include principal nested spheres (PNS) developed by [@jung_generalized_2010; @jung_analysis_2012] as a limit case of barycentric subspaces. It would be interesting to derive a similar procedure for hyperbolic spaces and to determine which types of subspaces could be obtained by such limits for more general non-local and higher order jets.
Barycentric subspace analysis {#Sec:BSA}
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PCA can be viewed as the search for a sequence of nested linear spaces that best approximate the data at each level. In a Euclidean space, minimizing the variance of the residuals boils down to an independent optimization of orthogonal subspaces at each level of approximation, thanks to the Pythagorean theorem. This enables building each subspace of the sequence by adding (resp. subtracting) the optimal one-dimensional subspace iteratively in a forward (resp. backward) analysis. Of course, this property does not scale up to manifolds, for which the orthogonality of subspaces is not even well defined.
Flags of barycentric subspaces in manifolds
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[@damon_backwards_2013] have argued that the nestedness of approximation spaces is one of the most important characteristics for generalizing PCA to more general spaces. Barycentric subspaces can easily be nested, for instance by adding or removing one or several points at a time, to obtains a family of embedded submanifolds which generalizes flags of vector spaces.
A flag of a vector space $V$ is a filtration of subspaces (an increasing sequence of subspaces, where each subspace is a proper subspace of the next): $\{0\} = V_0 \subset V_1 \subset V_2 \subset \cdots \subset V_k = V$. Denoting $d_i = \dim(V_i)$ the dimension of the subspaces, we have $0 = d_0 < d_1 < d_2 < \cdots < d_k = n$, where n is the dimension of V. Hence, we must have $k \leq n$. A flag is [*complete*]{} if $d_i = i$, otherwise it is a [*partial flag*]{}. Notice that a linear subspace $W$ of $V$ is identified to the partial flag $ \{0\} \subset W \subset V$. A flag can be generated by adding the successive eigenspaces of an SPD matrix with increasing eigenvalues. If all the eigenvalues have multiplicity one, the generated flag is complete and one can parametrize it by the ordered set of eigenvectors. If an eigenvalue has a larger multiplicity, then the corresponding eigenvectors might be considered as exchangeable in this parametrization in the sense that we should only consider the subspace generated by all the eigenvectors of that eigenvalue.
In an $n$-dimensional manifold ${\ensuremath{{\cal M}}}$, a strict ordering of $n+1$ independent points $x_0\prec x_1 \ldots \prec x_n$ defines a filtration of barycentric subspaces. For instance: $\operatorname{EBS}(x_0) = \{ x_0 \} \subset \cdots \operatorname{EBS}(x_0, x_1, x_k) \cdots \subset \operatorname{EBS}(x_0, \ldots x_n).$ The 0-dimensional subspace is now a points in ${\ensuremath{{\cal M}}}$ instead of the null vector in flags of vector spaces because we are in an affine setting. Grouping points together in the addition/removal process generates a partial flag of barycentric subspaces. Among the barycentric subspaces, the affine span seems to be the most interesting definition. Indeed, when the manifold ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$ is connected, the EBS of $n+1$ affinely independent points covers the full manifold ${\ensuremath{{\cal M}^*{(x_0, \ldots x_k)}}}$, and its completion covers the original manifold: ${\operatorname{Aff}}(x_0,\ldots x_n) = {\ensuremath{{\cal M}}}$. With the Fréchet or Karcher barycentric subspaces, we only generate a submanifold (the positive span) that does not cover the whole manifold in general, even in negatively curved spaces.
Let $x_0\preceq x_1 \ldots \preceq x_k$ be $k+1 \leq n+1$ affinely independent ordered points of ${\ensuremath{{\cal M}}}$ where two or more successive points are either strictly ordered ($x_i \prec x_{i+1}$) or exchangeable ($x_i \sim x_{i+1}$). For a strictly ordered set of points, we call the sequence of properly nested subspaces $FL_i(x_0\prec x_1 \ldots \prec x_k) = {\operatorname{Aff}}(x_0, \ldots x_i)$ for $0 \leq i \leq k$ the flag of affine spans $FL(x_0\prec x_1 \ldots \prec x_k)$. For a flag comprising exchangeable points, the different subspaces of the sequence are only generated at strict ordering signs or at the end. A flag is said complete if it is strictly ordered with $k=n$. We call a flag of exchangeable points $FL(x_0\sim x_1 \ldots \sim x_k)$ a pure subspace because the sequence is reduced to the unique subspace $FL_k(x_0\sim x_1 \ldots \sim x_k) = {\operatorname{Aff}}(x_0, \ldots x_k)$.
Forward and backward barycentric subspaces analysis
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In Euclidean PCA, the flag of linear subspaces can be built in a forward way, by computing the best 0-th order approximation (the mean), then the best first order approximation (the first mode), etc. It can also be built backward, by removing the direction with the minimal residual from the current affine subspace. In a manifold, we can use similar forward and backward analysis, but they have no reason to give the same result.
With a forward analysis, we compute iteratively the flag of affine spans by adding one point at a time keeping the previous ones fixed. The barycentric subspace $\operatorname{Aff}(x_0) = \{ x_0 \}$ minimizing the unexplained variance is a Karcher mean. Adding a second point amounts to compute the geodesic passing through the mean that best approximate the data. Adding a third point now differ from PGA, unless the three points coalesce to a single one. With this procedure, the Fréchet mean always belong to the barycentric subspace.
The backward analysis consists in iteratively removing one dimension. One should theoretically start with a full set of points and chose which one to remove. However, as all the sets of $n+1$ affinely independent points generate the full manifold with the affine span, the optimization really begin with the set of $n$ points $x_0, \ldots x_{n-1}$. We should afterward only test for which of the $n$ points we should remove. Since optimization is particularly inefficient in large dimensional spaces, we may run a forward analysis until we reach the noise level of the data for a dimension $k \ll n$. In practice, the noise level is often unknown and a threshold at 5% of the data variance is sometimes chosen. More elaborate methods exist to determine the intrinsic dimension of the data for manifold learning technique [@wang_scale-based_2008]. Point positions may be optimized at each step to find the optimal subspace and a backward sweep reorders the points at the end. With this process, there is no reason for the Fréchet mean to belong to any of the barycentric subspaces. For instance, if we have clusters, one expects the reference points to localize within these clusters rather than at the Fréchet mean.
Approximating data using a pure subspace
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Let ${Y} = \{ \hat y_i \}_{i=1}^N \in {\ensuremath{{\cal M}}}^N$ be $N$ data points and $X=\{x_0,\ldots x_k\}$ be $k+1$ affinely independent reference points. We assume that each data point $\hat y_i$ has almost surely one unique closest point $y_i(X)$ on the barycentric subspace. This is the situation for Euclidean, hyperbolic and spherical spaces, and this should hold more generally for all the points outside the focal set of the barycentric subspace. This allows us to write the residual $r_i(X) = {\ensuremath{\:\mbox{\rm dist}}}( \hat y_i,y_i(X))$ and to consider the minimization of the unexplained variance $\sigma^2_{out}(X) = \sum_j r_i^2(X)$. This optimization problem on ${\ensuremath{{\cal M}}}^{k+1}$ can be achieved by standard techniques of optimization on manifolds (see e.g. [@OptimizationManifold:2008]). However, it is not obvious that the canonical product Riemannian metric is the right metric to use, especially close to coincident points. In this case, one would like to consider switching to the space of (non-local) jets to guaranty the numerical stability of the solution. In practice, though, we may constraint the distance between reference points to be larger than a threshold.
A second potential problem is the lack of identifiability: the minimum of the unexplained variance may be reached by subspaces parametrized by several k-tuples of points. This is the case for constant curvature spaces since every linearly independent $k$-tuple of points in a given subspace parametrizes the same barycentric subspace. In constant curvature spaces, this can be accounted for using a suitable polar or QR matrix factorization (see e.g. \[suppB\]). In general manifolds, we expect that the absence of symmetries will break the multiplicity of this relationship (at least locally) thanks to the curvature. However, it can lead to very badly conditioned systems to solve from a numerical point of view for small curvatures.
A last problem is that the criterion we use here (the unexplained variance) is only valid for a pure subspace of fixed dimension, and considering a different dimension will lead in general to pure subspaces which cannot be described by a common subset of reference points. Thus, the forward and backward optimization of nested barycentric subspaces cannot lead to the simultaneous optimality of all the subspaces of a flag in general manifolds.
A criterion for hierarchies of subspaces: AUV on flags of affine spans
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In order to obtain consistency across dimensions, it is necessary to define a criterion which depends on the whole flag of subspaces and not on each of the subspaces independently. In PCA, one often plots the unexplained variance as a function of the number of modes used to approximate the data. This curve should decreases as fast as possible from the variance of the data (for 0 modes) to 0 (for $n$ modes). A standard way to quantify the decrease consists in summing the values at all steps, giving the Accumulated Unexplained Variances (AUV), which is analogous to the Area-Under-the-Curve (AUC) in Receiver Operating Characteristic (ROC) curves.
Given a strictly ordered flag of affine subspaces $Fl(x_0\prec x_1 \ldots \prec x_k)$, we thus propose to optimize the AUV criterion: $$\textstyle
AUV(Fl(x_0\prec x_1 \ldots \prec x_k)) = \sum_{i=0}^k \sigma^2_{out}( Fl_i(x_0\prec x_1 \ldots \prec x_k ) )$$ instead of the unexplained variance at order $k$. We could of course consider a complete flag but in practice it is often useful to stop at a dimension $k$ much smaller than the possibly very high dimension $n$. The criterion is extended to more general partial flags by weighting the unexplained variance of each subspace by the number of (exchangeable) points that are added at each step. With this global criterion, the point $x_i$ influences all the subspaces of the flag that are larger than $Fl_i(x_0\prec x_1 \ldots \prec x_k )$ but not the smaller subspaces. It turns out that optimizing this criterion results in the usual PCA up to mode $k$ in a Euclidean space.
\[THM8\] Let ${\hat Y} = \{ \hat y_i \}_{i=1}^N$ be a set of $N$ data points in ${\ensuremath{\mathbb{R}}}^n$. We denote as usual the mean by $\bar y = \frac{1}{N} \sum_{i=1}^N \hat y_i$ and the empirical covariance matrix by $\Sigma = \frac{1}{N} \sum_{i=1}^N (\hat y_i -\bar y) (\hat y_i -\bar y){^{\text{\tiny T}}}$. Its spectral decomposition is denoted by $\Sigma = \sum_{j=1}^n \sigma_j^2 u_j u_j{^{\text{\tiny T}}}$ with the eigenvalues sorted in decreasing order. We assume that the first $k+1$ eigenvalues have multiplicity one, so that the order from $\sigma_1$ to $\sigma_{k+1}$ is strict.
Then the partial flag of affine subspaces $Fl(x_0\prec x_1 \ldots \prec x_k)$ optimizing $$\textstyle
AUV(Fl(x_0\prec x_1 \ldots \prec x_k)) = \sum_{i=0}^k \sigma^2_{out}( Fl_i(x_0\prec x_1 \ldots \prec x_k ) )$$ is strictly ordered and can be parametrized by $x_0 = \bar y$, $x_i = x_0 + u_i$ for $1 \leq i \leq k$. The parametrization by points is not unique but the flag of subspaces which is generated is and is equal to the flag generated by the PCA modes up to mode $k$ included.
The proof is detailed in \[suppB\]. The main idea is to parametrize the matrix of reference vectors by the product of an orthogonal matrix $Q$ with a positive definite triangular superior matrix (QR decomposition). The key property of this Gram-Schmidt orthogonalization is the stability of the columns of $Q$ when we add or remove columns (i.e reference points) in $X$, which allows to write the expression of the AUV explicitly. Critical points are found for columns of $Q$ which are eigenvectors of the data covariance matrix and the expression of the AUV shows that we have to select them in the decreasing order of eigenvalues.
Sample-limited barycentric subspace inference on spheres
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In several domains, it has been proposed to limit the inference of the Fréchet mean to the data-points only. In neuroimaging studies, for instance, the individual image minimizing the sum of square deformation distance to other subject images has been argued to be a good alternative to the mean template (a Fréchet mean in deformation and intensity space) because it conserves the full definition and all the original characteristics of a real subject image [@lepore:inria-00616172]. Beyond the Fréchet mean, [@Feragen2013] proposed to define the first principal component mode as the geodesic going through two of the data points which minimizes the unexplained variance. The method named [*set statistics*]{} was aiming to accelerate the computation of statistics on tree spaces. [@Zhai_2016] further explored this idea under the name of [*sample-limited geodesics*]{} in the context of PCA in phylogenetic tree space. However, in both cases, extending the method to higher order principal modes was considered as a challenging research topic.
With barycentric subspaces, sample-limited statistics naturally extends to any dimension by restricting the search to (flags of) affine spans that are parametrized by data points. Moreover, the implementation boils down to a very simple enumeration problem. An important advantage for interpreting the modes of variation is that reference points are never interpolated as they are by definition sampled from the data. Thus, we may go back to additional information about the samples like the disease characteristics in medical image image analysis. The main drawback is the combinatorial explosion of the computational complexity: the optimal order-k flag of affine spans requires $O(N^{k+1})$ operations, where $N$ is the number of data points. In practice, the search can be done exhaustively for a small number of reference points but an approximated optimum has to be sought for larger $k$ using a limited number of random tuples [@Feragen2013].
![[**Left:**]{} Equi 30 simulated dataset. Data and reference points are projected from the 5-sphere to the expected 2-sphere in 3d to allow visualization. For each method (FBS in blue, 1-PBS in green and 1-BSA in red), the first reference point has a solid symbol. The 1d mode is the geodesic joining this point to the second reference point. The third reference point of FBS and 2-BSA (on the lower left part) is smaller. [**Middle:**]{} graph of the unexplained variance and AUV for the different methods on the Equi 30 dataset. [**Right:**]{} Mount Tom Dinosaur trackway 1 data with the same color code. 1-BSA (in red) and FBS (in blue) are superimposed.[]{data-label="Fig:Equi30"}](Figures/EquiTriangleBSA_30.png "fig:"){width="0.30\columnwidth"} ![[**Left:**]{} Equi 30 simulated dataset. Data and reference points are projected from the 5-sphere to the expected 2-sphere in 3d to allow visualization. For each method (FBS in blue, 1-PBS in green and 1-BSA in red), the first reference point has a solid symbol. The 1d mode is the geodesic joining this point to the second reference point. The third reference point of FBS and 2-BSA (on the lower left part) is smaller. [**Middle:**]{} graph of the unexplained variance and AUV for the different methods on the Equi 30 dataset. [**Right:**]{} Mount Tom Dinosaur trackway 1 data with the same color code. 1-BSA (in red) and FBS (in blue) are superimposed.[]{data-label="Fig:Equi30"}](Figures/EquiTriangleBSA_curves.pdf "fig:"){width="0.37\columnwidth"} ![[**Left:**]{} Equi 30 simulated dataset. Data and reference points are projected from the 5-sphere to the expected 2-sphere in 3d to allow visualization. For each method (FBS in blue, 1-PBS in green and 1-BSA in red), the first reference point has a solid symbol. The 1d mode is the geodesic joining this point to the second reference point. The third reference point of FBS and 2-BSA (on the lower left part) is smaller. [**Middle:**]{} graph of the unexplained variance and AUV for the different methods on the Equi 30 dataset. [**Right:**]{} Mount Tom Dinosaur trackway 1 data with the same color code. 1-BSA (in red) and FBS (in blue) are superimposed.[]{data-label="Fig:Equi30"}](Figures/DinoTrackBSA.png "fig:"){width="0.30\columnwidth"}
In this section, we consider the exhaustive sample-limited version of the Forward Barycentric Subspace (FBS) decomposition, the optimal $k$-dimensional Pure Barycentric Subspace with backward ordering (k-PBS), and the Barycentric Subspace Analysis up to order k (k-BSA). In order to illustrate the differences, we consider a first synthetic dataset where we draw 30 random points uniformly on an equilateral triangle of side length $\pi/2$ on a 6-dimensional sphere. We add to each point a (wrapped) Gaussian noise of standard deviation $\sigma = 10^{\circ}$. In this example, original data live on a 2-sphere: the ideal flag of subspaces is a pure 2d subspace spanning the first three coordinates. We illustrate in Fig.\[Fig:Equi30\] the different reference points that are found for the different methods. We can see that all methods end-up with different results, contrarily to the Euclidean case. The second observation is that the optimal pure subspace is not stable with the dimension: the reference points of the 0-PBS (the sample-limited Fréchet mean represented by the large blue solid diamond), the 1-PBS (in green) and the 2-PBS (identical to the red points of the 2-BSA in red) are all different. BSA is more stable: the first reference points are the same from the 1-BSA to the 3-BSA. In terms of unexplained variance, the 2-BSA is the best for two modes (since it is identical to the optimal 2-PBS) and reaches the actual noise level. It remains better than the 3-PBS and the FBS with three modes in terms of AUV even without adding a fourth point.
As a second example, we take real data encoding the shape of three successive footprints of Mount Tom Dinosaur trackway 1 described in [@small96 p.181]. For planar triangles, the shape space (quotient of the triad by similarities) boils down to the sphere of radius $1/2$. These data are displayed on the right of Fig.\[Fig:Equi30\]. In this example, the reference points of the 0-BSA to the 3-BSA are stable and identical to the ones of the FBS. This is a behavior that we have observed in most of our simulations when modes cannot be confused. This may not hold anymore if reference points were optimized on the sphere rather than on the data points only. The optimal 1-PBS (the best geodesic approximation) picks up different reference points.
Discussion
==========
We investigated in the paper several notions of subspaces in manifolds generalizing the notion of affine span in a Euclidean space. The Fréchet / Karcher / exponential barycentric subspaces are the nested locus of weighted Fréchet / Karcher / exponential barycenters with positive or negative weights summing up to 1. The affine spans is the metric completion of the largest one (the EBS). It may be a non-connected manifold with boundaries. The completeness of the affine span enables reconnecting part of the subspace that arrive from different directions at the cut-locus of reference points if needed. It also ensures that there exits a closest point on the submanifold for data projection purposes, which is fundamental for dimension reduction purposes. The fact that modifying the power of the metric does not change the affine span is an unexpected stability result which suggests that the notion is quite central. Moreover, we have shown that the affine span encompass principal geodesic subspaces as limit cases. It would be interesting to show that we can obtain other types of subspaces like principal nested subspheres with higher order and non-local jets: some non-geodesic decomposition schemes such as loxodromes and splines could probably also be seen as limit cases of barycentric subspaces.
Future work will address barycentric subspaces in interesting non-constant curvatures spaces. For instance, [@eltzner_dimension_2015] adaptively deforms the flat torus seen as a product of spheres into a unique sphere to allow principal nested spheres (PNS) analysis. A quick look at the flat torus shows that the the cut-locus of $k+1\leq n$ points in ${\cal S}_1^n$ divides the torus into $k^n$ cells in which the affine span is a $k$-dimensional linear subspace. The subspaces generated in each cell are generally disconnected, but when points coalesce with each others into a jet, the number of cells decreases in the complex and at the limit we recover a single cell that contain a connected affine span. For a first order jet, we recover as expected the restricted geodesic subspace (here a linear subspace limited to the cut locus of the jet base-point), but higher order jets may generate more interesting curved subspaces that may better describe the data geometry.
The next practical step is obviously the implementation of generic algorithms to optimize barycentric subspaces in general Riemannian manifolds. Example algorithms include: finding a point with given barycentric coordinates (there might be several so this has to be a local search); finding the closest point (and its coordinates) on the barycentric subspace; optimizing the reference points to minimize the residual error after projection of data points, etc. If such algorithms can be designed relatively simply for simple specific manifolds as we have done here for constant curvature spaces, the generalization to general manifolds requires a study of the focal set of the barycentric subspaces or guarantying the correct behavior of algorithms. We conjecture that this is a stratified set of zero measure in generic cases. Another difficulty is linked to the non-identifiability of the subspace parameters. For constant curvature spaces, the right parameter space is actually the $k$-Grassmanian. In more general manifolds, the curvature and the interaction with the cut-locus break the symmetry of the barycentric subspaces, but lead to a poor numerical conditioning of the system good renormalization techniques need to be designed to guaranty the numerical stability.
Finding the subspace that best explain the data is an optimization problem on manifolds. This raises the question of which metric should be considered on the space of barycentric subspaces. In this paper, we mainly see this space as the configuration space of $k+1$ affinely independent points, with convergence to spaces of jets (including non-local jets) when several points coalesce. Such a construction was named Multispace by [@olver_geometric_2001] in the context of symmetry-preserving numerical approximations to differential invariants. It is likely that similar techniques could be investigated to construct numerically stable implementations of barycentric subspaces of higher order parametrized by non-local jets, which are needed to optimize safely. Conversely, barycentric subspaces could help shedding a new light on the multispace construction for differential invariants.
Barycentric subspaces could probably be used to extend methods like the probabilistic PCA of [@tipping_probabilistic_1999], generalized to PGA by [@zhang_probabilistic_2013]. A first easy step in that direction is to replace the reference points by reference distributions on the manifold and to look at the locus of weighted expected means. Interestingly, this procedure soften the constraints that we had in this paper about the cut locus. Thus, following [@karcher77], reference distributions could be used in a mollifier smoothing approach to study the regularity of the barycentric subspaces.
