14468006
Comment on Pauli--Villars Lagrangian on the Lattice
It is interesting to superimpose the Pauli–Villars regularization on the lattice regularization. We illustrate how this scheme works by evaluating the axial anomaly in a simple lattice fermion model, the Pauli–Villars Lagrangian with a gauge non-invariant Wilson term. The gauge non-invariance of the axial anomaly, caused by the Wilson term, is remedied by a compensation between Pauli–Villars regulators in the continuum limit. A subtlety in Frolov–Slavnov’s scheme for an odd number of chiral fermions in an anomaly free complex gauge representation, which requires an infinite number of regulators, is briefly mentioned. ⋆ e-mail: hsuzuki@mito.ipc.ibaraki.ac.jp It seems interesting to put the Pauli–Villars type Lagrangian level regularization on the lattice. The interest is twofold: Firstly, the Pauli–Villars regularization [1] for fermion one-loop diagrams can be expressed as a Lagrangian of regulators (bosonic and fermionic spinors). In the actual perturbative calculation, however, the Lagrangian has to be supplemented with additional prescriptions, such that the momentum of propagators has to be assigned in the same way for all the fields, and the integrand in the momentum integral has to be summed before the integration. This point may conceptually be somewhat unsatisfactory, but once the Lagrangian is put on the lattice, those prescriptions are automatically implemented, providing the non-perturbative meaning. More interestingly and more importantly, ”superimposing” a different kind of regularization on the lattice regularization may give some clue to the lattice regularization of the chiral gauge theory. No manifestly gauge invariant lattice formulation of the chiral gauge theory, being consistent with the unitarity and the locality, is yet known [2]. In particular, for a chiral fermion in a complex gauge representation, it is impossible to introduce in a gauge invariant way the Wilson term [3] to eliminate the unwanted species doublers. † The difficulty of a manifestly gauge invariant lattice formulation of the chiral gauge theory is highlighted by the No-Go theorem [4]. The basic idea of “superimposing” is quite simple. Let us consider, for example, the naive momentum cutoff regularization applied to fermion one-loop diagrams in QED. This regularization breaks the gauge invariance, generating gauge noninvariant contributions. However we may use in addition say, the gauge invariant dimensional regularization. With this superimposed regularization, the infinite momentum cutoff limit can be taken and we are left with the gauge invariant expression in the dimensional regularization. Of course in this example there is no real need to break the gauge invariance by introducing the momentum cutoff, † For chiral fermions in real-positive gauge representation, and for even number of chiral fermions in pseudoreal representation, it is possible to introduce a gauge invariant (Majorana-type) Wilson term.
119414305
Anomalies, local counter terms and bosonization
We re-examine the issue of local counter terms in the analysis of quantum anomalies. We analyze two-dimensional theories and show that the notion of local counter terms need to be carefully defined depending on the physics contents such as whether one is analyzing gauge theory or bosonization. It is shown that a part of the Jacobian, which is apparently spurious and eliminated by a local counter term corresponding to the mass term of the gauge field in gauge theory, cannot be removed by a local counter term and plays a central role by giving the kinetic term of the bosonized field in the context of path integral bosonization.
119213970
Renormalizability of the gradient flow in the 2D O(N) non-linear sigma model
It is known that the gauge field and its composite operators evolved by the Yang--Mills gradient flow are ultraviolet (UV) finite without any multiplicative wave function renormalization. In this paper, we prove that the gradient flow in the 2D $O(N)$ non-linear sigma model possesses a similar property: The flowed $N$-vector field and its composite operators are UV finite without multiplicative wave function renormalization. Our proof in all orders of perturbation theory uses a $(2+1)$-dimensional field theoretical representation of the gradient flow, which possesses local gauge invariance without gauge field. As application of the UV finiteness of the gradient flow, we construct the energy--momentum tensor in the lattice formulation of the $O(N)$ non-linear sigma model that automatically restores the correct normalization and the conservation law in the continuum limit.