For applications where data live on Lie groups, generalizing barycentric subspaces to more general non-Riemannian spaces like affine connection manifolds is a particularly appealing extension. In computational anatomy, for instance, deformations of shapes are lifted to a group of diffeomorphism for statistical purposes (see e.g. [@lorenzi:hal-00813835; @lorenzi:hal-01145728]). All Lie groups can be endowed with a bi-invariant symmetric Cartan-Schouten connection for which geodesics are the left and right translation of one-parameter subgroups. This provides the Lie group with an affine connection structure which may be metric or not. When the group is the direct product of compact and Abelian groups, it admits a bi-invariant metric for which the Cartan-Schouten connection is the natural Levi-Civita connection. Other groups do not admit any bi-invariant metric (this is the case for rigid transformations in more than 2 dimensions because of the semi-direct product), so that a Riemannian structure can only be left or right invariant but not both. However the bi-invariant Cartan-Schouten connection continues to exists, and one can design bi-invariant means using exponential barycenter as proposed by [@pennec:hal-00699361]. Thus, we may still define exponential barycentric subspaces and affine spans in these affine connection spaces, the main difference being that the derivative of the log is not any more the Hessian of a distance function. This might considerably complexify the analysis of the generated subspaces.
The second topic of this paper concerns the generalization of PCA to manifolds using Barycentric Subspace Analysis (BSA). [@damon_backwards_2013] argued that an interesting generalization of PCA should rely on “nested sequence of relations”, like embedded linear subspaces in the Euclidean space or embedded spheres in PNS. Barycentric subspaces can naturally be nested by adding or removing points or equivalently by setting the corresponding barycentric coordinate to zero. Thus we can easily generalize PCA to manifolds using a forward analysis by iteratively adding one or more points at a time. At the limit where points coalesce at the first order, this amounts to build a flag of (restricted) principal geodesic subspaces. Thus it generalizes the Principal Geodesic Analysis (PGA) of [@fletcher_principal_2004; @sommer_optimization_2013] when starting with a zeroth dimensional space (the Fréchet mean) and the Geodesic PCA (GPCA) of [@huckemann_principal_2006; @huckemann_intrinsic_2010] when starting directly with a first order jet defining a geodesic. One can also design a backward analysis by starting with a large subspace and iteratively removing one or more points to define embedded subspaces.
However, the greedy optimization of these forward/backward methods generally leads to different solutions which are not optimal for all subspace jointly. The key idea is to consider PCA as a joint optimization of the whole flag of subspaces instead of each subspace independently. In a Euclidean space, we showed that the Accumulated Unexplained Variances (AUV) with respect to all the subspaces of the hierarchy (the area under the curve of unexplained variance) is a proper criterion on the space of Euclidean flags. We proposed to extend this criterion to barycentric subspaces in manifolds, where an ordering of the reference points naturally defines a flag of nested barycentric subspaces. A similar idea could be used with other iterative least-squares methods like partial least-squares (PLS) which are also one-step at a time minimization methods.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by the Erwin Schrödinger Institute in Vienna through a three-weeks stay in February 2015 during the program Infinite-Dimensional Riemannian Geometry with Applications to Image Matching and Shape. It was also partially supported by the Inria Associated team GeomStats between Asclepios and Holmes’ lab at Stanford Statistics Dept. I would particularly like to thank Prof. Susan Holmes for fruitful discussions during the writing of the paper.
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Appendix A: Proof of Theorem \[THM6\] {#ProofTHM6 .unnumbered}
=====================================
We first establish a useful formula exploiting the symmetry of the geodesics from $x$ to $y \not \in {\ensuremath{{\cal C}}}(x)$ with respect to time. Reverting time along a geodesic, we have: $\gamma_{(x,{\ensuremath{\overrightarrow{xy}}})}(t) = \gamma_{(y,{\ensuremath{\overrightarrow{yx}}})}(1-t)$, which means in particular that $\dot \gamma_{(x,{\ensuremath{\overrightarrow{xy}}})}(1) = - \dot \gamma_{(y,{\ensuremath{\overrightarrow{yx}}})}(0) = -{\ensuremath{\overrightarrow{yx}}}$. Since $\gamma_{(x,{\ensuremath{\overrightarrow{xy}}})}(t) = \exp_x(t {\ensuremath{\overrightarrow{xy}}})$, we obtain $ {\ensuremath{\overrightarrow{yx}}} = - D \left. \exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}} {\ensuremath{\overrightarrow{xy}}}.$ Now, we also have $ \left( D \left. \exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}} \right). D \left. \log_x \right|_y = {\ensuremath{\:\mathrm{Id}}}$ because $\exp_x( \log_x(y)) = y$. Finally, $D\exp_x$ and $D\log_x$ have full rank on ${\ensuremath{{\cal M}}}/{\ensuremath{{\cal C}}}(x)$ since there is no conjugate point before the cut-locus, so that we can multiply by their inverse and we end up with: $$\label{eq:symgeo}
\forall y \not \in {\ensuremath{{\cal C}}}(x), \quad {\ensuremath{\overrightarrow{xy}}} = - D \left. \log_x \right|_y {\ensuremath{\overrightarrow{yx}}}.$$
Let us first restrict to a convenient domain of ${\ensuremath{{\cal M}}}$: we consider a open geodesic ball $B(x_0, \zeta)$ of radius $\zeta$ centered at $x_0$ and we exclude all the points of ${\ensuremath{{\cal M}}}$ which cut locus intersect this ball, or equivalently the cut-locus of all the points of this ball. We obtain an open domain ${\cal D}_{\zeta}(x_0) = {\ensuremath{{\cal M}}}\setminus {\ensuremath{{\cal C}}}(B(x_0, \zeta))$ in which $\log_x(y)$ is well defined and smooth for all $x \in B(x_0, \zeta)$ and all $y\in {\cal D}_{\zeta}(x_0)$. Thanks to the symmetry of the cut-locus, $\log_y(x)$ is also well defined and smooth in the same conditions and Eq. (\[eq:symgeo\]) can be rephrased: $$\label{eq:symgeo2}
\forall x \in B(x_0, \zeta), y\in {\cal D}_{\zeta}(x_0), \quad {\ensuremath{\overrightarrow{xy}}} = - D \left. \log_x \right|_y {\ensuremath{\overrightarrow{yx}}}.$$
Let $\|w\|_{\infty} = \max_i \|w_i\|_{x_0}$ be the maximal length of the vectors $w_i$. For $\varepsilon < \zeta / \|w\|_{\infty}$, we have $\|\varepsilon w_i\|_{x_0} \leq \varepsilon \|w\|_{\infty} < \zeta$, so that all the points $x_i = \exp_{x_0}( \varepsilon w_i)$ belong to the open geodesic ball $B(x_0, \zeta)$. Thus, $\log_x(x_i)$ and $\log_{x_i}(x)$ are well defined and smooth for any $x \in {\cal D}_{\zeta}(x_0)$, and we can write the Taylor expansion in a normal coordinate system at $x_0$using Eq.\[eq:symgeo2\]: $$\textstyle
\log_x(x_i(\varepsilon))
= \log_x( x_0) + \varepsilon D\log_x|_{x_0} w_i + O(\varepsilon ^2)
= D\log_x|_{x_0} \left( \varepsilon w_i - \log_{x_0}(x) \right) + O(\varepsilon ^2)$$
Any point $x \in {\cal D}_{\zeta}(x_0)$ can be defined by $\log_{x_0}(x) = \sum_{j=1}^k \alpha_i w_i + w_{\bot}$ with ${\ensuremath{ \left<
\:w_{\bot}\:\left|\:w_i\right.\right> }} =0$ and suitable constraints on the $\alpha_i$ and $w_{\bot}$. Replacing $\log_x( x_0)$ by its value in the above formula, we get $$\textstyle
\log_x(x_i(\varepsilon)) = D\log_x|_{x_0} \left( \varepsilon w_i - \sum_{j=1}^k \alpha_j w_j - w_{\bot} \right) + O(\varepsilon ^2).$$ Since the matrix $D\log_x|_{x_0}$ is invertible, the EBS equation $\mathfrak{M}_1(x, \lambda)= \sum_{i=0}^k \lambda_i {\ensuremath{\overrightarrow{x x_i}}} =0$ is equivalent to $\textstyle \textstyle w_{\bot} + \sum_{j=1}^k \alpha_j w_j - \varepsilon\left(\sum_{i=1}^k {{\underaccent{\bar}{\lambda}}}_i w_i\right) = O(\varepsilon ^2).$ Projecting orthogonally to $W_{x_0}$, we get $w_{\bot} = O(\varepsilon^2)$: this means that any point of the limit EBS has to be of the form $x = \exp_{x_0}(\sum_{j=1}^k \alpha_i w_i)$. In other words, only points of the restricted geodesic subspace $GS^*(W_{x_0})$ can be solutions of the limit EBS equation.
Now, for a point of $GS^*(W_{x_0})$ to be a solution of the limit EBS equation, there should exists barycentric coordinates $\lambda$ such that $\sum_{j=1}^k (\alpha_j - \varepsilon {{\underaccent{\bar}{\lambda}}}_i) w_j = O(\varepsilon ^2)$. Choosing $\lambda = (\varepsilon - \sum_i \alpha_i: \alpha_1 : \ldots : \alpha_k)$, we obtain the normalized barycentric coordinates ${{\underaccent{\bar}{\lambda}}}_i = \alpha_i / \varepsilon$ for $1\leq i \leq k$ and ${{\underaccent{\bar}{\lambda}}}_0 = 1 - (\sum_i \alpha_i) / \varepsilon$ that satisfy this condition. Thus any point of $GS^*(W_{x_0}) \cap {\cal D}_{\zeta}(x_0)$ is a solution of the limit EBS equation with barycentric coordinates at infinity on ${\ensuremath{{\cal P}^*_k}}$. Taking $\zeta$ sufficiently small, we can include all the points of $GS^*(W_{x_0})$.
Riemannian manifolds
====================
A Riemannian manifold is a differential manifold endowed with a smooth collection of scalar products ${\ensuremath{ \left<
\:.\:\left|\:.\right.\right> }}_{x}$ on each tangent space $T_{x}{\ensuremath{{\cal M}}}$ at point $x$ of the manifold, called the Riemannian metric. In a chart, the metric is expressed by a symmetric positive definite matrix $G(x) = [ g_{ij}(x) ]$ where each element is given by the dot product of the tangent vector to the coordinate curves: $g_{ij}(x) = {\ensuremath{ \left<
\:\partial_i\:\left|\:\partial_j\right.\right> }}_x$. This matrix is called the [*local representation of the Riemannian metric*]{} in the chart $x$ and the dot products of two vectors $v$ and $w$ in $T_{x}{\ensuremath{{\cal M}}}$ is now ${\ensuremath{ \left<
\:v\:\left|\:w\right.\right> }}_x = v{^{\text{\tiny T}}}\: G(x)\: w = g_{ij}(x) v^i w^j$ using the Einstein summation convention which implicitly sum over the indices that appear both in upper position (components of \[contravariant\] vectors) and lower position (components of covariant vectors (co-vectors)).
Riemannian distance and geodesics
---------------------------------
If we consider a curve $\gamma(t)$ on the manifold, we can compute at each point its instantaneous speed vector $\dot{\gamma}(t)$ (this operation only involves the differential structure) and its norm $ \left\| \dot{\gamma}(t)\right\|_{\gamma(t)}$ to obtain the instantaneous speed (the Riemannian metric is needed for this operation). To compute the length of the curve, this value is integrated along the curve: $$\label{curve_length}
{\cal L}_a^b (\gamma) = \int_a^b \left\| \dot{\gamma}(t)\right\|_{\gamma(t)}
dt =
\int_a^b \left( {\ensuremath{ \left<
\: \dot{\gamma}(t)\:\left|\:\dot{\gamma}(t)
\right.\right> }}_{\gamma(t)} \right)^{\frac{1}{2}}dt$$ The distance between two points of a connected Riemannian manifold is the minimum length among the curves joining these points. The curves realizing this minimum are called geodesics. Finding the curves realizing the minimum length is a difficult problem as any time-reparameterization is authorized. Thus one rather defines the metric geodesics as the critical points of the energy functional ${\cal E}(\gamma) = \frac{1}{2}\int_0^1 \left\| \dot \gamma (t)\right\|^2\: dt$. It turns out that they also optimize the length functional but they are moreover parameterized proportionally to arc-length.
Let $[g^{ij}] = [g_{ij}]{^{\text{\tiny (-1)}}}$ be the inverse of the metric matrix (in a given coordinate system) and $\Gamma^i_{jk} = \frac{1}{2} g^{im}\left( \partial_k g_{mj} + \partial_j g_{mk} - \partial_m g_{jk} \right)$ the Christoffel symbols. The calculus of variations shows the geodesics are the curves satisfying the following second order differential system: $$\ddot{\gamma}^i + \Gamma^i_{jk} \dot{\gamma}^j \dot{\gamma}^k = 0.$$
The fundamental theorem of Riemannian geometry states that on any Riemannian manifold there is a unique (torsion-free) connection which is compatible with the metric, called the Levi-Civita (or metric) connection. For that choice of connection, shortest paths (geodesics) are auto-parallel curves (“straight lines”). This connection is determined in a local coordinate system through the Christoffel symbols: $\nabla_{\partial_i}\partial_j = \Gamma_{ij}^k \partial_k$. With these conventions, the covariant derivative of the coordinates $v^i$ of a vector field is $v^i_{;j} = (\nabla_j v)^i = \partial_j v^i +\Gamma^i_{jk} v^k$.
In the following, we only consider the Levi-Civita connection and we assume that the manifold is geodesically complete, i.e. that the definition domain of all geodesics can be extended to ${\ensuremath{\mathbb{R}}}$. This means that the manifold has no boundary nor any singular point that we can reach in a finite time. As an important consequence, the Hopf-Rinow-De Rham theorem states that there always exists at least one minimizing geodesic between any two points of the manifold (i.e. whose length is the distance between the two points).
Normal coordinate systems {#ExpMapIntro}
-------------------------
Let $x$ be a point of the manifold that we consider as a local reference and $v$ a vector of the tangent space $T_{x}{\ensuremath{{\cal M}}}$ at that point. From the theory of second order differential equations, we know that there exists one and only one geodesic $\gamma_{(x,v)}(t)$ starting from that point with this tangent vector. This allows to wrap the tangent space onto the manifold, or equivalently to develop the manifold in the tangent space along the geodesics (think of rolling a sphere along its tangent plane at a given point). The mapping $ \exp_{x}(v) = \gamma_{(x,v)}(1)$ of each vector $v \in
T_{x}{\ensuremath{{\cal M}}}$ to the point of the manifold that is reached after a unit time by the geodesic $\gamma_{(x,v)}(t)$ is called the [*exponential map*]{} at point $x$. Straight lines going through 0 in the tangent space are transformed into geodesics going through point $x$ on the manifold and distances along these lines are conserved.
The exponential map is defined in the whole tangent space $T_{x}{\ensuremath{{\cal M}}}$ (since the manifold is geodesically complete) but it is generally one-to-one only locally around 0 in the tangent space (i.e. around $x$ in the manifold). In the sequel, we denote by ${\ensuremath{\overrightarrow{xy}}}=\log_{x}(y)$ the inverse of the exponential map: this is the smallest vector (in norm) such that $y = \exp_{x}({\ensuremath{\overrightarrow{xy}}})$. It is natural to search for the maximal domain where the exponential map is a diffeomorphism. If we follow a geodesic $\gamma_{(x, v)}(t) = \exp_{x}(t\: v)$ from $t=0$ to infinity, it is either always minimizing all along or it is minimizing up to a time $t_0 < \infty$ and not any more after (thanks to the geodesic completeness). In this last case, the point $ \gamma_{(x,v)}(t_0)$ is called a [*cut point*]{} and the corresponding tangent vector $t_0\: v$ a [*tangential cut point*]{}. The set of tangential cut points at $x$ is called the [*tangential cut locus*]{} $C(x) \in T_{x}{\ensuremath{{\cal M}}}$, and the set of cut points of the geodesics starting from $x$ is the [*cut locus*]{} ${\ensuremath{{\cal C}}}(x) = \exp_{x}(C(x)) \in {\ensuremath{{\cal M}}}$. This is the closure of the set of points where several minimizing geodesics starting from $x$ meet. On the sphere ${\mathcal S}_2(1)$ for instance, the cut locus of a point $x$ is its antipodal point and the tangential cut locus is the circle of radius $\pi$.
The maximal bijective domain of the exponential chart is the domain $D(x)$ containing 0 and delimited by the tangential cut locus ($\partial D(x) = C(x)$). This domain is connected and star-shaped with respect to the origin of $T_{x}{\ensuremath{{\cal M}}}$. Its image by the exponential map covers all the manifold except the cut locus, which has a null measure. Moreover, the segment $[0,{\ensuremath{\overrightarrow{xy}}}]$ is mapped to the unique minimizing geodesic from $x$ to $y$: geodesics starting from $x$ are straight lines, and the distance from the reference point are conserved. This chart is somehow the “most linear” chart of the manifold with respect to the reference point $x$.
When the tangent space is provided with an orthonormal basis, this is called [*an normal coordinate systems at $x$*]{}. A set of normal coordinate systems at each point of the manifold realize an atlas which allows to work very easily on the manifold. The implementation of the exponential and logarithmic maps (from now on $\exp$ and $\log$) is indeed the basis of programming on Riemannian manifolds, and we can express using them practically all the geometric operations needed for statistics [@A:pennec:inria-00614994] or image processing [@A:pennec:inria-00614990].
The size of the maximal definition domain is quantified by the [*injectivity radius*]{} $\mbox{inj}({\ensuremath{{\cal M}}},x) = {\ensuremath{\:\mbox{\rm dist}}}(x,{\ensuremath{{\cal C}}}(x))$, which is the maximal radius of centered balls in $T_{x}{\ensuremath{{\cal M}}}$ on which the exponential map is one-to-one. The injectivity radius of the manifold $\mbox{inj}({\ensuremath{{\cal M}}})$ is the infimum of the injectivity over the manifold. It may be zero, in which case the manifold somehow tends towards a singularity (think e.g. to the surface $z=1/\sqrt{x^2+y^2}$ as a sub-manifold of ${\ensuremath{\mathbb{R}}}^3$).
In a Euclidean space, normal coordinate systems are realized by orthonormal coordinates system translated at each point: we have in this case ${\ensuremath{\overrightarrow{xy}}} = \log_{x}(y) = y-x$ and $\exp_{x}({\ensuremath{\overrightarrow{v}}}) = x+{\ensuremath{\overrightarrow{v}}}$. This example is more than a simple coincidence. In fact, most of the usual operations using additions and subtractions may be reinterpreted in a Riemannian framework using the notion of [*bipoint*]{}, an antecedent of vector introduced during the 19th Century. Indeed, vectors are defined as equivalent classes of bipoints in a Euclidean space. This is possible because we have a canonical way (the translation) to compare what happens at two different points. In a Riemannian manifold, we can still compare things locally (by parallel transportation), but not any more globally. This means that each “vector” has to remember at which point of the manifold it is attached, which comes back to a bipoint.
Hessian of the squared distance
===============================
Computing the differential of the Riemannian log
------------------------------------------------
On ${\ensuremath{{\cal M}}}/ C(y)$, the Riemannian gradient $\nabla^a = g^{ab} \partial_b$ of the squared distance $d^2_y(x)={\ensuremath{\:\mbox{\rm dist}}}^2(x, y)$ with respect to the fixed point $y$ is well defined and is equal to $\nabla d^2_y(x) = -2 \log_x(y)$. The Hessian operator (or double covariant derivative) $\nabla^2 f(x)$ from $T_x{\ensuremath{{\cal M}}}$ to $T_x{\ensuremath{{\cal M}}}$ is the covariant derivative of the gradient, defined by the identity $\nabla^2 f(v) = \nabla_v(\nabla f)$. In a normal coordinate system at point $x$, the Christoffel symbols vanish at $x$, so that the Hessian operator of the squared distance can be expressed with the standard differential $D_x$ with respect to the point $x$: $$\nabla^2 d^2_y(x) = -2 (D_x \log_x(y)).$$ The points $x$ and $y=\exp_x(v)$ are called conjugate if $D\exp_x(v)$ is singular. It is known that the cut point (if it exists) occurs at or before the first conjugate point along any geodesic [@A:LeeCurvature:1997]. Thus, $D\exp_x(v)$ has full rank inside the tangential cut-locus of $x$. This is in essence why there is a well posed inverse function ${\ensuremath{\overrightarrow{x y}}} = \log_x(y)$, called the Riemannian log, which is continuous and differentiable everywhere except at the cut locus of $x$. Moreover, its differential can be computed easily: since $\exp_x(\log_x(y)) =y $, we have $\left. D\exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}} D\log_x (y) = {\ensuremath{\:\mathrm{Id}}}$, so that $$D\log_x (y) = \left( \left. D\exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}} \right)^{-1}
\label{eq:Dylogxy}$$ is well defined and of full rank on ${\ensuremath{{\cal M}}}/C(x)$.
We can also see the Riemannian log $\log_x(y) = {\ensuremath{\overrightarrow{x y}}}$ as a function of the foot-point $x$, and differentiating $\exp_x(\log_x(y))=y$ with respect to it gives: $
\left. D_x \exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}} + \left. D\exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}}.D_x\log_x (y) =0.
$ Once again, we obtain a well defined and full rank differential for $x \in {\ensuremath{{\cal M}}}/C(y)$: $$D_x\log_x (y) = - \left( \left. D\exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}}\right)^{-1} \left. D_x \exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}}.
\label{eq:Dxlogxy}$$ The Hessian of the squared distance can thus be written: $$\frac{1}{2}\nabla^2 d^2_y(x) = - D_x \log_x(x_i) = \left( \left. D\exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}}\right)^{-1} \left. D_x \exp_x \right|_{{\ensuremath{\overrightarrow{xy}}}}.$$ If we notice that $J_0(t) = \left. D\exp_x\right|_{t {\ensuremath{\overrightarrow{xy}}}}$ (respectively $J_1(t) = \left. D_x \exp_x\right|_{t {\ensuremath{\overrightarrow{xy}}}}$) are actually matrix Jacobi field solutions of the Jacobi equation $\ddot J (t) + R(t) J(t) =0$ with $J_0(0)=0$ and $\dot J_0(0)={\ensuremath{\:\mathrm{Id}}}_n$ (respectively $J_1(0)={\ensuremath{\:\mathrm{Id}}}_n$ and $\dot J_1(0)=0$), we see that the above formulation of the Hessian operator is equivalent to the one of [@A:villani_regularity_2011]\[Equation 4.2\]: $\frac{1}{2}\nabla^2 d^2_y(x) = J_0(1){^{\text{\tiny (-1)}}}J_1(1)$.
Taylor expansion of the Riemannian log
--------------------------------------
In order to better figure out what is the dependence of the Hessian of the squared Riemannian distance with respect to curvature, we compute here the Taylor expansion of the Riemannian log function. Following [@A:brewin_riemann_2009], we consider a normal coordinate system centered at $x$ and $x_v = \exp_x(v)$ a variation of the point $x$. We denote by $R_{ihjk}(x)$ the coefficients of the curvature tensor at $x$ and by $\epsilon$ a conformal gauge scale that encodes the size of the path in terms of $\|v \|_x$ and $\| {\ensuremath{\overrightarrow{xy}}} \|_x$ normalized by the curvature (see [@A:brewin_riemann_2009] for details).
In a normal coordinate system centered at $x$, we have the following Taylor expansion of the metric tensor coefficients: $$\begin{split}
g_{ab}(v) = & g_{ab} - \frac{1}{3} R_{cabd}v^c v^d
- \frac{1}{6} \nabla_e R_{cabd} v^e v^c v^d
\\ & + \left( - \frac{1}{20} \nabla_e \nabla_f R_{cabd} + \frac{2}{45} R_{cad}^g R_{ebf}^h \delta_{gh} \right) v^c v^d v^e v^f + O(\epsilon^5). \end{split}
\label{eq:TaylorMetric}$$
A geodesic joining point $z$ to point $z+\delta z$ has tangent vector: $$\begin{aligned}
\left[ \log_z(z+\Delta z) \right]^a
&= &\Delta z^a +\frac{1}{3} z^b \Delta z^c \Delta z^d R^a_{cbd}
+ \frac{1}{12} z^b z^c \Delta z^d \Delta z^e \nabla_d R^a_{bce}
\\ && + \frac{1}{6} z^b z^c \Delta z^d \Delta z^e \nabla_b R^a_{dce}
+ \frac{1}{24} z^b z^c \Delta z^d \Delta z^e \nabla^a R_{bdce}
\\ && + \frac{1}{12} z^b \Delta z^c \Delta z^d \Delta z^e \nabla_c R^a_{dbe}
+ O(\epsilon^4).\end{aligned}$$
Using $ z= v$ and $z+\Delta z = {\ensuremath{\overrightarrow{xy}}}$ (i.e. $\Delta z = {\ensuremath{\overrightarrow{xy}}} -v)$ in a normal coordinate system centered at $x$, and keeping only the first order terms in $v$, we obtain the first terms of the series development of the log: $$\label{eq:TaylorLog}
\begin{split}
\left[ \log_{x +v}(y) \right]^a
& = {\ensuremath{\overrightarrow{xy}}}^a -v^a + \frac{1}{3} R^a_{cbd} v^b {\ensuremath{\overrightarrow{xy}}}^c {\ensuremath{\overrightarrow{xy}}}^d
+ \frac{1}{12} \nabla_c R^a_{dbe} v^b {\ensuremath{\overrightarrow{xy}}}^c {\ensuremath{\overrightarrow{xy}}}^d {\ensuremath{\overrightarrow{xy}}}^e
+ O(\epsilon^4).
\end{split}$$ Thus, the differential of the log with respect to the foot point is: $$\label{eq:Diff_logSupp}
- \left[ D_x \log_x(y) \right]^a_b = \delta^a_b - \frac{1}{3} R^a_{cbd} {\ensuremath{\overrightarrow{xy}}}^c {\ensuremath{\overrightarrow{xy}}}^d - \frac{1}{12} \nabla_c R^a_{dbe} {\ensuremath{\overrightarrow{xy}}}^c {\ensuremath{\overrightarrow{xy}}}^d {\ensuremath{\overrightarrow{xy}}}^e + O(\epsilon^3).$$ Since we are in a normal coordinate system, the zeroth order term is the identity matrix, like in the Euclidean space, and the first order term vanishes. The Riemannian curvature tensor appear in the second order term and its covariant derivative in the third order term. The important point here is to see that the curvature is the leading term that makes this matrix departing from the identity (i.e. the Euclidean case) and which may lead to the non invertibility of the differential.
Example on spheres {#sec:sphere}
==================
We consider the unit sphere in dimension $n \geq 2$ embedded in ${\ensuremath{\mathbb{R}}}^{n+1}$ and we represent points of ${\ensuremath{{\cal M}}}= {\cal S}_n$ as unit vectors in ${\ensuremath{\mathbb{R}}}^{n+1}$. The tangent space at $x$ is naturally represented by the linear space of vectors orthogonal to $x$: $T_x{\cal S}_n = \{ v \in {\ensuremath{\mathbb{R}}}^{n+1}, v{^{\text{\tiny T}}}x =0\}$. The natural Riemannian metric on the unit sphere is inherited from the Euclidean metric of the embedding space ${\ensuremath{\mathbb{R}}}^{n+1}$. With these conventions, the Riemannian distance is the arc-length $d(x,y) = \arccos( x{^{\text{\tiny T}}}y)= \theta \in [0,\pi]$. Denoting $f(\theta) = 1/ \mbox{sinc}(\theta) = { \theta}/{\sin(\theta)}$, the spherical exp and log maps are: $$\begin{aligned}
\exp_x(v) & = & \cos(\| v\|) x + \mbox{sinc}(\| v\|) v / \| v\| \\
\log_x(y) & = & f(\theta) \left( y - \cos(\theta) x \right)
\quad \text{with} \quad \theta = \arccos(x{^{\text{\tiny T}}}y).\end{aligned}$$ Notice that $f(\theta)$ is a smooth function from $]-\pi;\pi[$ to ${\ensuremath{\mathbb{R}}}$ that is always greater than one and is locally quadratic at zero: $f(\theta) = 1 +\theta^2/6 + O(\theta^4)$.
Hessian of the squared distance on the sphere
---------------------------------------------
To compute the gradient and Hessian of functions on the sphere, we first need a chart in a neighborhood of a point $x\in {\cal S}_n$. We consider the unit vector $x_v = \exp_x(v)$ which is a variation of $x$ parametrized by the tangent vector $v \in T_x{\cal S}_n$ (i.e. verifying $x{^{\text{\tiny T}}}v=0$). In order to extend this mapping to the embedding space to simplify computations, we consider that $v$ is the orthogonal projection of an unconstrained vector $w \in {\ensuremath{\mathbb{R}}}^{n+1}$ onto the tangent space at $x$: $v=({\ensuremath{\:\mathrm{Id}}}-x x{^{\text{\tiny T}}})w$. Using the above formula for the exponential map, we get at first order $x_v = x - v + O(\|v\|^2)$ in the tangent space or $x_w = x + ({\ensuremath{\:\mathrm{Id}}}-x x{^{\text{\tiny T}}})w + O(\|w\|^2)$ in the embedding space.
It is worth verifying first that the gradient of the squared distance $\theta^2 = d^2_y(x) = \arccos^2\left( {x{^{\text{\tiny T}}}y} \right)$ is indeed $\nabla d^2_y(x) = -2 \log_x(y)$. We considering the variation $x_w = \exp_x( ({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})w)= x +({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})w + O(\|w\|^2)$. Because $D_x \arccos(y {^{\text{\tiny T}}}x) = -y {^{\text{\tiny T}}}/ \sqrt{ 1 - (y {^{\text{\tiny T}}}x)^2}$, we get: $$D_w \arccos^2\left( {x_w{^{\text{\tiny T}}}y} \right)
= \frac{ -2 \theta}{\sin \theta} y{^{\text{\tiny T}}}({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})
= -2 f(\theta) y{^{\text{\tiny T}}}({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}}),$$ and the gradient is as expected: $$\nabla d^2_y(x) = -2 f(\theta) ({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})y = -2 \log_x(y).
\label{eq:GradDistSphere}$$
To obtain the Hessian, we now compute the Taylor expansion of $\log_{x_w}(y)$. First, we have $$f(\theta_w)
= f(\theta) - \frac{f'(\theta)}{\sin \theta} {y{^{\text{\tiny T}}}({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})w} + O(\|w\|^2),$$ with $ f'( \theta )
= (1-f(\theta)\cos \theta)/\sin \theta$. Thus, the first order Taylor expansion of $\log_{x_w}(y) = f(\theta_w) ( y - \cos(\theta_w) x_w )$ is: $$\begin{split}
\log_{x_w}(y)
& = f(\theta_w)
\left( {\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}}-({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})w x{^{\text{\tiny T}}}- x w{^{\text{\tiny T}}}({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}}) \right)y + O(\|w\|^2) \\
\end{split}$$ so that $$\begin{split}
-2 D_w \log_{x_w}(y) =
\frac{f'(\theta)}{\sin \theta} ( {\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})y y{^{\text{\tiny T}}}({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}}) - f(\theta) \left( x{^{\text{\tiny T}}}y {\ensuremath{\:\mathrm{Id}}}+ x y{^{\text{\tiny T}}}\right) ( {\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})
\end{split}$$ Now, since we have computed the derivative in the embedding space, we have obtained the Hessian with respect to the flat connection of the embedding space, which exhibits a non-zero normal component. In order to obtain the Hessian with respect to the connection of the sphere, we need to project back on $T_x{\cal S}_n$ (i.e. multiply by $({\ensuremath{\:\mathrm{Id}}}-x x{^{\text{\tiny T}}})$ on the left) and we obtain: $$\begin{split}
\frac{1}{2} H_x(y)
& = \left( \frac{1- f(\theta) \cos\theta}{\sin^2 \theta } \right) \left( {\ensuremath{\:\mathrm{Id}}}- x x{^{\text{\tiny T}}}\right) yy{^{\text{\tiny T}}}({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})
+ f( \theta )\cos \theta ({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}}) \\
& = \left( {\ensuremath{\:\mathrm{Id}}}- x x{^{\text{\tiny T}}}\right) \left( ( 1 - f(\theta) \cos\theta ) \frac{ yy{^{\text{\tiny T}}}}{ \sin^2\theta}
+ f( \theta )\cos\theta {\ensuremath{\:\mathrm{Id}}}\right) ({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}}).
\end{split}$$
To simplify this expression, we note that $\|({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})y\|^2 = \sin \theta$, so that $u = \frac{({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})y }{ \sin \theta} = \frac{ \log_x(y) }{\theta}$ is a unit vector of the tangent space at $x$ (for $y \not = x$ so that $\theta > 0$). Using this unit vector and the intrinsic parameters $\log_x(y)$ and $\theta = \| \log_x(y)\|$, we can rewrite the Hessian: $$\begin{aligned}
\qquad \frac{1}{2} H_x(y) & = & f( \theta )\cos\theta ({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}})
+ \left( \frac{ 1- f(\theta)\cos\theta}{\theta^2 } \right) \log_x(y) \log_x(y){^{\text{\tiny T}}}\\
& = & u u{^{\text{\tiny T}}}+ f( \theta )\cos\theta ({\ensuremath{\:\mathrm{Id}}}-xx{^{\text{\tiny T}}}- u u{^{\text{\tiny T}}}) \end{aligned}$$
The eigenvectors and eigenvalues of this matrix are now very easy to determine. By construction, $x$ is an eigenvector with eigenvalue $\mu_0=0$. Then the vector $u$ (or equivalently $\log_x(y) = f(\theta) ({\ensuremath{\:\mathrm{Id}}}-x x {^{\text{\tiny T}}}) y = \theta u$) is an eigenvector with eigenvalue $\mu_1=1$. Finally, every vector $u$ which is orthogonal to these two vectors (i.e. orthogonal to the plane spanned by 0, $x$ and $y$) has eigenvalue $\mu_2= f(\theta)\cos\theta = \theta \cot \theta$. This last eigenvalue is positive for $\theta \in [0,\pi/2[$, vanishes for $\theta = \pi/2$ and becomes negative for $\theta \in ]\pi/2 \pi[$. We retrieve here the results of [@A:buss_spherical_2001 lemma 2] expressed in a more general coordinate system.
Example on the hyperbolic space ${\ensuremath{\mathbb{H}}}^n$ {#example-on-the-hyperbolic-space-ensuremathmathbbhn}
=============================================================
We consider in this section the hyperboloid of equation $-x_0^2 + x_1^2 \ldots x_n^2 = -1$ (with $x_0 > 0$ and $n \geq 2$) embedded in ${\ensuremath{\mathbb{R}}}^{n+1}$. Using the notations $x=(x_0,\hat x)$ and the indefinite nondegenerate symmetric bilinear form ${\ensuremath{ \left<
\:x\:\left|\:y\right.\right> }}_* = x{^{\text{\tiny T}}}J y= \hat x{^{\text{\tiny T}}}\hat y -x_0 y_0$ with $ J = \mbox{diag}(-1, {\ensuremath{\:\mathrm{Id}}}_n)$, the hyperbolic space can be seen as the sphere $\|x\|^2_* =-1$ of radius -1 in the $(n+1)$-dimensional Minkowski space: $${\ensuremath{\mathbb{H}}}^n = \{ x \in {\ensuremath{\mathbb{R}}}^{n,1} / \|x\|^2_* = \|\hat x\|^2 -x_0^2 = -1 \}.$$ A point in ${\ensuremath{{\cal M}}}= {\ensuremath{\mathbb{H}}}^n \subset {\ensuremath{\mathbb{R}}}^{n,1}$ can be parametrized by $x=(\sqrt{1+\|\hat x\|^2}, \hat x)$ for $\hat x \in {\ensuremath{\mathbb{R}}}^n$ (Weierstrass coordinates). This happen to be in fact a global diffeomorphism that provides a very convenient global chart of the hyperbolic space. We denote $\pi(x)=\hat x$ (resp. $\pi{^{\text{\tiny (-1)}}}(\hat x)= (\sqrt{1+\|\hat x\|^2}, \hat x)$) the coordinate map from ${\ensuremath{\mathbb{H}}}^n$ to ${\ensuremath{\mathbb{R}}}^n$ (resp. the parametrization map from ${\ensuremath{\mathbb{R}}}^n$ to ${\ensuremath{\mathbb{H}}}^n$). The Poincarré ball model is another classical models of the hyperbolic space ${\ensuremath{\mathbb{H}}}^n$ which can be obtained by a stereographic projection of the hyperboloid onto the hyperplane $x_0 = 0$ from the south pole $(-1, 0 \ldots, 0)$.
A tangent vector $v=(v_0, \hat v)$ at point $x=(x_0,\hat x)$ satisfies ${\ensuremath{ \left<
\:x\:\left|\:v\right.\right> }}_* = 0$, i.e. $x_0 v_0 = \hat x{^{\text{\tiny T}}}\hat v$, so that $$T_x {\ensuremath{\mathbb{H}}}^n = \left\{ \left( \frac{\hat x{^{\text{\tiny T}}}\hat v}{\sqrt{1+\|\hat x\|^2}}, \hat v\right),\quad \hat v\in {\ensuremath{\mathbb{R}}}^{n} \right\}.$$ The natural Riemannian metric on the hyperbolic space is inherited from the Minkowski metric of the embedding space ${\ensuremath{\mathbb{R}}}^{n,1}$: the scalar product of two vectors $u=(\hat x{^{\text{\tiny T}}}\hat u / \sqrt{1+\|\hat x\|^2},\hat u)$ and $v=(\hat x{^{\text{\tiny T}}}\hat v / \sqrt{1+\| \hat x\|^2}, \hat v)$ at $x=(\sqrt{1+\|\hat x\|^2}, \hat x)$ is $${\ensuremath{ \left<
\:u\:\left|\:v\right.\right> }}_* = u{^{\text{\tiny T}}}J v = -u_0 v_0 + \hat u{^{\text{\tiny T}}}\hat v = \hat u{^{\text{\tiny T}}}\left( -\frac{\hat x \hat x{^{\text{\tiny T}}}}{1+\|\hat x\|^2} + {\ensuremath{\:\mathrm{Id}}}\right) \hat v$$ The metric matrix expressed in the coordinate chart $G={\ensuremath{\:\mathrm{Id}}}- \frac{ \hat x \hat x{^{\text{\tiny T}}}}{1+\|\hat x\|^2}$ has eigenvalue 1, with multiplicity $n-1$, and $1/(1+\|\hat x\|^2)$ along the eigenvector $x$. It is thus positive definite.
With these conventions, geodesics are the trace of 2-planes passing through the origin and the Riemannian distance is the arc-length: $$d(x,y) = \operatorname{arccosh}( - {\ensuremath{ \left<
\:x\:\left|\:y\right.\right> }}_* ).$$ The hyperbolic exp and log maps are: $$\begin{aligned}
\quad \exp_x(v) &=& \cosh(\| v\|_* ) x + {\sinh(\| v\|_* )} v / {\| v\|_* }
\\
\log_x(y) &=& f_*(\theta) \left( y - \cosh(\theta) x \right)
\quad \text{with} \quad \theta = \operatorname{arccosh}( -{\ensuremath{ \left<
\:x\:\left|\:y\right.\right> }}_* ),\end{aligned}$$ where $f_*(\theta) = { \theta}/{\sinh(\theta)}$ is a smooth function from ${\ensuremath{\mathbb{R}}}$ to $(0,1]$ that is always positive and is locally quadratic at zero: $f_*(\theta) = 1 - \theta^2/6 + O(\theta^4)$.