118409043
Euclidean lattice simulation for dynamical supersymmetry breaking
The global supersymmetry is spontaneously broken if and only if the ground-state energy is strictly positive. We propose to use this fact to observe the spontaneous supersymmetry breaking in Euclidean lattice simulations. For lattice formulations that possess a manifest fermionic symmetry, there exists a natural choice of a Hamiltonian operator that is consistent with a topological property of the Witten index. We confirm validity of our idea in models of the supersymmetric quantum mechanics. We then examine a possibility of a dynamical supersymmetry breaking in the two-dimensional $\mathcal{N}=(2,2)$ super Yang-Mills theory with the gauge group $\mathrm{SU}(2)$, for which the Witten index is unknown. Differently from a recent conjectural claim, our numerical result tempts us to conclude that supersymmetry is not spontaneously broken in this system.
119483534
Axial anomaly in the reduced model: Higher representations☆
Abstract The topological charge in the U( N ) vector-like reduced model can be defined by using the overlap Dirac operator. We obtain its large N limit for a fermion in a general gauge-group representation under a certain restriction of gauge field configurations which is termed U(1) embedding.
14282824
SUSY WT identity in a lattice formulation of 2D N = (2, 2) SYM
Abstract We address some issues relating to a supersymmetric (SUSY) Ward–Takahashi (WT) identity in Sugino's lattice formulation of two-dimensional (2D) N = ( 2 , 2 ) SU ( k ) supersymmetric Yang–Mills theory (SYM). A perturbative argument shows that the SUSY WT identity in the continuum theory is reproduced in the continuum limit without any operator renormalization/mixing and tuning of lattice parameters. As application of the lattice SUSY WT identity, we show that a prescription for the Hamiltonian density in this lattice formulation, proposed by Kanamori, Sugino and Suzuki, is justified also from a perspective of an operator algebra among correctly-normalized supercurrents. We explicitly confirm the SUSY WT identity in the continuum limit to the first nontrivial order in a semi-perturbative expansion.
118431737
A new baryonic equation of state at sub-nuclear densities for core-collapse simulations
We calculate a new equation of state for baryons at sub-nuclear densities meant for the use in core-collapse simulations of massive stars. The abundances of various nuclei are obtained together with the thermodynamic quantities. The formulation is the nuclear statistical equilibrium description and the liquid drop approximation of nuclei. The model free energy to minimize is calculated by relativistic mean field theory for nucleons and the mass formula for nuclei with atomic number up to ~1000. We have also taken into account the pasta phase, thanks to which the transition to uniform nuclear matter in our equation of state (EOS) occurs in the conventional manner: nuclei are not dissociated into nucleons but survive right up to the transition to uniform nuclear matter. We find that the free energy and other thermodynamical quantities are not very different from those given in the H. Shen's EOS, one of the standard EOSs that adopt the single nucleus approximation. On the other hand, the average mass is systematically different, which may have an important ramification to the rates of electron captures and coherent neutrino scatterings on nuclei in supernova cores. It is also interesting that the root mean square of the mass number is not very different from the average mass number, since the former is important for the evaluation of coherent scattering rates on nuclei but has been unavailable so far. The EOS table is currently under construction, which will include the weak interaction rates.
125991530
Neutrino Transport in Type II Supernovae: Boltzmann Solver vs Monte Carlo Method
In this paper we discuss the results of extensive comparison of the Boltzmann solver recently developed by us and the Monte Carlo method coded by Janka. The aim of our project is to improve the treatment of the neutrino transfer and to study the neutrino reactions systematically, since the numerical treatment of neutrino transfer and neutrino reaction rates in the hot dense medium are two of the greatest uncertainties in the supernova simulations.