Hessian of the squared distance on the hyperbolic space
-------------------------------------------------------
We first verify that the gradient of the squared distance $d^2_y(x) = \operatorname{arccosh}^2\left( -<x,y>_* \right)$ is indeed $\nabla d^2_y(x) = -2 \log_x(y)$. Let us consider a variation of the base-point along the tangent vector $v$ at $x$ verifying ${\ensuremath{ \left<
\:v\:\left|\:x\right.\right> }}_*=0$: $$x_{v} = \exp_x(v) = \cosh(\| v\|_* ) x + \frac{\sinh( \| v\|_* )}{ \| v\|_* } v
= x + v + O( \| v \|_*^2).$$ In order to extend this mapping to the embedding space around the paraboloid, we consider that $v$ is the projection $v=w + {\ensuremath{ \left<
\:w\:\left|\:x\right.\right> }}_* x$ of an unconstrained vector $w\in {\ensuremath{\mathbb{R}}}^{n,1}$ onto the tangent space at $T_x {\ensuremath{\mathbb{H}}}^n$. Thus, the variation that we consider in the embedding space is $$x_w = x + \partial_w x_w + O(\|w\|^2_Q) \quad \mbox{with} \quad
\partial_w x_w = w + {\ensuremath{ \left<
\:w\:\left|\:x\right.\right> }}_* x = ({\ensuremath{\:\mathrm{Id}}}+ x x{^{\text{\tiny T}}}J) w.$$
Now, we are interested in the impact of such a variation on $\theta_w = d_y(x_w) =\operatorname{arccosh}\left( - {\ensuremath{ \left<
\:x_w\:\left|\:y\right.\right> }}_* \right)$. Since $\operatorname{arccosh}'(t) = \frac{1}{\sqrt{t^2 -1}}$, and $\sqrt{\cosh(\theta)^2 -1} = \sinh(\theta)$ for a positive $\theta$, we have: $${d}/{dt} \left. \operatorname{arccosh}(t) \right|_{t=\cosh(\theta)} =
{ 1}/{{\sqrt{\cosh(\theta)^2 -1}}}
= {1}/{\sinh(\theta)},$$ so that $$\theta_w = \theta - \frac{1}{\sinh(\theta)} {\ensuremath{ \left<
\:w + {\ensuremath{ \left<
\:w\:\left|\:x\right.\right> }}_* x\:\left|\:y\right.\right> }}_* + O(\|v\|_*^2).$$ This means that the directional derivative is $$\partial_w \theta_w = - \frac{1}{\sinh(\theta)} {\ensuremath{ \left<
\:w + {\ensuremath{ \left<
\:w\:\left|\:x\right.\right> }}_* x\:\left|\:y\right.\right> }}_*
= - \frac{1}{\sinh(\theta)} {\ensuremath{ \left<
\:w\:\left|\:y -\cosh(\theta) x\right.\right> }}$$ so that $ \partial_w \theta_w^2 = -2 f_*(\theta) {\ensuremath{ \left<
\:w\:\left|\:y - \cosh(\theta) x \right.\right> }}_*.$ Thus, the gradient in the embedding space defined by $<\nabla d^2_y(x) , w>_* = \partial_w \theta_w^2$ is as expected: $$\nabla d^2_y(x) =
- 2 f_*(\theta) (y- \cosh(\theta) x) = - 2 \log_x(y).$$
To obtain the Hessian, we now compute the Taylor expansion of $\log_{x_w}(y)$. First, we compute the variation of $f_*(\theta_w) = \theta_w / \sinh(\theta_w)$: $$\partial_w f_*(\theta_w) = {f_*'(\theta)} \: \partial_w \theta_w
= - \frac{f_*'(\theta)}{\sinh(\theta)} {\ensuremath{ \left<
\:w\:\left|\:y -\cosh(\theta) x\right.\right> }}_*
= - \frac{f_*'(\theta)}{\theta} {\ensuremath{ \left<
\:w\:\left|\:\log_x(y)\right.\right> }}_*$$ with $ f_*'( \theta ) = (1-f_*( \theta)\cosh \theta)/\sinh \theta = (1- \theta \coth \theta)/\sinh \theta$. The variation of $\cosh \theta_w$ is: $$\partial_w \cosh \theta_w = \sinh \theta \: \partial_w \theta_w
= - {\ensuremath{ \left<
\:w\:\left|\:y -\cosh(\theta) x\right.\right> }}_*.$$ Thus, the first order variation of $\log_{x_w}(y)$ is: $$\begin{split}\partial_w \log_{x_w}(y)
&= \partial_w f_*(\theta_w) (y-\cosh \theta x ) - f_*(\theta) \left( \partial_w \cosh(\theta_w) x + \cosh(\theta) \partial_w x_w \right) \\
&= - \frac{f_*'(\theta)\sinh\theta}{\theta^2} {\ensuremath{ \left<
\:w\:\left|\:\log_x(y)\right.\right> }}_* \log_x(y) \\
&\:\: + f_*(\theta) \left( {\ensuremath{ \left<
\:w\:\left|\:y -\cosh(\theta) x\right.\right> }}_* x -\cosh(\theta) (w + {\ensuremath{ \left<
\:w\:\left|\:x\right.\right> }}_* x)\right) \\
&= - \frac{(1- \theta \coth \theta)}{\theta^2} {\ensuremath{ \left<
\:w\:\left|\:\log_x(y)\right.\right> }}_* \log_x(y) \\
&\:\: + {\ensuremath{ \left<
\:w\:\left|\:\log_x(y)\right.\right> }}_* x - \theta \coth(\theta) (w + {\ensuremath{ \left<
\:w\:\left|\:x\right.\right> }}_* x).
\end{split}$$ This vector is a variation in the embedding space: it displays a normal component to the hyperboloid $ {\ensuremath{ \left<
\:w\:\left|\:\log_x(y)\right.\right> }}_* x $ which reflects the extrinsic curvature of the hyperboloid in the Minkowski space (the mean curvature vector is $-x$), and a tangential component which measures the real variation in the tangent space: $$\begin{split}
({\ensuremath{\:\mathrm{Id}}}+ x x{^{\text{\tiny T}}}J) \partial_w \log_{x_w}(y) =
& - \frac{(1- \theta \coth \theta)}{\theta^2} {\ensuremath{ \left<
\:w\:\left|\:\log_x(y)\right.\right> }}_* \log_x(y) \\
& - \theta \coth(\theta) (J + x x{^{\text{\tiny T}}}) J w.
\end{split}$$ Thus the intrinsic gradient is: $$D_x \log_{x}(y) =
- \frac{(1- \theta \coth \theta)}{\theta^2} \log_x(y) \log_x(y){^{\text{\tiny T}}}J
- \theta \coth(\theta) ({\ensuremath{\:\mathrm{Id}}}+ x x{^{\text{\tiny T}}}J).$$ Finally, the Hessian of the square distance, considered as an operator from $T_x{\ensuremath{\mathbb{H}}}^n$ to $T_x{\ensuremath{\mathbb{H}}}^n$, is $H_x(y)(w) = -2 D_x \log_{x}(y) w$. Denoting $u= \log_x(y) / \theta$ the unit vector of the tangent space at $x$ pointing towards the point $y$, we get in matrix form: $$\frac{1}{2} H_x(y)
= u u{^{\text{\tiny T}}}J + \theta \coth \theta (J + x x{^{\text{\tiny T}}}-u u{^{\text{\tiny T}}}) J$$ In order to see that the Hessian is symmetric, we have to lower an index (i.e. multiply on the left by J) to obtain the bilinear form: $$H_x(y) (v,w) = {\ensuremath{ \left<
\:v\:\left|\:H_x(y)(w)\right.\right> }}_* = 2 v{^{\text{\tiny T}}}J \left( u u{^{\text{\tiny T}}}+ \theta \coth \theta (J + x x{^{\text{\tiny T}}}-u u{^{\text{\tiny T}}}) \right) J w.$$
The eigenvectors and eigenvalues of (half) the Hessian operator are now easy to determine. By construction, $x$ is an eigenvector with eigenvalue $0$ (restriction to the tangent space). Then, within the tangent space at $x$, the vector $u$ (or equivalently $\log_x(y) = \theta u$) is an eigenvector with eigenvalue $1$. Finally, every vector $v$ which is orthogonal to these two vectors (i.e. orthogonal to the plane spanned by 0, $x$ and $y$) has eigenvalue $\theta \coth \theta \geq 1$ (with equality only for $\theta=0$). Thus, we can conclude that the Hessian of the squared distance is always positive definite and does never vanish along the hyperbolic space. This was of course expected since it is well known that the Hessian stay positive definite for negatively curved spaces [@A:bishop_manifolds_1969]. As a consequence, the squared distance is a convex function and has a unique minimum.
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A QR decomposition of the reference matrix
==========================================
Let $X=[x_0, \ldots x_k]$ be a matrix of $k+1$ independent reference points in ${\ensuremath{\mathbb{R}}}^n$. Following the notations of the main paper, we write the reference matrix $$Z(x) = [x-x_0, \ldots x-x_k] = x\mathds{1}_{k+1} {^{\text{\tiny T}}}- X.$$ The affine span $\operatorname{Aff}(X)$ is the locus of points $x$ satisfying $Z(x)\lambda = 0$ i.e. $x = X \lambda / (\mathds{1}_{k+1}{^{\text{\tiny T}}}\lambda)$. Here, working with the barycentric weights is not so convenient, and in view of the principal component analysis, we prefer to work with a variant of the QR decomposition using the Gram-Schmidt orthogonalization process.
Choosing $x_0$ as the pivot point, we iteratively decompose $X - x_0 {\ensuremath{\mathds{1}}}_{k+1}{^{\text{\tiny T}}}$ to find an orthonormal basis of the affine span of $X$. For convenience, we define the zeroth vectors $v_0= q_0 =0$. The first axis is defined by $v_1 = x_1-x_0$, or by the unit vector $q_1 = v_1 / \| v_1\|$. Next, we project the second direction $x_2-x_0$ onto $\operatorname{Aff}(x_0, x_1) = Aff(x_0, x_0 + e_1)$: the orthogonal component $v_2 = ({\ensuremath{\:\mathrm{Id}}}- e_1 e_1{^{\text{\tiny T}}}) (x_2 -x_0)$ is described by the unit vector $q_2 = v_2 / \| v_2\|$. The general iteration is then (for $i\geq 1$): $$v_i = ({\ensuremath{\:\mathrm{Id}}}- \sum_{j=0}^{i-1} e_j e_j{^{\text{\tiny T}}}) (x_i - x_0), \qquad \text{and} \qquad q_i = v_i / \| v_i\|.$$ Thus, we obtain the decomposition: $$\begin{split}
X & = x_0 {\ensuremath{\mathds{1}}}_{k+1}{^{\text{\tiny T}}}+ Q T \\
Q & = [q_0, q_1, \ldots q_k] \\
T & = \left[
\begin{array}{ccccc}
q_0{^{\text{\tiny T}}}(x_0 -x_0) & q_0{^{\text{\tiny T}}}(x_1 -x_0) & q_0{^{\text{\tiny T}}}(x_2 -x_0) & \ldots & q_0{^{\text{\tiny T}}}(x_k -x_0) \\
0 & q_1{^{\text{\tiny T}}}(x_1 -x_0) & q_1{^{\text{\tiny T}}}(x_2 -x_0) & \ldots & q_1{^{\text{\tiny T}}}(x_k -x_0) \\
0 & 0 & q_2{^{\text{\tiny T}}}(x_2 -x_0) & \ldots & q_2{^{\text{\tiny T}}}(x_k -x_0) \\
0 & 0 & \ldots & \ldots & \ldots \\
0 & 0 & \ldots & \ldots & q_{k}{^{\text{\tiny T}}}(x_k -x_0)
\end{array}
\right]
\end{split}$$ With this affine variant of the QR decomposition, the $(k+1)\times (k+1)$ matrix $T$ is triangular superior with vanishing first row and first column (since $q_0=0$). The $n\times (k+1)$ matrix $Q$ also has a first null vector before the usual $k$ orthonormal vectors in its $k+1$ columns. The decomposition into matrices of this form is unique when we assume that all the points $x_0, \ldots x_k$ are linearly independent. This means that we can parametrize the matrix $X$ by the orthogonal (aside the first vanishing column) matrix $Q$ and the triangular (with first row and column zero matrix) $T$.
In view of PCA, it is important to notice that the decomposition is stable under the addition/removal of reference points. Let $X_i=[x_0, \ldots x_{i}]$ be the matrix of the first $i+1$ reference points (we assume $i<k$ to simplify here) and $X_i = x_0 {\ensuremath{\mathds{1}}}_{i+1}{^{\text{\tiny T}}}+ Q_i T_i$ its QR factorization. Then, the matrix $Q_i$ is made of the first $i+1$ columns of $Q$ and the matrix $T_i$ is the upper $(i+1) \times (i+1)$ bloc of the upper triangular matrices $T$.
Optimizing the $k$-dimensional subspace
=======================================
With our decomposition, we can now write any point of $x \in \operatorname{Aff}(X)$ as the base-point $x_0$ plus any linear combination of the vectors $q_i$: $ x = x_0 + Q \alpha$ with $\alpha \in {\ensuremath{\mathbb{R}}}^{k+1}$. The projection of a point $y$ on $\operatorname{Aff}(X)$ is thus parametrized by the $k+1$ dimensional vector $\alpha$ that minimizes the (squared) distance $d(x,y)^2 = \| x_0 + Q \alpha -y \|^2$. Notice that we have $Q{^{\text{\tiny T}}}Q = {\ensuremath{\:\mathrm{Id}}}_{k+1} -e_1 e_1{^{\text{\tiny T}}}$ (here $e_1$ is the first basis vector of the embedding space ${\ensuremath{\mathbb{R}}}^{K+1}$) so that $Q^{\dag} = Q{^{\text{\tiny T}}}$. The null gradient of this criterion implies that $\alpha$ is solving $Q{^{\text{\tiny T}}}Q \alpha = Q{^{\text{\tiny T}}}(y-x_0)$, i.e. $\alpha = Q^{\dag} (y-x_0) = Q{^{\text{\tiny T}}}(y-x_0) $. Thus, the projection of $y$ on $\operatorname{Aff}(X)$ is $$Proj(y, \operatorname{Aff}(X)) = x_0 + Q Q{^{\text{\tiny T}}}(y-x_0),$$ and the residue is $$\begin{split}
r^2(y) & = \| ({\ensuremath{\:\mathrm{Id}}}_{n} - Q Q{^{\text{\tiny T}}}) (y-x_0)\|^2 = {\mbox{\rm Tr}}\left( ({\ensuremath{\:\mathrm{Id}}}_{n} - Q Q{^{\text{\tiny T}}}) (y-x_0)(y-x_0){^{\text{\tiny T}}}\right).
\end{split}$$
Accounting now for the $N$ data points ${ Y} = \{ y_i \}_{i=1}^N$, and denoting as usual $\bar y = \frac{1}{N} \sum_{i=1}^N y_i$ and $\Sigma = \frac{1}{N} \sum_{i=1}^N ( y_i -\bar y) ( y_i -\bar y){^{\text{\tiny T}}}$, the unexplained variance is: $$\sigma_{out}^2(X)
= {\mbox{\rm Tr}}\left( ({\ensuremath{\:\mathrm{Id}}}_{n} - Q Q{^{\text{\tiny T}}}) ( \Sigma - (\bar y-x_0)(\bar y-x_0){^{\text{\tiny T}}}) \right)
.$$ In this formula, we see that the value of the upper triangular matrix $T$ does not appear and can thus be chosen freely. The point $x_0$ that minimizes the unexplained variance is evidently $x_0 = \bar y$. To determine the matrix $Q$, we diagonalize the empirical covariance matrix to obtain the spectral decomposition $\Sigma = \sum_{j=1}^n \sigma_j^2 u_j u_j{^{\text{\tiny T}}}$ where by convention, the eigenvalues are sorted in decreasing order. The remaining unexplained variance $\sigma_{out}^2(X)
= {\mbox{\rm Tr}}\left( ({\ensuremath{\:\mathrm{Id}}}_{n} - (U{^{\text{\tiny T}}}Q) (U{^{\text{\tiny T}}}Q){^{\text{\tiny T}}}) \mbox{Diag}(\sigma_i^2) \right)$ reaches its minimal value $ \sum_{i=k+1}^n \sigma_i^2$ for $[q_1, \ldots q_k] = [u_1, \ldots u_k] R$ where $R$ is any $k\times k$ orthogonal matrix. Here, we see that the solution is unique in terms of subspaces (we have $\text{Span}(q_1, \ldots q_k) = \text{Span}(u_1, \ldots u_k)$ whatever orthogonal matrix $R$ we choose) but not in terms of the matrix $Q$. In particular, the matrix $X = [\bar y, \bar y + u_1,\ldots \bar y + u_k ]$ is one of the matrices describing the optimal subspace but the order of the vectors is not prescribed.
The AUV criterion
=================
In PCA, one often plots the unexplained variance as a function of the number of modes used to approximate the data. This curve should decreases as fast as possible from the variance of the data (for 0 modes) to 0 (for $n$ modes). A standard way to quantify the decrease consists in summing the values at all steps. We show in this section that the optimal flag of subspaces (up to dimension $k$) that optimize this Accumulated Unexplained Variances (AUV) criterion is precisely the result of the PCA analysis.
As previously, we consider $k+1$ points $x_i$ but they are now ordered. We denote by $X_i=[x_0, \ldots x_i]$ the matrix of the first $i+1$ columns of $X=[x_0, \ldots x_k]$. The flag generated by $X$ is thus $$Aff(X_0)=\{x_0\} \subset \ldots \subset Aff(X_i) \subset \ldots \subset Aff(X) \subset {\ensuremath{\mathbb{R}}}^n.$$ The QR decomposition of $X$ gives $k$ orthonormal unit vectors $q_1$ …$q_k$ which can be complemented by $n-k$ unit vector $q_{k+1}, \ldots q_n$ to constitute an orthonormal basis of ${\ensuremath{\mathbb{R}}}^n$. Using this extended basis, we can write: $$\sigma_{out}^2(X)
= {\mbox{\rm Tr}}\left( W ( \Sigma - (\bar y-x_0)(\bar y-x_0){^{\text{\tiny T}}}) \right)$$ with $W= ({\ensuremath{\:\mathrm{Id}}}_{n} - Q Q{^{\text{\tiny T}}}) = \sum_{j=k+1}^n q_j q_j{^{\text{\tiny T}}}.$ Since the decomposition is stable under the removal of reference points, the QR factorization of $X_i$ is $X_i = x_0 {\ensuremath{\mathds{1}}}_{i+1}{^{\text{\tiny T}}}+ Q_i T_i$ with $Q_i=[q_0, \ldots q_i]$ and we can write the unexplained variance for the subspace $Aff(X_i)$ as: $$\sigma_{out}^2(X_i)
= {\mbox{\rm Tr}}\left( W_i ( \Sigma - (\bar y-x_0)(\bar y-x_0){^{\text{\tiny T}}}) \right)$$ with $W_i= ({\ensuremath{\:\mathrm{Id}}}_{n} - Q_i Q_i{^{\text{\tiny T}}}) = \sum_{j=i+1}^n q_j q_j{^{\text{\tiny T}}}.$ Plugging this value into the criterion $AUV(X) = \sum_{i=0}^k \sigma^2_{out}( X_i )$, we get: $$AUV(X_k) = {\mbox{\rm Tr}}\left( \bar W ( \Sigma - (\bar y-x_0)(\bar y-x_0){^{\text{\tiny T}}}) \right) $$ with $$\bar W = \sum_{i=0}^k W_i
= \sum_{i=0}^k ({\ensuremath{\:\mathrm{Id}}}_{n} - Q_i Q_i{^{\text{\tiny T}}})
= \sum_{i=0}^k \sum_{j=i+1}^n q_j q_j{^{\text{\tiny T}}}= \sum_{i=1}^k i q_i q_i{^{\text{\tiny T}}}+ (k+1) \sum_{i=k+1}^n q_i q_i{^{\text{\tiny T}}}.$$
PCA optimizes the AUV criterion
===============================
The minimum over $x_0$ is achieved as before for $x_0= \bar y$ and the AUV for this value it now parametrized only by the matrix $Q$: $$AUV(Q) = {\mbox{\rm Tr}}\left( U{^{\text{\tiny T}}}W_k U \mbox{Diag}(\sigma_i^2) \right)
= \sum_{i=1}^k i q_i{^{\text{\tiny T}}}\Sigma q_i + (k+1) \sum_{i=k+1}^n q_i{^{\text{\tiny T}}}\Sigma q_i.$$ Assuming that the first $k+1$ eigenvalues $\sigma_i^2$ ($1\leq i \leq k+1$) of $\Sigma$ are all different (so that they can be sorted in a strict order), we claim that the optimal unit orthogonal vectors are $q_i = u_i$ for $1\leq i \leq k$ and $[q_{k+1}, \ldots q_n] = [u_{k+1}, \ldots u_n] R$ where $R \in O(n-k)$ is any orthogonal matrix.
In order to simplify the proof, we start by assuming that all the eigenvalues have multiplicity one, and we optimize iteratively over each unit vector $q_i$. We start by $q_1$: augmenting the Lagrangian with the the constraint $\|q_1\|^2 =1$ using the Lagrange multiplier $\lambda_1$ and differentiating, we obtain: $$\nabla_{q_1} ( AUV(Q) + \lambda \|q_1\|^2) = \Sigma q_1 + \lambda_1 q_1 =0.$$ This means that $q_1$ is a unit eigenvector of $\Sigma$. Denoting $\pi(1)$ the index of this eigenvector, we have $q_1^* = u_{\pi(1)}$ and the eigenvalue is $-\lambda_1 = \sigma_{\pi(1)}^2$. The criterion for this partially optimal value is now $$AUV([q_1^*, q_2 \ldots q_n] ) = \sigma_{\pi(1)}^2 + \sum_{i=2}^k i q_i{^{\text{\tiny T}}}\Sigma q_i + (k+1) \sum_{i=k+1}^n q_i{^{\text{\tiny T}}}\Sigma q_i.$$ To take into account the orthogonality of the remaining vectors $q_i$ ($i > 1$) with $q_1^*$ in the optimization, we can project all the above quantities along $u_{\pi(1)}$. Optimizing now for $q_2$ under the constraint $\|q_2\|^2=1$, we find that $q_2$ is a unit eigenvector of $\Sigma - \sigma_{\pi(1)}^2 u_{\pi(1)} u_{\pi(1)}{^{\text{\tiny T}}}$ associated to a non-zero eigenvalue. Denoting $\pi(2)$ the index of this eigenvector (which is thus different from $\pi(1)$ because it has to be non-zero), we have $q_2^* = u_{\pi(2)}$ and the eigenvalue is $-\lambda_2 = 2 \sigma_{\pi(2)}^2$.
Iterating the process, we conclude that $q_i^* = u_{\pi(i)}$ for some permutation $\pi$ of the indices $1, \ldots n$. Moreover, the value of the criterion for that permutation is $$AUV([q_1^*, q_2^* \ldots q_n^*] ) = \sum_{i=q}^k i \sigma_{\pi(i)}^2 + (k+1) \sum_{i=k+1}^n \sigma_{\pi(i)}^2.$$ In order to find the global minimum, we now have to compare the values of this criterion for all the possible permutations.
Assuming that $i<j$, we now show that the permutation of two indices $\pi(i)$ and $\pi(j)$ give a lower (or equal) criterion when $\pi(i) < \pi(j)$. Because eigenvalues are sorted in strictly decreasing order, we have $\sigma_{\pi(i)}^2 > \sigma_{\pi(j)}^2$. Thus, $(\alpha-1) \sigma_{\pi(i)}^2 > (\alpha-1) \sigma_{\pi(j)}^2$ for any $\alpha \geq 1$ and adding $\sigma_{\pi(i)}^2 + \sigma_{\pi(j)}^2$ on both sides, we get $\alpha \sigma_{\pi(i)}^2 + \sigma_{\pi(j)}^2 > \sigma_{\pi(i)}^2 + \alpha \sigma_{\pi(j)}^2$. For the value of $\alpha$, we distinguish there cases:
- $i<j\leq k$: we take $\alpha = j/i > 1$. multiplying on both sides by the positive value $i$, we get: $i \sigma_{\pi(i)}^2 + j \sigma_{\pi(j)}^2 < i \sigma_{\pi(j)}^2 + j \sigma_{\pi(i)}^2$. The value of the criterion is thus strictly lower if $\pi(i) < \pi(j)$.
- $i \leq k< j$: we take $\alpha = (k+1)/i > 1$ and we get: $i \sigma_{\pi(i)}^2 + (k+1) \sigma_{\pi(j)}^2 < i \sigma_{\pi(j)}^2 + (k+1) \sigma_{\pi(i)}^2$. Once again, the value of the criterion is thus strictly lower if $\pi(i) < \pi(j)$.
- $k < i<j$: here permuting the indices does not change the criterion since $\sigma_{\pi(i)}^2$ and $\sigma_{\pi(j)}^2$ are both counted with the weight $(k+1)$.
In all cases, the criterion is minimized by swapping indices in the permutation such that $\pi(i) < \pi(j)$ for $i<j$ and $i<k$. The global minimum is thus achieved for the identity permutation $\pi(i) = i$ for the indices $1 \leq i \leq k$. For the higher indices, any linear combination of the last $n-k$ eigenvectors of $\Sigma$ gives the same value of the criterion. Taking into account the orthonormality constraints, such a linear combination writes $[q_{k+1}, \ldots q_n] = [u_{k+1}, \ldots u_n] R$ for some orthonormal $(n-k)\times (n-k)$ matrix $R$.
When some eigenvalues of $\Sigma$ have a multiplicity larger than one, then the corresponding eigenvectors cannot be uniquely determined since they can be rotated within the eigenspace. With our assumptions, this can only occur within the last $n-k$ eigenvalues and this does not change anyway the value of the criterion. We have thus proved the following theorem.
$ $\
Let ${\hat Y} = \{ \hat y_i \}_{i=1}^N$ be a set of $N$ data points in ${\ensuremath{\mathbb{R}}}^n$. We denote as usual the mean by $\bar y = \frac{1}{N} \sum_{i=1}^N \hat y_i$ and the empirical covariance matrix by $\Sigma = \frac{1}{N} \sum_{i=1}^N (\hat y_i -\bar y) (\hat y_i -\bar y){^{\text{\tiny T}}}$. Its spectral decomposition is denoted $\Sigma = \sum_{j=1}^n \sigma_j^2 u_j u_j{^{\text{\tiny T}}}$ with the eigenvalues sorted in decreasing order. We assume that the first $k+1$ eigenvalues have multiplicity one, so that the order from $\sigma_1$ to $\sigma_{k+1}$ is strict.
Then the partial flag of affine subspaces $Fl(x_0\prec x_1 \ldots \prec x_k)$ optimizing the AUV criterion: $$AUV(Fl(x_0\prec x_1 \ldots \prec x_k)) = \sum_{i=0}^k \sigma^2_{out}( Fl_i(x_0\prec x_1 \ldots \prec x_k ) )$$ is totally ordered and can be parameterized by $x_0 = \bar y$, $x_i = x_0 + u_i$ for $1 \leq i \leq k$. The parametrization by points is not unique but the flag of subspaces which is generated is and is equal to the flag generated by the PCA modes up to mode $k$ included.
[^1]: $p$-jets are equivalent classes of functions up to order $p$. Thus, a $p$-jet specifies the Taylor expansion of a smooth function up to order $p$. Non-local jets, or multijets, generalize subspaces of the tangent spaces to higher differential orders with multiple base points.
| ArXiv |
---
abstract: 'We describe Monte Carlo models for the dynamical evolution of the nearby globular cluster M4. The code includes treatments of two-body relaxation, three- and four-body interactions involving primordial binaries and those formed dynamically, the Galactic tide, and the internal evolution of both single and binary stars. We arrive at a set of initial parameters for the cluster which, after 12Gyr of evolution, gives a model with a satisfactory match to the surface brightness profile, the velocity dispersion profile, and the luminosity function in two fields. We describe in particular the evolution of the core, and find that M4 (which has a classic King profile) is actually a [*post-collapse*]{} cluster, its core radius being sustained by binary burning. We also consider the distribution of its binaries, including those which would be observed as photometric binaries and as radial-velocity binaries. We also consider the populations of white dwarfs, neutron stars, black holes and blue stragglers, though not all channels for blue straggler formation are represented yet in our simulations.'
author:
- |
Mirek Giersz$^{1}$[^1] and Douglas C. Heggie$^{2}$\
$^{1}$Nicolaus Copernicus Astronomical Centre, Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warsaw, Poland\
$^{2}$University of Edinburgh, School of Mathematics and Maxwell Institute for Mathematical Sciences, King’s Buildings, Edinburgh EH9 3JZ, UK
date: 'Accepted …. Received …; in original form …'
title: 'Monte Carlo Simulations of Star Clusters - V. The globular cluster M4'
---
\[firstpage\]
stellar dynamics – methods: numerical – binaries: general – globular clusters: individual: M4
Introduction
============
The present paper opens up a new road in the study of the dynamical evolution of globular clusters. We adopt the Monte Carlo method of Giersz [@Gi1998; @Gi2001; @Gi2006], which in recent years has been enhanced to deal quite realistically with the stellar evolution of single and binary stars, to study the dynamical history of the nearby globular cluster M4. An earlier version of the code had already been used to study the dynamical history of $\omega$ Cen [@GH2003], but at that time the treatment of stellar evolution was primitive and there were no binaries. The new code has been thoroughly tested on smaller systems, by comparison with $N$-body simulations and observations of the old open star cluster M67 [@GH2008]. There we showed that the Monte Carlo code could produce data of a similar level of detail and realism as the best $N$-body codes. Now for the first time we consider much richer systems, with about half a million stars initially, which are at present beyond the reach of $N$-body methods.
This paper has a place within a long tradition of the modelling of globular star clusters, but the place is a distinctive one. First, we are not concerned with a static model of a star cluster at the present day, like a King model. We are concerned with issues where the dynamical history of the star cluster is important, where static models are uninformative. Secondly, our aim is to construct a model of a specific star cluster, rather than trying to understand the general properties of the evolution of a population of star clusters. This has been done before, and a brief history is outlined in @GH2008, but the present work takes these efforts onto a new level of realism, in terms of the description of stellar evolution, and dynamical interactions involving binary stars.
This problem is not easy. Not only is it necessary to use an elaborate technique for simulating the relevant astrophysical processes, but it is necessary also to search for initial conditions which, after about 12Gyr of evolution, lead to an object resembling a given star cluster. By “resembling” we do not simply mean matching the overall mass, radius and binary fraction of a cluster, for example, for two reasons:
1. We have found that values found for data in the literature are highly uncertain, and different sources are contradictory. These data are usually derived, in some model-dependent way, from such data as surface-brightness profiles and velocity dispersion profiles, and we prefer to compare our models directly with this data, and not with inferred global parameters.
2. We have found that, even if one achieves a satisfactory fit to these profiles, the model may give a very poor comparison with the luminosity function.
From these considerations we conclude that a model which aims to fit only the mass and radius of a star cluster (say) may be very far from the truth.
Tackling this difficult problem is not just interesting, however. We have been motivated by a number of pressing astrophysical problems. For example, the two nearby star clusters M4 (the subject of this study) and NGC 6397 (which we shall consider in our next paper in this series) have rather similar mass and radius, and yet one has a classic King profile, while the other is a well-studied example of a cluster with a “collapsed core” [@Tr1995]. Among possible explanations one may consider differences in the population of binaries, which are known to affect core properties, or in tidal effects. Indeed the present paper will show that these two clusters may be much more similar than one would suppose from the surface brightness profiles alone.
A second motivation for our work is our involvement in observational programmes aimed at characterising the binary populations in globular clusters. What differences (e.g. in the distributions of periods and abundances) should one expect to find between the core and the halo? These issues are important in the planning of observations, and in their interpretation.
The cluster M4 is the focus of much of this effort because it is nearby, making it a relatively easy target for deep observational study. It was the first globular cluster to yield a deep sequence of white dwarfs [@Ri1995]. More recently it has been subjected to an intensive observational programme by the Padova group [@bedinetal2001; @bedinetal2003; @andersonetal2006], which includes searches for radial-velocity binaries in the upper main sequence [@Somm]. It also turns out to be a cluster which (we conclude) started with only about half a million stars, which facilitates the modelling. Along with the open cluster M67, M4 was chosen by the international MODEST consortium, at a meeting in Hamilton in 2005, as the focus for joint effort by theorists and observers, to cast light on its binary population and dynamical properties. M67 has been modelled very successfully by @Hu2005, using $N$-body techniques, and this paper represent the first theoretical step in a similar study of M4.
The paper is organised as follows. First, we summarise features of the code and the models, the data we used, and our approach to the problem of finding initial conditions for M4. Then we present data for our best models: surface brightness and velocity dispersion profiles, luminosity functions, the properties of the binary population, white dwarfs and other degenerate remnants, and the inferred dynamical state of the cluster. The final section summarises our findings and discusses them in the context of work on other clusters, including objects to which we will turn in future papers.
Methods
=======
The Monte Carlo Code
--------------------
The details of our simulation method have been amply described in previous papers in this series. Each star in a spherical star cluster is represented by its mass, energy and angular momentum, and its stellar evolutionary state may be computed at any time using synthetic formulae for single and binary evolution. It may be a binary or a special kind of single star that has been created in a collision or merger event.
Neighbouring stars interact with each other in accordance (in a statistical sense) with the theory of two-body relaxation. If one or both of the participants is a binary, the probability of an encounter affecting the internal dynamics is calculated according to analytic cross sections, which also determine the outcome. This is one of the main shortcomings of the code, as these cross sections are not well known in the case of unequal masses, and also the possibility of stellar collisions during long-lived temporary capture is excluded.
A star or binary may escape if its energy exceeds a certain value, which we choose to be lower than the energy at the nominal tidal radius, in order to improve the scaling of the lifetime with $N$, as explained in @GH2008. This is the second main shortcoming of the models, as it leads to a cutoff radius of the model that is smaller than the true tidal radius, and this lowers the surface density profile in the outer parts of the system.
A difficulty in applying the Monte Carlo code to M4 is that it employs a static tide, whereas the orbit of M4 appears to be very elliptical [@Di1999]. We have to assume that a cluster can be placed in a steady tide of such a strength that the cluster loses mass at the same average rate as it would on its true orbit. Some support for this procedure comes from $N$-body modelling. @BM2003 show that clusters on an elliptical orbit between about 2.8 and 8.5kpc dissolve on a time scale intermediate between that for circular orbits at these two radii, and that the dissolution time scales in almost the same way with the size of the system. @wilkinsonetal2003 show that the core radius of a cluster on an elliptic orbit evolves in very nearly the same way as in a cluster with a circular orbit at the time-averaged galactocentric distance.
All other free parameters of the code (e.g. the coefficient of $N$ in the Coulomb logarithm) take the optimal values found in the above study.
Initial Conditions
------------------
The initial models are as specified in Table \[tab:ics\]. Many of these features (e.g. the properties of the binaries, except for their overall abundance) were inherited from our modelling of the open cluster M67, and those were in turn mainly drawn from the work of @Hu2005. Some of the parameters were taken to be freely adjustable, and this freedom was exploited in the search for an acceptable fit to the current observational data.
Fixed parameters
--------------------------- ------------------------------
Structure Plummer model
Initial mass function$^1$ @Kr1993 in the
range \[0.1,50\]$\msun$
Binary mass distribution @Kr1991
Binary mass ratio Uniform (with component
masses restricted as for
single stars)
Binary semi-major axis Uniform in log, $2(R_1+R_2)$
to 50AU
Binary eccentricity Thermal, with eigenevolution
[@Kr1995]
Metallicity $Z$ 0.002
Age 12Gyr [@Ha2004]
Free parameters
Mass $M$
Tidal radius $r_t$
Half-mass radius $r_h$
Binary fraction $f_b$
Slope of the lower mass
function $\alpha$ (Kroupa = 1.3)
: Initial parameters for M4
\[tab:ics\]
Observational data and its computation {#sec:data_computation}
--------------------------------------
Our first task was to iterate on the initial parameters of our model in order to produce a satisfactory fit to a range of observational data at an age of 12Gyr. The data we adopted are as follows:
1. Surface brightness profile: here we used the compilation by @Tr1995, where the surface brightness is expressed in $V$ magnitudes per square arcsec.
2. Radial velocity profile: this came from [@Pe1995], and is the result of binning data the on the radial velocities of nearly 200 stars. Strictly we should refer to the line-of-sight velocity dispersion, as “radial” velocity has a different meaning in the Monte Carlo model.
3. The V-luminosity function (@Ri2004, from which we considered the results for the innermost and outermost of their four annuli). These data are lack correction for completeness, though the completeness factor for the outermost field is plotted in @richeretal2002. For main sequence stars it is almost 100% down to $V=15$, and drops steadily to less than 50% at $V=17$. For the innermost field the completeness correction would be larger.
Now we consider how to compare this data with the output of the Monte Carlo code. This includes a list of each particle in the simulation, along with data on its radius, radial and transverse velocities and absolute magnitude, and numerous other quantities. To construct the surface brightness, we think of each particle as representing a luminous shell of the same radius, and superpose the surface brightness of all shells. While this procedure involves a minimum of effects from binning, or randomly assigning the full position of the star, a shell has an infinite surface brightness at its projected edge. The effect of this, especially from the brightest stars, will be seen in some of the profiles to be presented in this paper. A similar problem arises in actual observations, and is often handled by simply removing the brightest stars from the surface brightness data presented. The output from the model is corrected for extinction (Table \[tab:m4dat\]. The distance of the line of sight from the cluster centre is converted between pc (as in the model) and arcsec (as in the observations) using the distance in the same table.
The line-of-sight velocity dispersion is computed in a similar way. For each particle we calculate the mean square line-of-sight velocity (because the orientation of the transverse component of the velocity is random), and then sum over all particles. In this sum, each particle is weighted by a geometrical factor proportional to the surface density of the particle’s shell along the line of sight. The result is an estimate of the velocity dispersion that is weighted by neither mass nor brightness, but only number. To check the effects of this, we have sometimes calculated velocity dispersions with various cutoffs in the magnitudes of the stars included. The effects are usually quite small.
Computation of the luminosity function is the least problematic. We count stars in each bin in $V$-magnitude, but lying above a line in the colour-magnitude diagram just below the main sequence. Again the contribution of each star is weighted by the same geometrical factor, and the $V$ magnitude is corrected for extinction.
----------------------------------- ------------------- --
Distance from sun $^a$1.72kpc
Distance from GC 5.9kpc
Mass $^a$63 000$\msun$
Core radius 0.53 pc
Half-light radius 2.3 pc
Tidal radius 21 pc
Half-mass relaxation time ($R_h$) 660 Myr
Binary fraction $^a$1-15%
$\left[\right.$Fe/H$]$ -1.2
Age $^b$12Gyr
$A_V$ $^a$1.33
----------------------------------- ------------------- --
: Properties of M4
\[tab:m4dat\]
References: All data are from the current version of the catalogue of @Ha1996, except $^a$ @Ri2004 (though this is not always the original reference for the quoted number) and $^b$ @Ha2004.
Finding initial conditions
--------------------------
Here we summarise our experience in approaching this problem. A number of studies (e.g. @BM2003, @La2005) give simple formulae for the evolution with time of the bound mass of a rich star cluster. It is possible to derive similar simple formulae for the evolution of the half-mass radius and other quantities. There are several problems with inverting these formulae, however, i.e. using them to infer the initial parameters of a cluster from its present mass and radius. First, these present-day global parameters are quite uncertain, even to within a factor of two. Second, these formulae depend on the galactic orbit and other parameters which are equally subject to uncertainty. Therefore we have adopted the more straightforward but more laborious approach of iterating on the initial parameters of our models (Tab \[tab:ics\], lower half); that is, we select values for the five stated parameters, run the model, find where the match with the observations is poor, adjust the parameters, and repeat cyclically.
We have employed two methods to facilitate this process to some extent. First, we have often carried out mini-surveys, i.e. small, coarse grid-searches around a given starting model, to find out how changes in individual parameters affect the results. Second, we have used scaling to accelerate the process, and we now describe this method in a little detail.
Suppose we wish to represent a star cluster which has mass $M$ and radius $R$ with a model representing a cluster with a (usually smaller) mass $M\ast$ and radius $R\ast$. Since two-body relaxation dominates much of the dynamics, we insist that the two clusters have the same relaxation time, and so $$\frac{R\ast}{R} =
\left(\frac{M}{M\ast}\right)^{1/3}\left(\frac{\log\gamma
N\ast}{\log\gamma N}\right)^{2/3},$$ where $N,N\ast$ are the corresponding particle numbers. Thus the model of lower mass has larger radius. The tidal radius is scaled in the same way. Then the observational results (surface brightness profile, etc) can be computed for the model of lower mass and then rescaled (by appropriate factors of the mass and radius) to give a result for the more massive cluster (assuming that the evolution is dominated by the processes of relaxation, stellar evolution and tidal stripping). In fact we have found that this is very successful, in the sense that the inferred best values for the initial conditions change little when runs are carried out with the “correct” (i.e. unscaled) initial mass, certainly when the proportion of binaries is 10% or less. Some aspects of the evolution are not well described by this scaling technique. For example, we do not change the distribution of the semi-major axes of the binaries. In this way the internal evolution of the binaries is correctly modelled (provided that the binaries remain isolated dynamically), though their dynamical interactions with the rest of the system are not. In principle one could scale the semi-major axes in the same way as $R$, but then the internal evolution of the binaries would be altered.
Models of M4
============
Finding initial parameters
--------------------------
Our starting point was our work on the old open cluster M67 [@GH2008], but with a larger initial mass and radius. (@BM2003, for example, suggested that the initial mass of M4 (NGC 6121) was of order $7.5\times10^5\msun$, though we used somewhat smaller values.) To begin with, our choice of initial tidal radius was inferred from the initial mass and the present-day estimates of mass and tidal radius given in Table \[tab:m4dat\], assuming that $r_t\propto M^{1/3}$. At first we adopted similar values for the “concentration” ($r_t/r_h$) and binary fraction as in the modelling of M67, but found that the surface brightness profile fitted poorly (with too large a core) unless the binary fraction was much smaller (5 to 10%) and the concentration much higher.
Before describing our best models, it is worth briefly mentioning one which provided a satisfactory fit to the surface brightness profiles. The fit to the velocity dispersion profile was tolerable, but indicated a model that was too massive by a factor of about 1.4. Its main flaw, however, was in the luminosity function, which was generally too large by a factor in the range 2–3. There are two reasons why this is interesting. One is that, for a long time, models of star clusters were constructed entirely on the basis of the surface brightness and velocity dispersion profiles. It should be realised that such models may be misleading in other ways. Second, this experience underscores the importance of properly normalised luminosity functions. In other words, it is important to know the area of the field where the stars have been counted, or some equivalent representation of properly normalised data. Very often, the emphasis is solely on the [*shape*]{} of the luminosity function, but we have found that the absolute normalisation is an essential constraint.
@BM2003 show that the lower mass function becomes flatter as the fraction of mass lost by the cluster increases (i.e. towards the end of its life). Therefore we could perhaps have improved the fit with the luminosity function by starting with a more massive model and somehow ensuring a larger escape rate so as to leave a similar mass at the present day. Instead, we elected to change the slope of the low-mass IMF from the canonical value of $\alpha=1.3$ [@Kr2007a] to $\alpha=0.9$. (There is some justification for a lower value for low-metallicity populations, though it has been argued [@Kr2007b] that there is no pristine low-metallicity population where the IMF can be inferred securely.)
A Monte Carlo model of M4
-------------------------
By some experimentation we arrived at a model which gave a fair fit to all three kinds of observational data; see Table 3 (where we compare with a King model developed by @Ri2004, and Figs 1-4. It is worth noting that no arbitrary normalisation has been applied in these comparisons between our model and the observations. The surface brightness profile, for example, is computed directly from the V-magnitudes of the stars in the Monte Carlo simulation, as described in Sec.\[sec:data\_computation\]. In the construction of a King model, by contrast, it is often assumed that the mass-to-light ratio is arbitrary.