118647169
New equations of state based on the liquid drop model of heavy nuclei and quantum approach to light nuclei for core-collapse supernova simulations
We construct new equations of state for baryons at subnuclear densities for the use in core-collapse simulations of massive stars. The abundance of various nuclei is obtained together with thermodynamic quantities. A model free energy is constructed, based on the relativistic mean field theory for nucleons and the mass formula for nuclei with the proton number up to ~1000. The formulation is an extension of the previous model, in which we adopted the liquid drop model to all nuclei under the nuclear statistical equilibrium. We reformulate the new liquid drop model so that the temperature dependences of bulk energies could be taken into account. Furthermore, we extend the region in the nuclear chart, in which shell effects are included, by using theoretical mass data in addition to experimental ones. We also adopt a quantum-theoretical mass evaluation of light nuclei, which incorporates the Pauli- and self-energy shifts that are not included in the ordinary liquid drop model. The pasta phases for heavy nuclei are taken into account in the same way as in the previous model. We find that the abundances of heavy nuclei are modified by the shell effects of nuclei and temperature dependence of bulk energies. These changes may have an important effect on the rates of electron captures and coherent neutrino scatterings on nuclei in supernova cores. The abundances of light nuclei are also modified by the new mass evaluation, which may affect the heating and cooling rates of supernova cores and shocked envelopes.
118538455
Neutrino oscillation and expected event rate of supernova neutrinos in the adiabatic explosion model
We study how the influence of the shock wave appears in neutrino oscillations and the neutrino spectrum by using the density profile of the adiabatic explosion model of a core-collapse supernova, which is calculated in an implicit Lagrangian code for general relativistic spherical hydrodynamics. We calculate expected event rates of neutrino detection at Super-Kamiokande (SK) and Sudbury Neutrino Observatory (SNO) for various ${\ensuremath{\theta}}_{13}$ values and both normal and inverted hierarchies. The predicted event rates of ${\overline{\ensuremath{\nu}}}_{e}$ and ${\ensuremath{\nu}}_{e}$ depend on the mixing angle ${\ensuremath{\theta}}_{13}$ for the inverted and normal mass hierarchies, respectively, and the influence of the shock wave appears for about 2\char21{}8 s when ${sin﻿}^{2}2{\ensuremath{\theta}}_{13}$ is larger than ${10}^{\ensuremath{-}3}$. These neutrino signals for the shock-wave propagation is decreased by $\ensuremath{\lesssim}30%$ for ${\overline{\ensuremath{\nu}}}_{e}$ in inverted hierarchy (SK) or by $\ensuremath{\lesssim}15%$ for ${\ensuremath{\nu}}_{e}$ in normal hierarchy (SNO) compared with the case without shock. The obtained ratio of the total event for high-energy neutrinos ($20\text{ }\text{ }\mathrm{MeV}\ensuremath{\leqq}{E}_{\ensuremath{\nu}}\ensuremath{\leqq}60\text{ }\text{ }\mathrm{MeV}$) to low-energy neutrinos ($5\text{ }\text{ }\mathrm{MeV}\ensuremath{\lesssim}{E}_{\ensuremath{\nu}}\ensuremath{\leqq}20\text{ }\text{ }\mathrm{MeV}$) is consistent with the previous studies in schematic semianalytic or other hydrodynamic models of the shock propagation. The time dependence of the calculated ratio of the event rates of high-energy neutrinos to the event rates of low-energy neutrinos is a very useful observable which is sensitive to ${\ensuremath{\theta}}_{13}$ and mass hierarchies. Namely, the time-dependent ratio shows a clearer signal of the shock-wave propagation that exhibits a remarkable decrease by at most a factor of $\ensuremath{\sim}2$ for ${\overline{\ensuremath{\nu}}}_{e}$ in inverted hierarchy (SK), whereas it exhibits a smaller change by $\ensuremath{\sim}10%$ for ${\ensuremath{\nu}}_{e}$ in normal hierarchy (SNO). Observing the time-dependent high-energy to low-energy ratio of the neutrino events thus would provide a piece of very useful information to constrain ${\ensuremath{\theta}}_{13}$ and mass hierarchy and eventually help understand how the shock wave propagates inside the star.
118908804
Multigroup simulation of protoneutron star cooling and neutrino spectra