--------------------------------- ------------------ ------------------ -------------------------
Quantity MC model MC model King model
($t = 0$) ($t=12$Gyr) [@Ri2004]
Mass ($\msun$) $3.40\times10^5$ $4.61\times10^4$
Luminosity ($L_\odot$) $6.1\times10^6$ $2.55\times10^4$ $6.25\times10^4$
Binary fraction $f_b$ 0.07 0.057 0
Low-mass MF slope $\alpha$ 0.9 0.03 0.1
Mass of white dwarfs ($\msun$) 0 $1.81\times10^4$ $3.25\times10^{4^\ast}$
Mass of neutron stars ($\msun$) 0 $3.24\times10^3$
Tidal radius $r_t$ (pc) 35.0 18.0
Half-mass radius $r_h$ (pc) 0.58 2.89
--------------------------------- ------------------ ------------------ -------------------------
\[tab:mc\_king\]
$\ast$: this is the quoted mass of “degenerates”
### Surface brightness
While the overall surface brightness profile is slightly faint, the most noticeable feature of Fig.\[fig:sbp\] is that the model has a somewhat smaller limiting radius than the observational data. The reason for this is explained in @GH2008: in short, we impose a smaller tidal radius than the nominal tidal radius, in an $N$-dependent way, which is intended to ensure that the overall rate of escape from the model behaves in the same way as in an $N$-body simulation. Another point to notice is the disagreement between the total luminosity of our model and that of the King model quoted in the final column of Table \[tab:mc\_king\]. @Tr1995 give an analytic fit to the surface brightness profile, and we have checked that the integrated value is close to ours.
The data for the Monte Carlo model in Table \[tab:mc\_king\] would be consistent with a galactocentric radius of about 1.7kpc, in an isothermal galaxy model with a circular velocity of 220km/s. While this is certainly much smaller than its current galactocentric distance, a small value was also found (using a similar argument and published values of the mass and tidal radius) by @vdb1995. The orbit given by @Di1999 has a still smaller perigalactic distance, the galactocentric distance varying between extremes of 0.6 and 5.9 kpc.
### Velocity dispersion profile
This is illustrated in Fig.\[fig:vdp\], where it is compared with the observational data of @Pe1995. The shortfall at large radii is of doubtful significance.
### Luminosity Functions
These are shown for our model at the median radius of the innermost and outermost fields observed by @Ri2004. The disagreement in the inner luminosity function at faint magnitudes may be attributable to the fact that the theoretical result assumes 100% completeness, while the observational data are uncorrected for completeness. A plot of the completeness correction in the outer field is given by @Ha2002 [Fig.3] and discussed above in Sec.\[sec:data\_computation\]. The mismatch between model and data is smaller in the outer field, but from the discussion above it would seem that the mismatch in the faintest bin may be too large to be accounted for by the estimated value of the completeness correction. The error bars in the last two observational points on this plot (not shown) almost overlap, and much of the mismatch may be simply sampling uncertainty. It is worth comparing the multi-mass King model constructed by @Ri2004, which is also problematic in the outermost field.
### Core collapse
Fig.\[fig:rc\] shows the evolution with time of the theoretical core radius. There is an early period of very rapid contraction, associated with mass segregation, followed by a slower reexpansion, caused by the loss of mass from the evolving massive stars which are now concentrated within the core.
Our most surprising discovery from our model of M4 is the subsequent behaviour. M4 is classified as a King-profile cluster [@Tr1993], and such clusters are usually interpreted as being clusters whose cores have not yet collapsed. But the plot of the theoretical core radius (Fig.\[fig:rc\] reveals that the model exhibited core collapse at about 8Gyr. Subsequently its core radius is presumably sustained by binary burning. Even non-primordial binaries may be playing a role here. To the best of our knowledge it has not previously been suggested that M4, is a post-collapse cluster, though on statistical grounds @dempp2007 have suggested that some King-type clusters have already collapsed. This issue is often approached by reference to the half-mass relaxation time (Table \[tab:m4dat\], but for M4 this is not short enough to be decisive.
The presence of radial colour gradients is correlated with the presence of a non-King surface brightness profile and hence with core collapse. We have computed the colour profile of our model, and find that it is nearly flat except for the influence of one or two very bright stars within the innermost few arcsec. Most values lie around 0.65 in $B-V$, which is much less than the global value of 1.03 given in @Ha1996 [January 2008]. On the other hand roughly similar mismatches between observations are found in other clusters (e.g. M30, [@Pi1988]).
### The colour-magnitude diagram
Figure \[fig:colour-magnitude\] shows the colour-magnitude diagram of the model. This is of interest, not so much for comparison with observations, but for the presence of a number of interesting features. The division of the lower main sequence is simply an artifact of the way binary masses were selected (a total mass above 0.2$\msun$ and a component mass above 0.1$\msun$.) Of particular interest are the high numbers of merger remnants on the lower white dwarf sequence. There are very few blue stragglers. Partly this is a result of the low binary frequency, but it is also important to note that some formation channels are unrepresented in our models (in particular, collisions during triple or four-body interactions, though if a binary emerges from an interaction with appropriate parameters, it will be treated as merged.) These numbers also depend on the assumed initial distribution of semi-major axis, which is not yet well constrained by observations in globular clusters.
Note in Fig.\[fig:colour-magnitude\] that there are some white dwarf-main sequence binaries (below the main sequence). @Ri2004 drew attention to the possible presence of such binaries in their colour magnitude diagrams of this cluster.
### Binaries
Photometric binaries are visible in Fig.\[fig:colour-magnitude\], and these are compared with observations in the inner field of @Ri2004 in Fig.\[fig:photometrics\]. In this figure, the model histogram has been normalised to the same total number of stars as the observational one. We made no attempt to simulate photometric errors, but the bins around abscissa = -0.75 suggest that the binary fractions in the model (which is under 6% globally; see Table \[tab:mc\_king\]) and the observations are comparable. We note here that @Ri2004, using this same data, concluded that the binary frequency was approximately 2% in the innermost field. But they also note that the measured frequency of “approximately equal-mass binaries” is 2.2%, and we consider that the balance between this number and our binary fraction can be made of binaries with companions whose masses are more unequal.
Radial velocity binaries should also be detectable in M4, and will be part of a separate investigation.
Now we consider the distributions of the binaries, as predicted by the theoretical model. Fig.\[fig:sma\] shows that binaries have evolved dynamically as well as through their internal evolution. In particular the softest pairs been almost destroyed.
By 12 Gyr the binaries exhibit segregation towards the centre of the cluster, but perhaps in more subtle ways than might be expected (Figs.\[fig:segregation\],\[fig:brights\]). When [*all*]{} binaries are considered, there is little segregation relative to the other objects in the system. (Most binaries in our model are of low mass.) But if one restricts attention to bright binaries, which we here take to mean those with $M_V < 7$ (i.e. brighter than about two magnitudes below turnoff), the segregation is very noticeable (Fig.\[fig:brights\]), with a half-mass radius smaller by almost a factor of 2 than for bright single stars. Still, bright binaries are not nearly as mass-segregated as neutron stars (Fig.\[fig:segregation\]), which, incidentally, receive no natal kicks in our model.
The history of the binary fraction is, in effect, given in Fig.\[fig:degenerate-numbers\]. In the first Gyr this falls steadily from the initial value of 0.07 to about 0.06, and it remains close to this value until the present day.
### Escape velocity
We have already referred to the fact that, in our model, neutron stars receive no natal kicks. Escape is of course governed by the escape speed, and this is of much interest in connection with the possibility of retaining gas from the first generation of rapidly evolving stars. For this reason we plot the central escape velocity in Fig. \[fig:vesc\]. The remarkably high value in the first few tens of millions of years, if valid for M4, could have interesting consequences for the early evolution of the cluster and its stars. It draws attention to the very high initial density of our model (Table \[tab:mc\_king\]), whose average value within the half-mass radius is $2\times10^5\msun$/pc$^3$. The central density is about $10^6\msun$/pc$^3$, about an order of magnitude larger than in the central young cluster in the HII region NGC 3603 [@St2006]. In the absence of local young star clusters as massive as our M4 progenitor, it is difficult to be sure whether our initial model is implausibly dense.
### Degenerate components
We have already mentioned the spatial segregation of the population of neutron stars, and the presence of a number of degenerate binaries at the present day. Now we consider the historical evolution of this population over the lifetime of the cluster so far, according to our model.
Fig.\[fig:degenerate-mass\] shows the evolution of these populations. It has already become well established [@VH1997; @HS2003] that white dwarfs account for an increasing proportion of the mass of globular clusters, and indeed it is of order 39% at the present day, according to this model. The proportion of neutron stars is almost certainly excessive, because of our assumption of complete retention.
In order to separate two of the components in this figure (black holes and neutron stars), we show in Fig. \[fig:degenerate-numbers\] the numbers of these degenerate stars, along with the numbers of all stars and all binaries. This shows that no stellar-mass black holes are expected to be present in M4 now.
Conclusions and discussion
==========================
In this paper we have presented a Monte Carlo model for the nearby globular cluster M4. This model includes the effects of two-body relaxation, evaporation across the tidal boundary, dynamical interactions involving primordial and three-body binaries, and the internal evolution of single stars and binaries. By adjustment of the initial parameters (total mass, tidal radius, initial mass function, binary fraction) we have found a model which, after 12Gyr of evolution, leads to a model with a surface brightness profile, velocity dispersion profile, and luminosity functions at two radii, all of which are in tolerable agreement with observational data. It also leads to a number of photometric binaries roughly consistent with observation.
This model has a current mass of $4.6\times10^4\msun$, a $M/L_V$ ratio of 1.8, a binary fraction of almost 6%, and an almost flat lower mass function (Table \[tab:mc\_king\]). Almost 40% of the mass is in white dwarfs, and about 7% in neutron stars, though our model assumes a 100% retention rate. They are strongly mass-segregated to the centre of the cluster. There are no stellar-mass black holes left in the cluster. The binaries have experience significant dynamical evolution, almost all the soft pairs having been destroyed. The binary population as a whole is only slightly segregated towards the centre, but there is more evident segregation of bright binaries (such as those that would be more readily observed in a radial velocity search.)
The most significant new result from our model is the implication that M4 is a core collapse cluster, despite the uncollapsed appearance of the surface brightness profile.
Now we discuss a number of shortcomings of our model and other issues related to this study.
1. [*Uniqueness:*]{} Though we have arrived at a broadly satisfactory model, it is not at all clear how unique it is. Certainly we were unable to construct models with much larger numbers of primordial binaries, or with significantly larger initial radius, as such models produced an insufficiently concentrated surface brightness profile. Furthermore, the fact that the lower slope of the initial mass function is less steep than the canonical value of 1.3 should not be taken to imply that we have inferred the initial value.
2. [*Fluctuations*]{}: we have noticed that runs differing only in the initial seed of the random number generator can give surprisingly different surface brightness profiles. In broad terms this confirms the important finding of @Hu2007, though we have not yet established (as he did) that the presence or absence of black hole binaries is the underlying mechanism. We shall return to this issue at appropriate length in the next paper in this series, on the cluster NGC 6397.
3. [*The Initial Model:*]{} The initial model is astonishingly dense, the central density being of order $10^6\msun$pc$^{-3}$. It also has to be realised that we are imposing initial conditions at a time when the residual gas from the birth of the cluster has already dispersed. Various properties of a star cluster, including its binary population and mass function, may change significantly during the phase of gas expulsion, which further undermines the power of our model to establish the initial conditions. Finally, we have assumed no initial mass segregation, even though recent work suggests that this should be present already before the cluster has assembled into a roughly spherical object [@Ve2007]. This is an aspect of the modelling that could be readily improved.
4. [*The early evolution of the model:*]{} It is worth drawing attention here to the high initial escape velocity from the centre of our model, in the first few tens of millions of years. Clearly this is dependent on the small initial radius. The initial high density is also responsible for the very rapid initial evolution of the core.
5. [*Imperfections of the modelling:*]{} Several important improvements need to be made to our technique.
1. At present few-body interactions are handled with cross sections. The problem is not simply that these are not well known, especially in the case of unequal masses; it also means that we are unable to determine if a collision occurs during a long-lived interaction. For this reason, we have said almost nothing about collision products (e.g. blue stragglers) in this paper. This limitation could be overcome by direct integration of the interactions, as is done by @FR2007 in their version of the Monte Carlo scheme.
2. Long-lived triples are neglected in the model at present. These are commonly produced in binary-binary encounters [@Mi1984a], and it is desirable to include these as a third species (beyond single and binary stars). Their observable effects may be small, but of course there is one intriguing example in the very cluster we have focused on here [@Th1993].
3. We assume that the tide is modelled as a tidal cutoff. We have taken some care to ensure that the tidal radius is adjusted so that the model loses mass through escape at the same rate as an $N$-body model would, if immersed in a tidal [*field*]{} with the same tidal radius. This means, however, that the effective tidal radius of the model is somewhat too small. A better treatment of a steady tide may be possible, without this drawback.
4. We assume that the tide is steady, something which is not true for M4. The effects of tidal shocks have been studied by a number of authors [e.g. @Ku1995], and it would be possible to add the effects as another process altering the energies and angular momenta of the stars in the simulation. On the other hand @Ag1988 found, on the basis of a simple model, that tidal evaporation was the dominant mechanism in the evolution of the mass of M4 at the present day, and that other factors (such as disk and bulge shocking) contributed at a level less than 1%.
5. Rotation: it has been shown [@Ki2004] that, to the extent that rotating and non-rotating models can be compared, rotation somewhat accelerates the rate of core collapse. Rotation is hard to implement in this Monte Carlo model, however.
6. The search for initial conditions is still very laborious, and we are constantly seeking ways of expediting this. Our most fruitful technique at present is the use of small-scale models which relax at the same rate as a full-sized model. Not all aspects of the dynamics are faithfully rendered by a scaled model, but the technique appears to be successful as long as the fraction of primordial binaries is not too large.
Despite these caveats and shortcomings, it is clear that the Monte Carlo code we have been developing is now an extraordinarily useful tool for assessing the dynamics of rich star clusters. For the cluster M4 it has led to the conclusion that M4 can be explained as a post-collapse clusters. It also provides a wealth of information on the distribution of the binary population, which will be important for the planning and interpretation of searches for radial velocity binaries, now under way.
$N$-body models will eventually supplant the Monte Carlo technique, but at present are incapable of providing a star-by-star model of even such a small cluster as M4. They can of course provide a great deal of general guidance on the dynamical evolution of rich star clusters, and they underpin the Monte Carlo by providing benchmarks for small models. Even when $N$-body simulations eventually become possible, Monte Carlo models will remain as a quicker way of exploring the main issues, just as King models have continued to dominate the field of star cluster modelling even when more advanced methods (e.g. Fokker-Planck models) have become available.
Discussion
==========
Acknowledgements {#acknowledgements .unnumbered}
================
We are indebted to Jarrod Hurley for much help with the BSE stellar evolution package. We thank Janusz Kaluzny for his kind advice on observational matters, and Harvey Richer for comments on various matters, especially the observed luminosity functions. This research was supported in part by the Polish National Committee for Scientific Research under grant 1 P03D 002 27.
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\[lastpage\]
[^1]: E-mail: [email protected] (MG); [email protected] (DCH)
| ArXiv |
---
author:
- Li Xi
- 'Michael D. Graham'
title: Active and hibernating turbulence in minimal channel flow of Newtonian and polymeric fluids
---
| ArXiv |
---
abstract: 'We study the Dirichlet series $F_b(s)=\sum_{n=1}^\infty d_b(n)n^{-s}$, where $d_b(n)$ is the sum of the base-$b$ digits of the integer $n$, and $G_b(s)=\sum_{n=1}^\infty S_b(n)n^{-s}$, where $S_b(n)=\sum_{m=1}^{n-1}d_b(m)$ is the summatory function of $d_b(n)$. We show that $F_b(s)$ and $G_b(s)$ have continuations to the plane ${\mathbb{C}}$ as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions $d_b$ and $S_b$ to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.'
author:
- Corey Everlove
bibliography:
- 'sumofdigitsbibliography.bib'
title: 'Dirichlet series associated to sum-of-digits functions'
---
[^1]
Introduction
============
There has been a great deal of study of properties of the radix expansions to an integer base $b\geq 2$ of integers $n$. For each integer base $b\geq 2$, every positive integer $n$ has a unique base-$b$ expansion $$n = \sum_{i \geq 0} \delta_{b,i}(n)b^i$$ with digits $\delta_{b,i}\in\{0,1,\dotsc,b-1\}$ given by $$\delta_{b,i} = \Bigl\lfloor \frac{n}{b^i} \Bigr\rfloor - b \Bigl\lfloor \frac{n}{b^{i+1}} \Bigr\rfloor.$$ This paper considers two summatory functions of base $b$ digits of $n$:
1. The *base-$b$ sum-of-digits function* $d_b(n)$ is $$d_b(n) = \sum_{i\geq 0} \delta_{b,i}(n).$$
2. The *(base $b$) cumulative sum-of-digits function* $S_b(n)$ is $$S_b(n) = \sum_{m=1}^{n-1} d_b(m).$$ We follow here the convention of previous authors (including [@delange-75] and [@flajolet-94]), with the sum defining $S_b(n)$ running to $n-1$ instead of $n$.
We associate to the functions $d_b(n)$ and $S_b(n)$ the Dirichlet series generating functions $$F_b(s) = \sum_{n=1}^\infty \frac{d_b(n)}{n^s}$$ and $$G_b(s) = \sum_{n=1}^\infty \frac{S_b(n)}{n^s}.$$ These Dirichlet series have abscissa of convergence $\operatorname{Re}(s)=1$ and $\operatorname{Re}(s)=2$, respectively.
This paper studies the problem of the meromorphic continuation to ${\mathbb{C}}$ of Dirichlet series associated to the base-$b$ digit sums $d_b(n)$ and $S_b(n)$. Here we obtain the meromorphic continuation and determine its exact pole and residue structure. The pole structure contains half of a two-dimensional lattice and the residues involve Bernoulli numbers and values of the Riemann zeta function on the line $\operatorname{Re}(s)=0$. A meromorphic continuation of these functions was previously obtained in the thesis of Dumas [@dumas-thesis] by a different method, which specified a half-lattice containing all the poles but did not determine the residues; in fact infinitely many of the residues on his possible pole set vanish.
The asymptotics of $S_b(n)$ have been extensively studied, see Section \[sec-previous-work\]. We mention particularly work of Delange [@delange-75], given below as Theorem \[thm-delange\], which gives an exact formula for $S_b(n)$ in terms of a continuous nondifferentiable function with Fourier coefficients involving values of the Riemann zeta function on the imaginary axis. Using an interpolation of Delange’s formula we formulate a continuous interpolation of $S_b(n)$ in the base parameter $b$, permitting definitions of $d_\beta(n)$ and $S_\beta(n)$ for a real parameter $\beta>1$. We obtain a meromorphic continuation of the associated Dirichlet series $F_{\beta}(s)$ and $G_{\beta}(s)$ to the half-planes $\operatorname{Re}(s)>0$ and $\operatorname{Re}(s)>1$, respectively. We note apparent fractal properties of $d_{\beta}(n)$ as $\beta$ is varied.
Results
-------
Our first results concern the meromorphic continuation of the functions $F_b(s)$ and $G_b(s)$ to the entire complex plane ${\mathbb{C}}$.
\[thm-db\] For each integer base $b\geq 2$, the function $F_b(s) = \sum_{n=1}^\infty d_b(n)n^{-s}$ has a meromorphic continuation to ${\mathbb{C}}$. The poles of $F_b(s)$ consist of a double pole at $s=1$ with Laurent expansion beginning $$F_b(s)=\frac{b-1}{2\log b}(s-1)^{-2} + \biggl( \frac{b-1}{2\log b} \log(2\pi) - \frac{b+1}{4}\biggr)(s-1)^{-1} + O(1),$$ simple poles at each other point $s=1+2\pi i m / \log b$ with $m\in {\mathbb{Z}}$ ($m\neq 0$) with residue $$\operatorname*{Res}\biggl( F_b(s) , s=1+\frac{2\pi i m}{\log b} \biggr) = - \frac{b-1}{2\pi i m} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr),$$ and simple poles at each point $s=1-k+2\pi i m /\log b$ with $k=1$ or $k\geq 2$ an even integer and with $m\in{\mathbb{Z}}$, with residue $$\operatorname*{Res}\biggl( F_b(s) , s=1-k+\frac{2\pi i m}{\log b} \biggr) = (-1)^{k+1}\frac{b-1}{\log b}\zeta\biggl(\frac{2\pi i m}{\log b}\biggr)\frac{B_k}{k!} \prod_{j=1}^{k-1} \biggl(\frac{2\pi i m}{\log b} - j\biggr)$$ where $B_k$ is the $k$th Bernoulli number.
Theorem \[thm-db\] is proved by first considering the Dirichlet series $\sum \bigl(d_b(n)-d_b(n-1)\bigr) n^{-s}$ and then exploiting a relation between power series and Dirichlet series to recover $F_b(s)$. The proof is presented in Section \[sec-mero-cont\].
The meromorphic continuation of Dirichlet series attached to $b$-regular sequences, of which our Dirichlet series $F_b(s)$ is a particular example, was studied by Dumas in his 1993 thesis [@dumas-thesis]; this work also showed that the poles of $F_b(s)$ must be contained in a certain half-lattice, strictly larger than the half-lattice here.
A similar method allows us to meromorphically continue the series $G_b(s)$ to the complex plane.
\[thm-sb\] For each integer $b\geq 2$, the function $G_b(s)=\sum_{n=1}^\infty S_b(n)n^{-s}$ has a meromorphic continuation to ${\mathbb{C}}$. The poles of $G_b(s)$ consist of a double pole at $s=2$ with Laurent expansion $$G_b(s) = \frac{b-1}{2\log b}(s-2)^{-2} + \biggl(\frac{b-1}{2\log b}\bigl(\log(2\pi)-1\bigr)-\frac{b+1}{4}\biggr)(s-2)^{-1} + O(1),$$ a simple pole at $s=1$ with residue $$\operatorname*{Res}(G_b(s),s=1) = \frac{b+1}{12},$$ simple poles at $s=2 + 2\pi i m / \log b$ with $m\in{\mathbb{Z}}$ ($m\neq 0$) with residue $$\operatorname*{Res}\biggl( G_b(s) , s= 2 + \frac{2\pi i m}{\log b} \biggr) = - \frac{b-1}{2\pi i m}\biggl(1+\frac{2\pi i m}{\log b}\biggr)^{-1} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr)$$ and simple poles at point $s=2-k + 2\pi i m / \log b$ with $k\geq 2$ an even integer and $m\in{\mathbb{Z}}$ with residue $$\operatorname*{Res}\biggl( G_b(s) , s= 2 - k + \frac{2\pi i m}{\log b} \biggr) = \frac{b-1}{\log b} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr) \biggl(\frac{B_k}{k(k-2)!} \biggr) \prod_{j=1}^{k-2}\biggl(\frac{2\pi i m}{\log b} - j\biggr).$$
An interesting feature of the above theorems is the abundance of poles. Since each function $F_b(s)$ and $G_b(s)$ has $\asymp r^{2}$ poles in the disk $\lvert s \rvert<r$, we have the following corollary, which we discuss further in Section \[sec-meromorphic-functions\].
The functions $F_b(s)$ and $G_b(s)$ are meromorphic functions of order at least $2$ on ${\mathbb{C}}$.
The Riemann zeta function, the Dirichlet L-functions, and the Dirichlet series generating functions of many important arithmetic functions (such as the Möbius function $\mu(n)$, the von Mangoldt function $\Lambda(n)$, the Euler totient function $\phi(n)$, and the sum-of-diviors functions $\sigma_\alpha(n)$) analytically continue as meromorphic functions of order $1$ on the complex plane. The Dirichlet series $F_b(s)$ and $G_b(s)$ thus have a different analytic character than many other Dirichlet series considered in number theory.
In Section \[sec-non-int\], we use a formula of Delange [@delange-75] for $S_b(n)$ to define continuous real-valued interpolations of the functions $d_b(n)$ and $S_b(n)$ from integer bases $b\geq 2$ to a real parameter $\beta>1$. As before, we associate to these interpolated sum-of-digits functions the Dirichlet series $$F_\beta(s) = \sum_{n=1}^\infty \frac{d_\beta(n)}{n^s}$$ and $$G_\beta(s) = \sum_{n=1}^\infty \frac{S_\beta(n)}{n^s}.$$
We prove that these Dirichlet series each have a meromorphic continuation one unit to the left of their halfplane of absolute convergence. For the function $F_\beta(s)$ we have the following theorem.
For each real $\beta>1$, the function $F_\beta(s)$ has a meromorphic continuation to the halfplane $\operatorname{Re}(s)>0$, with a double pole at $s=1$ with Laurent expansion $$F_\beta(s) = \frac{\beta - 1}{2\log \beta}(s-1)^{-2} + \biggl(\frac{\beta-1}{2\log \beta} \bigl(\log(2\pi)\bigr) - \frac{\beta+1}{4}\biggr) (s-1)^{-1} +O(1)$$ and simple poles at $s=1+2\pi i m / \log \beta$ for $m\in{\mathbb{Z}}$ with $m\neq 0$ with residue $$\operatorname*{Res}\biggl(F_\beta(s),s=1+\frac{2\pi i m}{\log \beta}\biggr) = -\frac{\beta-1}{2\pi i m}\zeta\biggl(\frac{2\pi i m}{\log \beta} \biggr).$$
For the function $G_\beta(s)$ we have the following theorem.
For each real $\beta>1$, the function $G_\beta(s)$ is meromorphic in the region $\operatorname{Re}(s)>1$ with a double pole at $s=2$ with Laurent expansion $$G_\beta(s) = \frac{\beta-1}{2\log \beta}(s-2)^{-2} + \biggl(\frac{\beta-1}{2\log \beta} \bigl(\log(2\pi) - 1\bigr) - \frac{\beta+1}{4}\biggr)(s-2)^{-1} +O(1)$$ and simple poles at $s=2+2\pi i m / \log \beta$ for $m\in{\mathbb{Z}}$ with $m\neq 0$ with residue $$\operatorname*{Res}\biggl(G_b(s),s=2+\frac{2\pi i m}{\log \beta}\biggr) = -\frac{\beta-1}{2\pi i m}\biggl(1+\frac{2\pi i m}{\log \beta}\biggr)^{-1}\zeta\biggl(\frac{2\pi i m}{\log \beta} \biggr).$$
To prove these theorems, we start by obtaining the continuation of the series $G_\beta(s)$ by working directly with its Dirichlet series and then obtain the continuation of $F_\beta(s)$ by studying the relation between these two Dirichlet series.
Previous work {#sec-previous-work}
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There has been much previous work studying the functions $d_b(n)$ and $S_b(n)$. The function $d_b(n)$ exhibits significant fluctuations as $n$ changes to $n+1$. It can only increase slowly, having $d_b(n+1) \leq d_b(n)+1$ but it can decrease by an arbitrarily large amount. The sequence $d_b(n)$ is a $b$-regular sequence in the sense of Allouche and Shallit [@allouche-shallit-92 Ex. 2, Sec. 7] and is a member of the $b$-th arithmetic fractal group $\Gamma_b({\mathbb{Z}})$ of Morton and Mourant [@morton-mourant-89 p. 256]. Chen et al. [@chen-hwang-zacharovas-14] survey results on the sum-of-digits function of random integers, and give many references.
Concerning the cumulative sum-of-digits function, Mirsky [@mirsky-49] proved in 1949 that for any integer base $b\geq 2$, the function $S_b(n)$ has the asymptotic $$S_b(n) = \frac{b-1}{2\log b} n \log n + O(n).$$ In 1968 Trollope [@trollope-68] expressed the error term for the base-$2$ cumulative digit sum $S_2(n)$ in terms of a continuous everywhere nondifferentiable function, the Takagi function—see [@lagarias-12] for a survey of the properties of this function. In 1975 Delange [@delange-75] proved the following formula for $S_b(n)$, expressing the error term as a Fourier series with coefficients involving values of the Riemann zeta function on the imaginary axis.
\[thm-delange\] The cumulative sum-of-digits function $S_b(n)$ satisfies $$\label{eq-delange-formula}
S_b(n) = \frac{b-1}{2\log b} n \log n + h_b\biggl(\frac{\log n}{\log b} \biggr) n$$ where $h_b$ is a nowhere-differentiable function of period $1$. The function $h_b$ has a Fourier series $$h_b(x) = \sum_{k=-\infty}^\infty c_b(k)e^{2\pi i k x}$$ with coefficients $$c_b(k) = -\frac{b-1}{2\pi i k}\biggl(1+\frac{2\pi i k}{\log b}\biggr)^{-1}\zeta\biggl(\frac{2\pi i k}{\log b} \biggr)$$ for $k\neq 0$ and $$c_b(0) = \frac{b-1}{2\log b}\bigl(\log(2\pi) - 1\bigr) - \frac{b+1}{4}.$$
A complex-analytic proof of a summation formula for general $q$-additive functions, of which the base-$q$ sum-of-digits function is an example, was given by Mauclaire and Murata in 1983 [@mauclaire-murata-83a; @mauclaire-murata-83b; @murata-mauclaire-88]. A shorter complex-analytic proof of in the specific case of $S_2(n)$ was given by Flajolet, Grabner, Kirschenhofer, Prodinger, and Tichy in 1994 [@flajolet-94]. The method of Flajolet et al. is based on applying a variant of Perron’s formula to the Dirichlet series $$\sum_{n=1}^\infty \bigl(d_2(n)-d_2(n-1)\bigr) n^{-s}.$$ Grabner and Hwang [@grabner-hwang-05] study higher moments of the sum-of-digits function by similar complex-analytic methods.
Our formulas for the residues of $F_b(s)$ and $G_b(s)$ involve the Bernoulli numbers. Kellner [@kellner-17] and Kellner and Sondow [@kellner-sondow-17] investigate another relation between sums of digits and Bernoulli numbers, proving that the least common multiple of the denominators of the coefficients of the polynomial $\sum_{i=0}^n n^k$, which can be written in terms of a Bernoulli polynomial, can be expressed as a certain product of primes satisfying $d_p(n+1)\geq p$.
Sum-of-digits Dirichlet series
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First we consider basic properties of the Dirichlet series $$F_b(s) = \sum_{n=1}^\infty \frac{d_b(n)}{n^s}$$ attached to the base-$b$ digit sum of $n$ and the Dirichlet series $$G_b(s) = \sum_{n=1}^\infty \frac{S_b(n)}{n^s}$$ attached to the cumulative base-$b$ digit sum. For standard references on the basic analytic properties of Dirichlet series, see the books of Hardy and Riesz [@hardy-riesz] or Titchmarsh [@titchmarsh-tof Ch. IX].
Recall that each ordinary Dirichlet series $\sum a_n n^{-s}$ has an abscissa of conditional convergence $\sigma_c$ such that the Dirichlet series converges and defines a holomorphic function if $\operatorname{Re}(s)>\sigma_c$ and diverges if $\operatorname{Re}(s)<\sigma_c$. Each Dirichlet series also has an abscissa of absolute convergence $\sigma_a$ such that the Dirichlet series converges absolutely if $\operatorname{Re}(s)>\sigma_a$ and does not converge absolutely if $\operatorname{Re}(s)<\sigma_a$. For ordinary Dirichlet series, one always has $\sigma_a-1\leq \sigma_c \leq \sigma_a$, and $\sigma_c=\sigma_a$ if the coefficients $a_n$ are nonnegative reals.
For each integer $b\geq 2$, the Dirichlet series $$\label{eq-fb-def-2}
F_b(s) = \sum_{n=1}^\infty \frac{d_b(n)}{n^s}$$ converges and defines a holomorphic function for $\operatorname{Re}(s)>1$.
A positive integer $n$ has $[\log n / \log b]+1$ digits when written in base $b$, each of which is at most $b-1$, so $$\label{eq-db-estimate}
d_b(n) \leq (b-1) \biggl(\biggl\lfloor\frac{\log n}{\log b} \biggr\rfloor+1\biggr).$$ We then obtain the estimate $$\label{eq-sb-estimate}
S_b(n) \ll n \log n$$ with an implied constant depending on $b$. This implies that the Dirichlet series has abscissa of absolute convergence at most 1 and therefore defines a holomorphic function for $\operatorname{Re}(s)>1$.
For each integer $b\geq 2$, the Dirichlet series $$\label{eq-gb-def-2}
G_b(s) = \sum_{n=1}^\infty \frac{S_b(n)}{n^s}$$ converges and defines a holomorphic function for $\operatorname{Re}(s)>2$.
The estimate gives $$\sum_{m=1}^n S_b(m) \ll n^2 \log n,$$ which shows that the Dirichlet series converges for $\operatorname{Re}(s)>2$.
It follows from Delange’s formula that $F_b(s)$ and $G_b(s)$ have abscissa of absolute convergence $\operatorname{Re}(s)=1$ and $\operatorname{Re}(s)=2$, respectively, and this can be proven directly using a more careful estimate of the function $d_b(n)$. We can also obtain the exact values of the abscissas of convergence as a corollary of our theorems on the meromorphic continuation of $F_b(s)$ and $G_b(s)$, since $F_b(s)$ has a pole at $s=1$ and $G_b(s)$ has a pole at $s=2$.
As in previous work on Dirichlet series associated to $q$-additive sequences, it is advantageous to consider the Dirichlet series $$Z_b(s) =\sum_{n=1}^\infty \bigl(d_b(n)-d_b(n-1)\bigr) n^{-s}$$ obtained by differencing the coefficients of the series $F_b(s)$, setting $d_b(0)=0$. Identity in the following proposition appears in a more general form (for $q$-additive functions) in the work of Mauclaire and Murata [@mauclaire-murata-83a; @mauclaire-murata-83b; @murata-mauclaire-88] and is stated and proved explicitly for the sum-of-digits series by Allouche and Shallit [@allouche-shallit-90]. We give a more direct proof of this result.
\[prop-differenced-zeta\] For each integer $b\geq 2$, the Dirichlet series $Z_b(s)$ has abscissa of absolute convergence $\sigma_a=1$, abscissa of conditional convergence $\sigma_c=0$, and has a meromorphic continuation to ${\mathbb{C}}$, satisfying $$\label{eq-zb-formula}
Z_b(s) = \frac{b^s-b}{b^s-1}\zeta(s).$$
For bases $b\geq 3$, we have $\lvert d_b(n)-d_b(n-1) \rvert \geq 1$ for all $n$; if $b=2$, we have $\lvert d_b(n)-d_b(n-1)\rvert \geq 1$ for at least all odd $n$. Hence $\sigma_a\geq 1$. We also have $d_b(n)-d_b(n-1)\ll \log n$, so $\sigma_a\leq 1$. The abscissa of conditional convergence $\sigma_c=0$ follows from the bound $$\sum_{m\leq n} \bigl(d_b(m)-d_b(m-1)\bigr) = d_b(n) \ll \log n.$$
The effect of adding 1 on the digit sum in base-$b$ arithmetic depends on the divisibility of $n$ by $b$; in particular, we have $$d_b(n) - d_b(n-1) = 1 - k(b-1)$$ where $k$ is the largest integer such that $b^k \mid n$. We may also express this as $$d_b(n) - d_b(n-1) = \sum_{m\mid n} \alpha(m) \beta(m/n)$$ where $$\alpha(n) = \begin{cases} 1 &\text{if $n=b^k$ for some $k$}\\ 0 &\text{otherwise} \end{cases} \qquad \beta(n)= \begin{cases} 1-b &\text{if $b\mid n$} \\ 1 &\text{if $b \nmid n$} \end{cases}.$$ Then we have, for $\operatorname{Re}(s)>1$, $$Z_b(s) = \sum_{n=1}^\infty \bigl( d_b(n) - d_b(n-1) \bigr) n^{-s} = \sum_{n=1}^\infty \Bigl( \sum_{m\mid n} \alpha(m) \beta(m/n) \Bigr) n^{-s}$$ Writing the right side as a product of two Dirichlet series, we have $$Z_b(s) = \sum_{n=1}^\infty \alpha(n) n^{-s} \sum_{n=1}^\infty \beta(n) n^{-s} = \sum_{n=1}^\infty b^{-ns} \sum_{n=1}^\infty (1-b) (bn)^{-s}.$$ Summing the geometric series, we obtain $$Z_b(s) = \frac{1}{1-b^{-s}} \bigl( \zeta(s) - bb^{-s} \zeta(s)\bigr) = \frac{b^s-b}{b^s-1}\zeta(s)$$ as claimed. Equation then provides a meromorphic continuation of $Z_b(s)$ since the right side is meromorphic on ${\mathbb{C}}$.
We will obtain information about the mermorphic continuation of $F_b(s)$ and $G_b(s)$ by considering the relation between these series and the series $Z_b(s)$. For future use, we list the poles of the function $Z_b(s)$.
\[lem-poles-of-ndb-ds\] The function $Z_b(s)$ is meromorphic on ${\mathbb{C}}$, with simple poles at $s=2\pi i m / \log b$ for $m\in{\mathbb{Z}}$. The residue at each pole is $$\operatorname*{Res}\biggl(Z_b(s),s=\frac{2\pi i m}{\log b}\biggr) = - \frac{b-1}{\log b} \zeta\biggl(\frac{2\pi i m}{\log b}\biggr).$$ In particular, at $s=0$, the function $Z_b(s)$ has a Laurent expansion beginning $$Z_b(s) = \biggl(\frac{b-1}{2\log b}\biggr)s^{-1} + \biggl(- \frac{b+1}{4} + \frac{b-1}{2\log b} \log(2\pi) \biggr) + O(s).$$
The function $(b^s-b)/(b^s-1)$ has simple poles at $s=2\pi i m /\log b$ for each $m\in{\mathbb{Z}}$, with residue $$\operatorname*{Res}\biggl(\frac{b^s-b}{b^s-1},s=\frac{2\pi i m}{\log b}\biggr) = -\frac{b-1}{\log b}.$$ The Laurent expansion at $s=0$ follows from multiplying the expansions $$\frac{b^s-b}{b^s-1} = -\frac{b-1}{\log b}s^{-1} + \frac{b+1}{2} + O(s)$$ and $$\zeta(s) = -\frac{1}{2} -\frac{1}{2}\log(2\pi)s + O(s^2).$$ The function $\zeta(s)$ has only a simple pole at $s=1$, cancelled by a zero of $(b^s-b)$.
Meromorphic continuation of $F_b(s)$ and $G_b(s)$ {#sec-mero-cont}
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In this section, we show that for integers $b\geq 2$, the Dirichlet series $F_b(s)$ and $G_b(s)$ have a meromorphic continuation to ${\mathbb{C}}$ and determine the structure of the poles, proving Theorems \[thm-db\] and \[thm-sb\].
Bernoulli numbers
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Our formulas for the meromorphic continuation of $F_b(s)$ and $G_b(s)$ involve Bernoulli numbers. For standard facts about the Bernoulli numbers and their basic properties, see [@abramowitz-stegun Ch. 23]. For a thorough reference on Bernoulli numbers, their history, and their relation to zeta functions, see [@arakawa-et-al].
The Bernoulli numbers $B_k$ are the sequence of rational numbers defined by the generating function $$\label{eq-bernoulli-gf}
\frac{x}{e^x-1} = \sum_{k=0}^\infty \frac{B_k}{k!}x^k.$$ If $x$ is a complex variable, this series converges for $\lvert x \rvert < 2\pi$.
Note that there are several competing notations for the Bernoulli numbers; with our definition, we have $B_0=1$, $B_1=-1/2$, and $B_2=1/6$. Because the function $$\frac{x}{e^x-1} + \frac{1}{2}x$$ is an even function, we find that $B_{2k+1}=0$ for all $k\geq 1$.
Power series and Dirichlet series
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To prove the meromorphic continuation of $F_b(s)$ and $G_b(s)$, we make use of the following classical relation between Dirichlet series and power series.
\[prop-ps-ds-relation\] Let $\sigma_c$ be the abscissa of conditional convergence of the Dirichlet series $\sum_{n=1}^\infty a_n n^{-s}$. Then $$\Gamma(s) \sum_{n=1}^\infty a_n n^{-s} = \int_0^\infty \Bigl( \sum_{n=1}^\infty a_n e^{-nx} \Bigr) x^{s-1} \, dx$$ for $\operatorname{Re}(s) >\max(\sigma_c,0)$.
See [@montgomery-vaughan eq. 5.23].
Proposition \[prop-ps-ds-relation\] allows us to translate the additive relations between the arithmetic functions $d_b(n)-d_b(n-1)$, $d_b(n)$, and $S_b(n)$, which are easily expressed in terms of power series generating functions, into relations between their associated Dirichlet series.
Meromorphic continuation of $F_b(s)$
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We now prove the meromorphic continuation of the Dirichlet series $F_b(s)$ by combining the relation between the Dirichlet series and power series generating functions of $d_b(n)$ with the relation between the power series generating functions of $d_b(n)$ and $d_b(n)-d_b(n-1)$.
Let $$\label{eq-px-def}
p(x) = \sum_{n=1}^\infty \bigl(d_b(n)-d_b(n-1)\bigr) x^n.$$ We note that $$\sum_{n=1}^\infty d_b(n)x^n = \frac{p(x)}{1-x}.$$ Then by Proposition \[prop-ps-ds-relation\], we have $$\Gamma(s) F_b(s) = \int_0^\infty \frac{1}{1-e^{-x}} p(e^{-x})x^{s-1}\, dx$$ for $\operatorname{Re}(s)>1$. The series expansion $$\frac{x}{1-e^{-x}} = \sum_{k=0}^\infty \frac{(-1)^kB_k}{k!} x^k,$$ which follows from , holds for $\lvert x \rvert<2\pi$. Since $$\Gamma(s)Z_b(s) = \int_0^\infty p(e^{-x})x^{s-1}\, dx,$$ for $\operatorname{Re}(s)>0$, we can write $$\label{eq-first-fb-expansion}
F_b(s) = \sum_{k=0}^K \frac{(-1)^kB_k}{k!} \frac{\Gamma(s-1+k)}{\Gamma(s)} Z_b(s-1+k) + R_K(s)$$ with $$\label{eq-RK-def}
R_K(s) = \frac{1}{\Gamma(s)}\int_0^\infty \Bigl(\frac{x}{1-e^{-x}} - \sum_{k=0}^K \frac{(-1)^k B_k}{k!} x^k \Bigr) p(e^{-x})x^{s-2} \, dx.$$ Note that $$\frac{\Gamma(s-1)}{\Gamma(s)} = \frac{1}{s-1}$$ and $$\frac{\Gamma(s-1+k)}{\Gamma(s)} = (s)(s+1)\dotsm (s+k-2).$$ Since $$\frac{x}{1-e^{-x}} - \sum_{k=0}^K \frac{(-1)^k B_k}{k!} x^k \ll x^{K+1}$$ as $x\rightarrow 0^+$, the integral in converges and defines a holomorphic function in the region $\operatorname{Re}(s)>1-K$. From Lemma \[lem-poles-of-ndb-ds\] we know that $Z_b(s)$ has simple poles at $s=2\pi ik/\log b$ for $k\in{\mathbb{Z}}$. The $k=0$ term $$\frac{1}{s-1}Z_b(s-1)$$ has a double pole at $s=1$ with Laurent expansion beginning $$\frac{1}{s-1}Z_b(s-1) = \frac{b-1}{2\log b}(s-1)^{-2} + \biggl( \frac{b-1}{2\log b} \log(2\pi) - \frac{b+1}{4}\biggr)(s-1)^{-1} + O(1),$$ and simple poles at each other point $s=1+2\pi i m /\log b$. Each term $$(-1)^k\frac{B_k}{k!}\prod_{j=0}^{k-2}(s+j) \cdot Z_b(s-1+k)$$ with $k=1$ or with $k$ an even integer with $k\geq 2$ has a simple pole at $s=1-k+2\pi i m/\log b$ for $m\in{\mathbb{Z}}$ with residue $$(-1)^{k}\frac{B_k}{k!} \prod_{j=1}^{k-1} \biggl(\frac{2\pi i m}{\log b} - j\biggr) \cdot \biggl(-\frac{b-1}{\log b}\biggr)\zeta\biggl(\frac{2\pi i m}{\log b}\biggr).$$ Since $K$ can be taken arbitrarily large, this proves the theorem.
Meromorphic continuation of $G_b(s)$
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We continue the function $G_b(s)$ to the plane in a similar fashion, using the fact that $S_b(n)$ is a double sum of the differences $d_b(n)-d_b(n-1)$ appearing in the series $Z_b(n)$.
Define $p(x)$ by . We make use of the identity of power series $$\frac{x}{(1-x)^2} p(x) = \sum_{n=1}^\infty S_b(n)x^n.$$ Then by Proposition \[prop-ps-ds-relation\], we have $$\Gamma(s)G_b(s) = \int_0^\infty \frac{e^{-x}}{(1-e^{-x})^2} p(e^{-x}) x^{s-1} \, dx$$ for $\operatorname{Re}(s)>2$. From and noting that $$\frac{e^{-x}}{(1-e^{-x})^2}=-\frac{d}{dx}\biggl(\frac{1}{e^x-1}\biggr),$$ we have the power series expansion $$\frac{x^2e^x}{(e^x-1)^2} = 1-\sum_{k=2}^\infty \frac{B_k}{k(k-2)!}x^k.$$ Then for a fixed integer $K\geq 2$ can write $G_b(s)$ as $$\label{eq-gbs-expansion}
G_b(s) = \frac{\Gamma(s-2)}{\Gamma(s)}Z_b(s-2) - \sum_{k=2}^K \frac{B_k}{k(k-2)!} \frac{\Gamma(s-2+k)}{\Gamma(s)} Z_b(s-2+k) + R_K(s)$$ with remainder $R_K$ given by $$R_K(s) = \frac{1}{\Gamma(s)}\int_0^\infty \biggl( \frac{x^2e^{-x}}{(1-e^{-x})^2} - 1 + \sum_{k=2}^K \frac{B_k}{k(k-2)!} x^k \biggr) p(e^{-x}) x^{s-3} \, dx.$$ The function $R_K(s)$ is holomorphic for $\operatorname{Re}(s)>2-K$.
As before, we consider the poles of each term of . The first term $$\frac{1}{(s-1)(s-2)} Z_b(s-2)$$ has a double pole at $s=2$ with Laurent expansion $$\frac{b-1}{2\log b}(s-2)^{-2} + \biggl(\frac{b-1}{2\log b}\bigl(\log(2\pi)-1\bigr)-\frac{b+1}{4}\biggr)(s-2)^{-1}+ \dotsb,$$ a simple pole at each point $s=2+2\pi i m / \log b$ with $m\neq 0$, and a simple pole at $s=1$ with residue $(b+1)/12$. Each other term $$\frac{B_k}{k(k-2)!} \prod_{j=0}^{k-3} (s+j) \cdot Z_b(s-2+k).$$ has simple poles at $s=2-k+2\pi i m / \log b$ for all $m$.
Instead of relating $G_b(s)$ to the series $Z_b(s)$ as we did in this proof, we could have also used the relation $$\Gamma(s) G_b(s) = \int_0^\infty \frac{e^{-x}}{1-e^{-x}} \biggl(\sum_{n=1}^\infty d_b(n) e^{-xn}\biggr) x^{s-1} \, dx,$$ following the proof of Theorem \[thm-db\] to write $G_b(s)$ in terms of $F_b(s)$.
Order of $F_b(s)$ and $G_b(s)$ as meromorphic functions {#sec-meromorphic-functions}
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The functions $F_b(s)$ and $G_b(s)$ are meromorphic functions on the complex plane with infinitely many poles on a left half-lattice. We now raise a further question about the analytic properties of these functions. Recall that the *order* of an entire function $f(z)$ on ${\mathbb{C}}$ is $$\rho = \inf \bigl\{ \rho\geq 0 \, \bigm\vert \, f(z) = O\bigl(\exp(\lvert z \rvert^{\rho+\varepsilon})\bigr)\text{ as $\lvert z \rvert\rightarrow\infty$} \bigr\}.$$ An entire function is of *finite order* if $\rho<\infty$.
The order of a meromorphic function is defined as the order of growth of its associated Nevanlinna characteristic function. This definition is equivalent (by [@rubel] Lemma 15.6 and Theorem on p. 91) to the following.
The *order* of a meromorphic function $f(z)$ is the infimum of $\rho\geq 0$ such that $f$ can be written as $f(z) = g(z)/h(z)$ for entire functions $g(z)$ and $h(z)$ of order $\rho$.
Many of the common Dirichlet series of analytic number theory are meromorphic (or entire) functions of order $1$. Examples include the Riemann zeta function $\zeta(s)$, the Dirichlet $L$-functions $L(s,\chi)$, and their relatives; more generally, all Dirichlet series in the Selberg class (as introduced in Selberg [@selberg-92]) are meromorphic functions of order 1.
By Proposition \[prop-differenced-zeta\], the function $Z_b(s)$ is meromorphic of order $1$. The functions $F_b(s)$ and $G_b(s)$, however, must have greater order by the following fact.
Let $f(z)$ be a meromorphic function of order $\rho$ and let $n(r,a)$ be the number of zeros of $f(z)-a$ in the disc $\lvert z \rvert<r$. Then $n(r,a)=O(r^{\rho+\varepsilon})$.
By Theorems \[thm-db\] and \[thm-sb\], the functions $F_b(s)$ and $G_b(s)$ each have $\gg r^2$ poles in the disc $\lvert z \rvert<r$. Hence we have the following corollary.
The functions $F_b(s)$ and $G_b(s)$ are meromorphic functions of order at least $2$.
A meromorphic function of order greater than 2 could still have only $O(r^2)$ poles in $\lvert z \rvert<r$, so without further information, we cannot deduce that $F_b(s)$ and $G_b(s)$ have order 2.
Are the functions $F_b(s)$ and $G_b(s)$ mermorphic functions of order exactly $2$?
Such a question has been answered positively in the related setting of strongly $q$-multiplicative functions: the Dirichlet series attached to such functions are entire of order exactly $2$ (see Alkauskas [@alkauskas-04]).
Meromorphic continuation of Dirichlet series for non-integer bases {#sec-non-int}
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In this section, we consider the problem of extending the digit sums $d_b(n)$ and $S_b(n)$ from integer bases $b$ to real parameters $\beta>1$. There are a number of possible ways to do this. One natural approach concerns the notion of $\beta$-expansions introduced by Renyi [@renyi-57] and studied at length by Parry [@parry-60]. However, for non-integer values of $\beta$, the $\beta$-expansion of an integer generally has infinitely many digits, so the sum of the digits will generally be infinite. Digit sums related to a different digit expansion with respect to an irrational base were considered by Grabner and Tichy [@grabner-tichy-91].
The approach which we consider in this section is to use the formula of Delange to define the cumulative digit sum $S_\beta(n)$ for real parameters $\beta>1$, from which we can define a digit sum $d_\beta(n)$ by differencing. The resulting functions are continuous in the $\beta$-parameter.
Extension to non-integer bases by Delange’s formula
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We begin by replacing the integer variable $b$ in Theorem \[thm-delange\], which gives a formula for $S_b(n)$ for integer bases $b\geq 2$, by a real parameter $\beta>1$.
\[def-non-integer\] For $\beta\in{\mathbb{R}}$ with $\beta>1$, define a generalized cumulative sum-of-digits function $S_\beta(n)$ by $$\label{eq-sbeta-delange-def}
S_\beta(n) := \frac{\beta-1}{2\log \beta} n \log n + h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n,$$ where the function $h_\beta(x)$ is defined by the Fourier series $$\label{eq-hbeta-fourier-series}
h_\beta(x) = \sum_{k=-\infty}^\infty c_\beta(k)e^{2\pi i k x}$$ with coefficients $$c_\beta(k) = -\frac{\beta-1}{2\pi i k}\biggl(1+\frac{2\pi i k}{\log \beta}\biggr)^{-1}\zeta\biggl(\frac{2\pi i k}{\log \beta} \biggr)$$ for $k\neq 0$ and $$c_\beta(0) = \frac{\beta-1}{2\log \beta}(\log 2\pi - 1) - \frac{\beta+1}{4}.$$
Define the generalized sum-of-digits function $d_\beta(n)$ for real $\beta>1$ by $$d_\beta(n) := S_\beta(n+1) - S_\beta(n).$$
A plot of $S_\beta(10)$ as a function of $\beta$ for $1\leq\beta\leq 15$ is shown in Figure \[Sb10-plot\]. Note that $S_\beta(n)$ is approximately constant for $\beta\geq 10$.
\[Sb10-plot\] 
The function $h_\beta(x)$
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In this section, we study properties of the function $h_\beta(x)$ appearing in Definition \[def-non-integer\] as a function of the variable $\beta>1$ and as a function of the variable $x$. When $\beta=b\in{\mathbb{N}}$, Delange showed that $h_b(x)$ is a continuous but everywhere non-differentiable real-valued function of $x$ with period 1.
For each fixed $\beta>1$, the function $h_\beta(x)$ is a real-valued continuous function of $x$ on ${\mathbb{R}}$.
The zeta function satisfies the bound $$\lvert\zeta(it)\rvert\ll t^{1/2+\varepsilon}$$ for $t\in{\mathbb{R}}$ (see for example [@titchmarsh-zeta eq. 5.1.3], so the Fourier coefficients of $h_\beta$ satisfy $$c_\beta(k)= -\frac{\beta-1}{2\pi i k}\biggl(1+\frac{2\pi i k}{\log \beta}\biggr)^{-1}\zeta\biggl(\frac{2\pi i k}{\log \beta} \biggr) \ll_\beta k^{-3/2+\varepsilon}.$$ This estimate shows that the Fourier series is absolutely and uniformly convergent for $x\in{\mathbb{R}}$, so gives a continuous function of $x$.
The function $h_\beta(x)$ is real-valued for $x\in{\mathbb{R}}$ since the Fourier coefficients $c_\beta(k)$ satisfy $\overline{c_\beta(k)} = c_\beta(-k)$.
\[fb2-plot\] 
\[fbl2-plot\] 
A plot of $h_\beta(2)$ as a function of the real parameter $\beta$ for $1\leq \beta \leq 8$ is shown in Figure \[fb2-plot\]. From the plot, it also appears that $h_\beta$ might be non-differentiable as a function of the real parameter $\beta$.
For fixed $x\in{\mathbb{R}}$, is the function $h_\beta(x)$ everywhere non-differentiable as a function of the real variable $\beta$?
Meromorphic continuation of $G_\beta(s)$
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Our proofs of the meromorphic continuation of $F_b(s)$ and $G_b(s)$ for integer bases relied on the identity $$Z_b(s) = \sum_{n=1}^\infty \bigl( d_b(n) - d_b(n-1) \bigr) n^{-s} = \frac{b^s-b}{b^s-1}\zeta(s).$$ If for non-integer $\beta>1$ we define $$Z_\beta(s) \coloneq \sum_{n=1}^\infty \bigl( d_\beta(n) - d_\beta(n-1) \bigr) n^{-s},$$ then $Z_\beta(s)$ is *not* equal to $$\label{eq-zbeta-wrong}
\frac{\beta^s-\beta}{\beta^s-1}\zeta(s)$$ as is not an ordinary Dirichlet series. We must therefore take a different approach.
We first consider the Dirichlet series generating function $$G_\beta(s) := \sum_{n=1}^\infty \frac{S_\beta(n)}{n^s}$$ for $\beta\in{\mathbb{R}}$ with $\beta>1$. Since the coefficients satisfy $$S_\beta(n) \asymp n \log n,$$ the Dirichlet series $G_\beta(s)$ has abscissa of absolute convergence $\sigma_a=2$. We show that the function $G_\beta(s)$ can be analytically continued to a larger halfplane.
For each real $\beta>1$, the function $G_\beta(s)$ is meromorphic in the region $\operatorname{Re}(s)>1$ with a double pole at $s=2$ with Laurent expansion $$G_\beta(s) = \frac{\beta - 1}{2\log \beta}(s-2)^{-2} + c_\beta(0) (s-2)^{-1} +O(1)$$ and simple poles at $s=2+2\pi i k / \log \beta$ for $k\in{\mathbb{Z}}$ with $k\neq 0$ with residue $$\operatorname*{Res}\biggl(G_b(s),s=2+\frac{2\pi i k}{\log \beta}\biggr) = c_\beta(k),$$ where the numbers $c_\beta(k)$ are those in Definition \[def-non-integer\].
Using the definition of $S_\beta$, we have $$\begin{aligned}
G_\beta(s)&=\sum_{n=1}^\infty \Biggl(\frac{\beta-1}{2\log \beta} n \log n + h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n\Biggr)n^{-s}\\
&= -\frac{\beta-1}{2\log \beta} \zeta'(s-1) + \sum_{n=1}^\infty h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n^{-(s-1)}.\end{aligned}$$ The function $\zeta'(s-1)$ is meromorphic on ${\mathbb{C}}$ with only singularity a double pole at $s=2$ with Laurent expansion $\zeta'(s-1)=-(s-1)^{-2} + O(1)$. Using the Fourier series for $h_\beta$, we have $$\begin{aligned}
\sum_{n=1}^\infty h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n^{-(s-1)} &= \sum_{n=1}^\infty \sum_{k=-\infty}^\infty c_\beta(k) \exp\biggl(2 \pi i k \frac{\log n}{\log \beta}\biggr) n^{-(s-1)}\\ &= \sum_{n=1}^\infty \sum_{k=-\infty}^\infty c_\beta(k) n^{-(s-1 - 2\pi i k / \log \beta)}.\end{aligned}$$ This double sum is absolutely convergent, so we may exchange the sums, giving $$\sum_{n=1}^\infty h_\beta\biggl(\frac{\log n}{\log \beta} \biggr) n^{-(s-1)} = \sum_{k=-\infty}^\infty c_\beta(k) \zeta\biggl( s- 1 - \frac{2\pi i k}{\log \beta}\biggr).$$ If $\operatorname{Re}(s)>1$, then $$\zeta\biggl( s- 1 - \frac{2\pi i k}{\log \beta}\biggr) \ll k^{1/2+\varepsilon}$$ for any $\varepsilon>0$. On any compact set $K$ in the halfplane $\operatorname{Re}(s)>1$ not containing a point $s=2+2\pi i k / \log \beta)$ for any $k\in{\mathbb{Z}}$, the sum $$\sum_{k=-\infty}^\infty c_\beta(k) \zeta\biggl( s- 1 - \frac{2\pi i k}{\log \beta}\biggr)$$ is uniformly convergent on $K$; if the compact set $K$ contains a point of the form $s=2 + 2\pi i k_0 / \log \beta$, then one term of the sum has a simple pole with residue $c_\beta(k_0)$ while the remaining sum is uniformly convergent.
When $\beta\geq 2$ is an integer, we know that the function $G_\beta(s)$ has a meromorphic continuation to the entire complex plane.
For noninteger $\beta>1$, does the Dirichlet series $G_\beta(s)$ have a meromorphic continuation beyond $\operatorname{Re}(s)>1$?
Meromorphic continuation of $F_\beta(s)$
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We now consider the Dirichlet series $$F_\beta(s) = \sum_{n=1}^\infty \frac{d_\beta(n)}{n^s}$$ for real $\beta>1$. We already know that this series has a meromorphic continuation to ${\mathbb{C}}$ when $\beta\geq 2$ is an integer. We show that for each real $\beta>1$, the Dirichlet series $F_\beta(s)$ has a meromorphic continuation to the halfplane $\operatorname{Re}(s)>0$.
The function $F_\beta(s)$ has a meromorphic continuation to the halfplane $\operatorname{Re}(s)>0$, with a double pole at $s=1$ with Laurent expansion $$F_\beta(s) = \frac{\beta - 1}{2\log \beta}(s-1)^{-2} + \biggl(c_\beta(0) +\frac{\beta-1}{2\log \beta}\biggr) (s-1)^{-1} +O(1)$$ and simple poles at $s=1+2\pi i k / \log \beta$ for $k\in{\mathbb{Z}}$ with $k\neq 0$ with residue $$\operatorname*{Res}\biggl(F_\beta(s),s=1+\frac{2\pi i k}{\log \beta}\biggr) = \biggl(1+\frac{2\pi i k}{\log \beta}\biggr) c_\beta(k).$$
Let $$p(x) = \sum_{n=2}^\infty S_\beta(n)x^n,$$ so that $$\Gamma(s)\bigl(G_\beta(s)-S_\beta(1)\bigr) = \int_0^\infty p(e^{-x}) x^{s-1}\, dx.$$ By our definition of $d_\beta(n)$, we have $$\sum_{n=1}^\infty d_\beta(n) x^n + S_\beta(1)= (x^{-1}-1)p(x).$$ Hence by Proposition \[prop-ps-ds-relation\] we have $$\Gamma(s)\bigl(F_\beta(s)+S_\beta(1)\bigr) = \int_0^\infty (e^x-1)p(e^{-x})x^{s-1}\, dx$$ for $\operatorname{Re}(s)>1$. Using the power series expansion Then we write $$\label{eq-fbeta-formula-1}
\Gamma(s)\bigl(F_\beta(s)+S_\beta(1)\bigr) = \Gamma(s+1)\bigl(G_\beta(s+1)-S_\beta(1)\bigr) + \int_0^\infty (e^x-1 - x) p(e^{-x})x^{s-1} \, dx.$$ Dividing by $\Gamma(s)$ and rearranging, we obtain $$\label{eq-fbeta-gbeta-relation}
F_\beta(s) = -S_\beta(1)(s+1) + s G_\beta(s+1) + R(s)$$ where the remainder term $$R(s) = \frac{1}{\Gamma(s)} \int_0^\infty (e^x-1 - x) p(e^{-x})x^{s-1} \, dx$$ is holomorphic in $\operatorname{Re}(s)>0$ since $e^x-1-x\ll x^2$ as $x\rightarrow 0^+$. Since $G_\beta(s+1)$ is meromorphic in $\operatorname{Re}(s)>0$, we find that $F_\beta(s)$ is meromorphic in $\operatorname{Re}(s)>0$, with poles coming from the poles of $G_\beta(s+1)$. Since $$sG_\beta(s+1) = (s-1)G_\beta(s+1) + G_\beta(s+1),$$ we find that $F_\beta(s)$ has a double pole at $s=1$ with Laurent expansion as given in the theorem. At each other point $s=1+2\pi i m / \log \beta$, $F_\beta(s)$ has a simple pole.
Meromorphic continuation of $F_\beta(s)$ to a larger halfplane would follow from continuation of $G_\beta(s)$ to a larger halfplane; in particular, by using more terms of the power series for $e^x$ in formula , we find that if $G_\beta(s)$ is meromorphic in $\operatorname{Re}(s)>c$ for some $c$, then $F_\beta(s)$ is meromorphic in $\operatorname{Re}(s)>c-1$.
Acknowledgements
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The author thanks Jeffrey Lagarias for many helpful comments.
[^1]: Work partially supported by NSF grant DMS-1701576.
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